104 research outputs found
Mathematical study of degenerate boundary layers: A Large Scale Ocean Circulation Problem
This paper is concerned with a complete asymptoticanalysis as of the stationary Munk equation in a domain , supplemented with
boundaryconditions for and . This equation is a
simplemodel for the circulation of currents in closed basins, the variables
and being respectively the longitude and the latitude. A crudeanalysis
shows that as , the weak limit of satisfiesthe
so-called Sverdrup transport equation inside the domain, namely, 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 and on the forcing term . 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( vs. ). We explain in
detail how the superpositiontakes place, depending on the geometry of the
boundary.Moreover, when the domain is not connex in the
direction, is not continuous in , 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 . 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 ,
and not only near theboundary. On the second hand, the northern and southern
boundary layerprofiles obey a propagation equation, where the space variable
plays the role of time, and are therefore not local.Comment: http://www.ams.org/books/memo/1206/memo1206.pd
Weak convergence results for inhomogeneous rotating fluid equations
We consider the equations governing incompressible, viscous fluids in three
space dimensions, rotating around an inhomogeneous vector B(x): this is a
generalization of the usual rotating fluid model (where B is constant). We
prove the weak convergence of Leray--type solutions towards a vector field
which satisfies the usual 2D Navier--Stokes equation in the regions of space
where B is constant, with Dirichlet boundary conditions, and a heat--type
equation elsewhere. The method of proof uses weak compactness arguments
Mathematical study of the betaplane model: Equatorial waves and convergence results
We are interested in a model of rotating fluids, describing the motion of the
ocean in the equatorial zone. This model is known as the Saint-Venant, or
shallow-water type system, to which a rotation term is added whose amplitude is
linear with respect to the latitude; in particular it vanishes at the equator.
After a physical introduction to the model, we describe the various waves
involved and study in detail the resonances associated with those waves. We
then exhibit the formal limit system (as the rotation becomes large), obtained
as usual by filtering out the waves, and prove its wellposedness. Finally we
prove three types of convergence results: a weak convergence result towards a
linear, geostrophic equation, a strong convergence result of the filtered
solutions towards the unique strong solution to the limit system, and finally a
"hybrid" strong convergence result of the filtered solutions towards a weak
solution to the limit system. In particular we obtain that there are no
confined equatorial waves in the mean motion as the rotation becomes large.Comment: Revised version after referee's comments. Accepted for publication in
M\'{e}moires de la Soci\'{e}t\'{e} Math\'{e}matique de Franc
On the propagation of oceanic waves driven by a strong macroscopic flow
In this work we study oceanic waves in a shallow water flow subject to strong
wind forcing and rotation, and linearized around a inhomogeneous (non zonal)
stationary profile. This extends the study \cite{CGPS}, where the profile was
assumed to be zonal only and where explicit calculations were made possible due
to the 1D setting. Here the diagonalization of the system, which allows to
identify Rossby and Poincar\'e waves, is proved by an abstract semi-classical
approach. The dispersion of Poincar\'e waves is also obtained by a more
abstract and more robust method using Mourre estimates. Only some partial
results however are obtained concerning the Rossby propagation, as the two
dimensional setting complicates very much the study of the dynamical system
The Brownian motion as the limit of a deterministic system of hard-spheres
We provide a rigorous derivation of the brownian motion as the limit of a
deterministic system of hard-spheres as the number of particles goes to
infinity and their diameter simultaneously goes to , in the
fast relaxation limit (with a suitable
diffusive scaling of the observation time). As suggested by Hilbert in his
sixth problem, we rely on a kinetic formulation as an intermediate level of
description between the microscopic and the fluid descriptions: we use indeed
the linear Boltzmann equation to describe one tagged particle in a gas close to
global equilibrium. Our proof is based on the fundamental ideas of Lanford. The
main novelty here is the detailed study of the branching process, leading to
explicit estimates on pathological collision trees
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