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What Does Water Symbolize? Cultural & Literary Examples

Last Updated: October 23, 2023 Fact Checked

This article was co-authored by Mari Cartagenova and by wikiHow staff writer, Aly Rusciano . Mari Cartagenova is a Psychic Medium & Animal Communicator based in Westford, Massachusetts. Mari has been seeing and speaking to Spirits since she was a child, and she loves to share her psychic gifts and abilities to help others. She specializes in mediumship, intuitive psychic guidance, past life or soul reading, and animal communication. Mari is the 2019 Award-Winning Best American Psychic and a Certified Advanced Spirit Worker. There are 12 references cited in this article, which can be found at the bottom of the page. This article has been fact-checked, ensuring the accuracy of any cited facts and confirming the authority of its sources. This article has been viewed 16,785 times.

Water. You use and see it every single day. It’s there when you brush your teeth, eat, and walk home in the rain. But did you know that water is more than just something you drink? Water has been an important symbol for religious, spiritual, and cultural practices for centuries. Believe it or not, your favorite movie and book scenes have even used these symbols. So, what are they, and what does water actually represent? Keep reading to find out.

Things You Should Know

• Emotionally, water can symbolizes profound depth, changing feelings, or sadness.
• Water is also a universal representation of life since no plant or animal can live without it.
• Spiritually, water invokes ideas of purification, rejuvenation, and transformation.

• As the popular proverb says, “Still waters run deep.” [1] X Research source

• As a literary example, Toni Morrison uses phrases like “wave of grief” and “ice cake torn away from the solid surface of the stream” to convey her characters' deep emotions in the novel Beloved . [2] X Research source

• In this context, someone may say they’re feeling “watered down,” like they're “treading water,” or are in “troubled waters.” [3] X Research source

• In Egyptian culture, life and water are represented by the same symbol: the ankh. [4] X Trustworthy Source US National Park Service Agency responsible for the maintenance and promotion of national parks and monuments Go to source

• For instance, you may be familiar with the phrase “Having a ripple effect,” which depicts that one action can cause other things to occur. [5] X Research source

Purification

• Many bodies of water are considered religiously sacred for this very reason. In Hinduism, for example, bathing in the Ganges River is said to aid spiritual transformation and cleanse one of crimes. [7] X Research source

• For instance, Yann Martel’s Life of Pi showcases a boy who has a near-death experience out at sea. While on the water, he undergoes a personal enlightenment and reinvents himself.
• In the Bible, a giant flood destroys everything on Earth except for what Noah has in his ark. The flood washes the sins of the Earth away as a biblical renewal.

Rejuvenation

• In Native American culture, water is 1 of the 4 Directions on the Medicine Wheel or Sacred Hoop, which is used for traditional healing practices. [10] X Research source

Self-reflection

• In Greek mythology, Narcissus falls in love with his own reflection after looking in a pool of water, emphasizing the vanity of humanity.

Subconsciousness

• One of the most famous examples of this is the representation of water in Herman Melville’s Moby Dick . In the novel, the sea represents the churning, beautiful, and maddening concepts of the human psyche.

Transformation

• Along with this, water is a constant habitat for evolution. Frogs, for example, start in water as tadpoles and transform into frogs over time.
• In the Bible, the Lord transforms water into wine.

• In Irish folklore, the story of Sinann depicts a woman following a stream to pursue knowledge. [14] X Research source

Expert Q&A

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Expert Interview

Thanks for reading our article! If you’d like to learn more about spirituality, check out our in-depth interview with Mari Cartagenova .

• ↑ https://extension.usu.edu/waterquality/kidspage/watersayings
• ↑ https://blogs.dickinson.edu/403lit/2017/10/01/water-and-liquid-in-toni-morrisons-beloved/
• ↑ https://www.nps.gov/afbg/learn/historyculture/ankh.htm
• ↑ https://www.britannica.com/topic/holy-water
• ↑ https://www.webpages.uidaho.edu/~rfrey/116ganges.htm
• ↑ https://www.ipl.org/essay/Symbolism-Of-Water-In-A-Separate-Peace-99182B588D0B9E74
• ↑ https://www.history.com/news/the-myth-of-ponce-de-leon-and-the-fountain-of-youth
• ↑ https://www.nlm.nih.gov/nativevoices/exhibition/healing-ways/medicine-ways/medicine-wheel.html
• ↑ https://jungcurrents.com/carl-jung-symbolism-water
• ↑ https://www.theatlantic.com/entertainment/archive/2016/12/the-power-of-water-in-moonlight/511547/
• ↑ https://norse-mythology.org/cosmology/yggdrasil-and-the-well-of-urd/
• ↑ https://www.jstor.org/stable/40732065

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Water Symbolism and Symbols – A Guide

Table of Contents

Symbolic meanings of water, common water symbols, water symbolism in ancient cultures , native americans, ancient greeks , taoism , water symbolism in religions, christianity , hinduism , water in art, photography, and music , quick facts about water.

Water symbols are as ancient as humankind and have been around for thousands of years. They are deeply connected with all cultures, not just as a symbol of survival, but as a mystical element paving the way for life. Water symbols are the chief element in many spiritual rituals and reflect rejuvenation, purification, and healing. However, this is just a summary. Let’s take a closer look at water symbolism and symbols. 

Water has specific representations in various cultures and religions. But it also has general significations that are common to all. Let’s take a brief look at some of the prevalent meanings of water. 

• Symbol of Life: Many people across the globe believe water is a symbol of life because it’s closely associated with birth and rejuvenation. It’s both a symbol of physical and spiritual birth. A baby can only be born after the mother’s water breaks, and likewise, spiritual regeneration is only possible after individuals have cleansed themselves.
• Symbol of Change: Water is commonly depicted as a symbol of change due to its flow and movement. It’s never constrained to one location, and it changes its form to become a river, waterfall, sea, or ocean. This constant movement of water is often used by saints and holy people to inspire humankind to adapt to changes instead of fearing them. 
• Symbol of the Unconscious: Many psychologists and psychiatrists have used the symbol of water to represent the unconscious mind. The ocean is as vast and deep as the unconscious mind, and its bottom can’t be easily discerned. The ocean is also much larger than the conscious realm, which is easily visible and understandable. 
• Symbol of Femininity: Water bodies such as the ocean have been associated with femininity and womanhood. They symbolize rawness, mysteriousness, vastness, and irrationality.
• Symbol of Purification and Forgiveness: Water is used to cleanse oneself, which makes it a symbol of purification. On a spiritual level, this association makes it a symbol of forgiveness, especially in Christianity, as a person’s sins are washed away by water baptism.
• Symbol of Flexibility: Water moves easily, adapting its form to suit its environment. In this way, water is often used as a symbol of flexibility.

Water is represented and depicted through symbols and images. Some of the common ones will be examined in the list below.

• Curvy Lines: Water is often represented by two curvy and squiggly lines. The Native Americans used this to symbolize moving water. 
• Inverted Triangle:   The inverted triangle was used by early Greek philosophers and alchemists as a symbol of the water element.
• Vertical Lines: In ancient China, water was symbolized by vertical lines surrounded by dots on either side. 
• Wavy Lines/Spirals:   Many tribal communities such as the Navahos and Hopi used wavy and spiral-like illustrations to depict water. 
• Crab, Scorpion, and Fish: The Astrological symbols associated with Cancer, Scorpio, and Pisces, are Crab, Scorpion, and Fish. The Crab represents the rejuvenating and reviving aspects of water while Scorpio symbolizes the still and mysterious features. On the other hand, Pisces reflects wisdom, knowledge, and infinity. 

Water has been an integral part of every ancient culture, and no wonder, as it’s the essence of life itself. However, apart from its practical use, water has also held mystical, symbolic meaning to almost every culture.

Native American tribes had different meanings and interpretations of water, but they all agreed that it was something to be honored, revered, and cherished.

In the creation myth of the Lakota people , water was a symbol of purification and nourishment. According to this tale, the creator of the world sent a flood to cleanse and restore the planet. All animals perished, but the crow remained and persuaded the creator to rebuild the land. For this purpose, marine creatures were requested to bring mud from the depths of the ocean.

However, the land was very dry couldn’t be inhabited by living creatures. To nourish the land, the creator shed his very own tears. This myth is commonly found in various Native American cultures and represents water as an emblem of purification and nourishment. 

In Ancient Greek mythology, water was a symbol of power and invincibility. One of the best examples of such powerful water relates to the River Styx .

The River Styx possessed many mystical properties that were both awe-inspiring and fearsome. Achilles, one of the greatest Trojan heroes, had been dipped in the Styx River when he was a young boy, which made him invincible like the gods. However, as his heel hadn’t touched the water, this became his weak point and the cause of his eventual death (hence the term Achilles’ heel ).

The Styx River was also the place where divine Olympian deities made their Oaths. If any of the Gods refused to comply with their promise, they were to face the harshest punishment from the river’s waters. 

In Taoism, water was a symbol of humility, virtue, benevolence, and strength. Tao Te Ching the ancient founder of Taoism compared water with the highest goodness and virtue. According to him, water fulfilled its purpose without any pride and traveled to the lowest points on earth. It also expressed its benevolence by indiscriminately providing for all living creatures.  

But the water wasn’t only soft and kind but also tough and resilient. It could withstand any obstacles, rocks, or metals that came it’s way, and flowed right on. Water served as a great example of how human beings could lead their lives both as benevolent and assertive souls. 

There’s no denying the importance of the symbolic meanings of water in religions across the world. Water features prominently in most religions, representing various symbolic roles.

In Christianity, water is a symbol of transformation, purification, and destruction. Jesus Christ was able to transcend beyond the material world by performing miracles with water, transforming water into wine, and even walking on water.

Water was also a symbol of purification in Christianity, and the process of Baptism stood as a testament to this. When an individual got baptized, they were submerged in holy water to purify their mind, body, and souls. By doing this, the individual could connect with God at a deeper level. It represented a washing away of sins and wrongdoings, and being bathed in God’s forgiveness.

The Bible also describes water as a tool for cleansing and destruction. In the book of Genesis, God sent a flood to destroy everything and return the earth to its former state free from the evils of humans. When this happened, everything became a watery mass, but thanks to Noah’s virtue, he, his family and a pair of every animal were saved.

In Islam, water is a symbol of birth, life, healing, and purification. All living creatures originated from water, and rainwater was sent by God to cleanse and purify earth.

Additionally, Allah revealed the Zam Zam Well to Hagar, in order to save her infant son from thirst. Even today, the Well remains among the holiest places in Islam, and is believed to heal people from illness and disease.  

Water is also a symbol of purification. Even today, Muslims cleanse themselves with water before praying.

In the Hindu religion, water is the most important symbol of spiritual cleansing and purification. The river Ganges, which is said to flow from the head of Shiva, was personified as Goddess Ganga and contained many mystical powers and energies.

The Ganges River was a vehicle for transporting one’s soul to heaven, and many cremations happened along its banks. Water from the Ganges river was also used to wash away one’s sins and begin afresh.

In one Hindu creation myth, the universe was a wide expanse of water, from which Vishnu, the deity of preservation was born. He and Brahma together helped in the creation of the world. 

Many artists, photographers, and musicians have sought inspiration from nature. As a major element of the world, water was one of their greatest sources. 

• Many of Claude Monet’s most famous paintings included water, such as his series showcasing his water lily pond and the Japanese footbridge.
• Photographer Andrew Davidhazy has become famous for his images on water droplets and cups. 
• Franz Liszt composed individual pieces on water, inspiring many musicians to use it as a common theme in their songs.  

In contemporary times, water doesn’t have the same meaning and significance as it did in ancient society. Nowadays, water is mostly associated with relaxing and recreational activities. People love to go on a seaside vacation or visit a spa to be rejuvenated and healed. But most importantly, long baths and showers have become the easiest and most practical way to feel refreshed. 

Today, water is a precious, natural resource that is depleting at an alarming rate .  This is why it’s essential to be aware of ways to save water, follow sustainable practices, use water-saving products ,  and reduce and reuse water as much as possible.

Water and water symbols have been an integral part of ancient societies and cultures. It remains one of the most important elements of nature, and one that continues to hold its importance, both as a physical object and as a symbolic representation of various universal concepts.

Tags: Ancient Symbols

Dani Rhys has worked as a writer and editor for over 15 years. She holds a Masters degree in Linguistics and Education, and has also studied Political Science, Ancient History and Literature. She has a wide range of interests ranging from ancient cultures and mythology to Harry Potter and gardening. She works as the chief editor of Symbol Sage but also takes the time to write on topics that interest her.

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The Symbolism of Water (8 Surprising Meanings)

This article discusses water symbolism, including the following water symbols:

• Refreshment

Water is everywhere you look – the sea, the lake, river, ocean, rain, snow, kitchen, bathroom. It’s the one thing that separates us from the rest of the universe; it’s the difference between life and death. Because of that, it’s become a venerated commodity representing a wide variety of symbolisms.

Water symbolism everywhere. It represents life and fertility , allowing generation upon generation to survive and thrive. Water also represents circulation , which is crucial for our survival and for the recreation of water. It’s also one of the four elements of life, the signs that we connect with – Earth, Air, Fire, and Water.

Because of the importance of water, it has played an important role in many cultures and civilizations of the past and still does to this day. Water is part of many creation stories, which we’ll also take a look at in this article.

The Symbolism of Water

Here’s what water represents.

Water brings life. It is the driving force behind human civilization. Without water, humanity would not exist – life on Earth would not exist, for that matter. More than 70% of Earth’s surface is covered by water. For adult men, 60% of their body composition is composed of water.

What is more, water has been the main driving factor for the growth of human civilization for centuries. From irrigation to the Industrial Revolution, it’s been the key to almost every development throughout our history.

Without water, we can only survive a few days. So do plants, animals, and everything else alive and kicking. Water also separates us from the rest of the universe. It’s the main factor that determines whether a planet is viable for life or not, and whether there is any possibility of life on those planets.

It’s the defining factor that keeps us ticking, and that enables all forms of life to exist on planet Earth. Without water, there is no life.

See Also: Waterfall Symbolism

2. Fertility

Water enables us to grow plants and to reproduce at a relatively fast rate. Everything that we grow requires water, which in turn produces food. And when food is abundant, the conditions are perfect for humans to reproduce.

It’s the source of life that’s needed for humans, plants, and animals to grow and reproduce. A human can survive without food for up to 3 weeks, but one simply cannot survive without water for longer than a few days.

Throughout history, civilizations that had water and that knew how to use it were more prosperous and successful as a rule. Think about Egyptians, who knew exactly how to use the power of the Nile to build farms and create produce that would drive the civilization to prosper.

Or Romans, who built aqueducts in order to drive water to households, enabling each household to have a good stream of water which created prosperity and a good ground upon which the empire would grow.

> Read Also: Growth Metaphors

    More Drinks Symbolism:

• Symbolism of Milk
• Symbolism of Water
• Symbolism of Wine
• Symbolism of Whiskey

3. Refreshment

During the hot summer days, there’s nothing more you would like than to dip yourself into a cool stream or lake. Or to drink some cool water that would help you get through the heat more easily.

In the winter , water is present as snow and ice. It’s this snow and ice that provides perfect conditions during the spring for plants to start growing. As it melts, the ice and snow create streams and rivers that are crucial for the ecosystem.

Water is a symbol of motion, and almost every time you think about it, you envision it as this moving entity. Whether it’s an ocean, sea, river, stream, or even rain, water is almost always connected to motion (see also: ocean symbolism ).

And even when water is stationary, such as in lakes, it was created with the help of a moving type of water, such as rain or river.

This sort of motion was crucial for civilizations of the past and still is today. We use the water’s motion to move between continents and countries with boats and ships.

Sometimes, people used to take advantage of the river’s motion to power water mills, which would eventually lead to the creation of electricity using water’s raw moving power – which still plays an important role today.

See Also: Bubbles in Water Symbolism

5. Circulation

Water is created through circulation. As rain falls, it creates streams, rivers, lakes, and seas. And then these sources of water evaporate, causing even more rain, and so the cycle continues. This cycle is crucial for the survival of us as human species, but also for the survival of everything else living on this planet.

Another way we could connect water to the symbolism of circulation is that rivers usually end up merging into seas. Small rivers flow into larger rivers, and these large rivers end up in seas or oceans, or sometimes lakes. This circulation is important for keeping these waters alive.

Ever since the Roman era, we’ve used water to clean ourselves. Nowadays, water is treated as something that’s self-intuitive, but it was not always like this. In medieval times and even up until the 18th century, clean water was only accessible for the rich and for the upper classes.

Water cleans and purifies anything that comes in contact with. We use it to clean ourselves, to clean our homes, as well as almost everything else we have in our possession.

Rain also sometimes plays this role as the purifying element. After long and hot summer days, rain is always welcome to freshen up nature and clean up what has long been dusting up.

Read Also: What Does ‘Blood is Thicker than Water’ Mean?

Water is still an unexplored entity. We still don’t know how deep we can go to reach the bottom of the deepest ocean. We don’t know what sort of creatures are lurking there, creating an element of mystery to water.

Also, there is an element of mystery to oceans. They are vast and they seem like they’re never-ending, which is mysterious to many people who want to know what’s on the other side of the ocean.

There’s also a feeling of exploration when it comes to water and sea. In the 16th and 17th centuries, many explorers took on waters – many have failed, while others have discovered new continents and new lands.

As humans, we still haven’t mastered the power of water and probably never will. The power of water is shown in the natural disasters that are caused by tsunamis and floods. In a way, that’s also the power of nature itself, although water is the main driver behind those catastrophes.

Water has long been used as a source of creating power itself. We’re now able to master this power of water and use it to create electricity, which is made using dams and in the past, water mills.

Sometimes, we’re just helpless against the power of water. For example, when ships find themselves in storms in the open sea, there’s not much to do but wait out the storm. Or, the phenomenon of the water current taking you in the direction without you knowing – also called rip currents, which are the ocean’s “silent killer”.

Read Also: What does ‘Blood is Thicker than Water’ Mean?

OTHER WATER SYMBOLISM:

• Beach Meaning & Symbolism
• 21 Ocean Metaphors and Similes
• Water Meaning & Symbolism
• 7 Symbolic Meanings of Rain
• River Meaning & Symbolism

The Symbolism of Water in Different Cultures and Religions

Water is such an important entity that it’s always played an important role in our culture. Let’s see how different cultures perceive water.

In Christianity

In Christianity, water is commonly seen with two important acts that are closely tied to Jesus Christ himself:

• Walking on water
• Transforming water into wine

The first one is to demonstrate the power of Jesus in the New Testament. It’s one of the miracles of Jesus where his disciples see him walk on water.

The act of transforming water into wine is depicted in the Gospel of John when Jesus starts transforming water into wine at the Wedding at Cana. This is yet another miracle of Jesus and it’s a very popular symbolism of water in Christianity.

In Hinduism

For Hinduism, water is the symbol of everything that’s alive and it is the most important substance that keeps us and everything else on this planet alive.

In Upanishads, water is also often a symbol of purity and purification, as well as wisdom.

Water in Islam

The symbolism of water in Islam is also the symbolism of life. In the Koran, water is portrayed as the gift from God who gave water to living beings to survive and thrive on planet Earth.

Native Americans

For Native Americans, water was also a symbol of life. They believed that water is the main element that tied the world together and helped them as well as every other creature survive. Because of that, water is commonly present in many creation stories made by Native American tribes.

• Sand Symbolism
• Shell Symbolism

Wrapping Up

Water is everywhere. It is the main driving force of civilizations of the past and today. Without it, we wouldn’t be able to survive, let alone thrive as a species. Because of that, the main water symbolism is life. Water also represents fertility, purity, power, and even mystery. In most cultures and religions, water is also closely connected to life.

You might be interested in other water-related symbolism concepts, like symbolism of ice (frozen water!), or symbolism of water animals, such as flamingos, turtles , or dragonflies.

I’m Chris and I run this website – a resource about symbolism, metaphors, idioms, and a whole lot more! Thanks for dropping by.

The Art of Water: A Multifaceted Exploration of Water in Art and Culture

Water, with its ever-changing forms and symbolic richness, has inspired artists across time and space. Its presence in art is as varied as its physical states, from serene lakes to turbulent seas. This exploration delves into the artistic representations of water, reflecting on cultural appreciation, artistic innovation, and the universal symbolism of water.

Waves of Inspiration: Painting and Water

Timeless beauty: water’s enduring allure in art.

Water has long been a subject of fascination and reverence in the world of art.

Its timeless beauty transcends cultures, eras, and artistic mediums, making it a universal symbol of life, purity, and transformation. Here’s a closer look at how the timeless beauty of water has been captured and celebrated in art.

The Fluidity of Form

Water’s ever-changing form, from calm lakes to roaring oceans, offers endless possibilities for artistic expression. Its fluidity allows artists to explore various techniques and styles, capturing the essence of water’s movement, reflection, and depth. The way light dances on the water’s surface or how ripples form and dissolve creates a visual poetry that has captivated artists for centuries.

Symbolism and Emotion

Beyond its physical beauty, water carries profound symbolic meanings.

It can represent the flow of life, the cleansing of the soul, or the tumultuous nature of emotions. Artists have used water to convey feelings of tranquility, sorrow, joy, or turmoil. Its symbolic versatility adds layers of meaning to artworks, resonating with viewers on a deeply emotional level.

A Mirror to Nature

Water’s beauty is often depicted as a mirror to nature’s wonder. Whether it’s a serene pond reflecting a lush landscape or a stormy sea portraying nature’s fury, water becomes a canvas for the artist’s observation and admiration of the natural world. It invites viewers to reflect on their own connection to nature and the delicate balance that sustains life.

A Universal Muse

The timeless beauty of water has inspired artists from various cultural backgrounds and historical periods.

From ancient frescoes depicting mythical water deities to modern abstract paintings exploring water’s form and color, the artistic representations of water are as diverse as human creativity itself. Water’s universal appeal bridges cultural gaps, creating a shared artistic language that speaks to the human condition.

Modern Reflections: Water’s Contemporary Resonance in Art

Contemporary painters continue to explore water , experimenting with different styles and themes. Water becomes a canvas for expression, reflection, and innovation.

In the modern era, water continues to be a compelling subject for artists, but with new perspectives, techniques, and themes that reflect contemporary concerns and sensibilities. Here’s an exploration of how water’s beauty and symbolism are being reimagined in modern art.

Experimental Techniques

Modern artists are pushing the boundaries of traditional art forms to capture water’s essence.

From digital media to mixed materials, the experimental techniques allow for a more abstract and conceptual exploration of water. Artists play with texture, color, and form to create visual experiences that evoke water’s fluidity and dynamism.

Environmental Consciousness

With growing awareness of environmental issues, water has become a symbol of sustainability , conservation, and ecological balance. Artists are using water-themed artworks to raise awareness about pollution, climate change, and water scarcity . Water’s beauty is juxtaposed with its fragility, urging viewers to reflect on their relationship with nature and their responsibility to protect it.

Cultural Commentary

Water’s representation in modern art often extends beyond aesthetics to engage with social and cultural commentary. Whether addressing migration across seas, urbanization along waterfronts, or the commodification of bottled water, artists use water as a metaphor to explore complex societal issues. Water becomes a lens through which to view and critique contemporary life.

Interactive and Immersive Art

Modern art often seeks to engage the audience in more interactive and immersive ways.

Water-themed installations, virtual reality experiences, and participatory art projects invite viewers to physically interact with water or to experience it through different sensory dimensions. These innovative approaches create a more personal and profound connection with water’s artistic representations.

Global Perspectives

In a globalized world, water’s artistic representations are enriched by diverse cultural perspectives. Artists from different parts of the world bring their unique insights, traditions, and contexts to their exploration of water. The result is a multifaceted and inclusive artistic dialogue that celebrates water’s universal significance while honoring cultural diversity.

Fluid Dance: Water in Dance Forms

Rhythmic flow: water’s influence on dance.

The rhythmic flow of water has been a source of inspiration for dance, a performing art that embodies movement, grace, and expression. From traditional folk dances to contemporary performances, water’s fluidity and symbolism have found resonance in the world of dance. Here’s an exploration of how water’s rhythmic flow has shaped and enriched dance forms across cultures and eras.

Water’s Movement in Dance Choreography

The fluidity of water’s movement, its gentle waves, and sudden splashes have been translated into dance choreography. Dancers emulate the flow of water through their bodies, creating a visual poetry that captures water’s grace and dynamism. The undulating movements, the seamless transitions, and the rhythmic patterns all mirror water’s natural dance.

Symbolism and Storytelling

Water’s symbolic richness has been woven into dance narratives.

It can represent life’s journey, emotional depth, purity, or transformation. In many cultural dances, water is a central theme that tells stories of love, loss, celebration, or spiritual connection. Its symbolic versatility adds layers of meaning and emotional resonance to dance performances.

Dance Forms Inspired by Water

Various dance forms around the world have been inspired by water. From classical ballets depicting mythical water creatures to folk dances celebrating rivers and seas, water’s influence is diverse and profound. Some dances are performed near water bodies during specific festivals, strengthening the connection between dance, water, and cultural heritage.

Contemporary Interpretations

Modern dance has embraced water’s rhythmic flow in innovative ways.

Choreographers experiment with water-themed performances that incorporate multimedia, water installations, or underwater dance. These experimental approaches create sensory experiences that transcend traditional dance boundaries, offering new perspectives on water’s beauty and complexity.

Water and Dance Therapy

The therapeutic qualities of both water and dance have been combined in some therapeutic practices. Water’s calming effect and dance’s expressive power are used to promote physical and emotional well-being. The rhythmic flow of water and dance becomes a healing journey that nurtures the body and soul.

Dance and Water

Modern dance performances that interact with water offer a sensory experience that transcends traditional boundaries, creating a fusion of art and nature.

Sculpting the Sea: Water in Sculpture

Form and meaning: water’s influence in sculpture.

Water’s fluid form and rich symbolism have made it a compelling subject in the realm of sculpture.

Artists have explored water’s physical properties and metaphorical meanings, creating works that challenge perceptions, invite reflection, and engage with broader themes of life, nature, and human experience. Here’s an exploration of how water’s form and meaning have been captured and expressed in sculpture.

Water as Material

Water itself has been used as a sculptural material, creating temporary or interactive artworks. Its transparent, reflective, and fluid nature offers unique aesthetic possibilities. Artists have manipulated water’s form through fountains, ice sculptures, or water-based installations, playing with light, movement, and space to create visually stunning experiences.

Water as Metaphor

Beyond its physical form, water has been a powerful metaphor in sculpture.

It can symbolize the flow of time, the cycle of life, the complexity of emotions, or the interconnectedness of all beings. Sculptors have used various materials to represent water’s symbolic meanings, creating works that resonate with viewers on a deeply philosophical and emotional level.

Interaction with Other Elements

Water’s form in sculpture often interacts with other natural elements like stone, metal, or glass. The juxtaposition of water’s fluidity with the solidity of other materials creates dynamic contrasts and harmonies. These interactions explore themes like balance, transformation, or duality, reflecting the complex relationships within nature and human existence.

Public Water Sculptures

Water sculptures in public spaces have become iconic landmarks in many cities .

From grand fountains to contemporary water installations, these public artworks create social spaces that foster community engagement and cultural identity. They transform urban landscapes, adding beauty, meaning, and vitality to public life.

Environmental Art and Sustainability

Some sculptors have used water-themed artworks to address environmental issues and promote sustainability. Water’s form and meaning become a visual and conceptual exploration of ecological consciousness, conservation, and human responsibility. These artworks invite viewers to reflect on their relationship with nature and their role in preserving the planet’s delicate balance.

Innovation and Technology

Modern sculptors are incorporating technology and innovation to reimagine water’s form in sculpture.

From digital water projections to kinetic water sculptures, these innovative approaches expand the creative possibilities of water in art. They create immersive and interactive experiences that engage viewers in new and unexpected ways.

Art and Community: Water’s Role in Building Connections

Water, with its universal presence and multifaceted symbolism, has played a significant role in fostering art and community connections. From public art installations to community-driven projects, water-themed art has become a catalyst for social engagement, cultural expression, and community building. Here’s an exploration of how water’s influence in art has nurtured connections within communities.

Public Water Art Installations

Public water art installations, such as fountains, sculptures, or murals, have become gathering points in urban landscapes. These installations are not just aesthetically pleasing; they create communal spaces where people can interact, reflect, and celebrate. They often become symbols of local identity and pride, fostering a sense of belonging and community cohesion.

Cultural Celebrations and Festivals

Many cultures have festivals and celebrations centered around water bodies, where art plays a vital role. Whether it’s a river festival with floating lanterns or a coastal celebration with sand sculptures, water-themed art brings communities together in joy, reverence, and shared cultural heritage. These events strengthen communal bonds and preserve cultural traditions.

Environmental Advocacy through Art

Water-themed art has been used to raise awareness about environmental issues such as water conservation , pollution, and climate change. Community art projects that focus on water’s importance often engage local residents in creative expression and environmental stewardship. These projects foster a sense of responsibility and empower communities to take collective action.

Healing and Therapeutic Spaces

Artistic representations of water in community centers, hospitals, or therapeutic spaces create calming and healing environments. Water’s soothing imagery or sounds can facilitate relaxation, meditation, and emotional well-being. These artistic interventions support community health and provide spaces for solace and reflection.

Educational Initiatives

Water-themed art has been integrated into educational initiatives to engage children and adults in learning about water’s scientific, historical, and cultural aspects. Interactive art installations, workshops, or community murals can make learning about water engaging and accessible. These initiatives foster curiosity, creativity, and a sense of connection to the natural world.

Collaborative Art Projects

Community-driven water art projects encourage collaboration, creativity, and dialogue.

Whether it’s a community garden with water features or a collaborative mural depicting local water landscapes, these projects bring diverse community members together. They facilitate conversations, build relationships, and celebrate communal creativity.

Literary Rivers: Water in Literature

Water has long been a powerful symbol and motif in literature, flowing through the pages of novels, poems, and essays. Its presence is felt in the form of rivers, oceans, rain, and even tears. From ancient epics to contemporary novels, water’s influence in literature is as profound as it is diverse. Here’s an exploration of how water, especially in the form of rivers, has shaped literary landscapes and narratives.

Rivers as Metaphors

Rivers often serve as metaphors in literature, representing life’s journey, time’s passage, or the flow of consciousness. They can symbolize transformation, renewal, or the inexorable force of destiny. The winding path of a river may mirror a character’s inner struggles, growth, or moral dilemmas, adding depth and complexity to the narrative.

Setting and Atmosphere

Rivers create vivid settings and atmospheres in literary works.

They can be serene and idyllic, reflecting tranquility and beauty, or dark and turbulent, evoking conflict and danger. The description of rivers can set the tone, mood, and context of a story, immersing readers in a specific time, place, and emotional landscape.

Cultural and Historical Context

Rivers have played essential roles in various cultures and histories, and this significance is often reflected in literature. They may be depicted as sacred sites, trade routes, or natural boundaries. Literary rivers can carry historical, social, or cultural meanings, connecting readers to a broader context and enriching the narrative’s authenticity and resonance.

Characters and Relationships

In literature, rivers often influence characters and their relationships.

They can be places of contemplation, love, separation, or reconciliation. Characters’ interactions with rivers or each other near rivers can reveal insights into their personalities, desires, conflicts, or transformations. Rivers become stages where human dramas unfold.

Themes of Nature and Ecology

Rivers in literature often explore themes of nature, ecology, and human’s relationship with the environment. They can be symbols of nature’s beauty, power, or fragility. Literary works that focus on rivers may raise awareness about environmental issues, such as pollution, conservation, or climate change, engaging readers in ethical and philosophical reflections.

Spiritual and Mythical Dimensions

Rivers have spiritual and mythical dimensions in many literary traditions.

They can be pathways to the divine, sites of mythical events, or symbols of spiritual purification. Rivers in religious texts or mythological tales carry profound spiritual meanings, connecting readers to universal truths, moral lessons, or transcendent experiences.

Cinematic Oceans: Water in Film

Water’s captivating presence in film is as vast and deep as the oceans themselves. From thrilling seafaring adventures to profound philosophical explorations, water’s influence in cinema is multifaceted and enduring. Oceans, in particular, have been a recurring theme and setting in films across genres and cultures. Here’s an exploration of how oceans have shaped cinematic narratives, aesthetics, and emotions.

Oceans as Setting and Spectacle

The vast expanse of the ocean provides a stunning backdrop for cinematic storytelling.

Whether it’s a remote island, a bustling seaport, or the open sea, oceans create visually spectacular settings. They offer filmmakers a canvas to craft breathtaking visuals, from serene sunsets to raging storms, adding grandeur and visual richness to films.

Adventure and Exploration

Oceans have been the stage for thrilling adventures and explorations in cinema. From pirate tales to deep-sea expeditions, ocean-themed films often celebrate human courage, curiosity, and resilience. They take audiences on exciting journeys, filled with challenges, discoveries, and triumphs, capturing the adventurous spirit of the sea.

Mystery and Metaphor

The ocean’s depths are often used to symbolize mystery, the unknown, or the subconscious.

Films may use the ocean as a metaphor for human emotions, existential questions, or spiritual quests. The ocean’s vastness and depth can reflect characters’ inner complexities, desires, or conflicts, creating layered and thought-provoking narratives.

Human and Nature Relationship

Films that focus on oceans often explore the relationship between humans and nature. They may highlight themes of ecological awareness, conservation, or human impact on marine life. These films foster reflection on environmental ethics, sustainability, and our responsibility as stewards of the planet’s oceans.

Survival and Resilience

Oceans in cinema have been the setting for gripping survival stories.

Characters stranded at sea, battling nature’s fury, become symbols of human resilience, determination, and survival instinct. These films often delve into psychological and physical struggles, offering intense and inspiring cinematic experiences.

Romance and Emotion

The beauty and romance of oceans have been captured in love stories and emotional dramas. The sound of waves, the dance of light on water, or a walk on the beach can evoke emotions of love, longing, or nostalgia. Oceans become spaces for romance, reflection, and emotional connection, resonating with audiences’ hearts.

Technology and Innovation

Filming oceans has led to technological innovations and creative techniques in cinema.

From underwater cinematography to visual effects, filmmakers have pushed technological boundaries to depict oceans authentically and artistically. These innovations have expanded cinematic language and possibilities, enhancing storytelling and visual artistry.

Musical Raindrops: Music Inspired by Water

Water’s gentle flow, rhythmic waves, and soothing sounds have been a source of inspiration for musicians and composers throughout history. From classical symphonies to contemporary soundscapes, water’s influence in music is as diverse as it is profound.

Classical Compositions

Many classical composers have been inspired by water’s various forms and movements.

Pieces like Handel’s “Water Music” or Debussy’s “La Mer” evoke the beauty, power, and tranquility of water through orchestration and melody. These compositions create auditory landscapes that transport listeners to rivers, oceans, or rain-soaked gardens.

Jazz and Blues

Water’s fluidity and rhythm have found expression in jazz and blues. Whether it’s a song about a river or the use of water sounds in instrumentation, these genres have explored water’s emotional and musical dimensions. The improvisational nature of jazz resonates with water’s unpredictability and flow.

Contemporary and Ambient Music

Modern musicians and sound artists have used water’s sounds and symbolism to create ambient, electronic, or experimental music. Water’s natural sounds, such as raindrops, waves, or streams, are often integrated into compositions, creating immersive soundscapes that connect listeners to nature and inner calm.

Symphony 1 Music Playlists

Symphony 1 Music has curated playlists and songs that celebrate water’s influence in music. These tracks blend various musical styles and sonics, capturing water’s essence through sound.

• Water : A track of pieces inspired by the ocean’s majesty and mystery. Perfect for relaxation, meditation, or contemplation.
• Water 2 : A track that captures the flow and rhythm of rivers, suitable for focus, creativity, or simply enjoying a peaceful moment.
• Water Fountain Sounds : A soothing hour long track of water fountain sounds, ideal for unwinding, reading, or introspection.
• Rain Sounds : A peaceful track for well-being, featuring calm rain sounds that celebrate water’s healing and nurturing qualities.
• Ocean Waves : A relaxing track featuring soothing ocean wave sounds.

Music for Healing and Well-being

Water-inspired music has been used for healing, therapy, and well-being.

The calming sounds of water, combined with harmonious melodies, can facilitate relaxation , stress reduction, and mindfulness. Symphony 1 Music’s playlists are designed to enhance mental and emotional well-being, connecting listeners to water’s therapeutic qualities.

Photographing Water: A Visual Exploration

Essence and texture.

Photographing water reveals its texture, reflection, and mystery. It’s a visual exploration of nature’s beauty and the cultural relationships with water.

Water in Fashion: Design and Expression

Fluid creativity.

Water inspires fashion creativity, elegance, and sustainability. Its themes and motifs shape fashion choices, expressions, and lifestyles.

Sacred Waters: Water in Religion

Divine connection.

Water’s spiritual symbolism transcends religious boundaries. It’s a sacred element, a conduit of divine connection, purification, and blessing.

Digital Waves: Water in Digital Art

Virtual exploration.

Digital artists create realistic water effects, enhancing storytelling and aesthetics. Virtual water expands the creative possibilities of water in the digital age.

Conclusion: A Symphony of Art and Nature

The art of water is a symphony of creativity, culture, and connection. It invites us to reflect on the essence of life, to savor the nuances of artistic expression, and to celebrate the universal allure of water. Whether through painting, dance, literature, or digital media, water continues to inspire, challenge, and connect us in a dance of creativity and cultural appreciation.

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Designing Water-Smart Cities: Integrating Water Management into Urban Planning

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Water, tradition, and innovation: flowing through japan’s cultural history.

Water, a gift from nature, is an essential part of our daily lives. People use water every day, everywhere, for everything—often without much consideration for its significance in terms of the larger social, cultural, historical, economic, and environmental implications for the twenty-first century. As an island nation, Japan has a deep connection with water in various ways, creating a cultural history where water and life go hand in hand. In Japan, water bridges past and present, tradition and modernity. The usage of water and its connections to identity and lifestyles are highlighted in cities, religion, daily life, food, visual art, and technological innovations. Japan’s relationship with itself and the world—the harmonious unification of ancient Japanese tradition and modern innovation, which defines the country’s sense of its cultural heritage and history—is clearly evident through the significance of Japanese people not only using water wisely, but also in some very culturally specific ways.

Kyoto, City of Water

A chain of islands surrounded by the Sea of Japan to the west, the East China Sea to the southwest, and the Pacific Ocean to the east, Japan is a country with universal access to a saltwater supply. However, Japan’s rugged and mountainous terrain poses a challenge for the country’s freshwater supply. Although blessed with abundant rainfall and lush vegetation, the reason for Japan’s limited freshwater supply is the lack of natural water reservoirs, the short and swift runoff of rivers, and the engineering difficulties of constructing large-scale drainage basins in the rugged mountains. Japan’s largest freshwater lake is Lake Biwa, which is located in west-central Honshu, Shiga Prefecture, northeast of Kyoto. Lake Biwa is fed by small rivers flowing from the surrounding mountains and serves as a freshwater reservoir for Kyoto, making the city more freshwater- than saltwater-based. Kyoto, one of Japan’s oldest cities and well-known tourist destinations, has enjoyed the benefits of water for centuries and prospered as a result. Located in the southwestern part of Honshu, Japan’s largest island, Kyoto today has a population around 1.5 million and is famous for its rich cultural heritage. During the Heian period (794–1185), Kyoto was called the “Capital of Peace and Tranquility”, and it was the capital and seat of the Imperial court, as well as the intellectual and cultural center for the production of masterpieces of Japanese literature, art, and architecture such as The Tale of Genji and The Phoenix Pavilion(Byōdō-in) in Uji, which are still admired today as the fruits of Heian Japan’s Golden Age. Today, Kyoto is one of the few cities in Japan combining historic temples, traditional Japanese houses, and prewar architecture with innovation, as is visible in Kyoto’s modern central train station and the city’s expansive high-speed transit system.

Japan’s largest freshwater lake is Lake Biwa, which is located in west-central Honshu, Shiga Prefecture, northeast of Kyoto.

Situated in the Yamashiro Basin, Kyoto is surrounded by three rivers, the Uji River to the south, the Katsura River to the west, and the Kamo River to the east. These rivers form an alluvial fan that lets water percolate down, creating a supply of ample freshwater so vast that Kyoto is often described as a city sitting atop a large natural water table. In addition to being one of the main water supplies for Kyoto and its surrounding areas, the water flowing through the three rivers also functions in other significant ways—both symbolic and practical—demonstrating how Japanese people have been using water in contexts according to their beliefs and customs, from the past to the present. For example, during the Heian period, the Kamo River, which flows from northeast to south, was prone to overflowing, causing major problems for the people living along its banks in and around the capital. Therefore, in accordance with Japanese geomancy belief that the northeast direction is a source of evil and misfortune, Shintō shrines, collectively called Kamo jinja (Kamo shrines), were built along the Kamo River to pray for protection from its wrath. The most famous of these Shintō shrines are Kamo-wakeikazuchi Shrine and Kamo-mioya Shrine in Kyoto. The United Nations Educational, Scientific, and Cultural Organization (UNESCO) designates both as World Heritage Sites. 1

Today, during the hot summer months from June to August, Kyoto street scenes are still enlivened by old customs that use water effectively. Yuka are wooden decks that are common sights along Kyoto’s rivers where travelers can relax and gain relief from the heat of the sun. Another popular practice, dating back to the Tokugawa period (1600–1868), is uchimizu, which refers to the act of sprinkling water on sidewalks and streets for the immediate benefit of cleaning and cooling (Figure 1). Uchimizu is also customary in temples and gardens, where the harmony between humans, water, and nature has a more ritual and contemplative function, tracing its roots back to ancient Shintō purification rites. The key to understanding people’s growing interest in this ancient custom, as well as its present-day popularity, lies in its environmental implication. Interestingly, the water utilized for uchimizu practice is not freshwater taken from rivers or household taps, but water that people recycled from their baths, rain, and air-conditioning equipment. In this way, uchimizu practice successfully exemplifies traditional religious, aesthetic, and utilitarian Japanese values by combining ends while at the same time making Japanese people think about using wastewater efficiently. In 2003, Japan conducted its first Mission Uchimizu, a campaign that took the country by storm in an effort to address the issues of climate change and heat-island effect. This phenomenon refers to air temperatures in densely built urban areas being higher than the temperatures of the surrounding rural areas—an increasing problem in large cities like Tokyo. These higher urban temperatures have a serious impact on the electricity demand for air conditioning in buildings and increase smog production, as well as the emission of pollutants. By means of social media and other campaigns, 340,000 people participated in the 2003 Mission Uchimizu, and as a result, in places where uchimizu occurred, the temperature decreased by about one degree. The 2005 Mission Uchimizu became a national movement, with 7.7 million participants. 2

Japanese Religion and Water

Apart from being places of worship along the Kamo River, Shintō shrines offer a more in-depth look into the relationship between Japanese religion and water. Shintō, literally meaning “the way of the gods,” is the indigenous religion of Japan and focuses on the worship of nature deities called kami—invisible spirits believed to dwell in elements of nature such as trees, rocks, and waterfalls. Frequently, ropes are tied around trees, marking them as sacred places of the kami. There are many Shintō shrines throughout Japan, but one of the two most famous shrines is the Ise Grand Shrine. 3 The Ise Shrine is dedicated to the Sun goddess Amaterasu, who is also venerated as the ancestor of the Japanese Emperor. Prior to entering a Shintō shrine, people pass through a simple wooden gate called tori, which symbolically separates the secular realm from the religious one, and purify themselves with freshwater from basins at the shrine entrance before praying and giving offerings to the kami (Figure 2). Harae, or purification rituals, play a significant role in Shintō and involve cleansing one’s hands and mouth with water. Harae is performed at the beginning of the Shintō ceremony so that people can rid themselves of any kind of pollution that, according to popular folk belief, includes sins, uncleanliness, bad luck, and diseases.4 This ritual cleansing is achieved by symbolic washing with water or by having a Shintō priest wave a wooden wand with haraigushi (white zigzag paper streamers) attached to its end in a left-right-left movement over the person, object, or place to be purified. Harae purification rites take place in various contexts. For example, during the annual Seven-Five-Three (Shichi-go-san) festival, a Shintō priest symbolically purifies children (three- and five-year-old boys, three- and seven-year-old girls) to bestow them with health and prosperity. Shintō purification is also commonly performed at the groundbreaking ceremonies of homes and businesses, in a car before it is first driven, or in an airplane before it takes off in flight for the first time to ensure good luck and safety.

The origin of harae can be traced back to the mythological stories associated with the Shintō deity Amaterasu in the Kojiki ( Record of Ancient Matters ). The Kojiki , compiled in 712, is Japan’s oldest extant historical record recounting the country’s mythological origin legend. In the Kojiki account:

One day, while Amaterasu was supervising the weaving of the garments of the gods in the pure weaving hall, Susano-o broke through the roof and let fall a heavenly horse which had been flayed. This startled one of Amaterasu’s attendants who, in her agitation, accidentally killed herself with the loom’s shuttle. Amaterasu fled to the heavenly cave Amano-Iwato. Susano-o was subsequently expelled from heaven and Amaterasu’s sovereignty resumed. The traditional harae Shintō purification ritual is represented when Susano-o is removed from heaven. 5

Another distinctive ritual related to harae is the rite of water purification called misogi. Originally, misogi denoted a practice of purification where devotees stood under the pounding cascade of a waterfall while chanting prayers. Since waterfalls are viewed as natural habitats of Shintō kami, the misogi practice elevates the harae purification practice to an immediate harmonious union between devotees and Shintō gods. The origin of misogi, a purification rite using water to free body and mind from pollution, can be traced back to the mythological stories of Izanami and Izanagi in the Kojiki.

Izanagi, after visiting the “bottom country,” or place of afterlife, returned to Earth and ritually bathed in the Tachibana River to rid himself of impurities. 6

Today, the misogi ritual encompasses more meanings of cleansing one’s body and soul, includes drinking sacred water from a waterfall, standing in a river or in an artificially constructed pool in front of a popular Shintō shrine, and emerging one’s entire body in water. Women wear white kimono-like robes and men white loincloths; both wear headbands (Figures 3 and 4). 7 For example, at Kiyomizu Temple in Kyoto, people flock to the “Sound-of-Wings” waterfall, but the majority of visitors only drink the water instead of bathing in it, signaling a shift from tradition to modernity.

Japanese Cuisine and Water

The purity of water and its usage also plays an important role for traditional Japanese cuisine or washoku. The heart of traditional Japanese cuisine is dashi, a broth made by steeping a variety of dried foods in water to release their flavor (Figure 5). It is the base for a clear soup ( suimono ) and miso soup, which are essential parts of traditional Japanese meals. Once the dashi is made, ingredients for the soup are added, simmered in, and then seasoned with soy sauce or salt for suimono and miso for miso soup. As for the dashi itself, the most common ingredients are a type of seaweed called kombu (kelp) and katsuo-bushi shavings (bonito fish that has been heated to dry it). Other items used to make dashi are small dried sardines, or niboshi, and dried shitake mushrooms. Due to the dried ingredients, dashi contains no fat, setting it apart from the soup stock of Western and Chinese cuisine. Regarding the making of dashi, the katsuo-bushi shavings are added to the water after it begins to boil, whereas the kombu, niboshi, and dried shitake are generally steeped in cold water first, after which you have a choice depending on the meal you are preparing— either bring the temperature up to simmer or take the ingredients out before heating the water. Due to Japan’s topography, the water is very soft, which gives the dashi its light characteristic of washoku cuisine.

Among the many representations of water in ancient and modern Japanese art, the most famous motif is the wave.

Water and Art

In addition to constituting an indispensible element of traditional Japanese cuisine, water is also an integral part of the Japanese visual arts. Among the many representations of water in ancient and modern Japanese art, the most famous motif is the wave. The simplest depiction is that of concentric semicircles aligned in a staggered fashion to symbolize waves, a pattern known as seigaiha (blue ocean waves) (Figure 6). The origins of seigaiha can be traced back to the gagaku, ancient Japanese court music and dance, that flourished under the patronage of the Imperial Court in the Heian period. The seigaiha pattern is believed to have originated from a gagaku piece by the same name, where dance performers wore costumes decorated with this blue ocean wave pattern. Although simple, the gentle waves extend in all directions without end, creating a feeling of happiness and good luck that hopefully will last forever. This seigaiha motive is a favorite in Japan, a nation surrounded by the sea, as well as with foreign visitors. Today, items decorated with this blue ocean wave pattern include silk kimono, yukata (traditional Japanese cotton robes), hand towels, and tableware such as cups, plates, and bowls. Most recently, the seigaiha symbolic wave patterns have also become popular among graphic designers and tattoo artists, two growing art trends in twenty-first-century Japan bridging tradition and innovation.

Japan: Water Conservation and Sustainable Living

Water conservation and sustainable living in Japan have overcome numerous challenges. As mentioned in the beginning of this essay, Japan benefits from universal access of water due to its topography. In terms of sustainable living, Japan also has one of the lowest levels of water distribution losses in the world, as well as very high standards for the quality of its drinking water and treated wastewater. While Japan is not a country stressed for water, the country’s water availability varies greatly from year to year, season to season, regionally, and at times of natural disasters such as droughts or earthquakes. In an effort to promote water conservation and sustainable living, the Japanese have come up with innovative technologies for water conservation and sanitation. 8

The most frequently encountered and commonly used innovative technology of this sort is the modern Japanese toilet. Originally, Japanese toilets were simple squat toilets that looked like miniature urinals set horizontally into the floor. Today, this kind of toilet can still be found in some public places such as train station restrooms. However, after World War II, the introduction of the Western-style toilet, which has a pedestal for sitting, marked the beginning of what would become Japan’s revolutionary toilet industry, leading up to today’s high-tech, water-saving, and environmentally friendly “super toilets.” 9 Most of these modern super toilets are produced by Toto Ltd., which is the leading Japanese company in this business. 10 The indigenous Japanese word for toilet is ōtearai (hand washing), which is significant because it ties the simple cleansing act of washing with the innovative feature of water conservation. One of the unique features of the Japanese toilet is the sink mounted on top of the water tank. A spigot on top of the tank allows users to conserve water by washing their hands with clean water coming directly from the wall outlet (Figure 7). During hand washing, the water collects in the tank in order to be used for the next flush. Another feature of the Japanese toilet design that bridges tradition and modernity are the large and small flush buttons that conserve water on common, less advanced toilets. A more technologically advanced feature of the modern Japanese toilet is the control panel that lets the user choose from functions and settings such as automatic lid opener, flushing modes, water jet adjustments, sound, seat heating, and massage options. For example, the luxury Toto Neorest hybrid-style toilet conserves water by using three different flush buttons that use only one gallon of water per flush.

In addition, this model includes an innovative tornado flushing system that uses a whirlpool technique to force everything to the center of the bowl, and then the jet wash system takes over, combining two types of flow to clean the bowl thoroughly. 11 In terms of sustainable living, this water-saving and self-cleaning innovation reduces the amounts of detergent and toilet paper released into the environment. On the flipside, these toilets consume energy and are rendered useless in power outages.

The role of water in Japan, especially its contribution to aspects of daily life, religion, food, art, and innovation, is still closely tied to ancient Japanese tradition and is constantly evolving in often-creative ways. Increased attention to the usage of water in Japan promises an exciting outlook for future innovation and research concerning Japanese people not only using water efficiently, but also in some very culturally specific ways.

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1. See, for example, the “Links” page from the Kamigamojinja website, accessed April 1, 2017, http://tinyurl.com/yb6v3pjw .

2. For more information, see the Uchimizu website (in English), accessed June 20, 2017, http://tinyurl.com/ycmzu58w .

3. Izumo is the other most revered shrine.

4. Eyal Ben-Ari, “Transformation in Ritual, Transformation of Ritual: Audiences and Rites in a Japanese Commuter Village,” Ethnology 30, no. 2 (1991): 135–147.

5. This is a summary of the Kojiki story and its relevance to the harae purification ritual. For a detailed discussion about this story, its analysis, and its relevance to Shintō purification rituals, see Alan L. Miller, “Ame no miso-ori me [The Heavenly Weaving Maiden]: The Cosmic Weaver in Early Shintō Myth and Ritual,” History of Religions 24, no. 1 (1984): 27–48.

6. Ann Llewellyn Evans, Shintō Norito: A Book of Prayers (Victoria: Trafford, 2001), 167.

7. For further reading on the misogi purification rite, see Appendix C: Misogi and Spiritual Exercises from Guji Yukitaka Yamamoto’s “Kami no Michi: The Life and Thought of a Shintō Priest,” Tsubaki Shrine, accessed June 20, 2017, http://tinyurl. com/y82ryu6l .

8. Ninety-seven percent of the Japanese population receive their water supply from public utilities, and 3 percent, mainly people living in rural areas, receive water from their own wells. See “Overview of Water Supply Services” from the Japanese government Ministry of Health, Labor, and Welfare website, accessed June 20, 2017, https:// tinyurl.com/y9e5ngvt .

9. For more on these “super toilets,” see “The History of Toilets in Japan,” Web-Japan, accessed June 20, 2017, http://tinyurl.com/y8956ucp .

10. In addition to toilets, Toto Ltd. also makes water-saving showerheads and other bathroom furniture.

11. For more, see “Japan’s Smart, Gorgeous Toilets,” CNet, accessed June 20, 2017, http:// tinyurl.com/yaxm3sma .

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Liquid water

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Where does water come from?

Why do cold water bottles and soft-drink bottles sweat, when does water boil, why is water blue, when is water the most dense.

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Water is made up of hydrogen and oxygen , and it exists in gaseous , liquid , and solid states. Water is one of the most plentiful and essential compounds , occurring as a liquid on Earth’s surface under normal conditions, which makes it invaluable for human uses and as plant and animal habitat . Since water is readily changed to a vapour (gas), it can travel through the atmosphere from the oceans inland, where it condenses and nourishes life .

A cold water bottle appears to sweat because it’s a cooling source for the water vapour in the layer of air that surrounds the bottle. Air that is relatively warm can hold more water vapour than cooler air.  When the cold water bottle is introduced, the warm air near the bottle cools and some of the water vapour condenses into liquid water, which is then deposited on the outside of the bottle.

Boiling occurs when bubbles form within a liquid, marking a change from a substance’s liquid or solid phase into a gas. The normal boiling point is the temperature at which the liquid’s vapour pressure equals the standard sea level atmospheric pressure (760 mm [29.92 inches] of mercury). At sea level, atmospheric pressure is high, and water boils at 100 °C (212 °F); at higher altitudes it is lower, so water boils at a lower temperature.

Water appears blue for two important reasons. In small quantities water appears colourless, but water actually has an intrinsic blue colour caused by the slight absorption of light at red wavelengths . For larger bodies of water—ponds, rivers , lakes , and oceans—water appears blue on clear days because it mirrors the blueness of the sky. On overcast days, larger water bodies appear gray.

Water’s density is greatest at about 4 °C (39.2 °F), in the liquid phase. Ice , water’s solid phase, is more buoyant, so it forms at the surface of water bodies and freezes downward. Lakes and rivers rarely freeze completely, and the liquid water below can become a winter refuge for aquatic life. When ice melts in the spring, the slowly warming surface meltwater sinks, displacing the water below and mixing nutrients throughout the water column.

Should bottled water be banned?

Whether bottled water should be banned is hotly debated. Some say banning bottled water would reduce waste and protect the environment while saving money. Others say banning bottled water removes a healthy choice and leads to increased consumption of unhealthy sugary drinks. For more on the bottled water ban debate, visit ProCon.org .

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water , a substance composed of the chemical elements hydrogen and oxygen and existing in gaseous, liquid, and solid states. It is one of the most plentiful and essential of compounds . A tasteless and odourless liquid at room temperature, it has the important ability to dissolve many other substances. Indeed, the versatility of water as a solvent is essential to living organisms. Life is believed to have originated in the aqueous solutions of the world’s oceans , and living organisms depend on aqueous solutions, such as blood and digestive juices, for biological processes. Water also exists on other planets and moons both within and beyond the solar system. In small quantities water appears colourless, but water actually has an intrinsic blue colour caused by slight absorption of light at red wavelengths.

Although the molecules of water are simple in structure (H 2 O), the physical and chemical properties of the compound are extraordinarily complicated, and they are not typical of most substances found on Earth . For example, although the sight of ice cubes floating in a glass of ice water is commonplace, such behaviour is unusual for chemical entities. For almost every other compound, the solid state is denser than the liquid state; thus, the solid would sink to the bottom of the liquid. The fact that ice floats on water is exceedingly important in the natural world, because the ice that forms on ponds and lakes in cold areas of the world acts as an insulating barrier that protects the aquatic life below. If ice were denser than liquid water, ice forming on a pond would sink, thereby exposing more water to the cold temperature. Thus, the pond would eventually freeze throughout, killing all the life-forms present.

Water occurs as a liquid on the surface of Earth under normal conditions, which makes it invaluable for transportation, for recreation, and as a habitat for a myriad of plants and animals . The fact that water is readily changed to a vapour (gas) allows it to be transported through the atmosphere from the oceans to inland areas where it condenses and, as rain , nourishes plant and animal life. ( See hydrosphere: The hydrologic cycle for a description of the cycle by which water is transferred over Earth.)

Because of its prominence, water has long played an important religious and philosophical role in human history. In the 6th century bce , Thales of Miletus , sometimes credited for initiating Greek philosophy , regarded water as the sole fundamental building block of matter:

It is water that, in taking different forms, constitutes the earth, atmosphere, sky, mountains, gods and men, beasts and birds, grass and trees, and animals down to worms, flies and ants. All these are different forms of water. Meditate on water!

Two hundred years later, Aristotle considered water to be one of four fundamental elements, in addition to earth, air , and fire. The belief that water was a fundamental substance persisted for more than 2,000 years until experiments in the second half of the 18th century showed that water is a compound made up of the elements hydrogen and oxygen.

The water on the surface of Earth is found mainly in its oceans (97.25 percent) and polar ice caps and glaciers (2.05 percent), with the balance in freshwater lakes, rivers, and groundwater . As Earth’s population grows and the demand for fresh water increases, water purification and recycling become increasingly important. Interestingly, the purity requirements of water for industrial use often exceed those for human consumption . For example, the water used in high-pressure boilers must be at least 99.999998 percent pure. Because seawater contains large quantities of dissolved salts, it must be desalinated for most uses, including human consumption.

This article describes the molecular structure of water as well as its physical and chemical properties. For other major treatments of water, see climate ; environmental works ; hydrosphere ; ice ; and pollution .

Structure of water

The water molecule is composed of two hydrogen atoms , each linked by a single chemical bond to an oxygen atom . Most hydrogen atoms have a nucleus consisting solely of a proton . Two isotopic forms, deuterium and tritium , in which the atomic nuclei also contain one and two neutrons , respectively, are found to a small degree in water. Deuterium oxide (D 2 O), called heavy water , is important in chemical research and is also used as a neutron moderator in some nuclear reactors .

Although its formula (H 2 O) seems simple, water exhibits very complex chemical and physical properties. For example, its melting point , 0 °C (32 °F), and boiling point , 100 °C (212 °F), are much higher than would be expected by comparison with analogous compounds , such as hydrogen sulfide and ammonia . In its solid form, ice, water is less dense than when it is liquid, another unusual property. The root of these anomalies lies in the electronic structure of the water molecule.

Hydrogen atoms in water molecules are attracted to regions of high electron density and can form weak linkages, called hydrogen bonds , with those regions. This means that the hydrogen atoms in one water molecule are attracted to the nonbonding electron pairs of the oxygen atom on an adjacent water molecule. The structure of liquid water is believed to consist of aggregates of water molecules that form and re-form continually. This short-range order, as it is called, accounts for other unusual properties of water, such as its high viscosity and surface tension .

An oxygen atom has six electrons in its outer ( valence ) shell, which can hold a total of eight electrons. When an oxygen atom forms a single chemical bond, it shares one of its own electrons with the nucleus of another atom and receives in return a share of an electron from that atom. When bonded to two hydrogen atoms, the outer electron shell of the oxygen atom is filled.

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Water Symbolism in Literature: What Does Water Symbolize?

Most often, water represents cleansing, life and freedom. Water is a contextual symbol in literature, however, meaning that it can symbolize many things depending on how it is used in a story or poem. Often, the particular meaning hinges on the type of water being used as a symbol. For example, rivers often represent the flow of life because they are constantly moving, following a distinct path.

Significance of Water in Literature

In literature, water can symbolize ease, grace, and fluidity. It may also be the symbol of a driving force due to its continuous stream. Different types of water come with different meanings. For example, a fast-flowing river often indicates strength and focus. Moreover, throughout history, literary critics have pointed out that water — and often river water — typically represents both rebirth and healing.

So, what are some common associations in well-known works of literature? Here are some enduring examples of water symbolism:

• One of the most common examples of literature infused with water symbolism is Herman Melville’s Moby Dick . In this classic story, Ishmael sails under Captain Ahab on a whaling ship. The journey Ishmael takes changes him forever — and the sea symbolizes the unpredictability of that journey.
• In The Old Man and the Sea by Ernest Hemingway, water symbolizes rebirth as the old fisherman is able to catch fish again after being dragged into the sea.
• In The Tide Rises, The Tide Falls by H.W. Longfellow, water symbolizes fluidity; in the same way the tide rises and falls, time moves on again and again.
• Finally, playwright and poet William Shakespeare often used water as a symbol of cleansing; in Macbeth , Lady Macbeth desperately tries to wash away an invisible bloodstain, eager to cleanse herself of guilt.

Leonardo da Vinci believed that “water is the driving force in nature.” Henry David Thoreau shared a similar thought, saying, “The life in us is like the water in the river. It may rise this year higher than man has ever known it, and flood the parched uplands; even this may be the eventful year, which will drown out all our muskrats.”

Symbolic Meaning of Water

Water is a sacred symbol in most religious traditions, often referring to purity, rebirth, and fertility. Water’s significance as a spiritual and physical entity is highlighted in the Bible by stories about transformation and cleansing. Jewish tradition calls on people to immerse themselves in a bath on special occasions as a means of cleansing their spirits, while Muslim people use water to purify their spirits and prepare for daily prayer.

In the Eastern Orthodox Christian tradition, water possesses healing powers, while Taoism considers water an agent of wisdom, because of its ability to flow, regardless of any obstacles. Finally, the pagans of ancient Greece used water as the symbol of transition and metamorphosis, due in part to its many forms.

The water symbol can also be found in astrology, with the signs Pisces, Cancer, and Scorpio all being water signs. However, despite being linked by the element of water, each of these signs has its own features. For example, Cancer is associated with purity; Scorpio is considered mysterious; and Pisces is linked with depth and creativity.

Water Imagery

The image of water is often used in works to illustrate a point. For example, in The Penelopiad , author Margaret Atwood relies on the recurring image of water continuing to flow — no matter what challenges it faces. It’s a driving force in everything it does, and, perhaps more importantly, it keeps going.

Atwood then reminds the reader that they, too, are half water and have that same strength, writing, “Water is patient. Dripping water wears away a stone. Remember that, my child. Remember you are half water. If you can’t go through an obstacle, go around it. Water does.”

Interestingly, Hermann Hesse presents similar imagery in Siddhartha by equating the river with an endless flow of the present. He notes that the river is everywhere and it doesn’t focus on the past or future — all it can control is the present.

Hesse writes, “That the river is everywhere at the same time, at the source and at the mouth, at the waterfall, at the ferry, at the current, in the ocean and in the mountains, everywhere and that the present only exists for it, not the shadow of the past nor the shadow of the future.”

Life & Death Symbols

As we all know, water is a catalyst of life. It fuels, renews, and cleanses us. However, it also has an immense — and potentially dangerous — power. While water is essential to life, it also has the ability to drown.

In Moonglass , Jessi Kirby creates interesting imagery by connecting water with a character’s turbulent emotions. The character then comes to the conclusion that her mother was a victim of her own emotions, which became too much to handle — a kind of symbolic drowning. Kirby writes, “I read once that water is a symbol for emotions. And for a while now I’ve thought maybe my mother drowned in both.”

More specifically, the ocean or sea can also present unexpected, sometimes dangerous, challenges. At times, the ocean is full of turbulence or death, as seen in flood stories, from Gilgamesh to the tale of Noah’s ark. The point is, death can take many forms; it may not always be characterized by a skull, scythe or unlit candle.

But one thing is for certain: water is an important symbol in literature, one that can be interpreted in many ways. And, since water itself comes in many forms, this feels like an apt symbol in and of itself.

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• Published: 27 November 2023

Functional relationships reveal differences in the water cycle representation of global water models

• Sebastian Gnann   ORCID: orcid.org/0000-0002-9797-5204 1 , 2   na1 ,
• Robert Reinecke   ORCID: orcid.org/0000-0001-5699-8584 1 , 3   na1 ,
• Lina Stein   ORCID: orcid.org/0000-0002-9539-9549 1 ,
• Yoshihide Wada 4 , 5 ,
• Wim Thiery 6 ,
• Hannes Müller Schmied   ORCID: orcid.org/0000-0001-5330-9923 7 , 8 ,
• Yusuke Satoh   ORCID: orcid.org/0000-0001-6419-7330 9 ,
• Yadu Pokhrel   ORCID: orcid.org/0000-0002-1367-216X 10 ,
• Sebastian Ostberg   ORCID: orcid.org/0000-0002-2368-7015 11 ,
• Aristeidis Koutroulis   ORCID: orcid.org/0000-0002-2999-7575 12 ,
• Naota Hanasaki   ORCID: orcid.org/0000-0002-5092-7563 13 ,
• Manolis Grillakis   ORCID: orcid.org/0000-0002-4228-1803 12 ,
• Simon N. Gosling   ORCID: orcid.org/0000-0001-5973-6862 14 ,
• Peter Burek   ORCID: orcid.org/0000-0001-6390-8487 5 ,
• Marc F. P. Bierkens   ORCID: orcid.org/0000-0002-7411-6562 15 , 16 &
• Thorsten Wagener   ORCID: orcid.org/0000-0003-3881-5849 1  

Nature Water volume  1 ,  pages 1079–1090 ( 2023 ) Cite this article

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Global water models are increasingly used to understand past, present and future water cycles, but disagreements between simulated variables make model-based inferences uncertain. Although there is empirical evidence of different large-scale relationships in hydrology, these relationships are rarely considered in model evaluation. Here we evaluate global water models using functional relationships that capture the spatial co-variability of forcing variables (precipitation, net radiation) and key response variables (actual evapotranspiration, groundwater recharge, total runoff). Results show strong disagreement in both shape and strength of model-based functional relationships, especially for groundwater recharge. Empirical and theory-derived functional relationships show varying agreements with models, indicating that our process understanding is particularly uncertain for energy balance processes, groundwater recharge processes and in dry and/or cold regions. Functional relationships offer great potential for model evaluation and an opportunity for fundamental advances in global hydrology and Earth system research in general.

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Global water models—including hydrological, land surface and dynamic vegetation models 1 —have become increasingly relevant for policymaking and in scientific studies. The Sixth Assessment Report 2 of the Intergovernmental Panel on Climate Change draws heavily on results from global water models, which provide information about climate change impacts on hydrological variables including soil moisture 3 , streamflow 4 , terrestrial water storage 5 and groundwater recharge 6 . Some of these models are already embedded in global water information services to provide information to a wide array of stakeholders, such as the Global Groundwater Information System 7 or the African Flood and Drought Monitor 8 . Because measurements of many hydrological variables are very sparse and insufficient for large-scale analyses, global water models are regularly used in scientific studies to provide globally coherent estimates of variables such as groundwater recharge and groundwater storage change 9 , 10 . Global water models are also an integral part of Earth system models, and a realistic representation of the water cycle is essential for simulating the role of water within and across the different components of the Earth system 11 .

The Intergovernmental Panel on Climate Change’s Sixth Assessment Report 2 concludes from an analysis of currently available global water model projections that ‘uncertainty in future water availability contributes to the policy challenges for adaptation, for example, for managing risks of water scarcity’. Whereas some of this uncertainty stems from projected and observed climatic forcing, considerable uncertainty stems from global water models themselves 4 , 6 , 12 , 13 , 14 . For instance, Beck et al. 13 found distinct inter-model performance differences when comparing simulated and observed streamflow for ten global water models driven by the same forcing. To illustrate this uncertainty, we show how 30-year (climatological) averages of actual evapotranspiration, groundwater recharge and total runoff vary globally on the basis of outputs from eight models driven by the same forcing (Fig. 1a–c ; Methods). We find substantial disagreement among models, as indicated by high coefficients of variation, particularly for groundwater recharge and total runoff. We further show which model deviates most from the ensemble mean and find that there is not one model that consistently deviates the most (Fig. 1d–f ). Whereas this analysis cannot tell us which models perform better or worse, it suggests that it is not straightforward to single out a model for a certain flux or a certain region, which warrants a more in-depth evaluation.

a – c , Left: maps showing the coefficient of variation, calculated per grid cell as the ensemble standard deviation divided by the ensemble mean of eight global water models for different water fluxes: actual evapotranspiration ( a ), groundwater recharge ( b ) and total runoff ( c ). Lighter areas (‘blank spaces’) indicate high coefficients of variation (CoV) values and thus show where models disagree most. d – f , Right: maps showing which model deviates most from the ensemble mean for each grid cell for different water fluxes: actual evapotranspiration ( d ), groundwater recharge ( e ) and total runoff ( f ). Dark grey areas in d – f indicate that multiple models deviate similarly strongly from the ensemble mean. Empty, blank areas in d – f indicate that no model deviates strongly from the ensemble mean. The percentages shown in d – f refer to the fraction of grid cells (not land area) covered by each model. Greenland is masked out for the analysis.

Most evaluation strategies compare model outputs to historical observations over the area for which the observation is representative. This can be at the plot (for example, flux towers), catchment (for example, gauging stations) or grid cell (for example, gridded remote sensing products) scale. Such approaches are necessary but not sufficient to robustly evaluate global models 15 . First, these approaches compare simulated and observed values location by location and are therefore limited to potentially improving a model for that location; however, given that large fractions of the global land area are ungauged, we require methods that can extract and transfer information from gauged to ungauged locations 16 . Second, relevant information for model evaluation might not just lie in comparing the magnitudes of simulated and observed values in a single location but rather in how a variable varies along a spatial gradient 17 . And third, comparison with historical observations does not guarantee that a model reliably predicts system behaviour under changing conditions 18 . Rather than evaluating global models in essentially the same way as catchment-scale models, evidence of different large-scale hydrological relationships presents us with an opportunity for a different evaluation strategy that is inherently large-scale but so far rarely exploited.

Towards evaluation using functional relationships

Reviewing the hydrological literature reveals a range of relationships 19 that, if they appear in empirical data, should also appear in models (and vice versa). Such relationships often capture behaviour that is not prescribed by small-scale processes but rather emerges through the interaction of these processes (or model components) at large scales. The perhaps most prominent example is the Budyko framework 20 , which describes the long-term partitioning of precipitation into evapotranspiration and streamflow solely as a function of the aridity index. Another example are so-called elasticities of streamflow to changing climatic drivers (for example, precipitation or temperature), which provide an observation-based constraint on climate change effects on streamflow 21 , 22 . A third example are empirical relationships between annual rainfall and runoff, which can be affected differently by prolonged drought; in Australia, some catchments have shown similar rainfall–runoff relationships before and after the Millennium Drought, while other catchments have transitioned to a new stable state 23 . The search for robust relationships that characterize the functioning of hydrological systems is in itself a great scientific challenge 19 , but such functional relationships also provide an excellent yet poorly explored opportunity for the evaluation of global water models.

We define the term function as the actions of (hydrological) systems on the inputs that enter them, such as partition, storage and release of water and energy 24 , 25 . Accordingly, we define functional relationships as relationships between two or more variables that characterize these functions. Such relationships often focus on forcing, state and response variables that are expected to be causally related (for example, precipitation and runoff), and they can focus on both temporal variability at a single location and (as used here) spatial variability across multiple locations. Functional relationships need not be uniquely defined and are typically characterized by substantial scatter due to other (secondary) controlling variables, local variability or uncertainty.

Whereas functional relationships have been used before to evaluate land surface, forest and Earth system models—for example, by analysing relationships between soil moisture and evaporation and runoff 26 , 27 , 28 , 29 or between precipitation and other atmospheric drivers and vegetation productivity 30 , 31 , 32 —their potential for evaluating global water models has not yet been sufficiently explored. The use of functional relationships is currently scattered among the hydrological literature (for example, refs. 33 , 34 , 35 ) and has not been formalized into an evaluation framework. There is a pressing need to develop a ‘theory of evaluation’ 36 that does justice to the nature of global models, the purposes for which they are used and their growing relevance for society 37 . Functional relationships have the potential to be a central building block of such a theory of evaluation, and below we show how they can help shed new light on model behaviour.

Here we focus on functional relationships that capture the spatial co-variability of forcing and response variables. Rather than focusing on a process-by-process comparison that can quickly become unmanageable 28 , functional relationships can capture emergent patterns and shift the focus to identifying the dominant controls on the variables of interest. Especially the relationships between water and energy availability and the major water fluxes leaving the land surface—evaporation and runoff—have been frequently studied 20 , 38 , providing an excellent starting point for model evaluation. In addition, functional relationships that focus on spatial patterns offer several advantages. First, such relationships are well suited for the analysis of global models due to their spatially distributed nature, which means that these relationships can be readily obtained from comparing values from multiple grid cells. Second, spatial relationships can be calculated based on long-term averages, which for some variables are often the only observations available (for example, for groundwater recharge 39 , 40 ). And third, such relationships can capture how hydrological variables co-vary across large scales and thus offer the potential for model improvement over large areas, including locations that lack observations.

In this analysis, we investigate how long-term averages of two forcing and three response variables co-vary spatially, leading to six variable pairs overall. The forcing variables are precipitation P and net radiation N (the available water and energy, respectively), and the response variables are actual evapotranspiration E a , groundwater recharge R and total runoff Q (three key water fluxes). We analyse forcing–response relationships based on 30-year (climatological) averages (1975–2004; all in mm per year) from eight global water models (CLM4.5, CWatM, H08, JULES-W1, LPJmL, MATSIRO, PCR-GLOBWB and WaterGAP2) from phase 2b of the Inter-Sectoral Impact Model Intercomparison Project (ISIMIP 2b 41 ). In addition, we use observational datasets, observation-driven machine learning products and the semi-empirical equation introduced by Budyko 20 to calculate functional relationships between the same variables as for the models as benchmarks (Table 1 ). To explore regional variability in functional relationships 38 , we divide the world into four climatic regions: wet–warm (18% of modelled area), wet–cold (15%), dry–cold (24%) and dry–warm (43%), shown in Fig. 2d . Details can be found in the Methods section.

a , Scatter plots between precipitation and groundwater recharge for PCR-GLOBWB and WaterGAP2. Owing to space constraints, we focus on a few examples with differing relationships. Scatter plots for all variable pairs are shown in Supplementary Figs. 15 – 20 . Each dot represents one grid cell and is based on the 30-year average of each flux. Spearman rank correlations ρ s measure the strength of the relationship between forcing and response variables and are calculated for all grid cells within a climate region. The lines connect binned medians (ten bins along the x axis with equal amount of points per bin) for each region. b , The climate regions are shown. The grey dashed line shows the 1:1 line, indicating the water limit assuming all water is supplied by precipitation.

Disagreement in functional relationships between models

We can visually assess relationships between forcing ( P ,  N ) and response variables ( E a ,  R ,  Q ) by inspecting scatter plots where each point represents one grid cell (or observation); this is shown for precipitation and groundwater recharge in Fig. 2a . We first take a closer look at the shapes of the functional relationships, indicated by the coloured lines in Fig. 2a . Later we will also quantify the strength of the relationships using Spearman rank correlations ρ s . We limit ourselves to a qualitative discussion, given that fitting an equation would mean that we would have to assume a functional form. We report mean values and slopes (obtained via linear regression) for each region in Supplementary Tables 4 – 7 , which quantitatively support our visual assessment. Figure 3 shows connected binned median values for precipitation and the three water fluxes for all models and observational datasets (Table 1 ), separated by climate region. A similar plot for net radiation and the three water fluxes is shown in Extended Data Fig. 1 .

Average functional relationships based on models and benchmark datasets among precipitation P and actual evapotranspiration E a , groundwater recharge R and total runoff Q , respectively. The coloured lines represent one model each, the grey-black lines represent different observational datasets, labelled on the outer-right panels. The MacDonald groundwater recharge dataset contains only enough data values for the dry–warm region and is thus only shown there. The lines connect binned medians (ten bins along the x axis with equal amount of points per bin) for each climate region. The grey dashed line shows the 1:1 line, indicating the water limit assuming all water is supplied by precipitation. Note that the graphs do not show the full range for some curves to better illustrate the model differences.

While the P – E a relationships look similar in shape, they can differ greatly in magnitude (Fig. 3 ). They increase rather linearly in dry (water-limited) regions and increase initially in wet (energy-limited) regions and then level off as they reach an energy limit that bounds actual evapotranspiration. The limit differs greatly between models, varying up to about 400 mm per year in wet–warm regions. Because all models are forced with the same total radiation, this difference is related to the way the models translate total radiation into net radiation and how they then use net radiation to calculate actual evapotranspiration. There is no obvious connection between this difference and the different potential evapotranspiration schemes used 42 , potentially because the models, while forced with the same climate inputs, differ in the way they parameterize the land surface (for example, land use, soils). In dry regions, actual evapotranspiration is mostly limited by precipitation, a forcing dataset that is the same for all models, resulting in less variability. The Budyko equation and the FLUXCOM 43 dataset suggest, in line with literature estimates 44 , that most models underestimate actual evapotranspiration, often greatly so (Supplementary Tables 4 and 5 ). However, we note that FLUXCOM probably overestimates actual evapotranspiration, especially in dry–warm regions, because it considers only vegetated areas 43 . Overall, the disagreement in modelled actual evapotranspiration, particularly visible in energy-limited regions, suggests substantial differences in the way models estimate the energy available for evapotranspiration.

Most P – R relationships increase monotonically, but the shape, the slope and the threshold at which some models start to produce groundwater recharge are very different (Fig. 3 ). For instance, in dry–warm regions, some models produce essentially no groundwater recharge even if precipitation is above 1,000 mm per year, while others produce over 200 mm per year. In dry–warm regions, we have by far the most extensive database on groundwater recharge 39 , 40 , and the observations fall (apart from those at very high precipitation values) within the range of the models. In wet–warm regions, we find the largest disagreement between models and observations, which suggest lower (higher) groundwater recharge rates for higher (lower) precipitation. Whereas this shows the benefit of using an ensemble rather than a single model, even a large ensemble spread does not always capture the observed relationships. The large spread further suggests that many models greatly over- or underestimate groundwater recharge rates and consequently greatly over- or underestimate how much groundwater contributes to evapotranspiration and streamflow 45 . These differences in slope, visible for all climate regions, reflect very different spatial sensitivities to changes in precipitation. Whether temporal sensitivities are similar can only be hypothesized given that no global observational dataset with groundwater recharge time series is available but would imply very different responses to projected changes in precipitation.

The P – Q relationships look similar in shape and mostly increase monotonically, especially for wet regions (Fig. 3 ). The relative differences are larger for dry places, commonly perceived as regions where runoff is more difficult to model 46 . The model and benchmark relationships disagree particularly strongly in dry–cold regions. There, the GSIM 47 , 48 dataset shows a variable relationship between total runoff and precipitation, whereas the GRUN 49 dataset shows almost no increase with increasing precipitation. Overall, GSIM, GRUN and the Budyko equation indicate, in line with an earlier evaluation 50 , that most models produce too much total runoff. This parallels recent findings that Earth system models predict higher runoff increases due to climate change than observations suggest 22 . The overestimation in total runoff is complementary to the underestimation of actual evapotranspiration and shows that most models partition too much precipitation into runoff rather than evapotranspiration.

Diverging dominance of forcing on response variables

To quantitatively compare the strength of the forcing–response relationships, we use Spearman rank correlations ρ s . A rank correlation close to 1 (or −1) indicates that the spatial variability in the forcing variable almost completely explains the spatial variability in the response variable, as can be seen in Fig. 2a for WaterGAP2. A rank correlation closer to 0 indicates that other factors control the response (for example, other input or model parameters describing the land surface), as can be seen in Fig. 2a for PCR-GLOBWB. We stress that a high correlation is not a measure of goodness of fit. Considerable scatter and correspondingly low correlations might indeed be characteristic for many relationships, and if models overestimate how strongly a forcing variable controls a model output, this also indicates unrealistic behaviour. Calculating rank correlations for all variable pairs, we find that the models differ substantially among each other and in comparison to observations (Fig. 4 ; rank correlations for all benchmark datasets and models are listed in Table 1 and Supplementary Table 3 , respectively).

a – f , Spearman rank correlations ρ s between forcing variables (precipitation ( a , c , e ), net radiation ( b , d , f )) and water fluxes (actual evapotranspiration ( a , b ), groundwater recharge ( c , d ) and total runoff ( e , f )), divided into different climate regions. Net radiation for LPJmL and PCR-GLOBWB is not available and is estimated as the median of the other models (per grid cell). The lines connecting the dots are only there as a visual aid. The numbered triangles show rank correlations based on benchmark datasets (grey background) and the Budyko equation, with numbers indicating the corresponding data source (Table 1 ). Observation-based rank correlations are shown only if they are based on more than 50 data points.

For precipitation and actual evapotranspiration (Fig. 4a ), the models show the same ranking between climate regions and rather small differences in magnitude, indicating that actual evapotranspiration is strongly constrained by the available water in all models. The model-based correlations are higher in dry regions ( ρ s  = 0.74–0.98) than in wet regions (0.57–0.83), reflecting water and energy limitations. The Budyko equation assumes complete dependence on aridity (here defined as N / P ). It thus predicts higher correlations overall and mainly distinguishes between wet (0.83–0.84) and dry (0.98–1.00) regions but, unlike models and FLUXCOM, not between cold and warm regions. The Budyko equation should thus be seen as a useful comparison but not as the ‘correct’ model, given that different studies have shown that snow 51 , climate seasonality 52 , vegetation type 53 , inter-catchment groundwater flow 54 and human impacts 55 can affect the long-term water balance beyond aridity.

We find much variability for net radiation and actual evapotranspiration (Fig. 4b ). There is no obvious correspondence between the potential evapotranspiration schemes used 42 (for example, Priestley–Taylor for LPJmL and WaterGAP2 or Penman–Monteith for JULES-W1 and CWatM) and the rank correlations, implying that other factors play a more important role (also, refs. 14 , 56 ). Both the Budyko equation and FLUXCOM show very high correlations for all wet places (0.93–0.99), indicating a strong energy limitation 57 , underestimated by many models (especially CWatM and MATSIRO). FLUXCOM shows a stronger N – E a relationship (Fig. 4b ) in dry–cold places than all models and the Budyko equation, while it shows a weaker P – E a relationship (Fig. 4a ) there. This could be due to an uncertain representation of energy balance processes in cold regions, possibly related to interactions between snow-affected albedo and evapotranspiration 58 , 59 , sublimation 60 or the aerodynamic component of potential evapotranspiration 61 .

For precipitation and groundwater recharge (Fig. 4c ), some models (CLM4.5, MATSIRO, WaterGAP2 and H08) show high to very high correlations (0.71–0.95) for all climate regions, suggesting that precipitation is the dominant control on groundwater recharge across all climate regions in these models. Other models (CWatM, JULES-W1, LPJmL, PCR-GLOBWB) show much lower and more variable correlations (0.35–0.85), suggesting different controls on groundwater recharge (for example, model structural decisions and parameterizations). H08 and WaterGAP2 use the same approach to calculate groundwater recharge 42 and they show almost identical rank correlations, indicating that the functional relationships might be relatable to the model structure in this case. Recent studies have shown a strong influence of precipitation and aridity on groundwater recharge 39 , 40 , 45 , and using the same datasets, we also find high to very high correlations in dry–warm regions (0.74–0.84). In these often highly water-limited regions, precipitation appears to be the dominant control on groundwater recharge. Besides climate, perceptual models of groundwater recharge generation usually include soil characteristics, topography, land use and geology 62 , 63 . This might explain why observations show a more scattered P – R relationship, particularly in wet–warm regions (−0.06).

For precipitation and total runoff (Fig. 4e ), WaterGAP2 and PCR-GLOBWB both show lower correlations (0.52–0.75) than the other models (0.58–0.95). WaterGAP2 is the only model here that is calibrated against streamflow observations 42 , which might explain why it shows the lowest rank correlations for total runoff. The Budyko framework assumes that long-term runoff only depends on aridity and thus shows higher correlations (0.87–0.99) than the benchmark datasets (0.27–0.94) and most models (0.52–0.95). Because factors other than aridity can influence total runoff 51 , 52 , 53 , 54 and given that GSIM tends to show lower correlations overall (0.32–0.80), models that show correlations as high as the Budyko equation probably overestimate how strongly precipitation controls total runoff. Similar to the shapes of the functional relationships (Fig. 3 ),we generally find the largest differences in both models and datasets in dry–cold regions, where GRUN and GSIM show particularly low correlations (0.27 and 0.32).

For net radiation and both groundwater recharge and total runoff (Fig. 4d,f ), we find high variability and mostly positive correlations. The models probably produce more groundwater recharge and total runoff in regions with higher net radiation because precipitation is also higher in these regions (Supplementary Fig. 1 ). Whereas it is difficult to interpret these correlations, the large variability still suggests considerable differences between models.

Focus areas for model improvement

Our analysis has revealed substantial disagreement between models and between models and observations, questioning the robustness of model-based studies and impact assessments, especially if only a single model is used. The energy balance, from total radiation to actual evapotranspiration, appears to be poorly represented, indicated by a different energy limit (Fig. 3 ), a general underestimation of actual evapotranspiration and widely varying N – E a relationships (Fig. 4 ). This warrants a closer look in future studies, as a realistic depiction of energy balance and evaporation processes is critical for climate change studies 57 , 58 . We find the largest disagreement for groundwater recharge, which is arguably the least understood process and poorly constrained by sparse observations 39 , 40 . The inter-model differences in groundwater recharge can be much larger than the differences in actual evapotranspiration and must therefore have other reasons. To better constrain the large variability between models, we need to improve our understanding of the dominant controls on groundwater recharge at large scales 64 . This knowledge is important for assessments of sustainable use of groundwater resources 9 , 10 , for groundwater modelling studies that use groundwater recharge from global water models as input 65 and for understanding the sensitivity of groundwater recharge to changing climatic drivers 6 . Most models overestimate total runoff and we find the largest disagreement for total runoff in dry–cold regions. This echoes existing literature 1 , 12 , 22 , 50 and highlights the need for model refinement in dry and/or cold regions, which are under-researched and strongly affected by climate change 46 . To explore more in-depth how snow processes affect the water balance, future studies could focus on functional relationships in snow-dominated regions by specifically delineating these regions using the fraction of precipitation falling as snow or snow cover extents.

Towards an inventory of robust functional relationships

We have used different observational datasets, observation-driven machine learning products and the Budyko equation 20 to derive empirical and theory-based functional relationships, but challenges remain. Observation-driven machine learning products 43 , 49 are not raw observations and may reflect their upscaling methods rather than the underlying natural distribution but serve as useful benchmarks in the absence of direct observations (for example, because of limited numbers of FLUXNET sites 66 ). The Budyko equation 20 is a climate-only model and thus provides a useful benchmark but neglects other influences on the long-term water balance. The observations themselves and the forcing data paired with them are also associated with uncertainty, even though most of the relationships used here appear to be relatively robust (Methods includes an extended discussion). Yet especially for variables with small numbers of observations, it is challenging to provide robust observation-based constraints for certain regions (Table 1 ). For example, groundwater recharge measurements have almost entirely been made in dry–warm regions (97% of MacDonald data 40 and 92% of Moeck data 39 ), leaving groundwater recharge in other regions poorly constrained. On the other hand, most streamflow measurements have been taken in wet regions (60% of GSIM data used here), and globally there is a placement bias of stream gauges towards wet regions 67 , even though—according to our classification—short of two-thirds of the global land area are defined as dry. Instead of taking new measurements to understand a specific place, new measurements would have much more leverage if they would help us to also understand other places, for example, by filling an observational gap along a climatic gradient (that is, in functional space). In addition, more quality-controlled datasets with uncertainty estimates 40 are critical to obtain realistic uncertainty estimates for functional relationships. This would ultimately allow us to obtain robust ranges of functional behaviour that we can benchmark our models against.

The functional relationships studied here appear to be robust with respect to modelled human impacts, probably because we investigated long-term averages over large regions where climatic controls on the selected hydrological variables dominate (Supplementary Figs. 26 – 30 ). Yet for different variables, especially when studied at shorter temporal and smaller spatial scales, human impacts might have a considerable effect on functional relationships. The effects of human impacts might be investigated by studying strongly managed and near-natural regions separately 68 . Indeed, comparing functional relationships between human impacted and natural regions would be an excellent strategy to assess the degree of human alteration of the natural water cycle. In addition, relationships that specifically focus on human impacts, such as relationships between irrigated areas and irrigation water withdrawals 69 , might be used to better understand the representation of human impacts in models.

Whereas visual comparison (focusing on the shape of the relationships) and rank correlations (focusing on the strength of the relationships) have exposed clear differences between models and observations, our approach here should be seen as a first step. There are other ways to describe the relationships analysed here, for example, by characterizing thresholds or nonlinearities (visible in Fig. 3 ). Metrics such as rank correlations also require careful interpretation. For example, positive correlations between net radiation and groundwater recharge probably arise because precipitation and net radiation are positively correlated and thus do not imply a causal relationship. The interpretation of empirical relationships should therefore be backed up by process knowledge or extended by methods that allow for discovery of causal relationships 70 . Physics-aware machine learning might be powerful in that respect, as it combines domain knowledge with versatile pattern recognition 71 . Beyond the relationships investigated here, we anticipate that exploring temporal relationships (for example, using elasticities 21 , 22 or shifts in P – Q relationships 23 ), dividing the landscape into additional categories (for example, hydrobelts 72 ) and including other variables, such as state variables or stores (for example, soil moisture, terrestrial water storage), will provide additional insights.

Conclusions

As our models grow in complexity, encompassing more processes and covering larger spatial and temporal scales, we need a concurrent development of model evaluation strategies: an evaluation framework for large-scale models. Central to such an evaluation framework should be functional relationships, which shift the focus away from matching historical records in specific locations to a more diagnostic and process-oriented evaluation of model behaviour 36 . Functional relationships allow us to focus on larger-scale assessments, to relate places to each other and to explore if dominant controls in models are consistent with observations, theory and expectations (that is, our perceptual model 73 ). This understanding is critical for ensuring that models faithfully represent real-world systems, ultimately leading to more credible projections of environmental change impacts. Eventually, expanding our range of functional relationships in hydrology, constrained by various observational datasets and expert knowledge, would give us a knowledge base of realistic system behaviour that could be used to evaluate models, diagnose model deficiencies and weight model ensembles, comparable to the use of emergent constraints in climate modelling 37 .

Both our approach and our findings have implications beyond hydrology. First, the terrestrial water cycle plays a central role in the Earth system and is often strongly coupled to other components, such as the biosphere, lithosphere and atmosphere and human activities (for example, refs. 74 , 75 , 76 ). More realistic simulations of the global water cycle therefore also enable us to better clarify how it influences and is influenced by other Earth system components. Methodologically, functional relationships are not limited to applications in hydrology. In fact, land surface, forest and Earth system models 26 , 27 , 28 , 29 , 30 , 31 , 32 have already been studied in similar fashions, though a broader application of this approach has so far been missing. As indicated by recent studies 76 , 77 , functional relationships provide an excellent opportunity to study the interactions between hydrology and, for example, terrestrial ecosystems, and thus represent a tool that can be used across disciplines.

Beyond model evaluation, functional relationships invite us to think about how the global water cycle functions, what we know, what we do not know and what that means for a future under climate change 73 . Our results suggest that improved process understanding will be particularly important for energy balance processes, groundwater recharge processes and generally in dry and/or cold regions. So how can we improve our process understanding? In 1986, Eagleson 78 stated that ‘science advances on two legs, analysis and experimentation, and at any moment one is ahead of the other. At the present time advances in hydrology appear to be data limited’. For some processes, this still seems to be the case. But clearly, we have a wealth of data available and might ask ourselves: are we extracting all of the information from the observations we have? On the basis of the data we have, what and where should we measure next? And are there functional relationships in hydrology yet to be found 19 ? Even if the search for such relationships is challenging, it will be a fruitful and exciting endeavour for global hydrology.

Model data retrieval and processing

We analysed 30-year (climatological) averages (1975–2004) from eight global water models 41 : CLM4.5 79 , CWatM 80 , H08 81 , JULES-W1 82 , LPJmL 83 , MATSIRO 84 , PCR-GLOBWB 85 and WaterGAP2 86 . The model simulations were carried out following the ISIMIP 2b protocol and here we used model outputs forced with the Earth system model HadGEM2-ES under historical conditions (historical climate and CO 2 concentrations). We note that the specific forcing chosen does not appear to influence model-based functional relationships (see below). We used precipitation P (ISIMIP variable name p r ), net radiation N (not an official ISIMIP output), actual evapotranspiration E a (ISIMIP variable name e v a p ), groundwater recharge R (ISIMIP variable name q r ) and total runoff Q (ISIMIP variable name q t o t ). Note that Q here refers to runoff generated on the land fractions (and not surface water bodies) of each grid cell and does not include upstream inflows, which allows for comparison to grid cell P . P , E a , R , and Q were downloaded from https://data.isimip.org/ . Net radiation N is not an official ISIMIP output and was provided by the individual modelling groups. It is not available for all models, so we used the ensemble median per grid cell for models without N data. We converted all fluxes to mm per year and removed E a values larger than 10,000 mm per year and set R values smaller than 0 to 0. Note that our analysis excludes Greenland and Antarctica. A more detailed description is given in the Supplementary Information .

CoV and most deviating model maps

For each grid cell, we used the 30-year averages of the eight models (that is, the model ensemble) and calculated the ensemble standard deviation divided by the ensemble mean. Maps of the standard deviation are shown in the Supplementary Information (Supplementary Figs. 8 – 10 ). To see which model dominates the ensemble spread, we checked for each grid cell which model shows the largest absolute difference (denoted by d 1 ) from the ensemble mean (denoted by μ ). To see if multiple models dominate the ensemble spread, we also checked for each grid cell which model shows the second-largest absolute difference (denoted by d 2 ) from the ensemble mean. If the relative difference between the largest and the second-largest difference is less than 20%, that is ( d 1  −  d 2 )/ d 1  < 0.2, the grid cell falls into the category ‘multiple’. If the relative difference between the most deviating model and the ensemble mean is less than 20%, that is d 1 / μ  < 0.2, the grid cell is counted as having no most deviating model (empty areas on Fig. 1d–f ).

Functional relationships

To visualize the shape of the functional relationships, we binned the data in each climate region into ten bins (along the x axis) with an equal amount of points, calculated the median per bin and connected the obtained median value. For groundwater recharge, we used only five bins because there are so few values. Note that the non-gridded observational datasets do not have the same spatial distribution as the gridded datasets and the models and thus do not have the same distribution of forcing variables. Their bins can therefore span different ranges of the forcing variables. As a metric for the strength of the functional relationships, we calculate Spearman rank correlations ρ s between model inputs and outputs per climate region, a measure of the monotonicity between two variables that is robust to outliers. We use the following categories for correlations: negative correlation (<0), no to low correlation (0 to 0.25), medium correlation (0.25–0.5), high correlation (0.5–0.75), very high correlation (0.75–1.0). We also show mean fluxes and slopes obtained through linear regression in Supplementary Tables 4 – 7 .

Climate regions

On the basis of the aridity index (here defined as N / P ; where N is model ensemble median), a place is categorized as either wet ( N / P  < 1) or dry ( N / P  > 1). On the basis of how many days per year fall below a 1 °C temperature threshold, a place is categorized as either cold (more than one month below 1 °C) or warm (less than one month below 1 °C). This results in four categories: wet–warm (15% of model grid cells/18% of modelled area), wet–cold (23%/15%), dry–cold (28%/24%) and dry–warm (34%/43%). To test how different decisions affect our climate region classification, we also used the ensemble median of potential evapotranspiration E p (partially downloaded, partially provided by the modelling groups) to calculate the aridity index ( E p / P ), and we used a different threshold for our warm/cold distinction. This resulted in little differences overall, as can be seen in the Supplementary Information (Supplementary Fig. 14 ).

Benchmark datasets and theory

To benchmark model performance, we used different observational datasets, observation-driven machine learning products and the Budyko equation 20 . If the datasets provide their own forcing data, we used these data. If not, we paired them with GSWP3 P data 87 to have one consistent forcing product. For E a , we used FLUXCOM data 43 (RS monthly 0.5° from 2001–2015) paired with GSWP3 P data 87 (downloaded from https://data.isimip.org/ ). For R , we used data from MacDonald et al. 40 , which include matching P data, and data from Moeck et al. 39 paired with GSWP3 P data 87 . For Q , we used GRUN data 49 from 1985–2004 paired with GSWP3 P data 87 (the dataset used in the creation of GRUN) and GSIM data 47 , 48 paired with GSWP3 P data 87 . For GSIM, we only used catchments with areas ranging from 250 to 25,000 km 2 with a minimum of ten years of data between 1985 and 2004 to ensure a sufficient number of catchments that do not differ too much in size from the model grid cells. To obtain theory-based estimates for E a and Q , we forced the Budyko 20 equation (equation ( 1 )) with HadGEM2-ES P (the same forcing as used for the models) and ensemble median N from the ISIMIP 2b models analysed here.

More details on data processing and quality checks can be found in the Supplementary Information .

Extended discussion on model forcing and scenario uncertainty

The choice of forcing product and differences in the treatment of human influences (for example, water use and dams) might affect the functional relationships exhibited by the models. To get an idea how much uncertainty this introduces, we compared our results to model runs using WATCH-WFDEI forcing with either variable historical conditions (varsoc) or no human influences (nosoc) for WaterGAP2 and PCR-GLOBWB, carried out following the ISIMIP 2a protocol. The results, shown in the Supplementary Information (Supplementary Figs. 26 – 30 ), stay essentially the same, showing that the model-based correlations are robust signatures of model behaviour.

Extended discussion on benchmark dataset uncertainty

Because not all datasets come with matching P data, we sometimes paired the observations with GSWP3 reanalysis data 87 . To get an idea how much uncertainty this introduces, we investigated how different P data sources affect the functional relationships. Correlations calculated using the MacDonald et al. 40 R data with either GSWP3 P data or the accompanying P data are very similar for dry–warm places (0.83 and 0.84; Supplementary Information ). Using HadGEM2-ES P (the model forcing) data instead of GSWP3 P data to calculate correlations with FLUXCOM E a 43 , Moeck R 39 , GRUN Q 49 and GSIM 47 , 48 , respectively, results in no notable differences. Because most datasets only contain a limited number of years of data, sometimes only one average value 39 , 40 , we used all available years in our analysis. The only observation-driven dataset that contains a long enough time series to analyse functional relationships for two independent 30-year periods is GRUN 49 . Using GRUN data from 1945–1974 instead of 1975–2004 results in virtually no differences. While we cannot rule out that other datasets would lead to different relationships, this analysis indicates that the functional relationships and the rank correlations are relatively robust (Supplementary Figs. 31 – 42 ).

Reporting summary

Further information on research design is available in the Nature Portfolio Reporting Summary linked to this article.

Data availability

The long-term averages created and used in this study are deposited at https://zenodo.org/record/7714885 . Correlations and other statistics are available in the Supporting Information. Data used in this study can be downloaded from the following links. ISIMIP 2b data (model outputs and GSWP3 precipitation data) are available from https://www.isimip.org/ . FLUXCOM data are available from http://www.fluxcom.org/ . MacDonald et al. recharge data are available from https://www2.bgs.ac.uk/nationalgeosciencedatacentre/citedData/catalogue/45d2b71c-d413-44d4-8b4b-6190527912ff.html (contains data supplied by permission of the Natural Environment Research Council (2022)). Moeck et al. recharge data are available from https://opendata.eawag.ch/dataset/globalscale_groundwater_moeck . GSIM data are available from https://doi.pangaea.de/10.1594/PANGAEA.887477 and https://doi.pangaea.de/10.1594/PANGAEA.887470 . MSWEP data can be requested for research purposes from http://www.gloh2o.org/mswep/ .

Code availability

Python and R codes used to perform the analyses are available at https://github.com/HydroSysPotsdam/GHM_Comparison .

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Acknowledgements

S.G., R.R., L.S. and T.W. acknowledge support from the Alexander von Humboldt Foundation in the framework of the Alexander von Humboldt Professorship endowed by the German Federal Ministry of Education and Research (BMBF). Y.S. was supported by the National Research Foundation of Korea (NRF) grant funded by the Korean Government (MSIT) (2021H1D3A2A03097768). Y.P. acknowledges the support from the National Science Foundation (grant number 1752729). A.K. and M.G. have received support from REACT4MED (GA 2122) PRIMA funded project, supported by Horizon 2020. N.H. is financially supported by JSPS KAKENHI grant number 21H05002. This publication is based upon work from COST Action CA19139—PROCLIAS, supported by COST (European Cooperation in Science and Technology, https://www.cost.eu ). We thank P. Döll for providing helpful comments on the manuscript. We also thank the ISIMIP team for their continued efforts within the ISIMIP project.

Open access funding provided by Universität Potsdam.

Author information

These authors contributed equally: Sebastian Gnann and Robert Reinecke.

Authors and Affiliations

Institute of Environmental Science and Geography, University of Potsdam, Potsdam, Germany

Sebastian Gnann, Robert Reinecke, Lina Stein & Thorsten Wagener

Chair of Hydrology, Faculty of Environment and Natural Resources, University of Freiburg, Freiburg, Germany

Sebastian Gnann

Institute of Geography, Johannes Gutenberg University Mainz, Mainz, Germany

Robert Reinecke

Climate and Livability Initiative, Center for Desert Agriculture, Biological and Environmental Science and Engineering Division, King Abdullah University of Science and Technology, Thuwal, Saudi Arabia

Yoshihide Wada

International Institute for Applied Systems Analysis, Laxenburg, Austria

Yoshihide Wada & Peter Burek

Department of Water and Climate, Vrije Universiteit Brussel, Brussels, Belgium

Institute of Physical Geography, Goethe University Frankfurt, Frankfurt am Main, Germany

Hannes Müller Schmied

Senckenberg Leibniz Biodiversity and Climate Research Centre (SBiK-F), Frankfurt am Main, Germany

Moon Soul Graduate School of Future Strategy, Korea Advanced Institute of Science and Technology, Daejeon, Republic of Korea

Yusuke Satoh

Department of Civil and Environmental Engineering, Michigan State University, East Lansing, MI, USA

Yadu Pokhrel

Potsdam Institute for Climate Impact Research (PIK), Member of the Leibniz Association, Potsdam, Germany

Sebastian Ostberg

School of Chemical and Environmental Engineering, Technical University of Crete, Chania, Greece

Aristeidis Koutroulis & Manolis Grillakis

National Institute for Environmental Studies, Tsukuba, Japan

Naota Hanasaki

School of Geography, University of Nottingham, Nottingham, UK

Simon N. Gosling

Department of Physical Geography, Utrecht University, Utrecht, Netherlands

Marc F. P. Bierkens

Unit Soil and Groundwater Systems, Deltares, Utrecht, Netherlands

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Contributions

S.G., R.R., L.S. and T.W. designed the study. Y.W., W.T., H.M.S., Y.S., Y.P., S.O., A.K., N.H., M.G. and P.B. conducted hydrological simulations under the ISIMIP 2b project, and S.N.G. and H.M.S. coordinated the ISIMIP global water sector. S.G. and R.R. processed the simulation results and conducted the analyses, and S.G., R.R. and L.S. prepared the graphics. S.G. wrote the first paper draft together with R.R., L.S. and T.W. Y.W., W.T., H.M.S., Y.S., Y.P., S.O., A.K., N.H., M.G., S.N.G., P.B. and M.F.P.B. contributed to discussions and interpretations of the results and edited the paper.

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Extended data

Extended data fig. 1 average functional relationships between net radiation and three key water fluxes..

Average functional relationships based on models and benchmark datasets between net radiation N and actual evapotranspiration E a , groundwater recharge R and total runoff Q , respectively. The colored lines represent one model each, the grey-black lines represent different observational datasets, labeled on the outer-right panels. The lines connect binned medians (10 bins along the x -axis with equal amount of points per bin) for each climate region. The grey dashed line shows the 1:1 line, indicating the water limit assuming all water is supplied by precipitation. Note that the graphs do not show the full range for some curves to better illustrate the model differences.

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Supplementary text, Tables 1–6, Figs. 1–39 and references.

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Gnann, S., Reinecke, R., Stein, L. et al. Functional relationships reveal differences in the water cycle representation of global water models. Nat Water 1 , 1079–1090 (2023). https://doi.org/10.1038/s44221-023-00160-y

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• States of Matter
• Phase Changes

Phase Diagram of Water

What is a Phase Diagram?

A phase diagram is a graphical representation of the various phases of a substance or mixture of substances that coexist in thermodynamic equilibrium, and undergo phase changes under different working conditions, such as temperature, pressure, or volume.

The water system is divided into three phases: ICE (S), WATER (L), and WATER VAPOUR (G)

Since H 2 O is the only chemical compound present, as a result, it is a one-component system.

Table of Contents

Depiction in phase diagram of water.

• Difference between General Phase Diagram and Phase Diagram of Water

Phase Diagram for Water

• Frequently Asked Questions – FAQs

At different temperatures and pressures, a phase diagram depicts the preferred physical states of matter. Water is a liquid at typical room temperatures and pressures, but it becomes solid (i.e. ice) if the temperature is reduced below 273 K, and gaseous (i.e. steam) if the temperature is increased above 373 K at the same pressure.

Each line describes the conditions that exist when two phases coexist, but a change in temperature or pressure can cause the phases to abruptly change from one to the other.

When three lines come together, there is a ‘triple point,’ which is where three phases coexist but can abruptly and completely change into each other due to a change in temperature or pressure. Four lines cannot intersect at the same point.

A ‘critical point’ is defined as the point at which the properties of two phases become indistinguishable from one another.

Difference between General Phase Diagram and Phase Diagram for Water

There is one significant difference between the general phase diagram and the water phase diagram. The slope of the line connecting the solid and liquid states in the water diagram is negative rather than positive. Water is an unusual substance in that its solid state is less dense than its liquid state. In liquid water, ice floats. As a result, a change in pressure has the opposite effect on those two phases. When ice is near its melting point, it can be converted into liquid water by applying pressure. In the liquid phase, water molecules are actually closer together than in the solid phase.

• OA- Water & Vapor

OA is the vaporisation curve. The curve OA comes to an end at A. It has a critical pressure of 218 atm and a temperature of 374℃. It represents the vapour pressure of a liquid at various temperatures.

Along the curve, two phases of water and water vapours coexist in equilibrium.

The vapour pressure is one atmosphere. The corresponding temperature in degrees Celsius is the boiling point of water, which is 100℃.

• OB- Ice and Vapor

The OB curve is a sublimation curve. The curve OB comes to an end at B, which is the absolute zero temperature of -273℃. It depicts the vapour pressure of solid ice at various temperatures.

In equilibrium, the two phases of solid ice and water vapour coexist.

• OC- Ice & Water

The OC curve is the fusion curve. The OC curve comes to an end at C, the critical pressure.

In equilibrium, the two phases of solid ice and liquid water coexist. The curve shows that the melting point of ice decreases as pressure increases. At one atm, the line intersects the curve at 0℃.

The area AOC, AOB, and BOC the curves or areas between the curves represent the conditions at temperature and pressure under which a single phase, i.e., ice, water, and water vapour, can exist indefinitely.

• BOC- Ice- denotes the solid phase.
• COA- Water- denotes the liquid phase.
• AOB- Vapour- denotes the gaseous phase

Triple Point:

All three curves, OA, OB, and OC, meet at point O, which is known as the triple point, where all three phases, solid, liquid, and vapour, are in simultaneous equilibrium.

At 0.0075℃ and 4.58 mm Hg pressure, the triple point occurs.

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Frequently Asked Questions on Phase Diagram of Water

What is the difference between the general phase diagram and the phase diagram for water.

The slope of the line connecting the solid and liquid states in the water diagram is negative rather than positive.

Define critical temperature.

A substance’s critical temperature (T C ) is the highest temperature at which it can exist as a liquid.

What does the critical point mean in the phase diagram of water?

The critical point is the point at which the critical temperature and pressure meet. Water particles in the gas phase are moving very quickly at 373.99°C. No matter how much pressure is applied to the gas, the gas phase cannot liquefy at any temperature higher than that.

Why does the melting point of ice get lower at higher pressures?

When pressure increases, the melting point of ice decreases because volume decreases, and the volume of water is less than that of ice.

Is it possible to compress steam to form a liquid if the temperature is higher than 374°C?

Water particles in the gas phase are moving very quickly at 373.99°C. No matter how much pressure is applied to the gas, the gas phase cannot liquefy at any temperature higher than that.

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The 5 Elemental Symbols: Fire, Water, Air, Earth, and Spirit

Learn Religions / Kelly Miller

• Mythological Figures
• Satanic Beliefs and Creeds
• M.A., History, University of Wisconsin, Milwaukee
• B.A., History, Kalamazoo College

The Greeks proposed the existence of five basic elements. Of these, four were the physical elements—fire, air, water, and earth—of which the entire world is composed. Alchemists eventually associated four triangular symbols to represent these elements. The fifth element, which goes by a variety of names, is more rarefied than the four physical elements. Some call it Spirit. Others call it aether or quintessence (literally " the fifth element " in Latin).

In traditional Western occult theory, the elements are hierarchical: spirit, fire, air, water, and earth—with the first elements being more spiritual and perfect and the last elements being more material and base. Some modern systems, such as Wicca , view the elements as equal.

Before we examine the elements themselves, it is important to understand the qualities, orientations, and correspondences that are associated with the elements. Each element is connected with aspects in each of these, and it helps correlate their relationship to one another.

Elemental Qualities

In classical elemental systems, each element has two qualities and shares each quality with one other element.

Warm or Cold

Each element is either warm or cold, and this corresponds with a male or female gender. This is a strongly dichotomous system, where male qualities are things like light, warmth, and activity, and female qualities are dark, cold, passive, and receptive.

The orientation of the triangle is determined by warmth or coldness, male or female. Male, warm elements point upward, ascending toward the spiritual realm. Female, cold elements point downward, descending into the earth.

Moist or Dry

The second pair of qualities is moistness or dryness. Unlike the warm and cold qualities, moist and dry qualities do not immediately correspond to other concepts.

Opposing Elements

Because each element shares one of its qualities with one other element, that leaves one element completely unrelated.

For example, air is moist like water and warm like fire, but it has nothing in common with earth. These opposing elements are on opposite sides of the diagram.

Hierarchy of Elements

There is traditionally a hierarchy of elements, although some modern schools of thought have abandoned this system. The lower elements in the hierarchy are more material and physical, with the higher elements becoming more spiritual, more rarefied, and less physical.

That hierarchy can be traced through this diagram. Earth is the lowest, most material element. Circling clockwise from earth you get water, then air, and then fire, which is the least material of the elements.

Elemental Pentagram

The pentagram has represented many diverse meanings over the centuries. Since at least the Renaissance, one of its associations is with the five elements.

Arrangement

Traditionally, there is a hierarchy among the elements ranging from the most spiritual and rarefied to the least spiritual and most material. This hierarchy determines the placement of elements around the pentagram.

Starting with spirit, the highest element, we descend to fire, then follow the lines of the pentagram over to air, across to water, and down to earth, the lowest and most material of the elements. The final line between earth and spirit completes the geometric shape.

Orientation

The issue of a pentagram being point-up or point-down only gained relevance in the 19th century and has everything to do with the arrangement of elements. A point-up pentagram came to symbolize spirit ruling over the four physical elements, while a point-down pentagram symbolizes spirit being subsumed by matter or descending into matter.

Since then, some have simplified those associations to represent good and evil. This is generally not the position of those who commonly work with point-down pentagrams, and is often not the position of those associating themselves with point-up pentagrams either.

The colors used here are those associated with each element by the Golden Dawn. These associations are commonly borrowed by other groups as well.

Elemental Correspondences

Ceremonial occult systems traditionally depend on systems of correspondence: collections of items that are all associated in some way with the desired goal. While the types of correspondences are nearly endless, associations between elements, seasons, time of day, elements, moon phases, and directions have become fairly standardized in the West. These are frequently the basis for additional correspondences.

The Golden Dawn's Elemental and Directional Correspondences

The Hermetic Order of the Golden Dawn codified some of these correspondences in the 19th century. Most notable here are the cardinal directions.

The Golden Dawn originated in England, and the directional and elemental correspondences reflect a European perspective. To the south are the warmer climates, and thus are associated with fire. The Atlantic Ocean lies to the west. The north is cold and formidable, a land of the earth but sometimes not a lot else.

Occultists practicing in America or elsewhere sometimes do not find these correspondences to work.

Daily, Monthly, and Yearly Cycles

Cycles are important facets of many occult systems. Looking at daily, monthly, and yearly natural cycles, we find periods of growth and dying, of fullness and barrenness.

• Fire is the element of fullness and life, and it is closely associated with the sun. Therefore, it's unsurprising that noon and summer would be associated with fire. By that same logic, the full moon should also be in the same category.
• Earth is in the opposite direction as fire and therefore corresponds with midnight, winter, and the new moon. While these things might represent barrenness, more often they are representative of potential and transformation; the point where the old gives way to the new; the empty fertility readies to feed new creations.
• Air is the element of new beginnings, youth, increase, and creativity. As such, it is associated with spring, the waxing moon, and sunrise. Things are growing warmer and brighter, while plants and animals give birth to a new generation.
• Water is the element of emotion and wisdom, particularly the wisdom of age. It represents a time past the peak of livelihood, moving toward the end of the cycle.

FuatKose/Getty Images

Fire is associated with strength, activity, blood, and life force. It is also seen as highly purifying and protective, consuming impurities and driving back the darkness.

Fire is traditionally seen as the most rarefied and spiritual of the physical elements because of its masculine properties (which were superior to female properties). It also lacks physical existence, produces light, and has a transformative power when it comes in contact with more physical material.

• Qualities: Warm, Dry
• Gender: Masculine (active)
• Elemental: Salamander (Here referring to a mythological lizard creature which could burst into flames)
• Golden Dawn Direction: South
• Golden Dawn ​Color: Red
• Magical Tool: Sword, athame, dagger, sometimes wand
• Planets: Sol (Sun), Mars
• Zodiac signs: Aries, Leo, Sagittarius
• Season: Summer
• Time of Day: Noon

Air is the element of intelligence, creativity, and beginnings. Largely intangible and without permanent form, air is an active, masculine element, superior to the more material elements of water and earth.

• Qualities: Warm, Moist
• Elemental: Sylphs (Invisible beings)
• Golden Dawn Direction: East
• Golden Dawn Color: Yellow
• Magical Tool: Wand, sometimes a sword, dagger, or athame
• Planets: Jupiter
• Zodiac Signs: Gemini, Libra, Aquarius
• Season: Spring
• Time of Day: Morning, Sunrise

Water is the element of emotion and the unconscious, as opposed to the conscious intellectualism of air.

Water is one of two elements with a physical existence that can interact with all of the physical senses. Water is still considered less material (and thus superior) to Earth because it possesses more motion and activity than Earth.

• Qualities: Cold, Moist
• Gender: Feminine (passive)
• Elemental: Undines (water-based nymphs)
• Golden Dawn Direction: West
• Golden Dawn Color: Blue
• Magical Tool: Cup
• Planets: Moon, Venus
• Zodiac Signs: Cancer, Scorpio, Pisces
• Season: Fall
• Time of Day: Sunset

Jutta Kuss/Getty Images

Earth is the element of stability, groundedness, fertility, materiality, potential, and stillness. Earth can also be an element of beginnings and endings, or death and rebirth, as life comes from the ground and then decomposes back into the earth after death.

• Qualities: Cold, Dry
• Elemental: Gnomes
• Golden Dawn Direction: North
• Golden Dawn Color: Green
• Magical Tool: Pentacle
• Planets: Saturn
• Zodiac Signs: Taurus, Virgo, Capricorn
• Season: Winter
• Time of Day: Midnight

Raj Kamal/Getty Images

The element of spirit does not have the same arrangements of correspondences as the physical elements since spirit is not physical. Various systems may associate planets, tools, and so forth with spirit, but such correspondences are far less standardized than those of the other four elements.

The element of spirit goes by several names. The most common are spirit, ether or aether, and quintessence, which is Latin for " fifth element ."

There is also no standard symbol for spirit, although circles are common . Eight-spoked wheels and spirals are also sometimes used to represent spirit.

Spirit is a bridge between the physical and the spiritual. In cosmological models, the spirit is the transitory material between the physical and celestial realms. Within the microcosm, the spirit is the bridge between body and soul.

• Golden Dawn Direction: Above, Below, Within
• Golden Dawn Color: Violet, Orange, White
• Geometric Shapes and Their Symbolic Meanings
• The Rosy Cross or Rose Cross
• Pentagrams Meaning
• Alchemical Sulfur, Mercury and Salt in Western Occultism
• Associating Zodiac Signs With Elements
• Complicated Polygons and Stars
• Ancient Spirals
• Marriage of the Red King and White Queen in Alchemy
• Occult Symbols
• The Symbolism of Squares
• Rebis: the Result of the Great Work in Alchemy
• The Hexagram's Use in Religion
• Deciphering Eliphas Levi's Baphomet: The Goat of Mendes
• What You Should Know About Octagrams - Eight-Pointed Stars
• Vodoun Symbols for Their Gods

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6.1: Group Theory Principles Applied to H2O

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• Frank Rioux
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Symmetry Analysis for H 2 O

Water belongs to the C 2v symmetry group and has the following symmetry elements: E, C 2 z , σ xz , and σ yz . Its character table is shown below.

\[ \begin{array}{|c|c|c|c|c|c|} \hline \text{C}_{2v} & \text{E} & \text{C}_2^z & \sigma_{xz} & \sigma_{yz} & \text{h = 4} \\ \hline \text{A}_1 & 1 & 1 & 1 & 1 & z,~ x^2,~y^2,~z^2 \\ \hline \text{A}_2 & 1 & 1 & -1 & -1 & \text{xy, R}_z \\ \hline \text{B}_1 & 1 & -1 & 1 & -1 & \text{x, xz, R}_y \\ \hline \text{B}_2 & 1 & -1 & -1 & 1 & \text{y, yz, R}_x \\ \hline \Gamma_{tot} & -9 & -1 & 1 & 3 \\ \hline \end{array} \nonumber \]

The C 2v symmetry group has four symmetry elements and four associated symmetry operations. It can be thought of as a four dimensional space with the A 1 , A 2 , B 1 , and B 2 irreducible representations playing the role of unite vectors of vector algebra. The irreducible representations span the space and any other vector or representation in that space can be written as a linear combination of them. For simple groups like this character table can be generated by examining how translations along the x-, y-, and z-axes and rotations about these axes transform under the symmetry operations of the group. Using the figure show below you should be able to confirm the designations shown in the right-hand column of the character table. That is the translation in the z-direction transforms like A 1 , rotation about the z-axis transforms like A 2 , translation in the x-direction and rotation about the y-axis have symmetry properties represented by B 1 , and translation in the y-direction and rotation about the x-axis have B 2 symmetry. In doing this note (see Atkins) that you record +1 if something is transformed into itself, -1 if it is transformed into minus itself, and 0 if it is transformed into something else under the symmetry operations of the group.

Note that just like the unit vectors in the Cartesian coordinate system, these irreducible representations are orthogonal. That is they have zero overlap.

\[ \begin{matrix} \text{A}_1 \text{B}_2 = \dfrac{[(-1)(1) + (1)(-1) + (1)(-1) + (1)(1)]}{4} = 0 \\ \text{A}_2 \text{B}_1 = \dfrac{[(1)(1) + (1)(-1) + (-1)(1) + (-1)(-1)]}{4} = 0 \end{matrix} \nonumber \]

Dividing by 4, the order of the group, normalizes the calculation. We can also show that all of the irreducible representation are normalized or have an overlap of 1.

\[ \text{A}_2 \text{A}_2 = \dfrac{[(1)(1) + (1)(1) + (-1)(-1) + (-1)(-1)]}{4} = 1 \nonumber \]

Note that these vector operations are exactly equivalent to evaluating quantum mechanical overlap integrals \( \int \Psi_1 \Psi_2 d \tau\).

Since H 2 O is a triatomic molecule it has a total of 9 degrees of freedom - three for each atom in the molecule. These are the x-, y-, and z-coordinates of the individual atoms and are shown as small vectors located on each of the atoms in the figure below.

A symmetry analysis for water begins by determining how these 9 coordinates behave under the symmetry operations of the C 2v group. You should be able to show that this generates the reducible representation Γ tot , which is given in the last row of the character table shown above. Γ tot can also be calculated as Γ uma (Γ x + Γ y + Γ z ), where Γ uma is the behavior of the atoms under the symmetry operations of the group. Of the 9 degrees of freedom possessed by the water molecule, three are for translation of the center of mass in the x-, y- and z-directions, and three are related to rotation about the x-, y-, and z-axes. This leaves three vibrational degrees of freedom. To determine the symmetry of the vibrational modes we decompose Γ tot into the unit vectors or irreducible representations of the C 2v character table. This involves taking the dot product of Γ tot with each of the irreducible representations of the C 2v symmetry.

\[ \begin{matrix} \Gamma_{tot} \text{A}_1 = \dfrac{[(9)(1) + (-1)(1) + (1)(1) + (3)(1)]}{4} = 3 \\ \Gamma_{tot} \text{A}_2 = \dfrac{[(9)(1) + (-1)(1) + (1)(-1) + (3)(-1)]}{4} = 1 \\ \Gamma_{tot} \text{B}_1 = \dfrac{[(9)(1) + (-1)(-1) + (1)(1) + (3)(-1)]}{4} = 2 \\ \Gamma_{tot} \text{B}_2 = \dfrac{[(9)(1) + (-1)(-1) + (1)(-1) + (3)(1)]}{4} = 3 \end{matrix} \nonumber \]

This procedure has revealed the that the reducible representation, Γ tot , is composed of the following irreducible representations:

\[ \Gamma_{tot} = 3 \text{A}_1 + \text{A}_2 + 2 \text{B}_1 + 3 \text{B}_2 \nonumber \]

From the right side of the character table we see that translation and rotation have the following symmetry properties:

\[ \begin{matrix} \Gamma_{trans} = \text{A}_1 + \text{B}_1 + \text{B}_2 \\ \Gamma_{rot} = \text{A}_2 + \text{B}_1 + \text{B}_2 \end{matrix} \nonumber \]

From this information we can determine the symmetry properties of the vibrational modes of the water molecule.

\[ \Gamma_{vib} = \Gamma_{tot} - \Gamma_{trans} - \Gamma_{rot} = 2 \text{A}_1 + \text{B}_2 \nonumber \]

There are three vibrational fundamentals. And, since there are two bonds there will be two stretching vibrations and one bending vibration. To determine which symmetry classification the bend belongs to, examine how the H-O-H bond angle transforms under the symmetry operations of the C 2v group.

The vibrational modes are shown above. Convince yourself that their symmetry properties are captured in the table below.

\[ \begin{array}{|c|c|c|c|c|c|} \hline \text{C}_{2v} & \text{E} & \text{C}_2^z & \sigma_{xz} & \sigma_{yz} \\ \hline \Gamma_{bend} & 1 & 1 & 1 & 1 & \text{A}_1 \\ \hline \Gamma_{stretch} & 1 & 1 & 1 & 1 & \text{A}_1 \\ \hline \Gamma_{stretch} & 1 & -1 & -1 & 1 & \text{B}_2 \\ \hline \end{array} \nonumber \]

Whether any of these modes is active in the infrared region of the spectrum is another question. This is determined by evaluating the quantum mechanical transition probability integral,

\[ P = e \int \Psi^f (vib) \hat{q} \Psi^i (vib) d \tau \nonumber \]

where q is the spatial coordinate and is either x, y, or z. If this integral is non-zero the transition is allowed. Group theory enables us to determine if the integral is non-zero as follows. We evaluate the direct product (note the similarity to the quantum mechanical transition probability integral) Γ vib f Γ q Γ vib i and if it turns out to be equal to, or contain, the irreducible representation A 1 then the transition is allowed. If the direct product isn't equal to or contain A 1 then the transition is forbidden.

Γ vib i , the representation for the ground state is always equal to A 1 and Γ vib f is always equal to A 1 and Γ vib f has the symmetry of the vibrational mode being excited, which in our case is either A 1 or B 2 . Thus we can see that a vibrational mode will be infrared active if it belongs to the same symmetry species as one of the Cartesian coordinates. You should verify that this is correct and also be able to show that all three vibrational modes of the water molecule are infrared active.

Group theory is also very helpful in constructing molecular orbitals from linear combinations of atomic orbitals. For the water molecule the atomic orbitals are the hydrogen 1s orbitals and the 2s and three 2p orbitals on the oxygen atom. We proceed by examining how the orbitals transform under the symmetry operations of the group. This is shown in the table below.

\[ \begin{array}{|c|c|c|c|c|c|} \hline \text{C}_{2v} & \text{E} & \text{C}_2^z & \sigma_{xz} & \sigma_{yz} \\ \hline \text{H}_1 & \text{H}_1 & \text{H}_2 & \text{H}_2 & \text{H}_1 \\ \hline \text{O}_{2s} & \text{O}_{2s} & \text{O}_{2s} & \text{O}_{2s} & \text{O}_{2s} \\ \hline \text{O}_{2px} & \text{O}_{2px} & - \text{O}_{2px} & \text{O}_{2px} & - \text{O}_{2px} \\ \hline \text{O}_{2py} & \text{O}_{2py} & - \text{O}_{2py} & - \text{O}_{2py} & \text{O}_{2py} \\ \hline \text{O}_{2pz} & \text{O}_{2pz} & \text{O}_{2pz} & \text{O}_{2pz} & \text{O}_{2pz} \\ \hline \text{A}_1 & 1 & 1 & 1 & 1 \\ \hline \text{A}_2 & 1 & 1 & -1 & -1 \\ \hline \text{B}_1 & 1 & -1 & 1 & -1 \\ \hline \text{B}_2 & 1 & -1 & -1 & 1 \\ \hline \end{array} \nonumber \]

To find the molecular orbital with A 1 symmetry take the sum of the dot product of the A 1 irreducible representation with each of the representations in the top half of the table. This is 2(H 1 + H 2 ) + 4O 2s + 4O 2pz . This indicates that the molecular orbital will consist of a linear combination of four of the six orbitals shown in the table above.

\[ \Psi (A_1) = c_h \left[ \psi (H_1) + \psi (H_2) \right] + c_o \psi (O_{2s}) + c_o \psi (O_{2pz}) \nonumber \]

When this is done for A 2 symmetry it is found that the sum is zero, indicating that there is no molecular orbital with A 2 symmetry. Repeating this process for B 1 symmetry yields 4O 2px . This means Ψ(B 2 ) = Ψ(O 2px ) and we say that the oxygen 2p x orbital is non-bonding because there are hydrogen orbitals with the same symmetry. This is easily seen by inspection of the water molecule when placed in a Cartesian coordinate system as shown in the figure above.

For B 2 symmetry the result is 2(H 1 - H 2 ) + 4O 2py . This is clearly another bonding molecular orbital.

\[ \Psi (B_2) = c_h \left[ \psi (H_1) - \psi (H_2) \right] + c_o \psi (O_{2py}) \nonumber \]

The next step is to use the molecular orbitals from this symmetry analysis as the basis for a self-consistent field calculation on the water molecule. This will yield a set of coefficients for the molecular orbitals and a molecular orbital energy level diagram. This results are given below. You should compare these results with the traditional valence bond (Lewis structure) formulation of the bonding in H 2 O.

\[ \begin{matrix} \Psi (A_1) = 0.15 \left[ \psi (H_1) + \psi (H_2) \right] + 0.82 \psi (O_{2s}) + 0.13 \psi (O_{2pz}) \\ \Psi (B_2) = 0.42 \left[ \psi (H_1) - \psi (H_2) \right] + 0.62 \psi (O_{2py}) \\ \Psi (A_1) = 0.26 \left[ \psi (H_1) + \psi (H_2) \right] - 0.50 \psi (O_{2s}) + 0.79 \psi (O_{2pz}) \\ \Psi (B_1) = \Psi (O_{2px}) \\ \Psi (A_1) = 0.75 \left[ \psi (H_1) + \psi (H_2) \right] - 0.84 \psi (O_{2s}) - 0.70 \psi (O_{2pz}) \\ \Psi (B_2) = 0.89 \left[ \psi (H_1) - \psi (H_2) \right] - 0.99 \psi (O_{2py}) \end{matrix} \nonumber \]

The 1a 1 MO is, as would be expected, the oxygen non-valence 1s orbital. The 2a 1 MO is mainly a linear combination of the hydrogen 1s and the 2s on oxygen. The 1b 2 MO is entirely the oxygen 2p y and the hydrogen 1s atomic orbitals. The 3a 1 MO and the 1b 1 MO correspond to the two long pairs. The 3a 1 MO and the 1b 1 MO correspond to the two long pairs. The 3a 1 MO consists mainly of the oxygen 2s and 2p z and the hydrogen 1s atomic orbitals and has its maximum value in the +z direction. The 1b 1 is entirely oxygen 2p x and thus is directed at right angles to the plane of the molecule; the lone pairs thus have their maxima in the xz plane.

The first two electronic transitions in water occur at 170 and 130 nm. The molecular orbital diagram obtained previously and the principles of group theory can be used to intepret the electronic spectrum as is shown in the figure below.

Molecular orbital scheme and state diagram for H 2 O.

By analogy with IR spectroscopy those electronic transitions are allowed for which the transition moment integral is non-zero.

\[ M = e \int \Psi^f (elec) \hat{q} \Psi^i (elec) d \tau \nonumber \]

As in the case of IR spectroscopy, this integral can be non-zero if and only if Γ vib f Γ q Γ vib i is equal to or contains the totally symmetric irreducible representation, A 1 . The figure shows that the electronic ground state has A 1 symmetry while the electric dipole operator (μ=eq) has the symmetry of the Cartesian coordinates (A 1 , B 1 , B 2 ). Thus Γ vib f Γ q Γ vib i will equal A 1 if the excited state has A 1 , B 1 , or B 2 symmetry. If this is the case then the transition is said to be orbitally allowed. Note that several orbitally allowed transitions are spin-forbidden (involve a change in spin multiplicity) and therefore do not occur.

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The Spiritual Meaning of Water

Water is a representation of wisdom, peace, and purity. the spiritual meaning of water lies in its reflection of the human spirit and the interconnection of all creations..

We are all well aware of the necessity of water. From keeping us alive to quenching our thirst, to washing the dirt from our bodies and our food, not to mention its role in producing that food, our survival depends on water. But what does such a valuable life force mean spiritually? Is there any spiritual essence or spiritual meaning of water?

Sacred Importance of Water

As a symbol of life, water represents divine generosity. Indeed, it’s a personification of healing and suffering, protection and vulnerability, change and constancy, stillness and rage. Water in all its glory is considered a purifier of the soul.

Of course, water is the basis of purification in multiple religions, and many cultures use water as a sacred practice to heal internal turmoil. For instance, Hindu devotees take a holy dip in the River Ganga to purify the soul. Similarly, Islam has a daily ritual of ablution (called “Wudhu”) to cleanse inside and outside the body before prayers. In Christianity, the ritual of baptism through holy water symbolizes purity. Likewise, other cultures use water to cleanse the spirit and eliminate impurities from the soul.

[Read: “Water Rituals Around the World.” ]

Fluid Spirituality: Be Like Water

As Bruce Lee famously said: “Empty your mind. Be formless, shapeless, like water. If you put water into a cup, it becomes the cup. You put water into a bottle, it becomes the bottle. You put it in a teapot, it becomes the teapot. Water can flow or it can crash. Be water, my friend.”

This depicts the flexibility and the changing nature of water. To be like water, we need to be fluid; we need to change and grow from the circumstances of life. Furthermore, we need to imbibe the formless, calm, and conscious nature of the water.

Drinking in the Spiritual Meaning of Water

Generally, when we drink water, we tend to be unaware of its significance. Habitually, we drink it hastily. But if you were to drink water to absorb its benefits completely, how would you drink it?

We can drink water with intention—an intention of healing, purification, spiritual wellness, and inner peace. With an intention of gratitude towards the earth.

According to Ayurveda, there are certain ways to drink water:

• Sitting down. When you drink water standing, the fluids get drawn into your joints. However, when you drink water seated, it reaches different organs sufficiently.
• With a 45-minute gap . Sip water 45 minutes before and after food intake. If you drink water before food it gets diluted with gut enzymes and reduces digestion. Excess water right after food alters the digestion process and may cause obesity.
• 8 glasses . Too little water makes you dehydrated. Excess water makes you groggy and dizzy due to electrolyte imbalance. Drinking eight glasses of water a day is considered optimum.
• Drink it warm. Cold water disrupts the digestion process, whereas warm water improves blood supply to organs and promotes metabolism.
• In small sips. Taking small divided sips instead of big gulps strengthens the digestive system.
• On an empty stomach. Drink warm water as soon as you wake up. This assists the digestion process.

Meditating on the Source

The spiritual meaning of water is evident in its form and sound. Like the crashing ocean waves, the serene river, the roaring waterfall, and the drizzling rain. It’s all something to marvel at and use as a basis for meditation practices.

What we perceive the water to be is what we want to instill in our hearts. Try meditating on the source of water. Take a deep breath in and out and bring your awareness to the sound or form of the water. Then visualize the water and its features. Accordingly, form a connection with the water and your true self. This leads to a sense of clarity and peace of mind.

In essence, water is the life force that centers our spirit and our body, signifying wisdom and conscious awareness. Take time to ponder the spiritual meaning of water and celebrate its gifts on earth.

Keep the flow: Consider “11 Ways to Raise Your Vibration With Water.”

About the Author

Amina Shazi is a health and wellness enthusiast and freelance writer focused on inspiring change through words. When she’s not writing you'll find her watching anime, drawing, and enjoying her cup of tea. Learn more about her passion at amishazi.com.

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The Fascinating Symbolism of Water in the Bible: 9 Attributes

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Symbolism of water in the Bible is a significant part of our Christian walk. It has symbolized the Spirit of God, cleaning and sanctification, God’s Word and so much more.

Water is so important in God’s Word that it is mentioned 722 times  in the Bible, more often than faith, hope, prayer, and worship

Of course, it is also an important part of our physical life as we need water to survive.

If you are interested in Bible symbolism, you may wish to check out my posts on the symbolism of plants in the Bible , colors in the Bible , milk in the Bible , bread in the Bible or one of my other posts on symbolism in the Bible.

Are you ready to dive into the symbolism of water in the Bible ?

Meanings of Water in the Bible

Water as the spirit of god.

Water is mentioned in the second verse of the Bible , in Genesis 1:2

The earth was without form and void, and darkness was over the face of the deep. And the Spirit of God was hovering over the face of the waters. – Genesis 1:2

The symbolism of water in the Bible is used in numerous verses to refer to the Spirit of God. In John 7:37-39 Jesus gave a prophecy about the Spirit that would later come and referred to him as the living water.

On the last day of the feast, the great day, Jesus stood up and cried out, “If anyone thirsts, let him come to me and drink. Whoever believes in me, as the Scripture has said, ‘Out of his heart will flow rivers of living water.’” Now this he said about the Spirit, whom those who believed in him were to receive, for as yet the Spirit had not been given, because Jesus was not yet glorified. – John 7:37-39 (ESV)

The Spirit was poured on the day of Pentecost to empower members so they could go forth and make disciples.

A Symbol Of Spiritual Things

There are numerous verses of water in the Bible pointing to the correlation between water and spiritual things.

In one such instance, in Isaiah 8:6-7, the Bible talks of the waters that flow gently.

“Because this people has refused the waters of Shiloah that flow gently, and rejoice over Rezin and the son of Remaliah,therefore, behold, the Lord is bringing up against them the waters of the River, mighty and many, the king of Assyria and all his glory. And it will rise over all its channels and go over all its banks, – Isaiah 8:6-7 (ESV)

In this verse, we see a contrast between the gently flowing waters, which refer to spiritual things and the strong waters which point to falsities.

Difficulties

Water in the Bible is also used to denote difficult times in life, especially for us, God’s children .

The prophet Jeremiah talks of waters and rivers as symbols of falsities and difficulties:

And now what do you gain by going to Egypt     to drink the waters of the Nile? Or what do you gain by going to Assyria     to drink the waters of the Euphrates?- Jeremiah 2:18

Both the water of the Nile and the waters of the Euphrates are used symbolically to portray the falsities from reasonings that people were turning to and away from God. While there was an option for people to follow the right way, God through Jeremiah points out the faulty thinking.

A few other verses which reinforce the symbolism of water in the Bible as difficulties in our Christian lives:

Therefore let everyone who is godly     offer prayer to you at a time when you may be found; surely in the rush of great waters,     they shall not reach him. – Psalm 32:6

When you pass through the waters, I will be with you; and through the rivers, they shall not overwhelm you; when you walk through fire you shall not be burned, and the flame shall not consume you. – Isaiah 43:2

water closed over my head; I said, ‘I am lost.’ – Lamentations 3:54

Water As Eternal Life

In both the Old and New Testaments , the word “water” is used as a symbol for salvation and eternal life, which God offers us through faith in his Son.

In the well-known story of the Samaritan woman by the well, Jesus tells her:

Jesus answered her, “If you knew the gift of God, and who it is that is saying to you, ‘Give me a drink,’ you would have asked him, and he would have given you living water.” The woman said to him, “Sir, you have nothing to draw water with, and the well is deep. Where do you get that living water? Are you greater than our father Jacob? He gave us the well and drank from it himself, as did his sons and his livestock.” 

Jesus said to her, “Everyone who drinks of this water will be thirsty again, but whoever drinks of the water that I will give him will never be thirsty again. The water that I will give him will become in him a spring of water welling up to eternal life.” The woman said to him, “Sir, give me this water, so that I will not be thirsty or have to come here to draw water.” – John 4:10-15 (ESV)

Here, Jesus refers to the water that He is to give the woman as the living water, and He goes ahead to say He is the source of that water which springs up to eternal life.

In Revelation 22:1, the Bible talks of the river of the water of life that flows from the throne of God. This is a description of heaven.

Then the angel showed me the river of the water of life, bright as crystal, flowing from the throne of God and of the Lamb –

Revelation 22:1 (ESV)

Additional verses about Jesus as the living water in the Bible:

With joy you will draw water from the wells of salvation. – Isaiah 12:3 (ESV)

The Spirit and the Bride say, “Come.” And let the one who hears say, “Come.” And let the one who is thirsty come; let the one who desires take the water of life without price. – Revelation 22:17 (ESV)

“Come, everyone who thirsts,      come to the waters; and he who has no money,      come, buy and eat! Come, buy wine and milk      without money and without price. – Isaiah 55:1 (ESV)

Another symbolism of water in the Bible is that of spiritual cleansing that we undergo when we accept God’s offer of salvation.

Shortly after the creation of the world and the expulsion of Adam and Eve out of their Garden, Noah is introduced. Water again becomes a purifying agent to cleanse this new earth from wickedness in order for humans to start anew.

Water is a strong symbol in the Bible that represents cleansing from sin. For example, 1 John 5:7-9 talks of “the water and blood.” This passage usually refers to the Holy Spirit’s work as well as Christ’s sacrifice on our behalf when we are baptized into Him through faith.

In fact, in Ephesians 5:26, the “water” that does the cleansing of the bride, the church, is directly tied in with God’s Word , of which it is a symbol.

that he might sanctify her, having cleansed her by the washing of water with the word, – Ephesians 5:26 (ESV)

While baptism in water is not the magic bullet to become a Christian, but an outward symbol to recognize an inward change in us.

Additional verses about water in the Bible as a symbol for cleansing:

I will sprinkle clean water on you, and you shall be clean from all your uncleannesses , and from all your idols I will cleanse you. – Ezekiel 36:25 (ESV)

let us draw near with a true heart in full assurance of faith , with our hearts sprinkled clean from an evil conscience and our bodies washed with pure water. – Hebrews 10:22 (ESV)

Water in the Bible as the Word of God

The word of God is often referred to as water in the Bible because it acts like a cleansing agent which explains what Paul meant when he wrote about Jesus’ cleansing the church.

that he might sanctify her, having cleansed her by the washing of water with the word,so that he might present the church to himself in splendor, without spot or wrinkle or any such thing, that she might be holy and without blemish – Ephesians 5:26-27 (ESV)

There has been some mystery behind the meaning when John refers to water in 1 John 5:7-9.

For there are three that testify: the Spirit and the water and the blood; and these three agree.  If we receive the testimony of men, the testimony of God is greater, for this is the testimony of God that he has borne concerning his Son. – 1 John 5:7-9 (ESV)

The Holy Spirit is the third person of our Trinity. He convicts sinners and points them to their need for a Savior, just like Jesus did on earth during his earthly ministry. This conviction in turn leads people toward repentance where they acknowledge that there is no hope without God’s forgiveness through Christ’s blood sacrifice at Calvary ( Colossians 2:13-14). The Word then can be compared with water as John says these three testify – the spirit, the word, and the water;

Water as Freedom and Being Set Apart

For the people of Israel , it is only through water that they find freedom and are set apart.

Then Moses stretched out his hand over the sea, and the Lord drove the sea back by a strong east wind all night and made the sea dry land, and the waters were divided. – Exodus 14:21 (ESV)

The chosen people travel to enter the Promised Land but must first wander in the desert. Moses listens to the cries of his people for water and, “lifted up his hand and struck the rock with his rod twice; and water came forth abundantly, and the congregation drank, and their cattle” (Numbers 20:11).

Water not only purifies and sets free, but also  nourishes and gives new life.

Living Water in the Bible

Living water is a phrase that can be found in different books throughout the Bible. We see it twice more during Old Testament times and then two new ones when Jesus comes along with his ministry, but there are four main instances of this word’s interesting use:

In the first instance we find “living” or “flowing” waters as mentioned by God to Moses at Mount Sinai (Exodus 17-18). The text says specifically “the Lord said unto him [Moses]…I will give you for a law”.

One of the early examples of living water in the Bible comes from Jeremiah 2:13 where God says, “My people have committed two sins: They have forsaken me, the spring of living water, and have dug their own cisterns, broken cisterns that cannot hold water.”

Later in Jeremiah 17:13, Jeremiah says of God , “Lord, you are the hope of Israel; all who forsake you will be put to shame. Those who turn away from you will be written in the dust because they have forsaken the Lord, the spring of living water.”

In these two passages, God is referred to as “the spring of living water,” who His people have abandoned to try and do things their own way by digging their own cisterns and turning away from Him. The passages show that God gives His people everything they need, like water, and that when they turn away from Him, they are lost and hopeless.

Those two times in Jeremiah Yahweh is metaphorically identified as “the spring of living water” (  Jeremiah 2:13  ;  17:13  ). In both instances Israel is rebuked for having forsaken the Lord for other cisterns that could in no way satisfy their “thirst.”

In a very important passage, Jesus identifies the “streams of living water” that flow from within those who believe in him with the Holy Spirit (  John 7:37-39  ). The reception of the Holy Spirit is clearly the special reception that was going to come after Jesus had been glorified at the Father’s right hand and happened on the Day of Pentecost as described in Acts 2.

In the New Testament, Jesus speaks to a Samaritan woman in John 4:10, saying, “If you knew the gift of God, and who it is that is saying to you, ‘Give me a drink,’ you would have asked him, and he would have given you living water.”

Later in John 7: 37-39 Jesus once again speaks of living water, saying, “‘Let anyone who is thirsty come to me and drink. Whoever believes in me, as Scripture has said, rivers of living water will flow from within them.’ By this he meant the Spirit, whom those who believed in him were later to receive. Up to that time the Spirit had not been given, since Jesus had not yet been glorified.”

In the two passages from John, Jesus speaks of living water not just as water, but also as a symbol for the Holy Spirit and what is to come after Jesus’ death and resurrection for his followers who believe in Him.

So, what is living water? Living water can be understood in various ways, but the clearest way is that living water is a symbol for salvation and a true knowledge of God, Jesus, and the Holy Spirit. God provides us with everything we need and is the “living water” that continues to always give to us.

Water in the Bible as a Symbolism of Salvation or Eternal Life

Isaiah talks about the wells of salvation.

With joy you will draw water from the wells of salvation. – Isaiah 12:3 (ESV)

In Revelation, Jesus says:

And he said to me, “It is done! I am the Alpha and the Omega, the beginning and the end. To the thirsty I will give from the spring of the water of life without payment. – Revelation 21:6 (ESV)

In John 4:14 we read:

but whoever drinks of the water that I will give him will never be thirsty again. The water that I will give him will become in him a spring of water welling up to eternal life.” – John 4:14 (ESV)

These verses confirm the symbolism of water in the Bible as Jesus as our salvation and eternal life.

Are You a Foodie?  Do you want to learn more about food in the Bible and its symbolism?

A guide to the symbolism of many foods in the Bible , including milk, honey, bread, and more.  Looking at scripture to describe food in biblical times and what it represents in our spiritual life. 122 pages.

Other Symbolism for Water in the Bible

In other passages of Scripture, the following are said to symbolize water:

God’s judgment

And I will make justice the line, and righteousness the plumb line; and hail will sweep away the refuge of lies, and waters will overwhelm the shelter.” – Isaiah 28:17 (ESV)

Man’s words

The words of a man’s mouth are deep waters; the fountain of wisdom is a bubbling brook. – Proverbs 18:4 (ESV)

Man’s purposes

The purpose in a man’s heart is like deep water, but a man of understanding will draw it out. – Proverbs 20:5 (ESV)

An adulterous woman

“Stolen water is sweet, and bread eaten in secret is pleasant.” – Proverbs 9:17 (ESV)

A person’s posterity

Hear this, O house of Jacob, who are called by the name of Israel, and who came from the waters of Judah, who swear by the name of the LORD and confess the God of Israel, but not in truth or right. – Isaiah 48:1 (ESV)

You may enjoy this brief video about water in the Bible by the International Fellowship of Christians and Jews.

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One of the most well known qualities of the ocean is that it is salty. The two most common elements in sea water, after oxygen and hydrogen, are sodium and chloride. Sodium and chloride combine to form what we know as table salt.

Sea water salinity is expressed as a ratio of salt (in grams) to liter of water, It is written parts per thousand (ppt). In sea water, there is typically close to 35 grams of dissolved salts in each liter (35ppt), but ranges between 33-37 grams per liter (33ppt - 37ppt).

But as in weather, where there are areas of high and low pressure, the ocean has areas of high and low salinity. Of the five ocean basins, the Atlantic Ocean is the saltiest. On average, there is a distinct decrease in salinity near the equator and at both poles, although for different reasons.

Near the equator, the tropics receive the most rain on a consistent basis. As a result, the fresh rain water falling into the ocean decreases the salinity of the surface water in that region. Rain decreases further from the equator, and with less rain and more sunshine, evaporation increases. Evaporation of water vapor from the ocean to the atmosphere leaves behind the salt, resulting in higher salinity. Toward the poles, fresh water from melting ice decreases the surface salinity once again.

The saltiest locations in the ocean are the regions where evaporation is highest or in large bodies of water where there is no outlet into the ocean. The saltiest ocean water is in the Red Sea and in the Persian Gulf region (around 40ppt) due to very high evaporation and little fresh water inflow.

Take it to the MAX!   "A Funny Bath" - The Dead Sea

Learning Lesson:   "A Funny Taste"

Floating sea ice is the result of a unique property of water. As the temperature decreases to 40°F (4°C), the molecules slow and contract, and the density increases, same as other substances. Below 40°F (4°C), however, the water molecules begin to bond to each other; as they do, they are held apart, and the water expands again, decreasing the density. At 32°F (0°C), all molecules are locked into a crystalline structure, resulting in a nine percent expansion in size. This expansion and corresponding decrease in density is the reason ice floats.

Take it to the MAX!   Titanic Bergs

The amount of salt in sea water determines the temperature at which sea water freezes because adding salt to water lowers the freezing temperature. Water with a salinity of 17ppt freezes at about 30°F (-1°C) and 35ppt water freezes at about 28.5°F (-2°C). However, sea ice itself contains very little salt, about a tenth of the amount of salt that sea water has. This is because ice will not incorporate salt into its crystal structure. Newly formed sea ice can trap pockets of salty water, called brine, making the ice salty at first. Eventually, though, the brine gets pushed out of the ice’s crystal structure. Therefore, older sea ice is actually drinkable.

Learning Lesson:   We all Scream for Ice Cream

The density of sea water, however, is influenced by both its temperature and salinity. Density increases as salinity increases and as temperature decreases. Because salt lowers the freezing point of water, sea water does not start forming the lattice structure that lowers its density until it is much colder than fresh water. Therefore, when sea ice does form and loses salt, the salt is concentrated in the water beneath it, and the salinity (and therefore the density) of the underlying water continues to increase well after an area is iced over.

Learning Lesson:   Salt 'n Lighter

The "Average Salinity" map (right) shows the lowest salinity being in the polar regions. However, this image depicts surface salinity only. Polar surface salinity is lower than in the tropical regions due to ice melting each summer. However, winter ice formation increases the salinity below the ocean surface, causing the water below the ice to sink. That sinking motion governs the flow of the ocean's deep-water currents.

Learning Lesson:   Diet Light

Article  

• Volume 24, issue 1
• HESS, 24, 397–416, 2020
• Peer review
• Related articles

On the representation of water reservoir storage and operations in large-scale hydrological models: implications on model parameterization and climate change impact assessments

Thanh duc dang, a. f. m. kamal chowdhury, stefano galelli.

During the past decades, the increased impact of anthropogenic interventions on river basins has prompted hydrologists to develop various approaches for representing human–water interactions in large-scale hydrological and land surface models. The simulation of water reservoir storage and operations has received particular attention, owing to the ubiquitous presence of dams. Yet, little is known about (1) the effect of the representation of water reservoirs on the parameterization of hydrological models, and, therefore, (2) the risks associated with potential flaws in the calibration process. To fill in this gap, we contribute a computational framework based on the Variable Infiltration Capacity (VIC) model and a multi-objective evolutionary algorithm, which we use to calibrate VIC's parameters. An important feature of our framework is a novel variant of VIC's routing model that allows us to simulate the storage dynamics of water reservoirs. Using the upper Mekong river basin as a case study, we calibrate two instances of VIC – with and without reservoirs. We show that both model instances have the same accuracy in reproducing daily discharges (over the period 1996–2005), a result attained by the model without reservoirs by adopting a parameterization that compensates for the absence of these infrastructures. The first implication of this flawed parameter estimation stands in a poor representation of key hydrological processes, such as surface runoff, infiltration, and baseflow. To further demonstrate the risks associated with the use of such a model, we carry out a climate change impact assessment (for the period 2050–2060), for which we use precipitation and temperature data retrieved from five global circulation models (GCMs) and two Representative Concentration Pathways (RCPs 4.5 and 8.5). Results show that the two model instances (with and without reservoirs) provide different projections of the minimum, maximum, and average monthly discharges. These results are consistent across both RCPs. Overall, our study reinforces the message about the correct representation of human–water interactions in large-scale hydrological models.

• Article (PDF, 10516 KB)
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• Article (10516 KB)

Dang, T. D., Chowdhury, A. F. M. K., and Galelli, S.: On the representation of water reservoir storage and operations in large-scale hydrological models: implications on model parameterization and climate change impact assessments, Hydrol. Earth Syst. Sci., 24, 397–416, https://doi.org/10.5194/hess-24-397-2020, 2020.

Hydrological systems consist of multiple physical, chemical, and biological processes, most of which are profoundly altered by anthropogenic interventions ( Nazemi and Wheater ,  2015 a , b ) . Land cover modifications or hydraulic infrastructures, for instance, affect both surface and subsurface hydrological processes by redistributing water over time and space ( Haddeland et al. ,  2006 ; Bierkens ,  2015 ) . Such alterations are expected to amplify in the near future, owing to the increase in water and energy consumption ( Abbaspour et al. ,  2015 ) . In this context, hydrological models play a key role, as they help in the planning of the use of water resources in a sustainable way, so as to avoid adverse impacts on ecosystems and livelihoods ( Bunn and Arthington ,  2002 ; Yassin et al. ,  2019 ) . A detailed and accurate representation of the anthropogenic interventions within hydrologic models is thus of paramount importance: successful water management plans must necessarily build on reliable models.

Water reservoirs are arguably one of the most common infrastructures altering hydrological processes at the catchment scale; yet, their representation in hydrological and land surface models is challenged by multiple factors. First, the vast majority of the models currently available was initially conceived to study and understand the behavior of natural systems, so the added representation of water reservoirs entails the partial modification of the model structure. Second, the existing databases (e.g., GRanD; Lehner et al. ,  2011 ) provide details on dam design specifications, but no information on the management aspects, such as the operating rules or flood contingency plans. Third, the installation of dams is generally combined with impoundment (or filling) strategies, which may largely differ from the steady-state operating rules and last from a few months to several years ( Gao et al. ,  2010 ; Zhang et al. ,  2016 ) . Although the complexity of these factors varies with the study site at hand, one might imagine that the representation of water reservoir storage and operations is particularly challenging for large-scale models, simply because of the number of dams deployed over time in large river basins. It is perhaps not surprising to observe that water reservoirs – and their corresponding operations – have not been consistently accounted for across the broad number of large-scale hydrological modeling studies available in the literature.

A simple and popular approach is the exclusion of large impounds from the streamflow routing models, a modeling choice that has been adopted in many regions across the globe ( Maurer et al. ,  2002 ; Jayawardena and Mahanama ,  2002 ; Akter and Babel ,  2012 ; de Paiva et al. ,  2013 ; Leng et al. ,  2016 ) . Such an approach can support the investigation of various physical processes (e.g., emergence of new hydrological regimes, generation of land surface fluxes), but obviously prevents the application of the hydrological models to downstream water management problems, such as investigating the impact of regime shifts on hydropower production. Another potential issue with this approach lies in the model parameterization, which might be affected by a calibration process carried out with hydrological time series altered by anthropogenic interventions. de Paiva et al. ( 2013 ) , for instance, implemented the MGB-IPH hydrologic–hydraulic model to the Amazon River basin – a region characterized by the presence of hydroelectric dams ( Finer and Jenkins ,  2012 ) – and yet showed reliable calibration performance at multiple gauging stations. A similar example is represented by Abbaspour et al. ( 2015 ) , who simulated hydrological and water quality processes for the entire European continent. Despite neglecting the presence of hydraulic infrastructures, the model yielded acceptable values for the goodness-of-fit statistics. One may thus wonder whether the calibration process somehow compensates for a deficiency in the model structure.

With the goal of striking a balance between an accurate representation of reservoirs and the “costs” due to the modification of the model structure, several researchers have adopted a hybrid approach, in which the output of hydrologic–hydraulic models (e.g., runoff or streamflow at multiple locations) is postprocessed with the aid of water management (or reservoir operation) models. The very first efforts employed data on water uses to correct the output of global models, such as WaterGAP ( Alcamo et al. ,  1997 ) or WBM ( Vörösmarty et al. ,  1998 ) . Using a similar concept, Hanasaki et al. ( 2006 ) accounted for 452 reservoirs in a global river routing model. More sophisticated postprocessing techniques are based on optimization algorithms, which are used to design either reservoir operating rules or sequences of reservoir discharges that meet pre-defined objectives (e.g., hydropower production). Lauri et al. ( 2012 ) and Hoang et al. ( 2019 ) , for example, first calibrated the distributed hydrological model VMod for the Mekong river basin and then postprocessed its output using a linear programming algorithm that designed the discharge time series for 126 dams over a given simulation scenario. Similarly, Turner et al. ( 2017 ) and Ng et al. ( 2017 ) examined the vulnerability of global hydropower production to climate changes and El Niño–Southern Oscillation by correcting the discharge simulated by WaterGAP. In this case, the correction entailed designing bespoke reservoir operating rules through the use of a stochastic dynamic programming algorithm ( Turner and Galelli ,  2016 ) . Other recent applications of postprocessing techniques were adopted in Masaki et al. ( 2017 ) , Veldkamp et al. ( 2018 ) , Zhou et al. ( 2018 ) .

Naturally, the most suitable approach lies in the direct representation of water storage and operations within a large-scale hydrological model ( Bellin et al. ,  2016 ) . This approach requires not only the modification of the model structure (or to develop a new one), but also the gathering of information on the design specifications and operating rules of the water reservoirs. Because of these challenges, the number of large-scale hydrological modeling studies adopting such an approach is limited. A first attempt was carried out by Pokhrel et al. ( 2012 ) , who incorporated a water regulation module into the MATSIRO model to reproduce the dynamics of heavily regulated global river basins. More recently, Shin et al. ( 2019 ) integrated a reservoir storage dynamics and release scheme into the continental hydrological model Leaf-Hydro-Flood to simulate ∼1900 reservoirs within the contiguous United States. In both studies, the authors put particular emphasis on the calibration of the reservoir operating scheme and demonstrated that the hydrological model accurately represents some processes altered by human interventions, such as the reservoir-floodplain inundation.

While the relevance and needs for the description of human–water interactions in hydrological models are now well acknowledged ( Nazemi and Wheater ,  2015 a ) , less is known about the risks associated with a poor representation of such interactions. For example, can the estimation of some hydrological parameters be flawed by an inaccurate representation of water reservoir storage? What are the implications for the downstream applications of a flawed model? To answer these questions, we take the upper Mekong river basin as a case study, for which we develop a computational framework based on the Variable Infiltration Capacity (VIC) model ( Liang et al. ,  1994 ) and a multi-objective evolutionary algorithm (MOEA) tasked with the problem of calibrating the model. A key feature of the framework is a novel variant of VIC – named VIC-Res – that allows us to represent the reservoir storage dynamics and operating rules within the streamflow routing model. In a first experiment, we use this framework to calibrate two instances of VIC – with and without reservoirs. As we shall see, both model instances attain the same accuracy, a result obtained by the model instance without reservoirs by adopting a parameterization that compensates for the absence of these infrastructures. In turn, this leads to a poor representation of key hydrological processes, such as infiltration or baseflow. In our second experiment, we demonstrate the potential implications of these unintended consequences by applying two selected model instances (with and without reservoirs) to a climate change impact assessment, for which we obtain partially diverging expectations on the hydrological alterations caused by global warming.

In the remainder of the manuscript, we first describe the study area (Sect. 2) and then proceed by illustrating the computational framework (Sect. 3), including the data on dams and operating rules. In Sect. 4, we provide a detailed description of the results obtained for the aforementioned experiments, whose implications are further discussed in Sect. 5.

The Mekong is a trans-boundary river that flows through China, Myanmar, Thailand, and Laos before pouring into one of the world's largest deltas, located in Cambodia and Vietnam. The catchment area of about 795 000 km 2 can be divided into two parts, namely the upper Mekong, or Lancang, and the lower Mekong basins (Fig.  1 a). The upper Mekong stretches in a north-to-south direction and drains an area of 167 400 km 2 . As shown in Fig. 1b, the region is characterized by a complex orography, with high mountains and deep valleys (the elevation ranges from 362 to 6494 m). Because of these orographical conditions, the spatiotemporal variability of rainfall and temperature is remarkable. The average annual precipitation across the basin ranges from 752 to 1025 mm, 70 % of which is concentrated in the monsoon season (May to November). The precipitation in the northwestern part of the basin is sometimes lower than 250 mm yr −1 , making it dryer than the southeastern part, which receives an average of 1600 mm yr −1 ( Han et al. ,  2019 ) . The average annual temperature across the basin varies narrowly (from 12.3 to 14.3  ∘ C), but the latitudinal temporal gradient is much larger – about 2.2  ∘ C per 100 km ( Wang et al. ,  2014 ) . Climate changes are expected to modify both rainfall and temperature patterns, making the region warmer, wetter, and more susceptible to extreme weather events ( Tang et al. ,  2015 ) .

Figure 1 Mekong river basin (a) and elevation map and location of the hydropower dams in the upper Mekong basin (b) . The red squares denote dams built before 2005 (and therefore included in our study), while the yellow circles indicate dams built after 2005.

The favorable orography and abundant water availability have attracted massive investments in the hydropower sector (see the location of the dams in Fig.  1 b), with consequent impacts on the riverine ecosystems ( Lauri et al. ,  2012 ; Dang et al. ,  2018 ; Hoang et al. ,  2019 ) . The impact of these dams goes beyond the upper Mekong basin ( Zhao et al. ,  2012 ; Han et al. ,  2019 ) : the analysis of historical data shows that dams have already modified many indicators of hydrological alterations in the entire basin, including the Cambodian lowlands and Vietnamese river delta ( Hecht et al. ,  2018 ) . These alterations appear to be more evident since the early 1990s, when Xi'er He 1 and Manwan dams started storing water ( Cochrane et al. ,  2014 ; Lu et al. ,  2014 ; Dang et al. ,  2016 ) . Overall, the upper Mekong basin offers two desirable features for investigating the effect of water reservoir storage and operations on the parameterization of hydrological models. First, the catchment is heavily regulated ( Hecht et al. ,  2018 ) . Second, the catchment area is about 24 % of the whole Mekong river basin, so this helps reduce the computational requirements of the optimization-based calibration process. The location of the gauging station (Chiang Saen) used for the calibration process is illustrated in Fig.  1 a. This station provides a long and reliable daily time series, which has been adopted by several studies on the Mekong basin (e.g., Lauri et al. ,  2012 , 2014 ; Cochrane et al. ,  2014 ; Hoang et al. ,  2016 ). To validate the model, we use monthly discharge values at Jiuzhou station (see Fig.  1 a), retrieved from He et al. ( 2009 ) , Wang et al. ( 2018 ) , and Tang et al. ( 2019 ) . For both stations, we used data belonging to the period 1996–2005.

The aforementioned orography and climate conditions are not particularly suitable for agricultural activities, which are indeed limited. The basin is mountainous, with mostly rocks and a shallow Quaternary alluvium ( Carling ,  2009 ; Gupta ,  2009 ) . Due to the impermeability of bedrock underneath isolated valleys, only a very small fraction of water leaks into the ground through karst aquifer units ( Lee et al. ,  2017 ) . As a result, subsurface water is mostly generated in the shallow loam layer in the form of baseflow.

The first goal of our study is to investigate the role of water reservoir storage and operations on the parameterization of large-scale hydrological models. For this purpose, we adopt the computational framework illustrated in Fig.  2 , which consists of VIC's rainfall–runoff and routing models and the ε -NSGA-II MOEA. In Sect.  3.1 we provide a detailed description of VIC, including the proposed variant for representing reservoir storage dynamics. The data and experimental setup of the framework are outlined in Sect.  3.2 and 3.3 . In Sect.  3.4 , we describe the climate change data used for our second goal, that is, to demonstrate that different model parameterizations caused by the absence (presence) of water reservoirs can affect the results of a climate change impact assessment.

Figure 2 Computational framework adopted in the first part of this study. The framework consists of VIC's rainfall–runoff and routing models and the MOEA ε -NSGA-II. The output of the rainfall–runoff model (i.e., gridded baseflow and runoff) is used by the routing model, which simulates the streamflow at multiple locations within the upper Mekong basin. The simulated streamflow is then used to calculate goodness-of-fit statistics, whose value is optimized with ε -NSGA-II by calibrating the parameters of the rainfall–runoff model. In other words, these parameters and goodness-of-fit statistics represent the decision variables and objective functions used by ε -NSGA-II.

3.1  Hydrological-water resources management model

3.1.1  variable infiltration capacity model.

VIC is a large-scale, semi-distributed land hydrological model maintained and developed by the University of Washington ( http://www.hydro.washington.edu , last access: 22 January 2020). VIC consists of two core components, namely a rainfall–runoff and routing model (Fig.  2 ), which can be applied to multiple spatial scales and implemented with different temporal resolutions – daily, in our case. The rainfall–runoff model simulates the water and energy fluxes that govern the terrestrial hydrological cycle ( Liang et al. ,  1994 ) . For these purposes, it takes as input climate forcings (precipitation and temperature), land use and soil maps, leaf area index and albedo, and a digital elevation model (DEM). For each computational cell, the model uses one vegetation layer and two (or three) soil layers: the upper soil layer controls evaporation, infiltration, and runoff, while the lower layer controls the baseflow generation. These gridded variables are then used by the routing model ( Lohmann et al. ,  1996 , 1998 ) , which simulates discharge throughout the river network using a linearized version of the Saint-Venant equations. Specifically, the model first creates the impulse response functions for each grid cell, and then simulates the flow convolution by aggregating the flow contribution from all upstream cells at each time step lagged according the response functions (ibid).

Following the approach adopted in previous works on the calibration of VIC (e.g., Dan et al. ,  2012 ; Park and Markus ,  2014 ; Xue et al. ,  2015 ), we focus our attention on six main parameters that control the rainfall–runoff process (Table  1 ). These parameters are the thickness of the two soil layers ( d 1 and d 2 ), the infiltration parameter ( b ), and three baseflow parameters ( D s , D max , and W s ). The parameter b characterizes the shape of the VIC curve, and therefore influences the available infiltration capacity and quantity of runoff generated by each cell (for additional details, please refer to Ren-Jun ,  1992 , and Todini ,  1996 ). A higher value of b leads to a lower infiltration rate and higher surface runoff. The three parameters D s , D max , and W s determine the shape of the baseflow curve ( Franchini and Pacciani ,  1991 ) , which relates the soil moisture in the lower layer to the amount of baseflow. More specifically, D max is the maximum baseflow that can occur in the lower layer, while D s is the fraction of D max associated with the transition from linear to nonlinear (rapidly increasing) baseflow generation. W s is the fraction of the maximum soil moisture (in the lower layer) where nonlinear baseflow occurs. Hence, higher values of W s increase the water content needed for rapidly increasing baseflow. The thickness of the two soil layers affects several processes. In general, thicker layers delay the seasonal peak flow and increase the evaporation losses (since they increase the water storage capacity).

Table 1 Main parameters controlling the rainfall–runoff process in VIC. The third column contains the range of each parameter value considered during the calibration process. Note that these are the same ranges typically adopted for the implementation of VIC to large basins (cf., Dan et al. ,  2012 ; Xue et al. ,  2015 ; Wi et al. ,  2017 ).

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3.1.2  Water reservoir storage and operations

To represent the storage dynamics of water reservoirs, we modified VIC's routing model (version 4.2) using the following steps. First, we determine the location of all dams within the basin and directly add them to the model using a dam cell (Fig.  3 a–b). To avoid allocating multiple dams within the same cell, we adopt a high spatial resolution of 0.0625 ∘ (approximately 6.9 km). Then, we aggregate the reservoir storage in the dam cell, where we calculate the daily mass balance. From the dam cell, water is discharged using the rule curves described in the following paragraph. Since the construction of a dam is likely to create an impoundment with surface area larger than the dam cell, we proceed by estimating the maximum reservoir extent, a factor used to determine the so-called reservoir cells, namely cells that are at least half-covered by water (see Fig.  3 b). Although these cells do not contain the reservoir storage, they can affect the evaporation processes, so their number and location must be determined accurately. The flow routing in these cells follows the information provided in the flow direction map (described in Sect.  3.2.1 ). Further details about the implementation of this variant of VIC's routing model are given in Sect. S1 in the Supplement. We note that a more realistic way of representing a reservoir within a hydrological model is to spread the reservoir storage over multiple upstream cells from the dam location ( Shin et al. ,  2019 ) . However, a successful implementation of this method requires detailed bathymetry of all reservoirs within the basin (information that may not always be available) and a 2-D model of the reservoir, so as to accurately calculate the water fluxes between the different reservoir cells.

Figure 3 Graphical representation of VIC's spatial domain (adapted from http://www.hydro.washington.edu , last access: 22 January 2020) (a) , including the selection of dam cell (black), reservoir cells (blue), and cells with other land use (white and white with green lines). The black and pink arrows indicate the direction of the flow routing and discharge from the reservoir (b) . Seasonal rule curve (c) .

As for the reservoir operations, we adopt an approach similar to that of Piman et al. ( 2012 ) , which relies on rule curves conceived to maximize the hydropower production – an assumption justified by the fact that all dams within the upper Mekong are operated for hydropower supply ( Räsänen et al. ,  2017 ) . Determining the rule curve for a given reservoir means determining the daily target water levels. For the case of hydropower production in the Mekong basin, such a rule should allow (1) drawdown of the reservoir storage during the drier months (e.g., December to May) to maximize the production of electricity, (2) recharge of the depleted storage during the monsoon season, and (3) avoidance of the risks of spilling water at the end of the monsoon season (see the illustration in Fig.  3 c). Rule curves are tailored to each reservoir within the basin by determining the time at which the minimum and maximum water levels are reached (May and November, in the Mekong; Piman et al. ,  2012 ), setting the value of the minimum and maximum water levels (the minimum and maximum elevation levels of each reservoir, in our case), and finally connecting these points with a piecewise linear function that gives us the daily target level for each calendar day.

As shown in Fig.  3 c, there are three water levels that divide the storage into four zones. These levels are the dead water (or minimum elevation) level, the target water level, and the full (or maximum elevation) level. If the water level falls below the dead water level (Zone 1), the turbines are not operated. If the level is between the dead water and target level (Zone 2), the model first uses the information on the incoming daily inflow to solve a mass balance equation, in which the discharge from the dam is kept at zero. The aim is to understand whether the water level is expected to go beyond the target at the end of the day. If that is the case, the model discharges through the turbines the amount of water needed to keep the level close to the target. Otherwise, the turbines are not activated. In Zone 3 (between the target and full level), the turbines are used at their maximum capacity, until the water reaches the target level. In Zone 4 (i.e., level above the maximum elevation), both turbines and spillways are used. The key advantage of the rule curves adopted here is that they do not require the calibration of any parameter. Naturally, such an approach is less applicable when the information on the operating objectives is not available, or when dealing with multi-purpose water systems.

Overall, our model implements the following mass balance equation at each simulation time step t (and for each reservoir):

where S t is the reservoir storage at time t , Q t is the inflow volume, E t is the evaporation loss, and R t is the water released from the reservoir. Both storage S t and discharge R t are constrained by the design specifications of each reservoir. Specifically, the storage cannot exceed the reservoir capacity S cap (Eq.  1b ), while the discharge is bounded by the water availability and capacity of the turbines R max (Eq.  1c ). The excess water, if any, is spilled:

Equation ( 1d ) thus represents the release dynamics when the reservoir water level is in Zone 4. Zone 1, 2, and 3 are represented by the following equations:

where S d is the storage corresponding to the dead water level, and S ts , t modT the target storage at time t modT (in our study, we use a period T of 365 d).

3.2  Data and preprocessing

3.2.1  climate forcings and other input variables.

Climate forcings are represented by precipitation and air temperature (maximum and minimum), which must be provided at a daily time step. As far as precipitation is concerned, we use the APHRODITE dataset (Asian Precipitation – Highly-Resolved Observational Data Integration Towards Evaluation), developed by the University of Tsukuba, Japan, using rain-gauge data ( Yatagai et al. ,  2012 ) . APHRODITE is available with a spatial resolution of 0.25 ∘ and has been shown by Lauri et al. ( 2014 ) to be the most suitable precipitation dataset available for the Mekong basin. A similar observation applies to the CFSR (Climate Forecast System Reanalysis) maximum and minimum temperature dataset ( Saha et al. ,  2014 ) . These data are then interpolated to meet the spatial resolution of 0.0625 ∘ adopted in our implementation. More specifically, we use the bilinear interpolation method, which has found successful application in some recent studies (e.g., Hoang et al. ,  2016 ; Shin et al. ,  2019 ). We also bias correct the APHRODITE dataset (using a multiplying factor of 1.26), as recommended by Lauri et al. ( 2014 ) .

The monthly leaf area index and albedo are derived from the Moderate Resolution Imaging Spectroradiometer (Terra MODIS) satellite images, which represent changes in canopy and snow coverage over time. (It is worth noting that snowmelt only marginally contributes to the streamflow of the Mekong river; Räsänen et al. ,  2016 .) Land use and land cover data are obtained from the Global Land Cover Characterization (GLCC) dataset, developed by the United States Geological Survey. We choose this product because it was completed in 1993, close to the simulation period adopted in our study (1995–2005). With such a choice, we make sure that the influence of land use dynamics on the model parameterization is minimized. Soil data are extracted from the Harmonized World Soil Database (HWSD), developed by the International Institute for Applied System Analysis and Food and Agriculture Organization and last updated in 2013. Both land use and soil maps are generated with the majority resampling technique, since their original spatial resolution is 30 arcsec (approximately 1 km). This technique assigns the most common values found from the group of involved pixels to the new cell. The resulting maps are illustrated in Fig.  4 a–b. The land use map shows that the upper reaches of the basin are characterized by the presence of grassland, while the lower reaches – with complex terrain and large altitudinal variations – present mixed coniferous forest ecoregions. Soil characteristics are also heterogeneous: in the central and northern part of the basin, soil is characterized by a shallow layer consisting of loam, sandy loam, and clay. At the border between China, Myanmar, and Laos (near Chiang Saen station), soil characteristics are dominated by the presence of sandy clay loam.

Figure 4 Land use map derived from the Global Land Cover Characterization dataset (a) ; soil map (for the top layer) retrieved from the Harmonized World Soil Database (b) ; flow direction map (c) . The red triangles denote the position of the Chiang Saen and Jiuzhou gauging stations.

To estimate the flow directions, we use the Global 30 arcsec Elevation (GTOPO30) DEM, which has been adopted in several studies (e.g., Kite ,  2001 ; Wu et al. ,  2012 ; Li et al. ,  2013 ). First, we mask this DEM with the shape of the upper Mekong basin. Since GTOPO30 has a spatial resolution of 30 arcsec, we then resample the DEM to the resolution of our VIC model using the average resampling technique ( Hoang et al. ,  2019 ) . Finally, we manually correct the flow direction map generated by ArcGIS by comparing it to a detailed river network provided by the Mekong River Commission. Such correction is necessary, since errors are to some extent unavoidable when automatically generating a flow direction map – because overland runoff and interflow directions depend on the relation between hillslope characteristics and adopted spatial resolution. The resulting flow direction map is illustrated in Fig.  4 c.

3.2.2  Dams and reservoir information

Our model requires detailed information on the reservoirs, namely location, storage capacity, dam height, dead storage, turbine design discharge, and maximum and minimum elevation levels. Such information (summarized in Table  2 ) was retrieved by cross-checking the databases provided by the Mekong River Commission, the International Commission On Large Dams, and the Global Reservoir and Dam Database. Since data on reservoir bathymetry are not available, we modeled the storage–depth relationship with Liebe's method, which assumes that the reservoir is shaped like a top-down pyramid cut diagonally in half ( Liebe et al. ,  2005 ) . In other words, the relation between reservoir volume ( V ) and depth (or level, h ) is equal to V = a h 3 , where a is a shape factor equal to V cap / h max 3 ( V cap is the live storage capacity and h max the maximum water depth). This method has been adopted for regional and global studies (see Ng et al. ,  2017 ; Shin et al. ,  2019 ).

As for the maximum reservoir extent (needed to determine the reservoir cells), the existing databases do not provide detailed information, such as the reservoir polygon, so we proceeded by analyzing remotely sensed data. More specifically, we extracted surface water profiles from Landsat TM and ETM + imagery. Landsat images are raster grids with seven layers corresponding to seven bands (excluding the panchromatic band). The normalized difference water index (NDWI) was calculated using the near-infrared (NIR, Band 4) and shortwave infrared (SWIR, Band 5) bands: NDWI  =  (NIR  −  SWIR)  ∕  (NIR  +  SWIR). Water bodies have NDWI values greater than 0.3  ( McFeeters ,  2013 ) , so from the NDWI raster we can create a binary raster in which 1 denotes a reservoir cell (and 0 a non-reservoir cell). This process yields an accurate estimation of the reservoir cells, since Landsat images have a spatial resolution of 30 m  ×  30 m.

When calculating the daily mass balance for each reservoir, we consider three main processes, namely inflow, evaporation, and release. Infiltration and seepage (via dam body, abutment, and foundation) are neglected. That is because of two reasons. First, the upper Mekong basin is a mountainous region, with mostly rocks and a shallow Quaternary alluvium (see Sect.  2 ), so infiltration losses are to some extent marginal as compared to inflow, release, and evaporation. Second, the dams considered in our study are built with concrete (and with rocky abutments and foundations), so seepage is indeed limited.

3.3  Experimental setup

To carry out the calibration exercise (with and without reservoirs), we couple VIC with the ε -NSGA-II algorithm, which has found successful application in many water resources problems – including model calibration ( Reed et al. ,  2013 ) . In our case, the decision variables are represented by the six parameters controlling the rainfall–runoff process in VIC (Sect.  3.1.1 ), and whose range of variability is reported in Table  1 . As for the objective functions, we consider two goodness-of-fit statistics dependent upon the simulated streamflow, namely the Nash–Sutcliffe efficiency (NSE) and transformed root mean square error (TRMSE), which assess the model performance on high and low flows, respectively ( Dawson et al. ,  2007 ) . The NSE is defined as follows:

where n is the number of time steps, Q s t is the simulated streamflow (at time t ), Q o t is the observed streamflow (at Chiang Saen station), and Q o ‾ is the mean of the observed streamflow. The TRMSE is defined as follows:

where z s, t and z o, t represent the value of the simulated and observed streamflow (at time t ) transformed by the expression z = ( 1 + Q ) λ - 1 λ , ( λ = 0.3 ) . In other words, λ scales down the values of the streamflow, and TRMSE thus emphasizes the errors on the low flows. In this specific modeling problem, capturing both high and low flows is particularly important, since the riverine ecosystems are sensitive to both dry and wet conditions ( Hoang et al. ,  2016 ) .

Table 2 Design specifications of the dams implemented in our VIC model (simulation period 1995–2005). The column marked “Year” denotes the year in which each reservoir became operational.

Both objective functions are calculated for the period 1996–2005 – after a 1-year spin-up period, 1995 – and scaled between 0 and 1, so we set only one value of ε (equal to 0.001). The other ε -NSGA-II parameters to set up are the size of the initial population and number of function evaluations, which are equal to 10 and 250 – a setting that strikes a reasonable balance between the computational requirements of the calibration exercise and the quality of the solutions. Each calibration exercise (with and without reservoirs) is solved with 20 different random seeds, so as to characterize the variability in the ε -NSGA-II stochastic search process. The final set of Pareto-efficient solutions (i.e., alternative parameterizations of VIC) thus corresponds to the set of Pareto-efficient solutions identified across all 20 seeds. All experiments are carried out on an Intel (R) Xeon (R) W-2175 CPU 2.50 GHz with 128 GB RAM running Linux Ubuntu 16.04 (Xenial Xerus), using a Python implementation of various MOEAs (Platypus) that allows the optimization experiments to be parallelized. For each of the 20 seeds, we used four cores, taking approximately 200 h per core (wall-clock time).

Since 6 (out of 11) dams became operational during the study period (see Table  2 ), the VIC simulation with reservoirs is implemented in such a way to activate the reservoirs at the right time. In this specific implementation, we do not use filling strategies different from the rule curves described in Sect.  3.2.2 , because all six dams reach a steady-state operation within a few months (data not shown).

3.4  Climate change data

For our second experiment, we used the CMIP5 climate projections to derive climate change scenarios for the period 2050–2060. Since the data provided by the Coordinated Regional Climate Downscaling Experiment only cover one GCM for our study site ( Giorgi and Gutowski Jr. ,  2015 ) , we followed the approach taken by previous studies (e.g., Hoang et al. ,  2016 , 2019 ) and proceeded by using GCM projections as a basis for our scenarios. As far as the GCMs are concerned, we used ACCESS1-0, CCSM4, CSIRO Mk3.6, HadGEM2-ES, and MPI-ESM-LR, whose reliability for this region has been evaluated in a few previous studies ( Sillmann et al. ,  2013 ; Huang et al. ,  2014 ; Ul Hasson et al. ,  2016 ; Hoang et al. ,  2016 ) . The main characteristics of the GCMs are summarized in Table  3 . As for the RCPs, we chose RCPs 4.5 and 8.5. The former is a medium-to-low scenario that assumes a stabilization of radiative forcing to 4.5 W m −2 by 2100, while the latter is a high emission scenario based on an increase of the radiative forcing to 8.5 W m −2 by 2100. These two RCPs provide a broad range of climate variability for the region – and thus exclude RCP 2.6, which is characterized by the lowest radiative forcings.

Table 3 CMIP5 GCMs used for the climate change impact assessment.

To prepare the precipitation and temperature data used by VIC, we then re-gridded and bias corrected the GCMs outputs. The first step is necessary to overcome the limited spatial resolution of the GCMs (our VIC implementation uses a resolution of 0.0625 ∘ × 0.0625 ∘ ) and is carried out using the bilinear interpolation method. The bias correction is performed with the delta method ( Diaz-Nieto and Wilby ,  2005 ; Choi et al. ,  2009 ) , which has already been applied to our study site ( Lauri et al. ,  2012 ) . With this method, we calculate correcting factors for precipitation and temperature using the following expressions:

where P ‾ series , i and T ‾ series , i are the (11-year) average precipitation and temperature for month i produced by the GCM in our control period (1995–2005), P ‾ ref , i and T ‾ ref , i are the (11-year) average observed precipitation and temperature for month i in the period 1995–2005, and σ ref, i is the standard deviation of the monthly average temperature during the same period for month i . These factors were then used to correct the future climate projections for each time series (using the same factor for all daily data in a given month).

The impact of climate change on hydrological processes is often assessed by studying changes in the flow regime, and, in particular, changes in the monthly, seasonal, and annual river discharges ( Lauri et al. ,  2012 , 2014 ) . More recently, some studies have focussed on hydrological extremes, such as high ( Q 5 ) and low flows ( Q 95 ) ( Hoang et al. ,  2016 ) . Since our goal is to demonstrate that different model parameterizations caused by the absence (or presence) of water reservoirs can largely impact the results of climate change assessments – and not to push forward the boundaries of climate change impact assessments – we chose a simple and established criterion, namely the annual and monthly river discharges at Chiang Saen and Jiuzhou stations.

To discuss the impact of water reservoirs on the parameterization of hydrological models, we first compare the results of the calibration exercise carried out with and without reservoirs and then proceed by comparing the performance of two selected parameterizations on the climate change impact assessment.

4.1  Model parameterization

The optimization-based parameterization exercise yielded a total of 118 and 109 parameterizations (or Pareto-efficient solutions) for the VIC implementations with and without reservoirs, respectively. To prove our hypothesis that the calibration process may somehow compensate for a deficiency in the model structure – the absence of reservoirs, in our case – we begin by analyzing the values of the goodness-of-fit statistics, namely NSE and TRMSE. Figure  5 reports the probability plots of NSE and TRMSE values obtained for the two model setups: results show that the calibration exercise yields a reasonable modeling accuracy, with NSE and TRMSE varying in the ranges 0.68–0.79 and 8.10–16.69. More interestingly, these results show that the NSE and TRMSE values of both model setups belong to the same range of variability and follow an almost identical distribution. In addition, all NSE and TRMSE values of the models without reservoirs fall within the 95 % confidence limits calculated using the NSE and TRMSE values attained by the models with reservoirs. To corroborate this finding, we carried out a Kolmogorov–Smirnov two-sample test to reject the null hypothesis that the values of NSE (and TRMSE) produced by the two model setups come from the same distribution. For both goodness-of-fit statistics, the hypothesis cannot be rejected (with a 5 % significance level). Overall, this confirms that the accuracy of the models is not affected by the presence (absence) of the reservoirs.

Figure 5 Probability plots for the NSE (a) and TRMSE (b) obtained in the model calibration process. The blue circles and red stars specify the results obtained by the models with and without reservoirs, respectively. The dashed blue and red lines represent the theoretical distributions. In both plots, we also report the 95 % confidence limits for the models calibrated with reservoirs.

How does the parameterization compensate for the absence of water reservoirs? To answer this question, we visualize both goodness-of-fit statistics (NSE and TRMSE) and model parameters ( D s , D max , W s , b , d 1 and d 2 ) in a parallel-coordinate plot (Fig.  6 ). These eight variables are shown in eight parallel axes, so each line connecting the axes represents a parameterization (i.e., a solution of the optimization problem) along with the corresponding value of the goodness-of-fit statistics (i.e., the objectives). Blue and red lines denote solutions obtained with and without reservoirs, respectively. First of all, one can notice that while NSE and TRMSE spread over the same ranges (results discussed in the previous paragraph), the presence or absence of reservoirs consistently yields different parameterizations. Let's analyze them. The value of b – characterizing the shape of the VIC curve – belongs to two distinct ranges (0.319–0.495 and 0.002–0.195) for the model implementation with and without reservoirs, respectively, indicating that the model without reservoirs has higher infiltration and lower surface runoff than the model with reservoirs (recall that a higher value of b leads to a lower infiltration rate and higher surface runoff; Sect.  3.1.1 ). A similar observation applies to the parameters D s , D max , and W s , which determine the shape of the baseflow curve. In this case, the model without reservoirs has higher values of D max (i.e., maximum baseflow) and lower values of D s and W s (i.e., fraction of D max where rapidly increasing baseflow begins, and fraction of the maximum soil moisture in the lower layer where rapidly increasing baseflow occurs), suggesting that the absence of reservoirs leads to model parameterizations that favor the generation of baseflow in the lower layer. Finally, we can note that d 1 (the thickness of the first layer) in the models without reservoirs tends to be larger, indicating that these model instances increase the water storage capacity of the top layer. The only parameter that does not appear to depend on the absence (or presence) of water reservoirs is d 2 , the thickness of the second soil layer. This result is corroborated by a global sensitivity analysis, which shows that d 2 is indeed the parameter with the least influence on the model output (Fig. S2 in the Supplement). Overall, it appears that the calibration process compensates for the absence of water reservoirs by determining values of the soil parameters that can somehow `mimic' the alterations caused by water reservoirs, namely an increase in the evaporation and delay in the peak flows – obtained by increasing infiltration, baseflow, and soil water storage capacity.

Figure 6 Parallel coordinate plot illustrating the values of the goodness-of-fit statistics (NSE and TRMSE) and model parameters ( D s , D max , W s , b , d 1 and d 2 ) obtained through the optimization-based parameterization exercise. Each line connecting the axes represents a parameterization, along with the corresponding model performance. Blue and red lines denote parameterizations obtained with and without reservoirs.

To further understand the unintended consequences of the absence of water reservoirs, we select two model parameterizations (with and without reservoirs) characterized by the same performance over the period 1996–2005. The values of NSE, TRMSE, and model parameters are illustrated in Fig.  7 a, while the simulated daily discharges produced by both models are compared in the scatter plot of Fig.  7 b. In Fig.  8 , we contrast the average values of simulated baseflow and runoff during the dry (December–April) and wet (May–November) seasons of the period 1996–2005. Unsurprisingly, results show that during the dry season the model without reservoirs generates more baseflow and runoff than the model with reservoirs (left four panels of Fig.  8 ): during the dry months, hydropower reservoirs release part of the water stored during the monsoon (recall the rule curves described in Sect.  3.1.2 ) – a process simulated by the model without reservoirs by increasing both baseflow and runoff – and, therefore, the discharge at the catchment outlet. During the wet season, we find an opposite trend: in these months, hydropower reservoirs tend to store part of the water (thus reducing the discharge at the catchment outlet), so the model without reservoirs slightly decreases the discharge by reducing baseflow and runoff (right four panels of Fig.  8 ). We also note that the difference between the two models is clearer during the dry season, when a larger amount of the water volumes is controlled by the hydropower reservoirs.

The effect of such flawed representation of baseflow and runoff is further demonstrated by validating the simulated discharge at Jiuzhou station. Figure  9 shows a macroscopic difference between the models calibrated with and without reservoirs. In particular we note that the model calibrated without reservoirs largely overestimates the dry season flow and slightly underestimates the wet season one; a result confirmed by the values of NSE (equal to 0.82 and 0.79 for the model with and without reservoirs) and TRMSE (equal to 21.48 and 28.95). One may also suspect that these unintended consequences could further propagate in downstream applications of the models, such as a climate change impact assessment.

Figure 7 Radar chart illustrating the values of the Nash–Sutcliffe efficiency (NSE), transformed root mean square error (TRMSE), and model parameters ( D s , D max , W s , b , d 1 and d 2 ) of the two selected models (a) ; scatter plot comparing the daily discharges at Chiang Saen station simulated by the two models over the period 1996–2005 (b) .

Figure 8 Average values of simulated baseflow (a, b, c, d) and runoff (e, f, g, h) simulated by the selected models (with and without reservoirs) during the dry (December–April) and wet (May–November) seasons of the period 1996–2005.

4.2  Climate change impact assessment

To begin the climate change impact assessment, we compare the data produced by the GCMs over the reference and future period (1996–2005 and 2050–2060). In general, the total annual precipitation in the Lancang basin is projected to increase under almost all climate change scenarios – only the CSIRO MK3-RCP 8.5 scenario projects a − 3.12 % decrease in the total annual precipitation. Yet, we observe a large spatial variability in the total annual rainfall within each scenario (see Fig. S3). For example, in ACCESS-RCP 4.5, rainfall changes vary between − 2 % in the central part of the basin to more than + 10 % in the southern part. All scenarios (but for CSIRO MK3-RCP 8.5) tend to share a similar spatial pattern, in which the lower part of the basin exhibits an increase in the projected precipitation. As for the temperature, we observe an increase in both minimum and maximum temperature across all scenarios (see Fig. S4), with higher warming for the RCP 8.5. Also in this case, we can note some variability across the GCMs as well as the spatial domain. As discussed in Hoang et al. ( 2016 ) , these precipitation and temperature scenarios represent an improvement with respect to the CMIP3 ones, which show a broader variability. However, there still are some non-negligible differences across the scenarios that are likely to cause different projections of the annual and monthly river discharges.

Figure 9 Comparison between observed and simulated monthly discharges at Jiuzhou station over the period 1996–2005. Simulated data are produced by the two selected models with and without reservoirs (blue and red dots, respectively).

The expected climate change impacts on the annual river discharge at Chiang Saen are synthesized in Table  4 , where we report the relative changes in discharge with respect to the period 1996–2005. Interestingly, it appears that the projections are robust with respect to the representation of the water reservoirs. Indeed, the model with and without reservoirs yield comparable ensemble means and ranges for the two RCPs. Specifically, we find that the annual discharge is predicted to increase in the vast majority of the scenarios, in response to the increase in precipitation described above. Such similarity between the projections is arguably attributable to the calibration process, which generates models producing similar aggregate performance measures at Chiang Saen station.

Table 4 Relative changes in annual river discharges at the Chiang Saen station for the future period (2050–2060) relative to the reference one (1996–2005). The lowest and highest changes are presented with the corresponding scenarios. The results reported in the first and second rows were produced by the selected models without and with reservoirs.

What is perhaps more interesting is a comparison between the monthly discharges at Chiang Saen predicted by the models with and without reservoirs. While both models produce similar ensemble ranges (see Fig.  10 a–d), a closer analysis of the data reveals a non-negligible difference in the minimum, maximum, and average monthly discharges (across the GCM scenarios) produced by the two models (Fig.  10 e–f). In particular, the model with reservoirs predicts higher discharges in the July–September period and lower discharges in October and November. Note that such difference is consistent across both RCPs. Since both models share the same rainfall and temperature scenarios, the only cause for this stark difference must lie in the unintended consequences of the parameterization process. As explained in Sect.  4.1 , the model without reservoirs shows two “artifacts” that help compensate for the absence of the hydropower reservoirs: first, it increases both baseflow and runoff during the dry season (to account for the water discharged to sustain hydropower production in the dry months); second, it decreases baseflow and runoff (to account for part of the water stored by the dams during the wet months). The latter artifact is responsible for the macroscopic change in the hydrograph described above. In the wetter conditions depicted by the GCM and RCP scenarios, the hydropower reservoirs of the Lancang basin receive larger inflows, part of which is directly spilled into the downstream reaches (data not shown). This is an unprecedented situation for the model without reservoirs, which cannot simulate an increase in the use of the spillways. In fact, this model tends to reproduce the dynamics learned during the calibration process, that is, storing part of the water (in the lower soil layer) during the monsoon season and slowly discharging it in the following months.

Figure 10 Projected monthly discharges at Chiang Saen under five GCMs and two RCPs for the two selected models calibrated without and with reservoirs (a–d) . Box plots highlighting the variability in monthly discharges predicted by the two models under RCP 4.5 (e) and RCP 8.5 (f) .

Figure 11 Variability in monthly discharges at Jiuzhou station predicted by the two selected models (with and without reservoirs) under RCP 4.5 (a) and RCP 8.5 (b) .

Naturally, the difference between the monthly discharges predicted by the two models becomes even more apparent when we consider Jiuzhou station, which was not used in the calibration process. As depicted in Fig.  11 , the model without reservoirs consistently yields higher discharges in the pre-monsoon season and lower discharges in the monsoon season. Note that, in some months, the difference between the average monthly discharges produced by the two models causes an uncertainty larger than the one surrounding the downscaled climate projections. For instance, the average monthly discharge in March (under both RCPs) predicted by two models is about 500 and 750 m 3  s −1 , that is, a 50 % difference.

This work contributes to the existing literature on large-scale hydrological modeling by studying the effect of water reservoir storage and operations on the parameterization of process-based models. To this purpose, we developed a computational framework consisting of VIC and the multi-objective evolutionary algorithm ε -NSGA-II, which we used to calibrate the model parameters through a simulation–optimization process. Our framework also includes a novel variant of VIC that simulates both storage dynamics and operations of water reservoirs. Using the Lancang river basin as a case study, we calibrated two implementations of VIC, with and without reservoirs. In line with previous studies (e.g., de Paiva et al. ,  2013 ; Abbaspour et al. ,  2015 ), we found that the model without reservoirs attains a reasonable modeling accuracy. In fact, we found that the calibration process of both model implementations yields de facto the same values of the goodness-of-fit statistics (NSE and TRMSE), suggesting that the model parameterization helps compensates for a structural error, namely the absence of the water reservoirs. More specifically, this effect is achieved by determining the values of six soil parameters ( D s , D max , W s , b , d 1 and d 2 ) that let this model implementation emulate the presence of water reservoirs.

The first implication of a flawed parameter estimation stands in a poor representation of key hydrological processes, such as surface runoff, infiltration, and baseflow. In our case, we found that, during the dry months, the models calibrated without water reservoirs generate a higher amount of baseflow and runoff than the models with reservoirs. This is an artifact needed to reproduce the higher discharges of hydropower dams that sustain the production of hydro-electricity in the dry season. Vice versa, baseflow and runoff are reduced during the wet months, so as to account for the decrease in peak flows caused by the fact that dams store part of the water for the following dry season. A poor parameter estimation is also likely to affect several downstream applications of a hydrological model. In our second experiment we exemplify this concept through a climate change impact assessment, in which we contrasted the annual and monthly discharges projected by two selected models (with and without reservoirs). Both models show a similar trend in the flow regime – i.e., increased monthly discharges during the monsoon season, caused by the projected increase in precipitation – a result found in previous studies ( Lauri et al. ,  2012 ; Hoang et al. ,  2016 , 2019 ) . Yet, one cannot neglect the different nuances of the flow regime alterations predicted by the two models. In particular, the model with reservoirs presents higher discharges at the peak of the monsoon season than the model without reservoirs. These nuances may impact some of the conclusions of a climate change impact assessment as well as other model-based studies depending on a reliable estimation of the flow regime.

Like any hydrological modeling study, this work also builds on a few modeling assumptions that should be properly discussed. First, our model calibration focuses solely on six main parameters controlling the rainfall–runoff process and assumes that they are homogeneously distributed across the basin. As explained in Sect.  3.1.1 , the choice of these parameters is already established in the literature ( Dan et al. ,  2012 ; Park and Markus ,  2014 ; Xue et al. ,  2015 ) ; yet, it is reasonable to expect that the use of more parameters could further improve the model accuracy. As for the use of homogeneously distributed parameters, our modeling choice is justified by the fact that the use of heterogeneously distributed parameters would largely impact the computational requirements of the calibration process. We also note that there are no reasons to believe that the use of more (or spatially distributed) parameters would deeply alter the main findings of this work. Second, the large spatial domain – and associated soil water retention capacity – might be a factor controlling the capability of the calibration process to compensate for the absence of water reservoirs. In other words, such capability might be dependent on the relation between soil water retention capacity and total storage volume of the reservoirs. In a small basin regulated by a large dam, a modified representation of runoff, infiltration, and baseflow may not be sufficient to compensate fully for a poor representation of reservoir storage and operations. Third, we focussed our attention on water reservoirs, which are indeed the infrastructures affecting the flow regime in the Lancang. In the lower Mekong basin (not considered in our spatial domain), the flow regime has been modified not only by hydropower reservoirs, but also by withdrawals for irrigation supply ( Hoang et al. ,  2019 ) . Looking forward, it would thus be interesting to extend the spatial domain of our model and study how these withdrawals could affect its parameterization.

As the pervasiveness of water resources management in Earth system models expands, so too does the need for a deeper understanding of the mechanisms regulating the calibration process. The explicit representation of water reservoirs – and other infrastructure – is indeed likely to result in more realistic soil parameters, a hypothesis whose verification depends on the availability of observations about soil physical properties for large spatial domains. In turn, this highlights the importance of studies aimed to infer such properties from remotely sensed images ( Chang and Islam ,  2000 ; Chabrillat et al. ,  2019 ) . A related topic that may also deserve future research is the robustness of these models with respect to changes in the operations or physical characteristics of dams. Variability in water and energy demand is a key driver for multiple management and planning interventions (e.g., modifications of the operating rules, construction on new storage), so it is paramount to know the extent to which models can still capture key hydrological processes once these modifications are in place.

Overall, the findings of this study reinforce the message that water infrastructures – and their operational settings – play a key role in the reliability of a hydrological modeling exercise, like the quality of the hydro-meteorological data, the model structure, or the calibration process ( Francés et al. ,  2007 ; Madsen ,  2000 ) . These findings gain further prominence if one considers the expected increase in hydropower development in several regions of the world ( Zarfl et al. ,  2015 ) .

Precipitation and air temperature data were retrieved from APHRODITE and CFSR datasets, available at http://www.chikyu.ac.jp/precip/english/downloads.html (RIHN and MRI/JMA, 2018) and https://rda.ucar.edu/datasets/ds093.1/#! (National Center for Atmospheric Research, 2018) Land use and land cover data were obtained from the GLCC dataset ( https://www.usgs.gov/centers/eros/science/ , United States Geological Survey, 2018a), while the soil data were extracted from the HWSD database ( http://www.fao.org/soils-portal/soil-survey/soil-maps-and-databases/harmonized-world-soil-database-v12/en/ , Food and Agriculture Organization of the United Nations, 2018). The Terra MODIS satellite images (used to calculate the monthly Leaf Area Index and albedo) are available at https://earthexplorer.usgs.gov/ (National Aeronautics and Space Administration, 2018a). The Landsat TM and ETM+ imagery are available at https://earthexplorer.usgs.gov/ (National Aeronautics and Space Administration, 2018b). The global Digital Elevation Model (GTOPO30) is available at https://www.usgs.gov/centers/eros/science/ (United States Geological Survey, 2018b). The GCMs projections were retrieved from https://esgf-node.llnl.gov/search/esgf-llnl/ (Lawrence Livermore National Laboratory, 2018). All these data are publicly available. The daily discharge data at Chiang Saen and the design specifications of all dams were obtained by the authors from the Mekong River Commission and the International Commission On Large Dams, so they cannot be shared without their consent. Additional data about the dams were retrieved from the Global Reservoir and Dam Database, available at http://globaldamwatch.org/data/#core_global (Socioeconomic Data and Application Center, 2018). The variant of VIC used in this study – named VIC-Res – is available at https://github.com/thanhiwer/VICRes (Dang, 2019).

The supplement related to this article is available online at:  https://doi.org/10.5194/hess-24-397-2020-supplement .

TDD and SG conceptualized the paper and its scope. Data collection, model implementation, and experiments were carried out by TDD, with inputs from AFMKC and SG. All authors contributed to the manuscript preparation.

The authors declare that they have no conflict of interest.

We thank the editor and reviewers for the positive and constructive comments that helped us improve the paper.

This research has been supported by the Singapore's Ministry of Education (grant no. MOE2017-T2-1-143).

This paper was edited by Axel Bronstert and reviewed by Charles Rougé and one anonymous referee.

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Enhancing the representation of water management in global hydrological models

Guta wakbulcho abeshu, fuqiang tian, thomas wild, mengqi zhao, sean turner, a. f. m. kamal chowdhury, chris r. vernon, hongchang hu, yuan zhuang, mohamad hejazi.

This study enhances an existing global hydrological model (GHM), Xanthos, by adding a new water management module that distinguishes between the operational characteristics of irrigation, hydropower, and flood control reservoirs. We remapped reservoirs in the Global Reservoir and Dam (GRanD) database to the 0.5 ∘ spatial resolution in Xanthos so that a single lumped reservoir exists per grid cell, which yielded 3790 large reservoirs. We implemented unique operation rules for each reservoir type, based on their primary purposes. In particular, hydropower reservoirs have been treated as flood control reservoirs in previous GHM studies, while here, we determined the operation rules for hydropower reservoirs via optimization that maximizes long-term hydropower production. We conducted global simulations using the enhanced Xanthos and validated monthly streamflow for 91 large river basins, where high-quality observed streamflow data were available. A total of 1878 (296 hydropower, 486 irrigation, and 1096 flood control and others) out of the 3790 reservoirs are located in the 91 basins and are part of our reported results. The Kling–Gupta efficiency (KGE) value (after adding the new water management) is ≥  0.5 and ≥  0.0 in 39 and 81 basins, respectively. After adding the new water management module, model performance improved for 75 out of 91 basins and worsened for only 7. To measure the relative difference between explicitly representing hydropower reservoirs and representing hydropower reservoirs as flood control reservoirs (as is commonly done in other GHMs), we use the normalized root mean square error (NRMSE) and the coefficient of determination ( R 2 ). Out of the 296 hydropower reservoirs, the NRMSE is >  0.25 (i.e., considering 0.25 to represent a moderate difference) for over 44 % of the 296 reservoirs when comparing both the simulated reservoir releases and storage time series between the two simulations. We suggest that correctly representing hydropower reservoirs in GHMs could have important implications for our understanding and management of freshwater resource challenges at regional-to-global scales. This enhanced global water management modeling framework will allow the analysis of future global reservoir development and management from a coupled human–earth system perspective.

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Abeshu, G. W., Tian, F., Wild, T., Zhao, M., Turner, S., Chowdhury, A. F. M. K., Vernon, C. R., Hu, H., Zhuang, Y., Hejazi, M., and Li, H.-Y.: Enhancing the representation of water management in global hydrological models, Geosci. Model Dev., 16, 5449–5472, https://doi.org/10.5194/gmd-16-5449-2023, 2023.

Reservoirs are pivotal in fulfilling various societal needs, including irrigation, hydropower production, flood control, domestic water supply, and navigation, to list a few (Belletti et al., 2020; Biemans et al., 2011; Grill et al., 2019). There are 6862 large reservoirs ( ≥  0.1 km 3 ) globally, with a cumulative storage capacity of 6197 km 3 in the Global Reservoir and Dam (GRanD) dataset (Lehner et al., 2011). Many of these reservoirs serve multiple purposes. However, if we partition reservoirs into categories based on their primary purposes, 1789 are irrigation reservoirs, with a total storage capacity of ∼  1100 km 3 ; 1541 are hydropower reservoirs, with a total storage capacity of ∼  3880 km 3 ; 542 are flood control reservoirs, with a total storage capacity of ∼  509 km 3 ; and the rest are water supply, navigation, or recreation reservoirs. Water storage and release in any given reservoir are managed, based on the reservoir's purposes. It is, therefore, important in global hydrological models (GHMs) to represent how management strategies differ across reservoirs with different purposes in order to more accurately simulate water balances and explore the implications of alternative water management strategies. It is particularly important to distinguish the behavior of hydropower reservoirs from others because hydropower production represents the primary purpose for nearly 63 % (based on GRanD) of the total global reservoir storage capacity.

Hanasaki et al. (2006) proposed a generic reservoir simulation scheme for use in GHMs that has been widely used (denoted hereinafter as the Hanasaki scheme). This scheme categorizes all reservoirs into only two types, based on their primary purposes, namely irrigation and non-irrigation reservoirs. All non-irrigation reservoirs are essentially simulated as flood control reservoirs. The Hanasaki scheme determines reservoir release in two stages. First, the provisional release is estimated. For irrigation reservoirs, the provisional release is estimated as a function of the demand for water placed on the reservoir, while provisional release from non-irrigation reservoirs is the long-term mean inflow. The provisional release is then adjusted based on the reservoir's degree of regulation (i.e., the ratio of reservoir storage capacity to inflow).

Most existing GHMs (see Table 1) adopt the Hanasaki classification and treat reservoirs as irrigation or non-irrigation (Burek et al., 2020; Hanasaki et al., 2008; Pokhrel et al., 2012; Schaphoff et al., 2018; Müller Schmied et al., 2021; Sutanudjaja et al., 2018; van der Knijff et al., 2010; Wisser et al., 2010; Zhou et al., 2020). Several GHM studies have employed this scheme with some modifications, including H08 (Hanasaki et al., 2008), MATSIRO-TRIP (Pokhrel et al., 2012), WaterGAP2 (Müller Schmied et al., 2021), WBMplus (Wisser et al., 2010), and LPJmL4 (Schaphoff et al., 2018). For example, some studies have modified the technique for estimating the parameters for irrigation reservoirs (i.e., water demand and spatial extent of the dependent area of a specific reservoir; Biemans et al., 2011). LPJmL4 (Schaphoff et al., 2018) and PCR-GLOBWB (Sutanudjaja et al., 2018) estimate irrigation reservoir release based on water demand, and for all other primary purposes, these models use a default strategy, where they release a pre-determined value (e.g., average discharge) while maintaining levels between a minimum and maximum storage. LISFLOOD (van der Knijff et al., 2010) and CWatM (Burek et al., 2020) do not classify reservoirs based on their purposes. Instead, they use three pre-determined releases based on storage, namely minimum outflow, non-damaging outflow, and normal outflow.

Table 1 List of global hydrological models with reservoir representations. The domain column indicates the spatial scale at which the model has been applied. The representations column describes how each reservoir class (from the reservoir classification column) is simulated. In all cases, the reservoirs are integrated into the models. The reservoir classification column shows how the reservoirs were represented.

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Most GHMs, however, still largely follow the Hanasaki scheme in treating all non-irrigation reservoirs as flood control reservoirs. In the Hanasaki scheme, the inflow, minimum pool level, maximum static full level, and water stored at the beginning of the hydrological year are the only significant factors controlling the magnitude and timing of water release (Hanasaki et al., 2006; Yassin et al., 2019). In reality, among non-irrigation reservoirs, hydropower reservoirs are typically operated differently from flood control reservoirs (Turner et al., 2017; Loucks et al., 2017). An essential difference between them is that hydropower reservoirs mostly operate with the objective of storing water over certain target levels to maximize releases through turbines (Loucks et al., 2017). The minimum and maximum releases corresponding to the minimum and maximum storage levels are also pre-determined. Furthermore, in large storage hydropower reservoirs with a large degree of regulation, storage levels may vary significantly over the course of a year (between the minimum and maximum storage levels) to avoid significant spillage and enable reliable hydropower generation throughout the year. Conversely, an essential feature of flood control reservoirs is to provide a reliable capacity to retain a predicted or unforeseen future flooding event by emptying existing reservoir storage. The objective of flood control reservoirs is to reduce peak flow magnitude, and the storage level is only a concern when there is an incoming flood event (Votruba and Broza, 1989). Therefore, treating hydropower reservoirs as flood control reservoirs can significantly underestimate their operational benefits (Turner et al., 2017; Loucks et al., 2017).

The model performance implications of representing reservoirs as flood control versus hydropower reservoirs are evident at the individual reservoir level. However, there remains a gap in the literature regarding the regional-to-global model performance implications of the representation of hydropower reservoirs, given that GHMs are designed for applications at this spatial scale but have not yet explored this question surrounding the representation of hydropower (Best et al., 2011; Döll et al., 2009; Hanasaki et al., 2008; Pokhrel et al., 2012; Schaphoff et al., 2018; Wisser et al., 2010; Voisin et al., 2013). This study overcomes the aforementioned limitation by demonstrating an enhancement to how water management is employed in Xanthos, a global hydrological model. Xanthos is a relatively lightweight model designed to interact with the components of the Global Change Intersectoral Modeling System (GCIMS), which includes the Global Change Analysis Model (GCAM; Hejazi et al., 2013, 2014; Li et al., 2017) at its core, along with a broader suite of interacting energy, water, and land models. GCAM is an integrated tool for exploring the multisector dynamics of coupled human–earth systems and the response of these systems to global changes (Calvin et al., 2019). Aided by Xanthos, GCAM enables an internally consistent evaluation of time-evolving water supply (i.e., surface water, groundwater, and desalinated water) and demand dynamics across multiple sectors. As such, GCAM and Xanthos have been used in combination to study issues such as the relative contributions of humans and climate change to future global water scarcity (Graham et al., 2020), regional water scarcity (Birnbaum et al., 2022), and sub-national water scarcity (Khan et al., 2020; Wild et al., 2021b, c), as well as climate impacts on the future evolution of hydropower and the broader power sector (Arango-Aramburo et al., 2019; Santos da Silva et al., 2021). Nevertheless, the existing version of Xanthos, denoted here as Xanthos-original, focuses only on representing the natural global water balance without human interventions such as reservoirs (Hejazi et al., 2013; Liu et al., 2018; Vernon et al., 2019). Accounting for water management in the way we propose will ensure that the crucial role of reservoirs is represented in regulating streamflow by mediating water availability and demand (Wan et al., 2018, 2017; Zhang et al., 2020, 2019, 2018).

The specific objectives of this study are 3-fold: (1) to enhance Xanthos by adding a new water management module, where irrigation, hydropower, and flood control reservoirs are treated differently (this enhanced Xanthos is denoted as Xanthos-enhanced); (2) to evaluate the performance of Xanthos-enhanced in terms of reproducing observed streamflow variability; and (3) to understand the impacts of differentiating between flood control and hydropower reservoir operations on regional-to-global-scale water balance. The first two objectives represent improvements to Xanthos and, thus, potential improvements to a broad array of coupled human–earth system studies that rely on linkages between GCAM and Xanthos. The third objective has the potential to inform future improvements to a diverse array of GHMs (see Table 1) because our study is the first, to our knowledge, to explore the GHM performance improvements that can be gained by treating the operational characteristics of hydropower dams as distinct from those of irrigation and flood control dams.

Xanthos is a distributed global hydrological model with a spatial resolution of 0.5 ∘ . Xanthos is a framework that enables users to create customized configurations of potential evapotranspiration estimation, runoff generation and concentration, routing, and post-processing modules ( https://github.com/JGCRI/xanthos , last access: 1 August 2022). By accounting for reservoir operation and local water withdrawal, Xanthos-enhanced enables exploring the influence of water management (Fig. 1). This section focuses on the water management module but first briefly summarizes the runoff and river-routing components for completeness. For more details on the runoff and river-routing components, please refer to Li et al. (2017), Liu et al. (2018), and Vernon et al. (2019).

Figure 1 A detailed schematic of the river-routing and reservoir management module in Xanthos-enhanced.

2.1  Runoff-generation module

Runoff generation in Xanthos-original is based on the abcd model. First developed by Thomas (1981), abcd is a simple water balance model effective for capturing key hydrologic processes, and their interactions, in diverse climatic and landscape settings (Martinez and Gupta, 2010, 2011). Liu et al. (2018) introduced the abcd model into Xanthos as its runoff module for simulating direct runoff, baseflow, evapotranspiration, and soil moisture at a monthly time step. The sum of direct runoff and baseflow is denoted as total runoff, which feeds into the river-routing module. The five parameters in the abcd model are described in Table 2. Parameters a and b pertain to runoff characteristics, while c and d relate to shallow soil moisture and deeper groundwater storage. The fifth parameter is a snowmelt coefficient, denoted as m . Since Xanthos-original is a distributed model, each grid cell has its own set of abcd parameters, though these parameters can optionally have the same values for all grid cells within a given river basin. Xanthos classifies the global water system into 235 large water basins.

Table 2 List of model parameters, description, and ranges. The parameters a , c , d , m , β , and α are dimensionless, and the unit for parameter b is meters. The value of α is fixed at 0.85, following Hanasaki et al. (2006).

2.2  River-routing module

In Xanthos, the routing of water through river networks is simulated using a simple cell-to-cell river-routing scheme, a modified version of the river transport model (Branstetter and Erickson, 2003) and hereinafter denoted as MRTM. MRTM is essentially based on the linear reservoir-routing method. The channel flow rate is estimated as a function of channel water storage, channel velocity, and flow distance from one grid cell to another (Zhou et al., 2015). MRTM uses spatially variable but temporally constant channel velocities, which were derived by averaging the long-term channel velocity simulations from Li et al. (2015). The flow distance values were derived by tracing the natural dominant river channel between grid cells to account for the meandering nature of rivers (Wu et al., 2011). Here we add a channel velocity adjustment coefficient (Table 2) to account for the uncertainties in our channel velocity field. For more details about MRTM, please refer to Zhou et al. (2015).

2.3  Water management module

To enhance Xanthos, we add a water management module on top of the river-routing module. The water management module represents the two most common surface water management activities of local surface water extraction and reservoir operation. Local surface water extraction is water that is locally consumed within a particular grid cell. For example, some fraction of the water applied to irrigated agricultural land may evaporate and effectively become unavailable for use in a given grid cell. This local consumptive water use is subtracted from the total runoff produced by the abcd model. The remaining runoff is discharged into the channels and routed downstream using MRTM. If the consumptive water use is greater than the total runoff in a grid cell, then the remaining runoff is zero. In such a case, the grid cell is considered to have unmet water demand or access to supply from other external sources, such as desalination or groundwater pumping, which are not currently represented in Xanthos. If there is a reservoir in a grid cell, then local runoff (after removing water consumption) and upstream inflow are first intercepted and stored in the reservoir. Reservoir operation is then invoked to estimate the release from the reservoir to the downstream grid cells. Note that a grid cell can contain only one reservoir. That is, if there are multiple individual reservoirs co-located in the same grid cell, we first lump these individual reservoirs into a single reservoir with a storage capacity equivalent to all the combined reservoirs. The primary purpose of this lumped reservoir within a given grid cell is determined in the following two steps: (1) sum up the storage capacities of the individual reservoirs in four categories based on their primary purposes (irrigation, hydropower, flood control, and other); and (2) in each category, sum up the reservoir storage capacities. The aggregated reservoir's primary purpose is assigned to the category with the largest summed storage capacity, while the volume of the single lumped reservoir is equivalent to the sum of all individual reservoir storage capacities across all purposes. The reservoir operation rule is defined for each lumped reservoir based on its primary purpose. For reservoir purposes, if the estimated release is unavailable or less than 10 % of the mean annual inflow, then the monthly release is set to the minimum environmental flow requirement (i.e., 10 % of the mean annual inflow; Tennant, 1976; Hanasaki et al., 2008; Müller Schmied et al., 2021). Next, we provide more details on the operating rule for each reservoir type (Fig. 1).

2.3.1  Irrigation reservoirs

Irrigation reservoirs are represented by adapting the widely adopted Hanasaki et al. (2006) approach, which determines the reservoir release based on the upstream inflow and the total water demand from the downstream areas. More specifically, for each irrigation reservoir, the provisional release is given as

where R m y ′ is the provisional monthly reservoir release (m 3  s −1 ) in month m and year y ; d m , y is the monthly mean total water demand from the downstream areas that are dependent on this reservoir (m 3  s −1 ); d mean is the long-term mean monthly water demand from the downstream areas (m 3  s −1 ); and i mean is the mean annual inflow from upstream (m 3  s −1 ). Both the magnitude of long-term average water demands and the monthly timing of demands are used as inputs, so releases are responsive to the timing of typical demands. The Hanasaki scheme has an allocation coefficient, which is a coefficient for grid cells with more than one reservoir upstream, but here it is assumed to be one and is thus not shown in Eq. (1). This is because, in this study, the dependent areas of reservoirs on the same stream do not overlap.

Though deterministic by nature, the provisional release equation for irrigation reservoirs is demand-driven. d m , y is calculated based on the delineated downstream-dependent grid cells. If d mean is greater than or equal to 50 % of the mean annual inflow i mean , then 50 % of i mean is continually released as a baseline, while seasonal release dynamics are determined by the ratio of monthly demand to d mean . If d mean is less than 50 % of i mean , then the provisional release can be estimated as the mean annual inflow modified by the seasonal demand variation around the mean annual demand.

The provisional release is further adjusted based on the degree of regulation ( γ ), initial storage at the beginning of y th operational year ( S first, y ), and reservoir capacity reduction factor ( α ). The degree of regulation is the ratio of reservoir storage capacity ( C ) to the annual total inflow in cubic meters per year ( I mean ). The reservoir capacity reduction factor is a non-dimensional constant that reduces the total reservoir capacity reported in GRanD to account for surcharge storage and storage reduction due to sediment accumulation. It ranges between 0–1, where a lower value means the reservoir capacity may have been significantly reduced by sediment accumulation, and at 0, the reservoir is not operational. The final release is estimated as follows:

where R m , y is the monthly release (m 3  s −1 ); i m , y is the monthly inflow (m 3  s −1 ); and I mean is the annual inflow (m 3  yr −1 ).

The GRanD reservoirs can be classified into relatively large and small storage reservoirs, based on the degree of regulation. If a reservoir's total storage capacity is less than 50 % of its mean annual inflow, then it is considered a hydrologically small reservoir, whereas greater than 50 % indicates a hydrologically large reservoir. In relatively large reservoirs (upper part of Eq. 2), releases are relatively independent of their monthly inflows, while in relatively small reservoirs (lower part of Eq. 2), releases are dependent on their monthly inflows (Hanasaki et al., 2006).

The total water demand for each reservoir is estimated by summing up the water demand values from grid cells within the reservoir's downstream-dependent area. The reservoir-dependent area is determined, following Hanasaki et al. (2006), Haddeland et al. (2006), and Biemans et al. (2011). Specifically, the downstream spatial extent of reservoir dependency along the main stem is determined based on an average stream velocity and the study's temporal interval (monthly). Assuming an average velocity of 0.5 m s −1 , the total travel distance of water in 1 month is 0.5 m s −1   ×  (30  ×  24  ×  3600 s per month)  ×  (0.001 km m −1 ) =  1296 km per month. Therefore, the dependent downstream grid cells along the main stem are roughly 20 grid cells (0.5  ×  0.5 ∘ ; about 55 km along each direction) downstream. If other reservoirs are located within this travel distance, then we assume that the dependency on the current reservoir stops and is taken over by the other reservoir (the allocation coefficient in Hanasaki et al. (2006) is set to one for this reason). We then delineate a buffer zone within ranges of four grid cells from each side of the main stem. Finally, assuming water movement is by gravity only, those grid cells with a mean elevation that is lower than that of the reservoir are identified as the reservoir's dependent grid cells within the buffer zone.

2.3.2  Hydropower reservoirs

We represent the operation of hydropower reservoirs using a stochastic dynamic programming (SDP) approach (Loucks et al., 2017; Turner et al., 2017). The SDP approach extends the dynamic programming approach to account for the uncertain nature of reservoir inflows explicitly (Loucks et al., 2017). It executes sequential decisions for temporal stages with nonlinear objectives, while considering reservoir inflows as random variables (Loucks et al., 2017). For a known inflow i m , y and hydrologic state variables in the current period (Stedinger et al., 1984), the SDP formulation estimates the benefit function f m , y , resulting from each release decision R m , y as

where T is the current system period ( T =12 for a monthly operating scheme). The reservoir state at each decision-making time step, i.e., month m in the year y , is described by the storage S m , y and the current inflow i m , y . For each state and time step, the release decision R m , y is selected to maximize the immediate benefit B m , y S m , y , i m , y , R m , y plus future benefit function f m + 1 , y S m + 1 , y , i m , y , which depends on the resultant state of the system at time step m +1 , i.e., the succeeding month.

The method for simulating the hydropower reservoir operation is adopted from “reservoir”, an R package that contains several reservoir release decision-making tools, including the SDP techniques described above (Turner, 2016). The same method was also employed in a global-scale study of hydroelectric plants' vulnerability to climate change (Turner et al., 2017). We integrated the SDP approach from this package (Turner, 2016; Turner et al., 2017) into Xanthos for hydropower release simulation. Here the SDP approach is first trained using the naturalized inflow to each reservoir to represent hydrological uncertainty, which we obtain by running MRTM without the water management option. The objective function is set to maximize hydropower production over the long term. The SDP procedure is executed to develop an energy-maximizing release policy for each month as a function of storage levels (see Fig. 1).

The working concept for the SDP algorithm we implemented is summarized as follows. Power ( P in kilowatts) generated by a hydropower plant is given by P = η ρ g ⋅ R ⋅ H , where ρ g is the specific weight of water (kN m −3 ), R is turbine flow (m 3  s −1 ), H is the turbine head (m), and η is efficiency (a constant value of 0.9 is used in this study). ρ g term is a constant term and hence the power-generation variability is a function of R ⋅ H . Thus, maximizing the R ⋅ H translates to maximizing power production. The following four steps are used to identify an optimal policy (i.e., a hydropower-maximizing policy) from a given reservoir inflow realization. First, we discretize the maximum turbine flow (i.e., the maximum allowable flow rate through the turbine) into 10 increments (i.e., between 0 to maximum turbine flow) and the storage capacity into 1000 (i.e., between 0 to storage capacity) increments. Discretization of decision and state variable space is a common practice in implementing dynamic programming-based methods (Piccardi and Soncini-Sessa, 1991; Zeng et al., 2019). Second, we developed a depth–volume relationship, based on an assumed reservoir shape. Here we assume a wedge reservoir shape for all reservoirs globally in the absence of any global datasets to support more heterogeneous representations. The storage–volume relationship is employed to estimate storage depth ( y ) corresponding to 1000 discretized storage volume levels. The turbine head at each storage level was obtained from the sum of y and intake elevation. The intake elevation is computed as the maximum turbine head (i.e., the difference between reservoir pool level and turbine elevation) minus the maximum storage depth (equal to dam height in this study). Using the power equation, the maximum turbine head is computed from the plant-installed capacity and maximum turbine flow. Third, we have an array of releases and turbine heads from the discretization; multiplying them as a matrix yields a 1000×10 matrix of RH (i.e., 10 possible RH values for each storage level). In the present study, we select the policy that maximizes power generation. The best policy for each month (i.e., January to December) at all 1000 storage levels is obtained through backward recursive iterations (i.e., from December to January); this yields what we call the release policy, with a matrix with a size of 1000 (storage levels) ×12 (months). Last, during streamflow simulation, the storage volume and month are used to look up the optimal release policy table (i.e., the 1000×12 table), and the corresponding optimal release is determined. When a storage level is at the reservoir's maximum storage capacity, the release equals the maximum turbine flow that generates power at the power plant's installed capacity.

While the online integration of SDP with hydrological models brings considerable advantages, it also presents certain challenges. One such challenge is managing the uncertainties in the inflow data, as these directly influence the reservoir's operational policy. The effects of inflow uncertainty can lead to potential operational deviations, such as preemptive release of water due to overestimated inflows or undue conservation based on underestimated inflows. To lighten this challenge, a careful parameter selection process is implemented (see Sect. 2.4). The initial stage of this process prioritizes achieving a reliable long-term water balance that aligns closely with observations. By focusing on this balance, we aim to minimize the uncertainties inherent in the inflow data, thereby improving the reliability of operational decisions derived from the SDP model.

2.3.3  Flood control and other purpose reservoirs

The primary purpose of flood control reservoirs is to redistribute the floodwater from a flood season to a non-flood season. The operation of flood control reservoirs is also estimated, following Hanasaki et al. (2006).

where R m , y is the monthly release (m 3  s −1 ); and i m , y is the monthly inflow (m 3  s −1 ). In this study, release from reservoirs categorized as “others” is also determined as a function of inflow and storage characteristics only and is thus similar to flood control reservoirs. The logical reasoning for the equations employed here is in line with Eq. (2). For instance, as with irrigation reservoirs, the α and γ parameters are used to adjust the behavior of flood control reservoirs.

2.4  Model parameter determination strategy

In total, Xanthos-enhanced now includes seven parameters for the runoff and routing or water management modules. Typically, there are two strategies for determining the parameter values in a hydrologic model, namely calibration and estimation a priori (i.e., without calibration; Beven, 2012). Parameter calibration requires thousands of model runs and is only feasible for computationally inexpensive models. Feasibility can be compromised by parameter calibration efforts that require refactoring a model to run more efficiently, the budget required to scale simulations via high-performance computing resources, and the time needed for a comprehensive run. Furthermore, most hydrological models are subject to concerns surrounding equifinality, since the number of parameters, in most cases, far exceeds the number of observational variables available for calibration (Beven, 2006). Alternatively, parameter estimation a priori requires each parameter to be physically meaningful and have robust relationships with the existing climate or landscape information. These relationships are usually not readily available and have to be identified via sound prior knowledge (e.g., Li et al., 2015) or machine learning techniques (e.g., Abeshu et al., 2022; Li et al., 2021).

This study proposes a new, two-stage parameter determination strategy (described in Fig. 2) that seeks to overcome existing limitations by (1) screening out parameter sets that are not physically meaningful and (2) significantly reducing the overall computational burden associated with identifying optimal parameter sets. We seek to determine seven Xanthos parameters in total, namely five from the runoff module and two from the routing module, including water management. We determine runoff parameters in the first stage and routing parameters in the second stage. The runoff module runs separately from the routing and water management modules and is relatively lightweight, taking a standard personal computer less than 2 h to execute it at a global scale for one million simulations covering a 20-year duration. Meanwhile, the routing and water management modules are much more computationally intensive because they run at a 3 h time step to ensure numerical stability (Li et al., 2011, 2015). The first stage takes advantage of the lightweight runoff module to exhaustively explore the runoff parameter space before handing off favorable subsets of parameters to the second stage, which then limits its focus to the more computationally intensive search for the remaining two (routing) parameters. We describe the parameterization strategy in detail in the remainder of this section, whereas the results of implementing the strategy using a particular set of global data are detailed in Sect. 3.2. This strategy is designed based on the characteristics of the Xanthos modules, but we suggest that it has the potential to be useful in diverse global hydrological modeling contexts.

Figure 2 Runoff and routing parameters selection strategy for Xanthos-enhanced. Each component of the process is categorized as one of the following: (1) sampling, wherein parameter combinations are sampled; (2) downselection criteria, which are applied to downsample a larger parameter set into a smaller, more favorable subset; (3) outputs, which describes model outputs; and (4) modules, which describes Xanthos model methods (or sections of code).

In the first stage, we determine the optimal values for the five parameters in the runoff-generation module (see Table 2) in four steps. (1) We generate 1 million runoff parameter combinations using a Latin hypercube sampling (LHS) scheme (McKay et al., 1979; Fig. S1 in the Supplement). LHS is a statistical method for multidimensional parameter space sampling. The stratified sampling strategy employed by LHS ensures that all portions of the sampling space are represented (McKay et al., 1979). The user decides on the required number of parameter combinations and the upper and lower bounds of the individual parameters. Based on that, LHS simultaneously stratifies all input dimensions. (2) For each runoff parameter combination, we execute the runoff module to produce the simulated monthly total runoff time series at each grid cell in the study period. In this study, we uniformly apply the same parameter values to all the grid cells in a basin to generate a monthly runoff time series at each grid cell. Parameter values vary among basins, just not across grid cells within a basin. (3) We calculate the simulated annual runoff depth at each grid cell. We then take the spatial average across the grid cells within the upstream drainage area of a gauge station where observed streamflow data are available; this is denoted as Q sim_annual (mm yr −1 ). (4) At the river gauge station, we calculate the long-term mean of observed streamflow and divide it by the drainage area, Q obs_annual (mm yr −1 ). We then select the top 100 runoff parameter combinations that produce the smallest normalized root mean square error (NRMSE) between Q sim_annual (the annual water consumption) and Q obs_annual .

Before these 100 runoff parameter combinations are passed onto the second stage, the runoff generated by the top 100 parameters is further evaluated at the mean monthly scale to confirm that the selected parameter combinations yield reasonable runoff simulations in terms of timing. For this purpose, we compare the peak time of the simulated mean monthly runoff (i.e., the calendar month in which the mean monthly runoff is highest; hereafter denoted as simulated peak runoff time) with that of Global Runoff Data Center (GRDC) mean monthly flow (i.e., the calendar month when the mean monthly flow is highest; hereafter denoted as observed peak flow time; Fig. S2). Note that the mean monthly runoff employed here is a simple spatial average with no channel routing. Therefore, a reasonable simulated peak runoff time is expected to be earlier than the observed peak flow time by 0–3 months. The range of 0–3 months is estimated by applying a 1.0 m s −1 travel velocity to the longest river in the world, the Nile River, which yields a total travel time between 2.0 and 3.0 months.

The selected 100 parameter combinations are then passed on to the second stage, where we determine the final optimal parameter set in four steps. (1) We set the reservoir capacity reduction factor ( α ) to a value of 0.85, following Hanasaki et al. (2006). (2) The channel velocity adjustment coefficient ( β ) is sampled in a relatively uniform manner within the range of 0.1–10.0. In total, there are 19 possible β values to be considered (i.e., β =0.1 , 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 1.0, 2.0, 3.0, 4.0, 5.0, 6.0, 7.0, 8.0, 9.0, and 10.0). (3) For each of the 100 selected runoff parameter combinations, we use the corresponding simulated runoff time series as the inputs and run the river and water management modules 19 times (each time corresponds to one of the 19 β values and α =0.85 ) at a 3 h time step. (4) We validate the simulated streamflow time series at the grid cell (where the gauge station is located) against the observed monthly streamflow time series. From step (3), there are 1900 simulations for each basin, each corresponding to a combination of five runoff parameters and one routing parameter ( a , b , c , d , m , and β ). The final optimal parameter set is the one that produces the best model performance (per the performance metrics discussed in the following Sect. 2.5). Note that within each basin, we held the set of parameters constant across the cells, which is a reasonable simplification since, typically, there is no sufficient observational data to effectively capture the spatial heterogeneity of these parameters within each basin.

This new strategy has several benefits. First, it largely alleviates the equifinality issue by effectively sampling the whole parameter space. Our experimental design covers the full theoretical value range for each of the six parameters. Second, it reduces the computational burden to a reasonable level. Our suggested approach includes 1 million model runs for the runoff module at the monthly time step for each river basin and another 1900 runs for the river-routing and water management modules at the 3 h time step. We suggest that this new strategy applies to those hydrologic modeling frameworks where (1) some module(s) is (are) computationally much cheaper than the others, and (2) these modules must run sequentially instead of simultaneously. A demonstration of this parameter determination strategy is provided in Sect. 3.

2.5  Metrics for model assessment

To evaluate model performance, we use the Kling–Gupta efficiency (KGE; Gupta et al., 2009), which is given by

where σ sim , σ obs , μ sim , and μ obs are the standard deviation of streamflow value for a given simulation, the standard deviation of observed streamflow, the simulated mean, and the observed mean values, respectively. A higher KGE indicates a better degree of agreement between the simulated and observed variables, and a KGE value of 1.0 indicates perfect agreement. The KGE value is − 0.41 if the simulated monthly flow equals the observed long-term mean flow for all months (Knoben et al., 2019).

While KGE is a useful means of evaluating the skill of a particular set of model parameters in reproducing observed streamflow, we also wish to directly compare simulation outputs against one another across multiple model configurations and parameterizations for both reservoir storage and reservoir release. To enable this comparison, we employ the following indices that capture key aspects of the regulation behavior of reservoirs.

Reservoir impact index (RII) . RII is the ratio of a reservoir storage capacity ( C ) in meters cubed to annual mean flow ( Q mean ; López and Francés, 2013; Wang et al., 2017). RII is similar to the Hanasaki scheme's degree of regulation term, except that RII is computed at the GRDC site instead of the reservoir site. Low and high values of RII indicate that the stream is lightly and heavily regulated, respectively.

where  Q mean  is the observed annual mean flow at the GRDC site (in m 3  yr −1 ).

Seasonality index (SI) . SI represents the degree of variability in the monthly release or storage within a year and is computed with the Walsh and Lawler (1981) method.

where X m  is the mean monthly value for the month  m , and  X ¯  is the annual mean value. SI ranges between 0 and 1.833, indicating uniform distribution over the 12 months and a single-month occurrence, respectively. When applying this equation, use units that represent a measure of water quantity over a month, such as depth (e.g., millimeters per month) or volume (e.g., meters cubed per month).

Coefficient of variation (CV) . CV is the ratio of standard deviation to mean and is employed here to depict the extent of interannual variability in storage and release.

No reliable global observational datasets exist for reservoir storage levels and releases, so it is difficult to establish whether the metric values (for RII, SI, and CV) from one model configuration versus another are closer to reality. Despite this limitation, comparing metric values across simulations is still useful for understanding the effects of modeling assumptions (e.g., representing hydropower reservoirs as such instead of as flood control reservoirs). To enable comparison, we measure the difference or closeness between two alternative time series, representing two alternative model configurations or parameterizations, using normalized root mean square error (NRMSE) and coefficient of determination ( R 2 ). NRMSE typically captures the magnitude difference between two time series, while R 2 measures the proportion of the variance explained (Moriasi et al., 2007).

2.6  Metrics for sensitivity analysis

As we have discussed, equifinality is a crucial issue when calibrating a hydrological model that is highly parameterized. To assess model robustness, it is important to evaluate how sensitive the model's performance is to each model parameter. The sensitivity analysis approach we propose here is moderately different from traditional methods, since we implement a novel parameter determination strategy (see Sect. 2.4). The sensitivity analysis aims to identify the most and least influential model parameters. Such an understanding can help identify priorities of parameter estimation in future works and simplify or improve the model structure. Two separate sensitivity analyses are performed. The first sensitivity analysis is performed before the parameter selection, using results from all 1 million parameter sets. Here, an NRMSE for each of the 1 million parameter sets was computed between simulated annual runoff and observed annual runoff. The annual runoff is observed as annual streamflow converted to an equivalent depth over the upstream contributing basin area. The simulated annual runoff is calculated by subtracting the basin's annual water consumption from the total runoff. The correlation coefficient was then computed between an array of the computed NRMSE and each runoff parameter to evaluate the correlation between the change in parameter values and model performance. We computed five correlation coefficients from the 1 million runs (i.e., between model performance and the five runoff parameters for each basin).

After applying the new parameter selection strategy, the second sensitivity analysis is carried out on 1900 samples (i.e., samples generated from combining the 100 samples from the first stage with 19 discretized β values). Each sample includes the five runoff parameters and the velocity adjustment coefficient employed for streamflow simulation during the second stage. Here, the model performance (KGE) is computed between the monthly observed and simulated streamflow. We switched to KGE for the monthly time series evaluation, as we are interested in metrics that reflect the agreement in both magnitude and patterns of monthly flow. The correlation coefficient is computed between the KGE and each parameter. We obtain six correlation coefficient values for each basin, corresponding to the five abcd model parameters and the routing parameter (i.e., β ). As with its interpretation in stage 1 of the sensitivity analysis, here a higher correlation coefficient between a parameter's values and the corresponding performance metric (in this case, KGE) suggests that the variance in model simulation outcomes is more strongly related to the changes in the target parameter and hence more sensitive to this parameter.

We apply Xanthos-enhanced over the global domain at a 0.5 ∘ resolution and monthly time step. The study period is 1971–1990, based on the availability of forcing and observed streamflow data over all the basins. We divide the study period into a calibration period, 1971–1980, and a validation period, 1981–1990.

3.1  Data and numerical experiments

For this study, we obtain gridded global monthly climatic data, including precipitation, maximum temperature, and minimum temperature, from the WATer and global CHange (WATCH; Weedon et al., 2011) dataset, which covers the period 1971–2001. We obtain global reservoir data from the GRanD dataset (Lehner et al., 2011; Fig. 3a). Monthly water demand and consumptive water use data for various sectors at a 0.5 ∘ resolution are from Huang et al. (2018b, a), which are available from 1971 to 2010 (Fig. 3c). Observed streamflow data for model parameter identification and validation are obtained from the GRDC ( https://www.bafg.de/GRDC , last access: 1 August 2022). We begin by comparing Xanthos' corresponding MRTM upstream area (after locating each gauge station within a Xanthos grid cell) with the GRDC gauge contributing area. If the drainage area difference is larger than ± 20 %, then we look for an option to readjust the station to one of the eight neighboring grid cells. Here, only gauges within ± 20 % in area difference (3097 GRDC gauges) are retained for further use in this study. Temporal filtering of these gauges with the availability of 20 years (1971–1990) of continuous data reduced the number of stations to 1178. These gauge stations are located within 91 of the 235 Xanthos basins. For model validation purposes, we select the GRDC gauge with the largest upstream area within each basin, i.e., 91 GRDC gauges in total (Fig. 3b).

Figure 3 Global data used in this study. (a)  Global distribution of 6862 reservoirs from the GRanD database classified, based on the primary reservoir purpose. (b)  GRDC stream gauge stations in 91 basins where data were of sufficient length, quality, and upstream watershed contributing area for use in this study. (c)  Basin mean monthly water demand in those same 91 river basins.

The GRanD database we use here only considers reservoirs with storage capacity values greater than 0.1 km 3 . We also exclude reservoirs with missing storage capacity values and those identified with purposes such as tide control, which reduces the total GRanD reservoirs from 6862 to 6847. For any grid cell with more than one reservoir, we aggregate all of the reservoirs located locally (i.e., within the grid cell) into a single reservoir with a storage capacity equivalent to that of the local reservoirs combined. The purpose of the combined storage is determined by the two steps described in Sect. 2.3. As a result of this process, the 6847 GRanD reservoirs are remapped into 3790 reservoirs. Among the 3790 reservoirs, 1095, 598, and 2097 are categorized as irrigation, hydropower, and flood control and others, respectively (Fig. 1). Furthermore, out of the 3790 global reservoirs, only 1878 of them are located within the 91 basins simulated in this study. Out of these 1878 reservoirs, the primary purpose is hydropower for 296, irrigation for 486, and flood control or others for 1096. The reservoirs across these 91 basins make up approximately 66 % of the total dams within the GRanD dataset. The construction years of these dams pose a critical factor in deciding the starting year of the calibration. Here, we found that ∼  69.5 % of these dams were constructed before 1971, and an additional ∼  17.2 % were built between 1971 and 1981. We considered it reasonable to include all dams, regardless of their construction year, in the calibration starting from 1971, mainly for the following two reasons: (i) incrementally aggregating dams built during this period over time, in addition to the dams built before 1971, would significantly complicate the modeling process; and (ii) dams constructed before 1981 account for approximately 84 % of the total storage within these basins.

With the aforementioned data, we carry out three global simulations to explore the performance of the Xanthos-enhanced (see Table 3). These include (1) a simulation with Xanthos-original, denoted Xanthos-original-sim, where the simulated flow is obtained by routing-calibrated runoff data generated by Liu et al. (2018) with calibrated abcd model parameters but no water management; (2) a simulation with Xanthos-enhanced, denoted Xanthos-enhanced-sim, where we run the runoff, river-routing, and water management modules with the final optimal parameter values determined, following the new strategy as outlined in Sect. 2.4; (3) a simulation similar to Xanthos-enhanced-sim but one that treats all the hydropower reservoirs as flood control reservoirs (denoted Xanthos-enhanced-sim2). By comparing Xanthos-enhanced-sim with Xanthos-original-sim, we demonstrate the overall improvement of model performance from Xanthos-original to Xanthos-enhanced, due to a combination of the new parameter determination strategy and new water management module. Note that in Liu et al. (2018), the traditional, brute-force calibration strategy was invoked, since Xanthos-original only consists of a monthly runoff-generation module and runs very quickly. By comparing Xanthos-enhanced-sim with Xanthos-enhanced-sim2, we isolate the net difference between simulating hydropower reservoirs based on Eq. (3) and the traditional approach employed by GHMs (i.e., treating hydropower reservoirs as flood control reservoirs, based on Eq. 4).

Table 3 Types of modeling experiments performed in this study.

3.2  Parameter determination outcomes

We apply the two-stage model parameter determination strategy described in Sect. 2.4, using the global datasets described in Sect. 3.1. The LHS generates a parameter set using defined bounds (see Fig. S1 in the Supplement). Here, we describe the results from the implementation of the two-stage strategy, using the Amazon basin as an example. A subset of 100 good parameter sets (filtered) are identified among the 1 million parameter sets (raw; see Fig. S2a in the Supplement). The mean monthly runoff generated with the subset and the observed mean monthly runoff (see Fig. S2b in the Supplement) showed that the simulated runoff peak time is earlier than the streamflow peak time and within the 1–3-month range established in Sect. 2.4. The model's integrity is contingent upon the theoretical expectation of the streamflow peak trailing the runoff peak by a span of days to months. Any set of parameters resulting in a reversed pattern, where the runoff peak occurs later, is deemed unacceptable due to possible anomalies within the model.

The peak time differences (i.e., the difference between GRDC mean monthly peak flow time minus simulated runoff peak flow time) corresponding to the selected sets of parameters are among the best of the 1 million samples when ranked in an ascending order, based on an absolute value of the peak time difference (see Fig. S2c in the Supplement). The robustness of the implemented procedure is justified by the presence of a range of parameter values between their upper and lower bounds (see Fig. S2d in the Supplement), indicating that the selected parameters are not concentrated within a specific parameter space. These characteristics have also been observed in most of the basins evaluated for this study (figure not shown). We select one parameter combination for each basin that results in the best KGE value. The spatial maps of the final optimal parameter values are shown in Fig. S3 in the Supplement. In most cases, optimal values for parameter a are close to the upper bound, while those of parameter d are closer to the lower bound. Parameter b is low in basins in the high-latitude sub-region; to some degree, this may be attributed to the fact that, in general, evapotranspiration decreases towards most of the high-latitude regions. Parameter c seems to be lower in the eastern hemisphere and has relatively no distinct pattern in the western hemisphere basins. The snowmelt parameter m is only above zero in regions with significant snow contributions. The parameter β is higher in high-latitude basins. β was only introduced to readjust the global velocity data after noticing bias in the monthly flow timing at many sites; hence, applications of our methodology that use more reliable velocity data should consider setting β to a value of 1. The high values of β in the higher-latitude basins could be attributed to the original velocity estimation approach's systematic bias in cold regions (Li et al., 2015).

3.3  Global evaluation

Overall, Xanthos' performance has improved after adding the water management module. Figure 4 shows violin plots of KGE between the GRDC monthly observed streamflow and those simulated from the Xanthos-original-sim and Xanthos-enhanced-sim simulations for the 91 basins during the calibration (Fig. 4a) and validation (Fig. 4b) periods, respectively. In most cases, during both calibration and validation periods, the Xanthos-enhanced-sim simulation's KGE values are consistently higher than those of the Xanthos-original-sim simulation. For the Xanthos-enhanced-sim simulation, the KGE value is no less than 0.5 and 0.0 for 59 and 89 basins during the calibration period and 39 and 81 basins during the validation period, respectively.

Figure 4 Box plots of the KGE values for the Xanthos-original-sim and Xanthos-enhanced-sim simulations during (a)  the calibration period (1971–1980) and (b)  the validation period (1981–1990). In this plot, the outliers (KGE values lower than − 1) are not shown. For Xanthos-original-sim, 77 basins in calibration and 70 basins in validation are shown. For Xanthos-enhanced-sim, 91 basins during calibration and 89 during validation are shown.

Figure 5 Spatial maps of basin-specific difference between the KGE values from Xanthos-enhanced-sim and those from Xanthos-original-sim for the calibration period 1971–1980 (map at center), where a value greater than zero indicates improved performance from the addition of water management features to Xanthos. The time series plots are simulated and observed monthly streamflow for basins with the highest water demand in different global regions during the validation period (1981–1990). KGEcal is the KGE during the calibration period, while KGEval is the KGE during the validation period.

Figure 5 provides a comprehensive comparison of the Xanthos-enhanced-sim and Xanthos-original-sim, emphasizing the impact of water management integration. This is illustrated both spatially, through a map depicting KGE differences, and temporally, via time series plots for selected basins. Overall, by incorporating this element, improvements are noticeable in the KGE values of 75 basins, thus indicating a better simulation accuracy for these basins. The increase in KGE values is substantial and exceeds 0.05 when compared to those of the Xanthos-original-sim. However, it is also crucial to note that the water management integration negatively affected the KGE values (i.e., KGE values decreased by more than 0.05) in seven basins. In the remaining nine basins, KGE did not significantly change. For basins in which performance worsened, the decrease in performance is likely due to factors such as the uncertainties in the climate forcing data and GRDC streamflow observations (Moges et al., 2021) and the lack of spatial heterogeneity in the estimated parameters at the sub-basin scale (i.e., the parameters are uniform across all grid cells in a given basin). The distribution and operational patterns of reservoirs, particularly in relation to their closeness to gauge stations, could also represent a nontrivial contributing factor to this issue.

To further examine Xanthos' performance in more detail, Fig. 5 also shows the monthly time series of simulated and observed streamflow at the 12 GRDC stations (out of the 91 evaluated here) with relatively higher average annual water demand in their geographical region (see Fig. 3c) and hence stronger water management effects. The 12 basins are the Rhine, Po, Siberia north coast, Ziya He interior, Ganges–Brahmaputra, Chao Phraya, Murray–Darling, South Africa south coast, Uruguay–Brazil South Atlantic coast, east Brazil South Atlantic coast, California basin, and mid Atlantic. Compared to Xanthos-original, Xanthos-enhanced-sim better captures the seasonal variations in the streamflow, more closely matching the observed streamflow during the high-flow and low-flow periods. This highlights the importance of the reservoir regulation effect (e.g., attenuating high flows and augmenting low flows) that Xanthos-original has not captured.

3.4  Parameter sensitivity analysis

To identify which parameters are most critical (i.e., contribute most to the variance in key model outputs), we evaluate the sensitivity of the model's performance to the changes in a , b , c , d , m , and β , as shown in Fig. 6. Note that we fix the value of α at 0.85, following Hanasaki et al. (2006), in this study. We first carry out the sensitivity analysis on the runoff parameters based only on the first-stage parameter determination results (Fig. 6a). The results show a significant sensitivity (correlation coefficient >   | ± 0.4 | ) only for parameters a (which represents the propensity of runoff to occur before the soil is fully saturated) and b (which represents an upper limit on the sum of evapotranspiration and soil moisture storage). The correlation between parameter a and the model performance is negative, indicating that it is inversely related to the NRMSE computed from annual observed and simulated runoff. Parameter a controls the volume of runoff generation when soil is undersaturated, and the relationship suggests that annual runoff is estimated better when saturation excess runoff is not the primary process. Parameter b controls the soil saturation level. Hence, it is responsible for the memory of the basin. Therefore, the positive correlation indicates that the difference between simulated and observed annual runoff increases as the basin memory increases.

Figure 6 Parameter sensitivity analysis for Xanthos-enhanced in the form of the Pearson correlation coefficient (Pearson r ) (a)  between runoff parameters (i.e., 1 million parameter sets) and their corresponding normalized root mean squared error (NRMSE) computed with annual runoff; and (b)  between streamflow simulation parameters (i.e., the combination of the top 100 runoff parameter sets and sampled routing parameter) and KGE computed from streamflow simulated at monthly scale. A higher Pearson r implies that the model performance is more sensitive to the parameter. For instance, (a)  Pearson r less than zero indicates a decrease in annual runoff NRMSE as the parameter value increases, while Pearson r greater than zero indicates a positive association between annual runoff NRMSE and parameter value. Note that out of the 91 basins, only half of the basin labels (i.e., the x -axis labels) appear on the first panel, and the other half appears on the second panel, but all labels apply to both panels.

A similar analysis is made for the set of parameters generated by combining the 100 best abcd model parameter sets with the velocity adjustment parameter ( β ; Fig. 6b). Here, it appears that β has a stronger influence on model performance than the other parameters. This is expected because the differences among the 100 selected runoff parameter combinations are supposed to be small (e.g., see Fig. S2a (filtered) in the Supplement for the Amazon basin). For β , the sensitivity corresponds to an adjustment in the flow timing, leading to improved KGE. Note that this parameter can be avoided with a better estimate of spatially and temporally varying flow velocity. Considering these observations, it becomes evident that an enhanced parameterization of this variable, along with the other variable used to estimate the grid's water residence time (i.e., channel length), warrants increased attention. Hence, in the future, proper generation and integration of these components are crucial for boosting the model's accuracy and robustness, given their pivotal role in the MRTM-flow-routing process.

3.5  Hydropower reservoirs

Among the 91 basins we studied here, 51 have one or more hydropower reservoirs included in GRanD and hence in our simulations. Recall that these reservoir counts reflect our lumping of multiple reservoirs together within any given grid cell, so 296 reservoirs in our methodology reflect 433 actual reservoirs. At each of the 296 reservoirs, the simulated release and storage time series from Xanthos-enhanced-sim2 are compared with those from Xanthos-enhanced-sim to identify the benefit of capturing hydropower operations.

Figure 7 compares Xanthos-enhanced-sim and Xanthos-enhanced-sim2 with regard to intra-annual (Fig. 7a and b) and interannual (Fig. 7c and d) variability. The seasonality index (SI) summarizes the intra-annual variability; weak seasonality (i.e., low SI) indicates that most months contribute significantly to the annual magnitude, and strong seasonality (i.e., high SI) indicates that very few months contribute to the annual flux magnitude. Although SI showed more difference, both the SI and coefficient of variation (CV; which summarizes interannual variability) for release fall on the 1:1 line for most reservoirs (Fig. 7a and c), indicating that in most cases, the two scenarios have less impact on the intra- and interannual variability in the release. On the other hand, both SI and CV values for storage at most reservoirs show a significant difference, indicating that the two experiments significantly disagree in terms of the interannual and intra-annual variability in the storage. We emphasize that the significant takeaway from this comparison is not that one experiment's storage and release simulations are more variable than the other but that the two experiments led to substantially different seasonal and annual patterns. This highlights the drawbacks of representing hydropower reservoirs as flood control reservoirs.

Figure 7 Comparing intra-annual and interannual variability in the storages and releases simulated with the hydropower reservoirs of Xanthos-enhanced-sim and Xanthos-enhanced-sim2 experiments distributed over the 91 basins. (a)  Seasonality index (SI) for release, (b)  seasonality index for storage, (c)  coefficient of variation (CV) for release at the annual scale, and (d)  coefficient of variation (CV) for storage at the annual scale. The red line is a 1:1 line, where both scenarios are equal.

Figure 8 Empirical cumulative distribution function (CDF) of reservoir storage and release R 2   (a) and NRMSE (b)  between Xanthos-enhanced-sim and Xanthos-enhanced-sim2 monthly simulations across all hydropower reservoir sites. The R 2 CDF plot demonstrates that we produce very different storage and release patterns by accounting explicitly for hydropower reservoir functionality in Xanthos-enhanced-sim, although the difference in magnitude, as indicated by NRMSE CDF, is small. A higher R 2 of 1.0 and NRMSE of 0.0 represent a perfect agreement between the two simulations, indicating that distinguishing between the representation of hydropower and flood control behavior was less important for basins with those values. The CDF plot is made of 296 hydropower reservoirs distributed across 51 basins. The labels on the plots (HP-1, HP-2, HP-3, and HP-4) correspond to hydropower reservoirs located upstream of the Yenisey basin's GRDC site (see Fig. S5 in the Supplement), as discussed in Sect. 3.5. The markers for these labels are similar for both panels  (a) and (b) .

Figure 8 summarizes the reservoir storage and release comparisons between Xanthos-enhanced-sim and Xanthos-enhanced-sim2 with an empirical cumulative distribution function (CDF) that plots R 2 (and/or NRMSE) values across all 296 hydropower reservoirs, according to their rank-ordered exceedance probabilities. The spatial map for the comparisons is also shown in Fig. S4. Recall that a high NRMSE value means a significant magnitude difference between the two different time series, and a low R 2 value means a significant timing difference. Of the 296 reservoirs, the simulated reservoir releases differ significantly between the two model configurations in ∼  45 % of reservoirs in terms of magnitude (if we set a threshold at NRMSE >  0.25; Fig. 8b) and at only ∼  28 % in terms of timing (if we set a threshold at R 2 <0.5 ; Fig. 8a). According to Fig. 8a and b, treating hydropower reservoirs as flood control reservoirs does not significantly impact the model simulated reservoir releases from most reservoirs, which partly supports the lack of differentiation between hydropower and flood control reservoirs in previous studies. However, the simulated reservoir storages are significantly different for ∼  44 % of the 296 reservoirs in terms of magnitude (NRMSE  >  0.25; Fig. 8b) and ∼  90 % in terms of timing ( R 2 <0.5 ; Fig. 8a). Treating hydropower as flood control reservoirs thus has much more impact on the simulation of reservoir storage than release, particularly in terms of timing. The NRMSE and R 2 values in Fig. 8 do not appear to relate to the reservoir sizes (figure not shown).

To explore the dynamics responsible for these broad patterns in Fig. 8, we select the Yenisey basin here to study them in more detail. Here, the Yenisey basin is selected for demonstration because it has a mix of only flood control and hydropower reservoirs and has just six reservoirs upstream of the GRDC site. In the Yenisey basin, the upstream area of the GRDC station is dominated by hydropower reservoirs, i.e., four hydropower reservoirs and two flood control, as shown in Fig. S5a in the Supplement. Note that one of the two flood control reservoirs is located downstream of the hydropower reservoirs (Fig. S5a). This spatial arrangement allows us to evaluate the effects of simulating hydropower reservoirs as flood control reservoirs without interference from the third purpose (i.e., in cases where an irrigation reservoir is located downstream of a hydropower reservoir). Figure S5b (see the Supplement) shows the total simulated storage (sum of all six reservoirs) from Xanthos-enhanced-sim2 and Xanthos-enhanced-sim. The difference in the magnitude of total simulated storage between the two simulations is very significant (KGE between them is 0.44). In Xanthos-enhanced-sim2, where all reservoirs are simulated as flood control, the storage is relatively more variable from month to month, while Xanthos-enhanced-sim changes are more smooth, likely because the release aims to maintain mean annual flow in Xanthos-enhanced-sim2, which leads to releases that exceed inflow during the drier seasons and quick fill-up during the wet seasons. The streamflow comparison at the GRDC site (Fig. S5c in the Supplement) indicates that the difference in the simulated reservoir releases is also significant. The KGE values drop from 0.366 to 0.152 during the calibration period (1971–1980) and from 0.293 to 0.008 during the validation period (1981–1990) when simulating the hydropower reservoirs as flood control.

For those basins where hydropower reservoirs serve a secondary purpose compared to irrigation, flood control, or other types of reservoirs, there is no significant difference in the KGE values between Xanthos-enhanced-sim2 and Xanthos-enhanced-sim, suggesting that treating hydropower reservoirs as flood control reservoirs will not lead to a significant difference in streamflow simulations at the regional or basin level. Figure 9 depicts the RII of the hydropower reservoirs on flow at GRDC stations for basins with one or more hydropower reservoirs (i.e., 51 of the 91 basins). The RII, shown here, corresponds to a hydropower reservoir with the largest storage within the basin. Figure 9 also shows a time series plot of the relative difference between Xanthos-enhanced-sim2 and Xanthos-enhanced-sim storage and release for 10 basins with relatively higher RII within different geographic regions.

Figure 9 The difference between reservoir storage and release monthly time series between Xanthos-enhanced-sim and Xanthos-enhanced-sim2 simulations at hydropower reservoirs demonstrates the value added by explicitly accounting for hydropower reservoir functionality in Xanthos-enhanced-sim. The reservoir impact index (RII) is the ratio of reservoir storage capacity (in cubic meters) to annual mean flow (in cubic meters). Low and high values of RII indicate that the stream is lightly and highly regulated, respectively. For instance, RII  >  1.0 shows the reservoir's capacity to shift the downstream flow below the annual mean flow. The map at the center of the figure displays the RII for the hydropower reservoir with the largest RII in each basin. In other words, each basin is represented by one hydropower reservoir with the largest impact on flow at the GRDC site. The times series plots show the storage relative difference (S-RD) and release relative difference (R-RD) between the two scenarios. For example, S-RD represents Xanthos-enhanced-sim storage minus Xanthos-enhanced-sim2 storage scaled by the mean of the two storages.

The time series plots in Fig. 9 show the storage relative difference (S-RD) and release relative difference (R-RD) between the two scenarios. S-RD represents Xanthos-enhanced-sim storage minus Xanthos-enhanced-sim2 storage scaled by the mean of the two storages. Similarly, R-RD is the scaled difference in the simulated releases. From the time series plots of S-RD, one can see that, in some example basins, S-RD is >  0 (Fig. 9). This characteristic implies that when a reservoir is simulated as a hydropower reservoir, it generally maintains high storage with less variation than when simulated as a flood control reservoir. This can be attributed to our release policy for hydropower simulation, which targets maximum long-term revenue, where reservoir storage level is an essential component. Out of the 296 reservoirs, about 150 of them demonstrate this type of behavior for at least 50 % of the study period (1981–1990).

Flow downstream of hydropower reservoirs is also influenced by the change in the reservoir purpose from hydropower to flood control (Fig. S5c in the Supplement). Similarly, a comparison of simulated releases (Fig. S6 in the Supplement) shows the difference between the simulated monthly releases in the peak and low-flow periods. On the one hand, the release from the flood control reservoirs is high during peak flow periods because they aim to create excess storage capacity to attenuate inflow during the next flood event. On the other hand, the release from the hydropower reservoirs can only go up to the maximum turbine flow plus spillover. The Hanasaki et al. (2006) approach readjusts the mean annual flow, depending on the reservoir's degree of regulation (i.e., capacity ratio to mean annual inflow). Therefore, in Xanthos, given that the readjusted mean annual flow is greater than the environmental flow (10 % of the mean annual flow), release remains constant during the low-flow periods. For hydropower reservoirs, low-flow releases are determined by a release policy intended to maximize revenue. Because of the changes in reservoir purpose, downstream reservoir releases are also modified.

Taken together, Figs. 7–9 and Figs. S4–S6 (in the Supplement) suggest that individual hydropower and flood control reservoirs behave very differently under the same climate and upstream conditions, particularly in terms of the simulated reservoir storage variations. Regarding regional-scale simulations, treating hydropower reservoirs as flood control leads to noticeably different simulated streamflow only in the basins, where hydropower reservoirs dominate over the other types of reservoirs. For instance, for the lower Colorado (RII  =  1.92), Caspian Sea southwest coast (RII  =  1.2), Yenisey (RII  =  1.02), Hudson Bay coast (RII  =  1.32), and São Francisco (RII  =  0.67) basins, KGE improvement was >  0.1 over the calibration period. The indicated RII corresponds to the total effect of hydropower reservoirs located upstream of the basin's GRDC site. This observation will have critical implications in studies for which freshwater storage is the core interest, since reservoir storage is a critical component of terrestrial freshwater storage. For instance, the number of hydropower reservoirs in many global basins is rapidly increasing (Zarfl et al., 2015). Hence, the potential of simulating them in GHMs is vital, as the water use characteristics in many of these basins with hydropower reservoirs could change in the next decade or two if hundreds of new dams are built. Furthermore, the observed distinct characteristics between hydropower and flood control reservoir storages have substantial implications for reservoir sedimentation, which is another essential feature that the GHMs are increasingly looking to capture.

The results in this paper highlight some promising potential outcomes from accounting explicitly for hydropower objectives and operational behavior in GHMs. However, we note that it is premature to conclude from the above analysis that treating hydropower reservoirs as flood control leads to poor hydrological simulations, and vice versa. Many reservoirs, particularly large ones, serve multiple purposes, so their behavior is controlled by multiple factors. This study takes the same simplification strategy adopted by all existing GHMs, i.e., treating each reservoir as having a single purpose. Overcoming this simplification in a GHM setting is beyond the scope of this study and is left for the future.

This study adds a new water management module into the Xanthos model to improve its representation of global hydrological systems. The new water management module enhances Xanthos mainly by introducing reservoir regulation and local surface water withdrawal. We represent unique reservoir operation behavior for each reservoir based on its primary purpose, which can fall into the following three categories: irrigation, hydropower, and flood control and others. In particular, hydropower reservoirs have been treated as flood control reservoirs in previous GHM studies, while here we determined the operation rules for hydropower reservoirs via optimization that maximizes long-term hydropower production. We apply the enhanced Xanthos (Xanthos-enhanced) globally at a 0.5 ∘ spatial resolution and monthly time step. Validation against observed streamflow in 91 river gauge stations demonstrates improved performance over the original Xanthos (Xanthos-original) version. At the individual reservoir level, we show that hydropower and flood control reservoirs indeed behave quite differently, particularly in terms of reservoir storage variations. At the regional level, we show that treating hydropower reservoirs as flood control reservoirs leads to a noticeable impact on the simulated streamflow only in the basins where hydropower reservoirs are dominant. The model's performance improved by more than the KGE of 0.1 for some of the basins with a significant reservoir impact index (RII; e.g., the lower Colorado basin, the Caspian Sea southwest coast basin, Yenisey basin, and the Hudson Bay coast). The RII value corresponds to the total effect of hydropower reservoirs located upstream of a basin's GRDC site. Adding this new hydropower reservoir module can improve the analysis of finer-scale energy–water–land dynamics within frameworks capable of ingesting Xanthos outputs to capture water sector supply–demand dynamics (e.g., Graham et al., 2020; Khan et al., 2020; Birnbaum et al., 2022; Wild et al., 2021c, b). The benefits of distinguishing the unique behavior of hydropower reservoirs in GHMs may become more prominent if hydropower expansion in the coming decades occurs as planned (Zarfl et al., 2015).

There are several opportunities to improve Xanthos-enhanced further. First, in this study, we only determine optimal parameters for Xanthos-enhanced in 91 out of 235 large river basins globally, due to the availability of observed streamflow data, and we assume each set of basin parameter values is uniform across grid cells within a basin. For future global applications of Xanthos-enhanced, one candidate approach for estimating the parameter values in the remaining river basins is to simply use average parameter values from the 91 basins that are gauged. Another possible approach is to estimate the parameters over these ungauged basins by invoking a hydrologic parameter regionalization strategy; i.e., estimating the parameter values a priori from existing climatology and landscape data based on multivariable regression techniques (Ye et al., 2014) or machine learning methods (Abeshu et al., 2022). Second, the groundwater storage (both above and below confined aquifers) could be represented more explicitly, in line with advancements in the representation of groundwater made by other GHMs (Gleeson et al., 2021), which will enable a more realistic representation of water supply with groundwater pumping as an additional source and potentially better streamflow simulation. Third, natural lakes should be represented in the model in addition to reservoirs. Lakes are an essential source of water supply, although they are not as heavily managed as reservoirs. They also have important impacts on the regional climate through their water and energy exchanges with the atmosphere. Fourth, hydrologically small reservoirs (i.e., those with a storage capacity less than 0.1 km 3 ; Lehner et al., 2011) are currently not accounted for due to data limitations, but they potentially play an important role in the regional and global water supply. Last, but not least, the representation of reservoirs could be enhanced by accounting for reservoir sedimentation, given that reservoir storage is being lost globally at a rate of 0.5 % yr −1 (Mahmood, 1987; White, 2001). Relatively simple empirically based approaches to capture these dynamics for reservoirs globally have been shown to be effective and can be borrowed from other open-source modeling frameworks (e.g., Wild et al., 2021a).

While our current two-stage calibration framework provides substantial insights, we anticipate its evolution towards a more comprehensive multigauge calibration approach. The existing framework, which relies on a single gauge per basin (typically the most downstream one), could potentially be expanded to a multigauge calibration. Theoretically, this process would calibrate the model parameters using multiple gauges scattered throughout the basin and accommodate the spatial variability inherent in these parameters. Such an expansion could incorporate hierarchical and multiobjective optimization methods into the present two-stage framework. The hierarchical approach initiates calibration with the smaller, upstream sub-basins. The parameters determined at these stages subsequently inform the calibration of the larger, downstream basins, continuing in this fashion until the calibration of the most downstream gauge. This method capitalizes on the detailed information accessible at smaller scales, thereby assuring the consistency of large-scale simulations with those on smaller scales. Incorporating multiobjective optimization, with objectives set at multiple gauges, is another approach that could augment the fidelity of the simulation within the two-stage calibration framework. This approach could mitigate discrepancies between simulated and observed discharges at multiple gauges simultaneously. Consequently, the model could represent a comprehensive array of hydrological behaviors across space, especially in large and heterogeneous basins where significant spatial variability in hydrological processes is common.

Even with the above limitations, the water management module we introduce here offers a more realistic representation of river systems in global hydrological models like Xanthos. The model has the potential to provide insight into the competition between changes in water availability (primarily affected by climate variability) and water demand (controlled mainly by human activities) at regional or global scales and support scientific analysis and planning in a complex socioeconomic system setting under various future climate change and management scenarios.

The current version of the model source code is available at https://github.com/gutabeshu/xanthos-wm/tree/updatev1 (last access: 1 August 2023) and https://doi.org/10.5281/zenodo.7581990 (Abeshu, 2023c), and the steps for reproducing the model results and figures in the paper are available at https://github.com/gutabeshu/Abeshu-etal_2023_GMD/tree/v1.01 (last access: 1 August 2023) and https://doi.org/10.5281/zenodo.7557380 (Abeshu, 2023b).

The supporting input data required to conduct the experiment are available at https://doi.org/10.5281/zenodo.7557403 (Abeshu, 2023a). The model output data for reproducing the figures in this paper are also available at https://doi.org/10.5281/zenodo.7557380 (Abeshu, 2023b).

The supplement related to this article is available online at:  https://doi.org/10.5194/gmd-16-5449-2023-supplement .

GWA developed the code, carried out the simulations, prepared the figures, and wrote the draft. GWA, HYL, MH, TW, ST, and MZ contributed to Sect. 2. GWA, HYL, TW, and MZ performed the analysis. All the co-authors contributed to the writing and discussion.

The contact author has declared that none of the authors has any competing interests.

Publisher's note: Copernicus Publications remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

This research has been supported by the U.S. Department of Energy, Office of Science, through the MultiSector Dynamics, Earth and Environmental System Modeling Program. Guta Wakbulcho Abeshu and Hong-Yi Li also acknowledge the support from the U.S. National Science Foundation (EAR; grant no. 1804560) and the Research Computing Data Core at the University of Houston for assistance with the computations carried out in this work. The Pacific Northwest National Laboratory (PNNL) is operated for the U.S. Department of Energy by Battelle Memorial Institute (grant no. DE-AC05-76RL0183). The source code and input data for Xanthos used in this study can be freely downloaded at https://github.com/JGCRI/xanthos (last access: 1 August 2022, https://doi.org/10.5281/zenodo.5177210 , Braun et al., 2021).

This research has been supported by the U.S. Department of Energy via the Pacific Northwest National Laboratory (grant no. 551981) and the National Science Foundation (EAR; grant no. 1804560).

This paper was edited by Charles Onyutha and reviewed by two anonymous referees.

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Wild, T. B., Khan, Z., Clarke, L., Hejazi, M., Bereslawski, J. L., Suriano, M., Roberts, P., Casado, J., Miralles-Wilhelm, F., Gavino-Novillo, M., Muñoz-Castillo, R., Moreda, F., Zhao, M., Yarlagadda, B., Lamontagne, J., and Birnbaum, A.: Integrated energy-water-land nexus planning in the Colorado River Basin (Argentina), Reg. Environ. Change, 21, 62, https://doi.org/10.1007/s10113-021-01775-1 , 2021b. 

Wild, T. B., Khan, Z., Zhao, M., Suriano, M., Bereslawski, J. L., Roberts, P., Casado, J., Gaviño-Novillo, M., Clarke, L., Hejazi, M., Miralles-Wilhelm, F., Muñoz-Castillo, R., Vernon, C., Snyder, A., Yarlagadda, B., Birnbaum, A., Lamontagne, J., White, D., and Ojeda-Matos, G.: The Implications of Global Change for the Co-Evolution of Argentina's Integrated Energy-Water-Land Systems, Earth's Future, 9, e2020EF001970, https://doi.org/10.1029/2020EF001970 , 2021c. 

Wisser, D., Fekete, B. M., Vörösmarty, C. J., and Schumann, A. H.: Reconstructing 20th century global hydrography: a contribution to the Global Terrestrial Network- Hydrology (GTN-H), Hydrol. Earth Syst. Sci., 14, 1–24, https://doi.org/10.5194/hess-14-1-2010 , 2010. 

Wu, H., Kimball, J. S., Mantua, N., and Stanford, J.: Automated upscaling of river networks for macroscale hydrological modeling, Water Resour. Res., 47, 1–18, https://doi.org/10.1029/2009WR008871 , 2011. 

Yassin, F., Razavi, S., Elshamy, M., Davison, B., Sapriza-Azuri, G., and Wheater, H.: Representation and improved parameterization of reservoir operation in hydrological and land-surface models, Hydrol. Earth Syst. Sci., 23, 3735–3764, https://doi.org/10.5194/hess-23-3735-2019 , 2019.  

Ye, S., Li, H. Y., Huang, M., Ali, M., Leng, G., Leung, L. R., Wang, S. W., and Sivapalan, M.: Regionalization of subsurface stormflow parameters of hydrologic models: Derivation from regional analysis of streamflow recession curves, J. Hydrol., 519, 670–682, https://doi.org/10.1016/j.jhydrol.2014.07.017 , 2014. 

Yoshida, T., Hanasaki, N., Nishina, K., Boulange, J., Okada, M., and Troch, P. A.: Inference of Parameters for a Global Hydrological Model: Identifiability and Predictive Uncertainties of Climate-Based Parameters, Water Resour. Res., 58, e2021WR030660, https://doi.org/10.1029/2021WR030660 , 2022. 

Zarfl, C., Lumsdon, A. E., Berlekamp, J., Tydecks, L., and Tockner, K.: A global boom in hydropower dam construction, Aquat. Sci., 77, 161170, https://doi.org/10.1007/s00027-014-0377-0 , 2015. 

Zeng, X., Hu, T., Cai, X., Zhou, Y., and Wang, X.: Improved dynamic programming for parallel reservoir system operation optimization, Adv. Water Resour., 131, 103373, https://doi.org/10.1016/j.advwatres.2019.07.003 , 2019.Please provide article number or page range. 

Zhang, X., Li, H.-Y., Deng, Z. D., Ringler, C., Gao, Y., Hejazi, M. I., and Leung, L. R.: Impacts of climate change, policy and Water-Energy-Food nexus on hydropower development, Renew. Energ., 116, 827–834, https://doi.org/10.1016/j.renene.2017.10.030 , 2018. 

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• Introduction
• Methodology
• Global application and results
• Discussion and conclusions
• Code availability
• Data availability
• Author contributions
• Competing interests
• Acknowledgements
• Financial support
• Review statement

Physical Review Applied

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Ground-state-energy calculation for the water molecule on a superconducting quantum processor

Michael a. jones, harish j. vallury, and lloyd c.l. hollenberg, phys. rev. applied 21 , 064017 – published 7 june 2024.

• No Citing Articles
• Supplemental Material

The accurate computation of properties of large molecular systems is classically infeasible and is one of the applications in which it is hoped that quantum computers will demonstrate an advantage over classical devices. However, due to the limitations of present-day quantum hardware, variational-hybrid algorithms introduced to tackle these problems struggle to meet the accuracy and precision requirements of chemical applications. Here, we apply the quantum computed moments (QCM) approach combined with a variety of noise-mitigation techniques to an eight qubit and spin-orbital representation of the water molecule ( H 2 O ). A noise-stable improvement on the variational result for a four-excitation trial state (circuit depth 25, 22 cz gates) was obtained, with the ground-state energy computed to be within 1.0 ± 0.8 mHa of exact diagonalization in the 14 spin-orbital basis. Thus, the QCM approach, despite an increased number of measurements and noisy quantum hardware ( cz error rates c.0.3% corresponding to expected error rates on the trial-state circuit of order 7%), is able to determine the ground-state energy of a nontrivial molecular system at the required accuracy (c.0.05%). To the best of our knowledge, these results are the largest calculations performed on a physical quantum computer to date to achieve millihartree agreement with exact diagonalization, and a promising indicator of how such hybrid approaches might scale to problems of interest in the low-error and fault-tolerant regimes as quantum computers develop.

• Received 2 February 2024
• Revised 11 April 2024
• Accepted 13 May 2024

DOI: https://doi.org/10.1103/PhysRevApplied.21.064017

© 2024 American Physical Society

Physics Subject Headings (PhySH)

• Research Areas
• Physical Systems

Authors & Affiliations

• School of Physics, University of Melbourne , Parkville 3010, Australia
• * Corresponding author: [email protected]

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Vol. 21, Iss. 6 — June 2024

Subject Areas

• Atomic and Molecular Physics
• Chemical Physics
• Quantum Information

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Flow chart of the overall approach employed in this work. (a) Definition of the molecular system. (b) Calculation of the molecular orbitals and electronic Hamiltonian. (c) Hybrid minimization loop to approximate the ground state on the quantum device. (d) Powers of the Hamiltonian can be evaluated independently of (i.e., in parallel with) the minimization loop. (e) Calculation of the Hamiltonian moments, ⟨ H p ⟩ , and hence the Lanczos cluster expansion-corrected energy estimate, E L . For additional detail on the above steps see Sec. 4 and the Appendices.

Structure of the trial-state circuits used in this work [ 44 ]. (a) Minimal circuit to prepare the Hartree-Fock state. Four-excitation trial-state circuit compiled to the base gateset of (b) ibmq_kolkata and (c) ibm_torino each with depth D = 25 . Six-excitation trial-state circuit compiled to the base gateset of (d) ibmq_kolkata and (e) ibm_torino with depths D = 74 and D = 87 , respectively. For each circuit, there exist a set of parameters such that the Hartree-Fock state is produced. These circuits are followed by linear depth basis rotations (see Appendix pp2 ), then computational basis measurements.

Energy-estimation results for the water molecule based on measuring ⟨ H ⟩ and E L for the four-excitation ( D = 25 ) and six-excitation ( D = 74 and D = 87 ) circuits presented in Figs.  2 – 2 . The noiseless simulation results are shown for the trial state (in green) and the reference state (in dashed black). Noise-mitigated results, taken from the quantum devices ibmq_kolkata ( 10 5 shots per basis) and ibm_torino ( 2.5 × 10 4 shots per basis), are marked by magenta and orange crosses, respectively. The cumulative distributions indicate 500 statistical bootstrapping results and the error bars represent one standard deviation. Note that the horizontal axis uses a linear scale in the target accuracy window [ − 1.6 , 1.6 ] mHa and a logarithmic scale outside this window.

(a) Conversion of the nonlocal fermionic double excitation to a local operation using fermionic swap operations, f , on linear connectivity. The θ gate is a Y rotation by angle θ and when combined with the cnot gates, the controlled version can be decomposed to respect the linear connectivity as in Fig.  5 . The cz interactions are between each qubit in the bundle and one of the target qubits. The output of the two circuits is identical up to reordering with fermionic swaps . (b) Decomposition of the fermionic swap gate. Note that when many swaps are chained together in this way, the fermionic swap is cheaper to implement than the standard swap in both number of cnots and depth.

Decomposition of the local double excitation to linear connectivity. The ± θ / 8 gates are Y rotations by ± θ / 8 . This can be improved by prior knowledge of the input state, e.g., correlations between input qubits can be used to remove the long-range controls allowing for more efficient decomposition. Including the cost of performing fermionic swaps , the worst-case cnot count is 4 n + 3 for n qubits and spin orbitals. If it is required that the logical qubits return to their original physical qubit then the cost is 8 n − 13 cnot gates.

Measurement circuits for single-excitation operators, a j † a k . When k ∉ { j , j ± 1 } , fermionic swap operations can be used to reorder the logical qubits, then the circuits above can be applied. The π / 4 gates are rotations about the y axis of the Bloch sphere by π / 4 .

Example circuits for logical-qubit routing from the integer linear program for 8 (left) and 12 (right) qubits. The problem is to interact each qubit with its partner (matching color and superscript) in as few timesteps as possible, with a secondary objective to minimize the number of swaps . The f gates are fermionic swaps while the U gates are the interactions given in Fig.  6 for converting the joint measurement into single-qubit measurements.

The reference-state calibration method can be visualized on a plot of expectation value versus circuit failure rate. The maximally noisy (a) and noiseless (b) energies of the reference state are classically tractable, so using quantum computation of the reference energy (c) allows estimation of the effective noise level in the device (d). By evaluating the energy of the trial state on a quantum computer (e) the noiseless energy of the trial state (f) can be estimated by extrapolation. The linear relationship is due to the white-noise model, with more sophisticated noise models requiring more reference-state calculations to fit [ 90 ]. The reference-state error-mitigation scheme differs from the usual zero-noise extrapolation method [ 11, 92 ] since the additional information is obtained through reference states instead of artificially increasing the noise.

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A detailed analysis of the origin of deep-decoupling oscillations

• Bailie, John
• Dijkstra, Henk A.
• Krauskopf, Bernd

The variability of the strength of the Atlantic Meridional Overturning Circulation is influenced substantially by the formation of deep water in the North Atlantic. In many ocean models, so-called deep-decoupling oscillations have been found, whose timescale depends on the characteristics of convective vertical mixing processes. Their precise origin and sensitivity to the representation of mixing have remained unclear so far. To study this problem, we revisit a conceptual Welander model for the evolution of temperature and salinity in two vertically stacked boxes for surface and deep water, which interact through diffusion and/or convective adjustment. The model is known to exhibit several types of deep-decoupling oscillations, with phases of weak diffusive mixing interspersed with strong convective mixing, when the switching between them is assumed to be instantaneous. We present a comprehensive study of oscillations in Welander's model with non-instantaneous switching between mixing phases, as described by a smooth switching function. A dynamical systems approach allows us to distinguish four types of oscillations, in terms of their phases of diffusive versus convective mixing, and to identify the regions in the relevant parameter plane where they exist. The characteristic deep-decoupling oscillations still exist for non-instantaneous switching, but they require switching that is considerably faster than needed for sustaining oscillatory behaviour. Furthermore, we demonstrate how a gradual freshwater influx can lead to transitions between different vertical mixing oscillations. Notably, the convective mixing phase becomes shorter and even disappears, resulting in long periods of much reduced deep water formation. The results are relevant for the interpretation of ocean-climate variability in models and (proxy) observations.

• Mathematics - Dynamical Systems;
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80th Anniversary of D-Day at Normandy American Cemetery

June 6, 2024, will marked the 80th anniversary of the D-Day landings along the Normandy coast during World War II. This event, which ultimately led to the liberation of Europe, was commemorated at  Normandy American Cemetery . 

Individuals wishing to view the replay of the 80th anniversary of D-Day livestream may do so by clicking on ABMC's YouTube channel , one of ABMC's social profiles or by copying and pasting the social media URL in web browser. 

YouTube : https://www.youtube.com/live/IoXxto2ZLS4?si=ZFEwbNwSuvUrjIHK Facebook : https://facebook.com/usabmc X : https://x.com/usabmc LinkedIn : https://linkedin.com/company/american-battle-monuments-commission

Tickets: The D-Day 80 commemorative event ticket request portal has now closed as capacity has been reached. All visitors will need tickets to access the ceremony. Visitors should send their requests for tickets to [email protected]  and they will be placed on a waiting list.

Please follow the below steps if you have not received your ticket for the D-Day 80 ceremony:

• All ceremony tickets have been sent from [email protected]. Please verify the email has not been deleted by mistake.
• Check spam/junk folder.
• Contact [email protected] for issuance of your ticket. Our staff will verify if you are registered for the event and will then send you your ticket.

Media: Media registration for this event is now closed. For additional information regarding D-Day, Normandy American Cemetery, and ABMC leadership biographies, please see the Press Kit.  

Please note that Normandy American Cemetery’s ceremony on June 6, 2024, is not the international ceremony. The ABMC is not the correct point of contact for this event.  Date:  June 5, 2024

Cemetery Operation Status:  The Normandy American Cemetery will be closed to the public. 

Additional Information:  Guests should schedule visits on a different date.

Date:  June 6, 2024

Cemetery Operation Status:  The Normandy American Cemetery will be closed to the general public for the entire day on June 6. Only ticketed guests will have access to the cemetery on June 6.

For ticketed guests who will attend the ceremony at Normandy American Cemetery on June 6, take note of the following restrictions:

•    Prohibited Items: Weapons of any kind, explosives, firearms, aerosols, ammunition, camera tripods, selfie sticks, umbrellas larger than 18 inches or 0.5 meters, balloons, animals, drones, glass/thermal/metal containers, laser pointers, mace/pepper spray, signs, chairs, knives including Swiss army knives, large bags/luggage/suitcases, whole fruit, any potentially hazardous items.  

•  Picnics and coolers are NOT allowed, but guests should bring sufficient water and snacks for an all-day event. Water and snacks should be in disposable/recyclable containers, but NOT in glass or metal containers. Bottled water will be made available before the start of the event. 

•    Recommended items: Hat, rain poncho, sunscreen, sunglasses, water (no glass/metal).

•    Reenactors and vintage military vehicles will not be allowed inside the cemetery grounds on June 6, 2024.

A regulated traffic zone will be in effect along the Normandy coastline from 06:00am to 10:00pm on June 6, 2024. To travel in the zone on June 6, you must request a vehicle sticker from the local town halls whether you reside in the zone or not. Only vehicles displaying a sticker issued by the authorities will be allowed to enter into the zone. Access to the zone will be controlled by the police throughout the day on June 6, although vehicles will be free to leave the zone. More information about how to request the vehicle sticker can be found at:  FAQ - Restricted Traffic Zone (RTZ) and sticker application

All guests must arrive by shuttle buses provided by the French government from three hub locations: “Parc des expositions” in Caen, parking of “Bayeux Aventure” in Cussy, near Bayeux, and “Parc des expositions” in Saint-Lô. There are no other ways to access the ceremony site.

Admission hours at the shuttle hubs:

•    Parc des expositions in Caen (Calvados): from 06:00am to 07:15am (public) – 06:00am to 08:30am (tour operators). •    Parking of « Bayeux Aventure » in Cussy, near Bayeux (Calvados): in Cussy near Bayeux: from 06:45am to 07:30am (public) – 07:15am to 08:45am (tour operators). •    Parc des expositions in Saint-Lo (Manche): from 06:30am to 07:15am (public) – 06:00am to 08:30 (tour operators).

After security checks at these staging areas, these buses will be escorted by French police directly to the Cemetery. Buses will drop off passengers and then depart the cemetery to park at locations away from the cemetery. Buses will return for passenger pickup after the end of the ceremony.

Seating at the cemetery will be provided based on precedence and on order of arrival for the public. Given security measures in and around the cemetery, departures from the cemetery of all guests by shuttles to the hub locations may take two to three hours.

Please, use the hub closest to your place of accommodation before the event. Priority should be given to the shuttle hub located in Caen if you have not precisely indicated your place of accommodation in Normandy on June 6 during your registration process.

If you are coming as part of a tour group, please direct any inquiries directly to your tour leader. For additional information, please consult: www.abmc.gov/d-day80.

Any individual cars or vehicles that have been provided access to the restricted coastal zone by the French authorities, or which were already inside the restricted zone prior to the closure, will not be able to access the cemetery unless they have been to a centralized security staging area for security checks. All vehicles are subject to these security checks and must then join a police-escorted convoy to access the cemetery. None of these vehicles will be allowed to remain inside the cemetery parking lot after drop-off, so drivers will have to remove the vehicles to sites away from the cemetery. These rules have been mandated by French security authorities to ensure the safety of the public.

ABMC is only responsible for issuing tickets to visitors for the U.S. ceremony inside the Normandy American Cemetery, and for the management of visitors once they are INSIDE the cemetery. All ground transportation between the cemetery and any other location is the responsibility of French authorities, from pre-selected staging areas.

Additional Information: D-Day 80 commemorative event tickets are not available for purchase. Entry will be limited to ticketed guests only and is free of charge. Please continue to monitor this page frequently for further updates.

For questions regarding the commemoration of the 80th anniversary of D-Day, please contact us at: [email protected].  If you are the family member of an individual who is buried or memorialized in Normandy American Cemetery or traveling with a World War II veteran, please include that information in your email.

Email Notification: ABMC employees are unable to send communication via email to "@free.fr" email accounts. Please use an alternate email address when communicating with Normandy American cemetery staff, provide an alternate method to contact you within your email or contact the cemetery by telephone.

On June 6, 2019, we remember the stories and sacrifices of those who were a part of the Allied landings in Normandy.

This World War II cemetery in France contains the graves of nearly 9,400 war dead, and nearly 1,600 names on the Walls of the Missing.

Directions to site

Standard travel routes and access to the site will be restricted on Jun 6, 2024.

Travel directions for The Normandy American Cemetery D-Day 80 Commemoration will be provided to all registrants/ticket holders. Please monitor this webpage frequently for updates.

Drone policy

A film permit is required to use a drone.

ABMC cemeteries, monuments, and memorials are pristine national shrines, for the purpose of remembrance and commemoration.

Picnics and coolers are NOT allowed, but guests should bring sufficient water and snacks for an all-day event. Water and snacks should be in disposable/recyclable containers, but NOT in glass or metal containers. Bottled water will be made available before the start of the event. 

Handicap access

The cemetery is wheelchair accessible.

ABMC cannot provide lodging recommendations but there are various lodging options within a 30-minute drive of the cemetery.

Due to high visitation in June 2024, lodging options will be limited along the coastline.

Personal transportation

Parking will not be authorized inside the Normandy American Cemetery on June 6, 2024. Attendees must take shuttles to access the cemetery grounds.

Photography/Videography

Personal photos are allowed throughout the cemetery grounds. We remind visitors that our cemeteries are sacred places of remembrance and commemoration, therefore, please make sure that the photos you take are respectful of the site and the men and women buried and commemorated within. Photos of a non-commemorative nature including tik-tok videos, wedding photos, political advocacy, publicity, or for influencer promotion are not acceptable. Any videos or photography for commercial/media purposes must be requested in advance through the ABMC public affairs office through a film permit. A film permit is also required for official film and photography shoots, and for any visitors, media or organizations wishing to use a drone.

Reenactors and vintage military vehicles will not be allowed inside the cemetery grounds on June 6, 2024.

Porta Potty restrooms are available before and during the D-Day 80 ceremony. The visitors center will NOT be available to the public after the ceremony.

Service dogs

Our cemeteries are the resting places of thousands of fallen American service members. To ensure our sacred sites are kept at the highest standard possible, the only animals allowed within cemeteries and memorials are specially trained guide dogs or other service dogs accompanying individuals with disabilities. We ask that owners of these animals maintain proper decorum. Animals needing to relieve themselves must be taken outside of the viewshed of the plot areas and the owner must remove any waste from the site. Ultimately, the cemetery staff will determine whether a service dog may access the cemetery, therefore, visitors wishing to bring service dogs should contact the cemetery prior to visiting. Visitors may contact the cemetery staff via phone or [email protected] to get available updates and information.

Water fountains

Water fountains are part of the restroom complexes and cannot be used before and during the D-Day 80 ceremony. The visitor center and public toilet buildings are accessible only after the ceremony.