Lagoons modeling is intrinsically a multi-physics problem. Beside the basic hydrodynamical models that can be either the simple shallow water models or more sophisticated multi-layered shallow water or three dimensional models, one has to consider the representation of sediment transport, deposition and erosion as well as the modeling of the chemical/biological processes driven by the presence of these sediments/nutrients. The coupling of these different types of models involves multiple time and spatial scales and require the development of innovative and robust numerical methods able to take into account many different regimes. An easy to read (we hope) introduction to lagoon modeling can be found here : Des mathématiciens à la rescousse des lagunes méditerranéennes (sorry only in french)

Hydrodynamical models.

In this project, we will focus on the use of the shallow water model or its multi- layered variants as the basic hydrodynamical tool to represent the water circulation in basins. The reason to choose these models is two folds : First it provide a level of description of the water circulation compatible with the accuracy of the simulations envisioned in this project for a reasonable computer and human cost and second a large number of interesting and challenging questions in numerical analysis have to be solved in the framework of this model. Actually, while the approximation of the shallow water system on a flat bottom is a classical problem, its extension in presence of variable topography or multi-layers still pose numerical and theoretical problems (Weak solutions are not defined) that will be addressed in this project in the framework of finite element or finite volume methods.

Sediment transport and bed-load

The coupling between sediment transport erosion and deposition with the water circulation is of fundamental importance for basin modeling. To describe the dynamics of the basin, we have to consider in its simpler form a system composed of three different but coupled components : (a) the shallow water system that represent the dynamics of the water flow, (b) a morphodynamical equation (usually called the Exner equation) describing the dynamics of the bed-load and the evolution of the bed topography and (c) a set of convection/diffusion/reaction equations defining the transport/dispersion/deposition of the suspended sediments. It is therefore clear that this complex coupled system exhibits many different regimes, space and time scales. One of the objectives of this project will be therefore to propose numerical techniques able to cope with the existence of these different regimes.

Biological/chemical models

The introduction of biological/chemical models in the previous system adds a supplementary level of complexity. Indeed if from a mathematical point of view, biological models (a prototype of them being the WQRRS model developed in the seventies by the US Corp of Engineers) are represented by a set of convection / diffusion / reaction equations of the same type than considered for sediment modeling, the number of variables describing the system can be extremely large (26 in WQRRS) and the time scales representative of the evolution of the system can be totally different from the ones describing the water and sediment dynamics.