Mathematical study of degenerate boundary layers: A Large Scale Ocean Circulation Problem

Abstract

This paper is concerned with a complete asymptoticanalysis as $\mathfrak{E} \to 0$ of the stationary Munk equation $\partial\_xψ-\mathfrak{E} Δ^2 ψ=τ$ in a domain $Ω\subset \mathbf{R}^2$, supplemented with boundaryconditions for $ψ$ and $\partial\_n ψ$. This equation is a simplemodel for the circulation of currents in closed basins, the variables$x$ and $y$ being respectively the longitude and the latitude. A crudeanalysis shows that as $\mathfrak{E} \to 0$, the weak limit of $ψ$ satisfiesthe so-called Sverdrup transport equation inside the domain, namely$\partial\_x ψ^0=τ$, while boundary layers appear in the vicinity ofthe boundary.These boundary layers, which are the main center of interest of thepresent paper, exhibit several types of peculiar behaviour. First, thesize of the boundary layer on the western and eastern boundary, whichhad already been computed by several authors, becomes formally verylarge as one approaches northern and southern portions of the boudary,i.e. pieces of the boundary on which the normal is vertical. Thisphenomenon is known as geostrophic degeneracy. In order to avoid suchsingular behaviour, previous studies imposed restrictive assumptionson the domain $Ω$ and on the forcing term $τ$. Here, we provethat a superposition of two boundary layers occurs in the vicinity ofsuch points: the classical western or eastern boundary layers, andsome northern or southern boundary layers, whose mathematicalderivation is completely new. The size of northern/southern boundarylayers 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 superpositiontakes place, depending on the geometry of the boundary.Moreover, when the domain $Ω$ is not connex in the $x$ direction,$ψ^0$ is not continuous in $Ω$, and singular layers appear inorder to correct its discontinuities. These singular layers areconcentrated in the vicinity of horizontal lines, and thereforepenetrate the interior of the domain $Ω$. Hence we exhibit some kindof boundary layer separation. However, we emphasize that we remainable to prove a convergence theorem, so that the singular layerssomehow remain stable, in spite of the separation.Eventually, the effect of boundary layers is non-local in severalaspects. On the first hand, for algebraic reasons, the boundary layerequation is radically different on the west and east parts of theboundary. As a consequence, the Sverdrup equation is endowed with aDirichlet condition on the East boundary, and no condition on the Westboundary. Therefore western and eastern boundary layers have in factan influence on the whole domain $Ω$, and not only near theboundary. On the second hand, the northern and southern boundary layerprofiles obey a propagation equation, where the space variable $x$plays the role of time, and are therefore not local.