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