Maths and the beach

07.23.2026, by

Reading time: 8 minutes

Holiday-makers cross paths at the beach, but so do multiple facets of physics and mathematics. Three research projects are working to better describe this environment so highly sensitive to climate change.

When sitting on the beach with one’s feet in the wet sand, nobody would suspect the expansive physics behind this apparently peaceful landscape, including wave generation, coastal erosion, and seawater seepage into soil, together forming a complex reality that scientists are striving to grasp.

Three Maths-Vives [8] Priority Research Programme and Equipment (PEPR)1 projects are focusing on this environment, which is as critical as it is beautiful. With climate change, coasts will be subject to rising waters, increasingly frequent extreme climate events such as tsunamis, and accelerating coastal erosion. Disciplines such as geophysics, oceanography, and mathematics will be used to convert beach dynamics into equations, in an effort to improve coastal management in the future.

Mathematics helps model coastal erosion, such as that seen here in Soulac-sur-Mer along the Aquitaine coast (southwestern France), where the coastline has receded to the point of endangering homes.

“We want to enlighten policy-makers. We are not saying that we will be able to anticipate events, but that we will understand them better. And understanding better means deciding better,” asserts Didier Bresch, CNRS research professor at Université Savoie Mont-Blanc2.

Anticipating tsunamis

For over 200 years, scientists have taken an interest in the beach, specifically water. In the 19th century, physicists and mathematicians developed a system of equations for describing the motion of fluid, water, and air, among others. Now known as the Navier-Stokes [12] partial differential equations, these are mathematical tools for describing “a phenomenon in motion, in space and time”.

“Ocean dynamics involve, in essence, water molecules that collide,” explains Catherine Choquet, mathematics professor at Université de La Rochelle3. “However, at our scale of observation, multiple effects are not perceptible, and have therefore been excluded from this model.” It is impossible to describe the interactions between billions of molecules on the microscopic scale when attempting to model a wave. Instead, Navier-Stokes equations concentrate on what is important for depicting the phenomenon as it is perceived on the macroscopic scale.

©A. G. Filippini et. al. "An operational discontinuous Galerkin shallow water model for coastal flood assessment" Ocean Modelling, 2024.

3D simulation of a tsunami (reading direction: left to right, top to bottom). The tsunami wave advances until it breaks over the city’s seafront.

These equations are still used today, especially in the context of climate change, in which extreme episodes may become more frequent and destructive. “The impact that these events will have on society will increase, hence the importance of understanding the mechanisms underpinning them, in order to provide local governments with forecasting tools,” states David Lannes, CNRS research professor at Université de Bordeaux4.

The scientist is taking part in the Climaths5 project from the Maths-Vives PEPR, which studies the appearance and behaviour of extreme waves in coastal settings. There are many reasons for doing so, such as to develop prediction tools for improved risk management by local or regional authorities (evacuation plans, urban planning regulations), and to secure offshore renewable energy installations (wind and wave power).

However, many phenomena are involved, from the ocean-atmosphere coupling behind wave generation, to swell propagation and the submersion of entire towns. “As all of these occurrences are complex, we are trying to eliminate the details that are less useful for applications,” indicates Lannes, with a view to developing a model for the phenomenon that is simple enough for practical application.

The science of sand castles

The beach is a place where water meets grains of sand. The behaviour of water has been studied for a long time, as has that of sand grains [14]. This environment nevertheless represents a new source of complexity: “Grains are not all the same, they can vary in size, shape, and surface properties,” reminds Farhang Radjai, CNRS research professor at Université de Montpellier6.

Three areas can be made out on the beach: one where the sand is totally saturated with water, another where it is dry, and between them a third that is unsaturated, which is to say the empty spaces between the grains of sand are partially filled with water.

Interaction between these worlds of water and sand is much less familiar, despite the fact that astonishing properties result from it, such as the ability to build castles with wet sand. “Liquid water glues sand grains together, just as it does hair to our head. This is referred to as capillary action,” Radjai explains. On the beach, capillarity joins the other forces that are present, such as hydrodynamics for the behaviour of water, and friction between grains of sand.

Radjai is one of the coordinators of the ComplexFlows7 project, which seeks to describe this granular and humid world where two types of material – solid grains and liquid water – interact with each other, thereby complicating the modelling of their behaviour. “If you look at a pile of wet sand, it is made up of water and sand. We have equations for water, and others for solids. But how to describe the wet solid? How does the system change when the quantity of liquid increases or decreases?” asks Bresch, who is also a coordinator for the programme.

©J.-Y. Delenne, V. Richefeu, F. Radjai (2015), “Liquid clustering and capillary pressure in granular media"

Capillary action, modelled here using computer code, can assemble drops of water and sand grains into a coherent and balanced whole.

It may seem trivial and even anecdotal to focus on granular flows, yet they are involved in highly concrete phenomena that transform our coasts: coastal erosion of rocks from heavy rain and the action of the sea. “With this project we hope to enhance our understanding of the coastline’s evolution, to achieve even a rudimentary grasp,” indicates Bresch. Another goal is to monitor the tendency of beaches to lose their sand, “for one might believe that sand is an inexhaustible source, but it is actually a fairly rare material”, adds Radjai.

Beneath the beach, saltwater

The beach is not just what we see, but also what we do not: the seepage of seawater in the depths, all the way down to the water table. “When I arrived in La Rochelle (western France, Ed’s note), I saw that the inhabitants had clean water wells in their gardens to supply the water network, but did not use them. Their wells had in fact been contaminated by saltwater,” recounts Catherine Choquet, the leader of the Hydraumath8 Project, which studies interactions between salt water and groundwater.

The risk of groundwater salinisation rises in the summer. “As clean water is drawn, saltwater rises to the surface. When the water table is salty, the well can no longer be used,” Choquet adds. Understanding this dynamic helps to identify the appropriate rate for drawing well water, whilst minimising risk.

Near La Rochelle (western France), seawater seeps into the depths, making groundwater unfit for human consumption.

The problem is that part of the interaction between seawater and groundwater happens out of view, far down below. “We have to use mathematics in order to model what we cannot see,” points out Choquet. “We can’t turn the ground into Swiss cheese to observe the water table.” This makes it difficult to identify the driving forces behind salinisation, as well as the impact of storms and rising waters on this phenomenon.

The other stumbling block is timescales: the movement of the ocean is very fast, whereas water seepage into groundwater occurs very slowly. “We realised that seawater interacts with water in the depths, where we did not expect it to. That is true in Perpignan (southwestern France, Ed’s note), where we would instead have thought it to interact on the surface. There is therefore a transition we are not aware of,” points out Choquet.

Between models and reality on the ground

“Until recently, to better grasp such occurrences, we conducted tests in the laboratory. Yet this does not always shed light on the physical mechanisms at play, and we cannot wait for disasters to happen in order to act,” Radjai stresses. In all three projects, mathematical modelling takes over from laboratory experiments and field sampling (videos, sensor readings, photos, etc.), with a view to getting a better grasp of such complex events.

“The data is often incomplete, and mathematics enables us to complement it,” Choquet observes. Artificial intelligence is increasingly included in the statistical tools used to analyse material, helping mathematicians to improve existing models (Navier-Stokes equations, groundwater flow models, sand grain models, etc.). These can also be refined via digital simulation, as is the case with the ComplexFlows project: simulating friction between sand grains can shed light on the overall behaviour of granular flows.

For all that, maths is not enough on its own. “What is important in these projects is to collaborate right from the formalisation of the problem, to constantly explore the issues and data types we are dealing with, in order to fashion a model that is close to physical reality,” affirms Lannes.

“The issues explored by the PEPR are highly varied and complex. This requires a great deal of time and a broad range of expertise to advance our analyses of such phenomena,” Bresch adds. With all this in mind, beaches may not be such quiet places after all.