Madalina Petcu

Some Mathematical Problems in Geophysical Fluid Dynamics

Abstract This chapter reviews the recently developed mathematical setting of the primitive equations (PEs) of the atmosphere, the ocean, and the coupled atmosphere and ocean. The mathematical issues that are considered here are the existence, uniqueness, and regularity of solutions for the time-dependent problems in space dimensions 2 and 3, the PEs being supplemented by a variety of natural boundary conditions. The emphasis is on the case of the ocean that encompasses most of the mathematical difficulties. This chapter is devoted to the PEs in the presence of viscosity, while the PEs without viscosity are considered in the chapter by Rousseau, Temam, and Tribbia in the same volume. Whereas the theory of PEs without viscosity is just starting, the theory of PEs with viscosity has developed since the early 1990s and has now reached a satisfactory level of completion. The theory of the PEs was initially developed by analogy with that of the incompressible Navier Stokes equations, but the most recent developments reported in this chapter have shown that unlike the incompressible Navier-Stokes equations and the celebrated Millenium Clay problem, the PEs with viscosity are well-posed in space dimensions 2 and 3, when supplemented with fairly general boundary conditions. This chapter is essentially self-contained, and all the mathematical issues related to these problems are developed. A guide and summary of results for the physics-oriented reader is provided at the end of the Introduction ( Section 1.4 ).

STOCHASTIC SOLUTIONS OF THE TWO-DIMENSIONAL PRIMITIVE EQUATIONS OF THE OCEAN AND ATMOSPHERE WITH AN ADDITIVE NOISE

The aim of this article is to establish the existence and uniqueness of stochastic solutions of the two-dimensional equations of the ocean and atmosphere. White noise is additive, and the solutions are strong in the probabilistic sense. Finally, from the point of view of partial differential equations, they are of the type z-weak, that is, bounded in L∞(L2) together with their derivative in z.

SINGULAR PERTURBATION ANALYSIS ON A HOMOGENEOUS OCEAN CIRCULATION MODEL

In this article, we consider the barotropic quasigeostrophic equation of the ocean in the context of the β-plane approximation and small viscosity (see, e.g., [21, 22]). The aim is to study the behavior of the solutions when the viscosity goes to zero. To avoid the substantial complications due to the corners (see, e.g., [25]) which will be addressed elsewhere, we assume periodicity in one direction (0y). The behavior of the solution in the boundary layers at x = 0, 1 necessitate the introduction of several correctors, solving various analogues of the Prandtl equation. Convergence is obtained at all orders even in the nonlinear case. We also establish as an auxiliary result, the regularity of the solutions of the viscous and inviscid quasigeotrophic equations.

A numerical analysis of the Cahn---Hilliard equation with non-permeable walls

In this article we consider the numerical analysis of the Cahn–Hilliard equation in a bounded domain with non-permeable walls, endowed with dynamic-type boundary conditions. The dynamic-type boundary conditions that we consider here have been recently proposed in Ruiz Goldstein et al. (Phys D 240(8):754–766, 2011) in order to describe the interactions of a binary material with the wall. The equation is semi-discretized using a finite element method for the space variables and error estimates between the exact and the approximate solution are obtained. We also prove the stability of a fully discrete scheme based on the backward Euler scheme for the time discretization. Numerical simulations sustaining the theoretical results are presented.

The nonlinear 2D supercritical inviscid shallow water equations in a rectangle

In this article we consider the inviscid two-dimensional shallow water equa- tions in a rectangle. The flow occurs near a stationary solution in the so called supercritical regime and we establish short term existence of smooth solutions for the corresponding initial and boundary value problem. Motivated by the study of the inviscid primitive equations, we consider in this article the inviscid two-dimensional shallow water equations in a rectangle in the so-called supercritical regime. It has been shown that a certain vertical expansion of the inviscid primitive equa- tions leads to a system of coupled nonlinear equations similar to the inviscid shallow water equations; see (RTT08b) and (HT14a). Hence beside their intrinsic interest, the nonlinear shallow water equations can be seen as one mode of the vertical expansion of the primitive equations. The issue of the boundary conditions to be associated with the primitive or shallow water equations has been emphasized as a major problem and limitation for the so-called Local Area Models for which weather predictions are sought and simulations are performed within a domain for which the boundary has no physical significance, so that there are no phys- ical laws prescribing the boundary conditions (see (WPT97) and e.g. (RTT08a, RTT08b), (CSTT12, SLTT)). The choice of the boundary conditions relies then on mathematical considerations (derivation of a well-posed mathematical problem), and on general compu- tational considerations and physical intuition. The boundary conditions suitable for the one-dimensional shallow water equations were derived in an intuitive context in the book of Whitham (Whi99) and in (NHF08); see (PT13) for a rigorous study. For general results on boundary value problems for quasilinear hyperbolic system in space dimension one see (LY85); for initial and boundary value problems for hyperbolic equations in smooth domain see the thorough book (BS07). The present article follows the study of the one-dimensional inviscid shallow water equations in (PT13, HPT11) and the study of the linearized shallow water equations in (HT14a). In the study of the linearized inviscid shallow water equations in (HT14a) we have shown that five cases can occur depending on the respective values of the velocity and the height (not counting the non-generic cases and the symmetries). The nonlinear case that we consider in this article relates to what was called the supercritical case in (HT14a); see (HT14b) for the study of a subcritical case.

Well-posedness and long time behavior of a perturbed Cahn-Hilliard system with regular potentials

The aim of this paper is to study the well-posedness and long time behavior, in terms of finite-dimensional attractors, of a perturbed Cahn-Hilliard equation. This equation differs from the usual Cahn-Hilliard by the presence of the term e(−Δu + f (u)). In particular, we prove the existence of a robust family of exponential attractors as e goes to zero.

Exponential decay of the power spectrum and finite dimensionality for solutions of the three dimensional primitive equations

In this article we estimate the number of modes, volumes and nodes, sufficient to describe well the solution of the three dimensional primitive equations; the physical meaning of these estimates is also discussed. We also study the exponential decay of the spatial power spectrum for the three dimensional primitive equations.