A good understanding of waves in shallow water, typically in coastal regions, is important for several environmental and societal issues: submersion risks, protection of harbors, erosion, offshore structures, wave energies, etc. The goal of this serie of lectures is to show how efficient asymptotic models can be derived from the full fluid equations (Navier-Stokes and Euler) and to point out several modelling, numerical and mathematical challenges that one still has to understand in order to describe correctly and efficiently such complex phenomena as wave breaking, overtopping, wave-structures interactions, etc.

I Derivation of several shallow water models

We will show how to derive several shallow water models (nonlinear shallow water equations, Boussinesq and Serre-Green-Naghdi systems) from the free surface Euler equations. We will consider first the case of an idealized configuration where no breaking waves are involved, where the water height does not vanish (no beach!), and where the flow is irrotational – this is the only configuration for which a rigorous justification of the asymptotic models can be justified.

II Brief analysis of these models.

We will briefly comment the mathematical structure of these equations, with a particular focus on the properties that are of interest for their numerical implementation. We will also discuss how these models behave in when the water height vanishes, since they are typically used in such configurations (see the lecture by P. Bonneton).

III Vorticity and turbulent effects

We will propose a generalization of the derivation of the main shallow water models in the presence of vorticity, and show that the standard irrotational shallow water models must be coupled with an equation for a ”turbulent” tensor. We will also make the link with a modelling of wave breaking proposed by Gavrilyuk and Richard in which wave breaking is taken into account as a source term in this additional equation.

IV Floating objects.

This last section will be devoted to the description of a new approach to describe the interaction of waves in shallow water with floating objects, which leads to several interesting mathematical and numerical issues.[-]

A good understanding of waves in shallow water, typically in coastal regions, is important for several environmental and societal issues: submersion risks, protection of harbors, erosion, offshore structures, wave energies, etc.

The goal of this serie of lectures is to show how efficient asymptotic models can be derived from the full fluid equations (Navier-Stokes and Euler) and to point out several modelling, numerical and mathematical challenges that one still has to understand in order to describe correctly and efficiently such complex phenomena as wave breaking, overtopping, wave-structures interactions, etc.

I Derivation of several shallow water models

We will show how to derive several shallow water models (nonlinear shallow water equations, Boussinesq and Serre-Green-Naghdi systems) from the free surface Euler equations. We will consider first the case of an idealized configuration where no breaking waves are involved, where the water height does not vanish (no beach!), and where the flow is irrotational – this is the only configuration for which a rigorous justification of the asymptotic models can be justified.

II Brief analysis of these models.

We will briefly comment the mathematical structure of these equations, with a particular focus on the properties that are of interest for their numerical implementation. We will also discuss how these models behave in when the water height vanishes, since they are typically used in such configurations (see the lecture by P. Bonneton).

III Vorticity and turbulent effects.

We will propose a generalization of the derivation of the main shallow water models in the presence of vorticity, and show that the standard irrotational shallow water models must be coupled with an equation for a ”turbulent” tensor. We will also make the link with a modelling of wave breaking proposed by Gavrilyuk and Richard in which wave breaking is taken into account as a source term in this additional equation.

IV Floating objects.

This last section will be devoted to the description of a new approach to describe the interaction of waves in shallow water with floating objects, which leads to several interesting mathematical and numerical issues.[-]

In this course I will be covering three main topics. The first part will be concerning the NavierStokes and Euler equations - a quick survey. The second part will discuss the question of global regularity of certain geophysical flows. The third part about coupling the atmospheric models with the microphysics dynamics of moisture in warm clouds formation.
The basic problem faced in geophysical fluid dynamics is that a mathematical description based only on fundamental physical principles, which are called the 'Primitive Equations', is often prohibitively expensive computationally, and hard to study analytically. In these introductory lectures, aimed toward graduate students and postdocs, I will survey the mathematical theory of the 2D and 3D Navier-Stokes and Euler equations, and stress the main obstacles in proving the global regularity for the 3D case, and the computational challenge in their direct numerical simulations. In addition, I will emphasize the issues facing the turbulence community in their turbulence closure models. However, taking advantage of certain geophysical balances and situations, such as geostrophic balance and the shallowness of the ocean and atmosphere, I will show how geophysicists derive more simplified models which are easier to study analytically. In particular, I will prove the global regularity for 3D planetary geophysical models and the Primitive equations of large scale oceanic and atmospheric dynamics with various kinds of anisotropic viscosity and diffusion. Moreover, I will also show that for certain class of initial data the solutions of the inviscid 2D and 3D Primitive Equations blowup (develop a singularity).[-]