Laws of water part 2 : Bridging hydrology, climate, and ecorestoration with the Budyko curve
We looked at a first law of water a couple of essays ago, which was how water seeped through the soil and land. In this essay we will talk about a second law of water which is about how rain, evapotranspiration, the ability of the land to hold water, and the sun’s energy are connected. It links the downward direction of water (rain) with the upward direction of water (evapotranspiration). It helps explain why certain parts of the world have particular biomes. It is a law that bridges hydrology, climate science, and ecology, and intriguingly could provide a roadmap for what kind of ecorestoration to do to restore the water cycle, as well as a metric to track the effects of both individual restoration efforts and larger collective efforts on the water cycle, in the air and on the ground.
First, though, let us clarify what a law is in earth science. It is different from the fundamental laws of nature you find in physics. At the Earth level, laws are emergent. They arise from the interaction of many variables and describe patterns of behavior. These laws are more statistical in nature; they tell us what is most likely to happen given a set of parameters.
Like learning about many laws, this one is built on a number of concepts that one must first understand. Some of these concepts might be unfamiliar, but do not worry, take your time, chew on them, and apply them to different situations. Learning a new concept is like learning new vocabulary. It gives you new ways to talk about phenomena, in this case the water cycle, climate, and eco-restoration.
At the turn of the 20th century, the French geographer De Martonne was trying to understand the world’s biomes and how they relate to climate. He wanted to know why one region is grassland while another is forest, why one is scrubland while another is desert. The first idea might be to think of water availability and rainfall. Consider California and Germany: they have similar rainfall, but rainfall alone does not determine vegetation or biome. Temperature also matters. California is hotter than Germany, and so the air “sucks up” more water. Or consider Senegal and Scotland: both receive similar rainfall, but in Senegal the sun is ferocious, leaving the landscape parched, while Scotland is boggy and marshy. The sun reduces water availability in Senegal more than in Scotland. So De Martonne divided rainfall by temperature. This metric, his dryness index, helped determine which biome occurs where. It worked well enough to produce climate maps approximating actual vegetation.
But De Martonne’s equation had a problem. It assumed a linear relationship: double the temperature, double the drying effect. That was not what the data showed. As temperatures rise, evaporation does not just increase proportionally; it accelerates. The atmosphere’s ability to extract water from soil and plants grows faster than temperature alone predicts, especially at higher heats. The linear equation was an approximation, and in hot climates, it broke down.
In the 1940s, American climatologist Thornthwaite also studied how climate affected biomes. He wanted something that captured the atmosphere’s actual “sucking power” more precisely. He called it potential evapotranspiration (PET), which directly quantified how much water the atmosphere would pull out if water were freely available. Consider a cloth periodically sprayed with water and left under a heat lamp. The amount of water sprayed divided by the potential evapotranspiration the lamp can induce gives a dryness index. If the lamp can evaporate more water than is in the cloth, the system is arid; if not, it is humid. In Senegal, potential evapotranspiration is high, whereas in Scotland, it is low. Dividing rainfall by potential evapotranspiration gives an aridity index, providing a better sense of water availability. This improved upon De Martonne’s method.
Mikhail Budyko, from the Soviet Union, was a pioneer of modern climate science. He helped us understand Earth’s heat balance and climate. He was curious about what drove rainfall and evaporation worldwide. Budyko was interested in “energy limits” as a universal principle, not just for Earth’s water cycle but as a way to understand ecosystems on other planets. He also was not just a climate scientist; he had a deep interest in long-term human adaptation to climate and in the 1970s, he developed one of the earliest models estimating how global warming could affect human settlements, agriculture, and freshwater availability, decades before climate change became a mainstream concern.
Budyko took Thornwaithe’s ideas to the next stage. He defined the idea of water-limited landscapes and energy-limited landscapes. Consider a terrarium where you can vary the water supply and the heat lamp shining on it. If you have little water, the system is water-limited because the heat lamp can evaporate more than is available. If it is a cold room, the lamp is weak, and there is plenty of water, the system is energy-limited because circulation is constrained by energy. Senegal is water-limited, Scotland is energy-limited, the Amazon is energy-limited, and Arizona is water-limited. Budyko sought a universal law of how the sun drives water around the world. Using the terrarium analogy, he wanted to understand how water behaves as you vary the heat lamp temperature and water quantity.
He took the evaporative index (actual evapotranspiration divided by precipitation) and utilized it in his theory. This evaporative index gives you a sense of how much evapotranspiration is happening in an area relative to the rain that falls in the area. In Scotland, this index is low; in Senegal, it is high.
He then plotted this evaporative index vs the aridity index. And saw it had a pattern. So in the plot below each red dot is a place in the world, and the x-axis is the aridity index and the y-axis is the evaporative index. To the left hand side is where aridity index is below one, so the climate is more arid, it is water limited. To the right hand side is where the aridity index is above one, so the climate is more humid, it is energy limited.
The red dots in the plot below are not just scattered randomly. There is a pattern to them. This is an emergent law of behavior of the water cycle. Now if you are interested in a puzzle before reading on, try to figure out why the red dots are distributed this way. What is the connection of the aridity (in the x axis) with the evaporative index, the ratio of evaporatranspiration to rain (in the y axis)?
Budyko’s curve shows that Scotland, with low aridity, is energy-limited, while Senegal, with high aridity, is water-limited and has a higher evaporative index. In essence, the climate’s “thirst,” captured by the aridity index, largely controls how much water evaporates. So you don’t see lower red dots on right hand side, because when you have more ‘sucking power’ then there will be more evaporation and a higher evaporative index. When you have a lower aridity index, meaning a lower sucking power, on the left hand side of the plot, the red dots will be have a lesser y axis value because the lower sucking power leads to a lower evaporative index. In a terrarium with higher heat lamp but lesser water you would be on the right hand side of the diagram, with a higher sucking power (higher aridity index) leading to a higher evaporation (higher evaporative index). In a terrarium with lower heat lamp to water availabity ratio, you would be on the left hand side, and have lower aridity and sucking power leading to lower evaporative index. The x-axis determines the y-axis, providing a universal framework linking global climate dryness to the fraction of rainfall cycling back into the atmosphere. This is the law of the global water cycle.
Budyko viewed the Earth as a self-regulating system, where diverse ecosystems, from the Sahara to the Amazon, emerge from a balance of energy. If you crank up the heat on a stove without adding water, the pan dries and becomes scorching, which represents the water-limited regime of a desert. If you flood the pan, the temperature is capped by how fast water turns to steam, which represents the energy-limited regime of a rainforest. Budyko’s insight was realizing the Earth constantly seeks a balance between these extremes.
Budyko’s curve impacted hydrology and climate science. Before Budyko, hydrology was site-specific. After him, hydrologists recognized universal patterns: runoff and water budgets could be inferred from the aridity and evaporative indices alone. In climate science, the curve helped categorize regions as water-limited or energy-limited. It helped scientists determine evapotranspiration and precipitation based on more general principles.
As important, and as universal as Budkyo’s curve turned out to be, it wasn’t done evolving. As more research was done on this curve, researchers realized there were actually many curves. Soil and vegetation influence water storage and evapotranspiration, represented by a parameter w. Well-restored forests and soils increase w, degraded landscapes reduce it. This allows finer predictions of how landscapes respond to precipitation. So curves with higher w are higher up in Budyko’s curve plot.
This framework is useful for ecorestoration. Seasonal shifts can move a site from water-limited (dry season) to energy-limited (wet season). Reducing floods in the wet season involves moving an energy-limited landscape rightward on the curve, enhancing evapotranspiration through planting trees, expanding wetlands, and increasing vegetation cover. Mitigating drought involves moving water-limited landscapes leftward, restoring soil structure, increasing organic matter, and creating retention features like ponds and swales.
Here is the Budyko curve for the Okefenokee swamp area in Georgia [Corak 2026] which experienced fire. The AET/P is actual evapotranspiration divided by precipitation which is the evaporative index. The PET/P is the potential evapotranspiration divided by precipitation which is the aridity index. The blue triangles are pre-fire. After the fire you can see they move to a lower curve. If we want to stop the drought-fire-flood fire cycle its important that we move lower points back up to higher curves through restoring soil and the water cycle.
The Budyko curve clarifies which restoration actions are most important where. In arid areas like the Sahel, capturing rainfall with zai pits (which are indentations in the land) and planting grasses is critical before tree planting. As areas become less arid, tree planting can proceed with less concern about water capture. Restoration gradually moves landscapes toward curves with higher water retention and evapotranspiration, increasing resilience to drought and floods.
The Budyko curve can track eco-restoration progress worldwide, monitoring individual sites and observing global shifts in the constellation of points on the plot. It is a unifying law of water, bridging eco-restorers, permaculturists, regenerative agriculture practitioners, ecologists, hydrologists, and climate scientists.
References
A video about Budyko curve
Berghuijs, W., and P. Greve. "A review of the Budyko water balance framework." Proceedings of the EGU General Assembly, Vienna, Austria (2015): 12-17.
Nicholas K. Corak, Ana P. Barros, Lauren E.L. Lowman, Budyko scatter reveals interactions between wildfire, land cover change, and climate, Journal of Hydrology, Volume 669, Part A,2026,135096,ISSN 0022-1694,https://doi.org/10.1016/j.jhydrol.2026.135096
Reaver, Nathan George Frederick, David Kaplan, Harald Klammler, and James Jawitz. "Explicit Analytical Inversion of the Parametric Budyko Equations." Available at SSRN 4949309 (2024)
Wang, Cong, Shuai Wang, Bojie Fu, and Lu Zhang. "Advances in hydrological modelling with the Budyko framework: A review." Progress in Physical Geography 40, no. 3 (2016): 409-430.
Zhang, L., Dawes, W. R., and Walker, G. R.: Response of mean annual evapotranspiration to vegetation changes at catchment scale, Water Resour. Res., 37, 701–708, 2001.
Actual catchments have points moving around curves, as shown in diagram below. Explained in this video
[Reaver 2022]
Dear Alpha,
I have been reading your Laws of Water essays with considerable attention (with my partner Claude!) — both Part 1 on Darcy's and Richards' equations, and Part 2 on the Budyko curve.
I wanted to share an analytical observation that struck me reading them together, and to introduce a project I believe you will find genuinely relevant.
Reading the two essays in sequence, I noticed they form a vertical analytical chain that I have not seen made explicit elsewhere. Darcy's law and Richards' equation describe what happens in soil during a specific rainfall event — the mechanics of infiltration, the K(θ) function, the difference between a soil that welcomes rain and one that sheds it. The Budyko curve describes the long-term equilibrium of a landscape between precipitation and evapotranspiration. What connects them is the parameter w in the Budyko framework. That parameter is, in effect, the aggregate long-run consequence of K(θ): restore soil structure through regenerative practice, raise K, and the landscape gradually moves to a higher Budyko curve — more water retained, more evapotranspiration, more contribution to the small water cycle. The chain runs from the pore structure of a single field all the way to the regional water balance. I think making this chain explicit — in a single analytical piece — would be a significant contribution.
The reason I am writing now is that I am leading a project called the World Transformation Programme (WTP), a 100-year (analytical and investment) initiative focused on preventing civilisational collapse by returning humanity's aggregate burden to within planetary boundaries. Water systems is one of our six core transformation domains. We have recently completed a Comprehensive Review of the Global Water Crisis — and I would very much like to share it with you, because your work on the small water cycle, groundwater, and now soil hydrology sits directly at the analytical core of what we are building.
Our review makes an argument that parallels the framing you and John Cherry have developed: that the water crisis is not primarily a climate consequence but the result of three distinct mechanisms being dismantled — the atmospheric water cycle through deforestation, groundwater through systematic mining, and green water through soil degradation. The "slow it, spread it, sink it" framework from your Part 1 essay maps directly onto our investment programme logic for Wave 3–4 countries building their water infrastructure from scratch.
I would be very glad to send you the review (White Paper) and to explore whether there is a basis for closer collaboration. Anastassia Makarieva is already part of our scientific community, and I believe your work belongs in the same conversation.
I was having this discussion in regards to tree planting and failures in the Sahal with the primary focus on using slow release spreader levees across flat land to restore the ground water before or along with the tree planting for greater success. The primary advantage we have as a species is to adjust the water cycle to maximize output and although the impact on global warming as a whole may be limited, the restoration of our flat degraded and semi arid land which cover over 20% of the globe will lower temperature extremes and create a much more habitable environment for all species. They are already doing this with great success in Australia and now I see pilot programs in Chad showing great success. You can think of these as Zia pits that cover thousands of acres at once which have the added benefit of lifting down stream water out of confining erosion gullies and spreading these for water infiltration even where there has been no rain.