Water cycle: Estimate for the first time the flow of rivers on a planetary scale

Fresh water is vital for life, for our societies and cannot be substituted. Economic developments, agriculture, industries and growing demography are creating tensions around this limited resource, all the more so because of climate change.

Agriculture represents 70% of fresh water consumption worldwide (90% in some countries), industry 19% and domestic use only 12% (of which a large share is due to household uses such as washing machine, dishwasher, watering, bathroom and a very small part is related to the drink). Nowadays, 1 out of 2 inhabitants of our planet lives in areas affected by severe water scarcity for at least one month a year. France is also experiencing tensions around the use of water on its own scale. Following the dry and hot winter and then spring of 2022, the prefecture of the Alpes-Maritimes, for example, had to place two thirds of its municipalities (including several large cities) on "drought alert" for several months, from the end of Winter.

Seen from the angle of natural risks (which overall are three times more frequent in 2020 than in the 1975s), "too much water" also represents a consequent danger. Floods represent almost the half the risk with, for the year 2021 alone, more than 50 major events and more than 80 billion dollars in damage. These findings are worsening and will continue to worsen with climate change.

The water cycle is still poorly understood and flows poorly quantified, including surface flows (runoffs) that flow towards the seas via hydrographic networks (rivers). Rivers and rivers act as the veins of our territories.

Human activities greatly depend on the amount of water available and also modify the resource. Typically, the policy of managing a dam to develop agriculture in a region can cause a shortage in a region further downstream. Tensions between regions or users (agriculture, industry, populations) can then arise. Examples include the various conflicts between Israel and the Arab states which are exacerbated by water scarcity (e.g. the disputes around the Litani in Lebanon or the small Yarmouk river in the Golan Heights), tensions between the countries bordering the Nile or the tensions between Mexico and the United States around the Rio Grande and the Colorado.

Estimating the flow of our planet's rivers is a major challenge both from a scientific point of view and from a socio-economic point of view. Contrary to what we might think at first sight, flows are far from being well estimated on a planetary scale and the task is absolutely not obvious.

Mathematical measurements and models

The key variable to quantify surface water fluxes is the river flow Q (m³/s), Q=AU, A (m²) being the section across the river, U (m/s) the average speed in this section.

Flow measurements are available daily or even hourly in industrialized or densely populated regions of the world, for example in France via the Vigicrues network. Conversely, in the less developed regions, the data are non-existent; the flows are therefore very poorly estimated.

The estimation of the flow of a river is possible via mathematical models and numerical calculations. On the other hand, to do this, it is necessary to know the depth of the river (h in the figure), the shape and nature of its bed and the topography of the surrounding terrain. Without field measurement, the depth of a river remains unknown (how deep is the bottom?). In addition, physical parametrizations such as the coefficient of friction of the flow on the ground are required to be able to use these numerical models.

A challenge then consists in knowing how to estimate mathematically, numerically the depth of the river h, its physical parameterizations and finally its flow Q, from the available measurements which are generally only the height of the water surface (and not its depth) at some points of the river.

Measure water levels from space

To make up for our lack of field measurements, which is the case in the vast majority of regions of the globe, the spatial observation of rivers should soon be a solution.

The SWOT Satellite which will be launched in the fall of 2022 will make it possible to measuring for the first time the height of the water surface of rivers, for rivers wider than 100 m and over 90% of the globe, i.e. 213 sections of around ten kilometres. The frequency of measurements will be around ten days (depending on the latitude of the river). The spatial density of the measurement points will be approximately 500 m.

From these measurements of water surface heights H (m), the scientific challenge consists in transforming these measurements into values ​​of flow Q (m³/s), knowing that in non-instrumented areas, the speed of the flow and the depth of the river are unknown!..

Multidisciplinary research carried out in applied mathematics, computational sciences, hydraulics and hydrology, at INSA – Institute of Mathematics of Toulouse, INRAe, the University of Strasbourg – ICUBE and the CS group (CNES funding) aim to meet this challenge scientific: determining the depth of rivers and their flow from satellite measurements of water levels. This challenge is in the process of being taken up on the basis of mathematical models of fluid mechanics (for example, the equations of Saint-Venant XIX^e century) revisited in this particular multi-scale and observational context, mathematical methods of the optimal control type similar to those used to control the trajectory of a robot or determine the initial state of the atmosphere before a weather forecast, and deep learning (“artificial intelligence”).

These scientific advances are then implemented to obtain computational algorithms. Our algorithm entitled HiVDI for Hierarchical Variational Discharge Inference is available within our calculation software, which is certainly technical, but open to all (search software DassFlow).

The estimates currently obtained are based on purely numerical measurements from a CNES-NASA simulator of the future SWOT instrument and also three comparative algorithms (including two United States) with different methodologies.

The results of the calculations make it possible to hope for obtaining an approximate estimate of the depth of non-instrumented rivers and above all a relatively precise estimate of the flow (to within about 30%), in near real time. Such estimates should be available after a full year of satellite overflight, the time for model calibration and learning.

Will these global river flow estimates help improve our knowledge of the water cycle? on the interaction between large non-instrumented rivers and local ocean currents? Will we be able to better estimate the impact of the different uses of certain large rivers (poorly or not at all instrumented to date) and therefore better manage them in the future?

This article is part of the series "The great stories of science in open access", published with the support of the Ministry of Higher Education, Research and Innovation. For more information, please visit the page Openthescience.fr.

Jerome Monnier, University Professor, Applied Mathematics, INSA Toulouse

This article is republished from The Conversation under Creative Commons license. Read theoriginal article.

Image credit: Shutterstock.com / Gareth Kirkland