Mathematical Study of Degenerate Boundary Layers: A Large Scale Ocean Circulation Problem

This paper is concerned with a complete asymptotic analysis as $\mathfrak{E} \to 0$ of the stationary Munk equation $\partial_x\psi-\mathfrak{E} \Delta^2 \psi= \tau$ in a domain $\Omega\subset \mathbf{R}^2$, supplemented with boundary conditions for $\psi $ and $\partial_n \psi$. This equation is a simple model for the circulation of currents in closed basins, the variables $x$ and $y$ being respectively the longitude and the latitude. A crude analysis shows that as $\mathfrak{E} \to 0$, the weak limit of $\psi$ satisfies the so-called Sverdrup transport equation inside the domain, namely $\partial_x \psi^0=\tau$, while boundary layers appear in the vicinity of the boundary. These boundary layers, which are the main center of interest of the present paper, exhibit several types of peculiar behaviour. First, the size of the boundary layer on the western and eastern boundary, which had already been computed by several authors, becomes formally very large as one approaches northern and southern portions of the boudary, i.e. pieces of the boundary on which the normal is vertical. This phenomenon is known as geostrophic degeneracy. In order to avoid such singular behaviour, previous studies imposed restrictive assumptions on the domain $\Omega$ and on the forcing term $\tau$. Here, we prove that a superposition of two boundary layers occurs in the vicinity of such points: the classical western or eastern boundary layers, and some northern or southern boundary layers, whose mathematical derivation is completely new. The size of northern/southern boundary layers is much larger than the one of western boundary layers ($\mathfrak{E}^{1/4}$ vs. $\mathfrak{E}^{1/3}$). We explain in detail how the superposition takes place, depending on the geometry of the boundary. Moreover, when the domain $\Omega$ is not connex in the $x$ direction, $\psi^0$ is not continuous in $\Omega$, and singular layers appear in order to correct its discontinuities. These singular layers are concentrated in the vicinity of horizontal lines, and therefore penetrate the interior of the domain $\Omega$. Hence we exhibit some kind of boundary layer separation. However, we emphasize that we remain able to prove a convergence theorem, so that the singular layers somehow remain stable, in spite of the separation. Eventually, the effect of boundary layers is non-local in several aspects. On the first hand, for algebraic reasons, the boundary layer equation is radically different on the west and east parts of the boundary. As a consequence, the Sverdrup equation is endowed with a Dirichlet condition on the East boundary, and no condition on the West boundary. Therefore western and eastern boundary layers have in fact an influence on the whole domain $\Omega$, and not only near the boundary. On the second hand, the northern and southern boundary layer profiles obey a propagation equation, where the space variable $x$ plays the role of time, and are therefore not local.