Maths for understanding water flow

Some places have higher levels of precipitation while others have higher levels of dryness, so it is no surprise that maps showing climatic variations will vary greatly. The frequency of rainfall and its partial evaporation, the nature of soils and the way they are used and water flow into rivers are all parameters that are impacted. These parameters initiate a cause and effect chain of events that influence the capacity of our rivers, industrial production, and agriculture and could extend to the management of water, flood-risk areas and periods of dryness. Having the capability to predict these evolutions has become a major challenge for hydrology. Each river basin has individual characteristics that have a unique effect on the water cycle. A colossal amount of data is required in order to correctly calibrate ecosystem models. However, there may be another way of predicting the behaviour of river basins across the world while at the same time eliminating the need for vast and tedious amounts of field data. This “universal response” can be found in a simple equation, the maximum power principle. Intuition is bold but nature tends to respond favourably to empirical methods. Put another way, up to now there is nothing to prove that this equation doesn’t work.  

There are three main possible outcomes for rainwater once it reaches a river basin. It can evaporate, it can flow into rivers or seep into the ground. The extent of these three outcomes depends on factors such as temperature, the frequency of precipitation and the properties of basins (type of ground cover, presence of vegetation, etc.). All of these factors are naturally influenced by weather variations. Global warming will naturally exacerbate trends that are already unbalanced. Rainfall will become more intense in regions where it is already high and will become rarer in areas that are already endlessly subjected to periods of drought. These developments will have important hydrological and environmental consequences, mainly due to modifications in river flow volumes. Higher, more extreme flow volumes will increase the risk of flooding. Conversely, lower flow volumes will limit access to drinking water, agriculture, industrial exploitation and navigation. “The urgent question is now to succeed in determining the impact of climate change on river basins and river flow volumes”, states Martijn Westhoff, a post-doctoral student at the HECE unit (Hydraulics in Environmental and Civil Engineering) in the Faculty of Applied Sciences of the University of Liege and the leading author of an article published in Hydrology and Earth System Sciences (HESS) (1). “In order to do this, we usually create models that are calibrated by means of surveyed data from the field. We can then modify certain variables and simulate changes to water flow. But these models have significant limitations. Most of them only take into account variations in precipitation and temperature. They do not take account of the evolution of river basin properties”. However, changes in soil structure but also their cover (presence of dense vegetation, etc.) will have a significant impact on evaporation and seepage and therefore also on water flow. “The main problem with these complex models involves their calibration, explains Benjamin Dewals, a lecturer in hydraulic engineering at ULg and co-author of the publication. “In order to accurately reflect reality, they must be extremely detailed. In order to achieve this, an enormous amount of data over several years must be gathered in order to be integrated into the models and only then can simulations even begin. And in the case of many river basins, which are often barely accessible, we have very few measurements. At the same time, there is a real need to predict their evolution for the coming decades. And hydrological models are the only tools that make it possible to assess the risk of flooding or drought, or to improve water quality management based on water flow levels”.

From the particular to the universal

Certainly, the creation of a complex and detailed model, capable of reproducing all the processes while taking into account the countless parameters such as the type of vegetation, erosion or the chemical nature of soils is an attractive prospect. But the approach is a laborious one and Martijn Westhoff chose a second, complementary option, which, though not as precise, seems to be a more realistic method at the current time. “The idea is to search for a model that will provide a greater understanding and more global answers”, says the engineer. “It is no longer a question of studying the particularities of every river basin and their individual responses, or the evolution of each plant in relation to an increase in temperature, but rather to identify an overall ‘universal’ principle to which nature responds, a principle that each basin will follow in order to adapt to later changes”.

This principle has not yet actually been established, but there are several theories currently. Martijn Westhoff, has focussed on a particular case, the maximum power principle, which implies that nature, despite its numerous complex responses, tends towards a state of maximum power transfer according to the energy and resistance present in the system. “Therefore, in the case of water flow, there is an optimal state in accordance with temperature, precipitation, soil etc. In response to certain variations, the river basin adapts and evolves until it reaches this state”. This principle has not been chosen by chance, but because it has already worked in other contexts. In the case of hydrology, the principle remains a theory. It is a theory that has not yet been demonstrated in practice, but which provides a focus for the researcher. By coupling analyses of data from river basins across the world and the study of samples in the laboratory, the researcher verifies whether or not nature responds to the predictions that can be made by using this principle.

The maximum power principle in thermodynamics

A good way to understand this maximum power principle is to return to its source. In 1979, the Australian atmospheric physicist Garth Paltridge suggested a theory to explain heat transfer in the atmosphere. His model can be reduced to two sets, two simplified blocks. The equator on one side, the poles on the other. The equator is more exposed to solar radiation and gathers more heat than the poles. Part of this heat at the Equator returns to space. If, in fact, this energy input was conserved in the system, the temperature of the Earth would soar, continually. The remaining heat, is redistributed to the poles by an entire series of mechanisms, the winds in particular. Paltridge did not try to understand these mechanisms in detail, but rather to quantify the transfer of energy that resulted from it. This measure would enable him to verify his principle which stated that the power associated with the transfer of heat from the equator to the poles was naturally and continually optimal. And what else?

“On a graph”, explains Benjamin Dewals, “the x axis represents the facility of the heat transfer”, and the y axis the power of the transfer. In the extreme case of a very strong resistance, at point zero, there would be no transfer of heat, therefore no deployment of energy. The two blocks, equator and poles, would keep a very high temperature differential and would not communicate. Conversely, if there was no resistance, the heat would be instantly transferred to the poles. It is because of the temperature difference that a continuous transfer is possible. If the temperature was equal everywhere, there would also not be any transfer of energy, a state which corresponds to the right limit of the x axis. Between zero and infinity, an entire series of possibilities condition the power of this heat transfer”. Somewhere among these intermediaries, a resistance linked to a certain temperature differential will enable a maximum power transfer of heat. In theory, the principle states that the atmosphere will naturally tend towards this balance which conditions the maximal deployment of energy. And the measurements taken since then corroborate this prediction.

This is not the only natural phenomenon that has been predicted using this principle. There have been others, such as the vertical circulation of air from the ground surface to the atmosphere. Martijn Westhoff continues, “Let’s imagine a room with a basin of water. When the water evaporates, the temperature of the room drops proportionately. Because this transformation from the liquid state to the gaseous state requires energy and is translated by heat transfer. This loss of heat rebalances the distribution of heat vertically. This observation creates the possibility of predicting the evaporation from the soil surface by taking very few measurements”. The principle therefore works for the atmosphere and for other systems too, but how to verify its capacity to predict water flow in river basins?

An empirical approach to prove the theory

As in the case with analogies made with the atmosphere, there is no question of entering into fine details, into the mechanisms that facilitate or restrict the transfer of water (gradient, ground, erosion, vegetation etc.), but to envisage the dynamic on an overall basis. The angle is very large and the method is empirical. The researchers gathered and compared the average partitioning of rainfall into evaporation and run-off in river basins all over the world (America, Africa, more local data also such as the Ourthe), to search and verify if this distribution corresponded to the maximum or not. The method was backed up by laboratory work. “The principle is meant to be universal”, says Benjamin Dewals. “That it has been verified in large spaces is one thing but it must also be able to be observed on very small scales. We therefore experimented with water flow in small samples over half a cubic metre of ground”. This approach made it possible to observe flow in a very detailed way, while seeking to verify again whether this distribution operates in such a way that the power is at a maximum level.

An enigma that has not been resolved for half a century

At the end of the 1960s, a Russian climatologist by the name of Mikhail Budyko observed the evaporation from a large series of river basins in order to place them on a graph according to two units of measurement. The evaporation was divided by the total precipitation on the vertical axis and the potential for maximum evaporation on the horizontal axis. In other words, if an infinite amount of water was available, this axis would represent what could have been evaporated in accordance with solar radiation. “The scope of possibilities is constrained by two asymptotes. On one hand, the evaporation can be close to an infinitesimal level of the total quantity of precipitation, but would never be able to exceed it. There cannot be more evaporation than the existing water. On the other axis, the evaporation cannot be higher than the thermal energy that the sun allows”.

Mikhail Budyko therefore represented, by points, the average data for each of the river basins studied. These points placed on the graph progressively formed a curve which today bears the name of the researcher. What is astonishing, is that these points aggregate around this curve which is also relatively close to the two asymptotes. As explained above, it is normal that no point should be found beyond the two asymptotes. “But the big question is why there is no point present in the space located between the curve and the horizontal axis. Theoretically, it is quite possible. We could observe all kinds of river basins which would have very heterogeneous evaporation properties that would not be found on this curve”.

To date, nobody has yet succeeded in explaining why the evaporation properties of river basins are so neatly aligned. This is precisely the reason why the young civil engineer suggested the maximum power principle as an answer. “By mathematically imposing the constraints linked to the asymptotes and by applying this maximum power principle, we found that the Budyko curve applied to our river basins. There is therefore a coherence between this principle and our observations which we must continue to prove”, say the two jubilant researchers.

On studying the Budyko graph, the observer notices quite quickly that the points form a cluster rather than an alignment according to a single curve. The theories put forward were up to now based on ground cover. The extent of forests, for example should have an effect on the level of evaporation. But these theories have never been conclusive up to now. “In our mathematical model, we have explored other avenues. First and foremost, we tried to place the asymptotes in a situation of constant rainfall. A diagram of reality to simplify our calculations and already verify whether we were going in the right direction. But there is no such place as a region where it rains all the time. We therefore introduced a seasonal dynamic of rainfall and temperature to simulate rainy and sunnier periods where the evaporation would vary accordingly”. As soon as this dynamic was introduced into the model, the results tended to point toward the cluster. This could signify that the slight variations around the Budyko curve can be explained by changes in precipitation and temperature, implying that soil cover and climate variable are correlated.  

Return from the universal to the particular

The tool which closely associates the Budyko curve to the maximum power principle, could therefore help to better predict the evaporation from river basins. However, it is not yet operational. Each point on the graph corresponds to the average precipitation and evaporation from a river basin over several years. Resolution in time and space rem    ains imprecise. “Our publication is only a first step. If we succeed in verifying that the maximum power principle is correct, we will be able to apply it on a much smaller scale while taking account of the real dynamic of each region and particular events etc.” A second drawback for the moment is the fact that the model only makes it possible to predict one of the three factors relative to the cycle of precipitation and evaporation. With two unknowns, it is currently impossible to determine the part played by the flow and seepage factors. It is therefore difficult to predict the impact of climate variations on the supply of rainwater to rivers.

Two birds with one stone

Apart from the desire to demonstrate the universal character of the principle, it was precisely to compensate for these two approximations that the researchers carried out a series of very small-scale laboratory experiments. These ground samples offer the possibility of observing their behaviour in detail and therefore water flow, as well as evaporation. The experiment, which is quite simple, is nothing but a hydrological version of the atmospheric transfer of heat between the equator and the poles.

Two reservoirs were placed side by side and were separated by a ground sample a metre and a half in length. Only the left reservoir was supplied with water from the top, in order to simulate rainfall. On the bottom on the external edges of the reservoirs natural water loss (by evaporation and seepage) was simulated by a tap. The equivalent of thermal energy rediffused into space and which is not kept in the atmospheric system. At the start of the experiment, the water strikes a compact ground. A high resistance prevents the deployment of maximum power transfer. As in the atmosphere, mechanisms will ease this resistance. Here it is not a question of winds or properties associated with air. The processes are different. “One of them, which we are trying to simulate in this experiment, is internal erosion. The water will progressively displace the grains of soil and dig channels. The bigger these arteries are, the more rapid the transfer to the other reservoir will be and the more powerful the transfer will be. Conversely, if these channels become too big, the resistance will become almost non-existent. The water level progressively balances out between the two basins and the volume diminishes until it disappears completely. Between the two extremes, at a given moment, there is a situation where the water flow reaches a maximum power level. The objective of the experiment is to verify whether, as the maximum power principle predicts, these erosion mechanisms will stop developing when the system gets close to this state”.

Currently the results are promising, but experiments need to be repeated to avoid all possible biases. If the approach develops in such a way as to systematically involve the predictions of the maximum power principle, the researchers will obtain an extra element of proof that their theory works. More importantly, they will be able to develop a means of analysing very precise facts about river basins in detail on a case-by-case basis, and to combine the evaluation of two of the ways in which water is transferred. Evaporation on one side with the help of the Budyko curve and water flow on the other following laboratory experiments. “We will then be able to predict the evolution of evaporation and flow of rainwater, but also seepage, because we will only need to subtract from the total precipitation evaporation and water flow to obtain the seepage level. The methodological and theoretical work will still be long, but we will then be able to envisage a practical application of the model”.

(1) Westhoff, M., Zehe, E., Archambeau, P., and Dewals, B.: Does the Budyko curve reflect a maximum-power state of hydrological systems? A backward analysis, in Hydrology and Earth System Sciences, 20, 479-486, doi:10.5194/hess-20-479-2016, 2016.