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

About this Title

Anne-Laure Dalibard and Laure Saint-Raymond

Publication: Memoirs of the American Mathematical Society
Publication Year: 2018; Volume 253, Number 1206
ISBNs: 978-1-4704-2835-8 (print); 978-1-4704-4407-5 (online)
Published electronically: April 11, 2018
Keywords:Boundary layer degeneracy, geostrophic degeneracy, Munk boundary layer, Sverdrup equation, boundary layer separation

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Table of Contents


  • Chapter 1. Introduction
  • Chapter 2. Multiscale analysis
  • Chapter 3. Construction of the approximate solution
  • Chapter 4. Proof of convergence
  • Chapter 5. Discussion: Physical relevance of the model
  • Appendix


This paper is concerned with a complete asymptotic ÌĎanalysis as of the stationary Munk equation in a domain , supplemented with boundary ÌĎconditions for and . This equation is a simple ÌĎmodel for the circulation of currents in closed basins, the variables ÌĎ and being respectively the longitude and the latitude. A crude ÌĎanalysis shows that as , the weak limit of satisfies ÌĎthe so-called Sverdrup transport equation inside the domain, namely ÌĎ, while boundary layers appear in the vicinity of ÌĎthe boundary. ÌĎ ÌĎThese boundary layers, which are the main center of interest of the ÌĎpresent paper, exhibit several types of peculiar behaviour. First, the ÌĎsize of the boundary layer on the western and eastern boundary, which ÌĎhad already been computed by several authors, becomes formally very ÌĎlarge as one approaches northern and southern portions of the boudary, ÌĎi.e. pieces of the boundary on which the normal is vertical. This ÌĎphenomenon is known as geostrophic degeneracy. In order to avoid such ÌĎsingular behaviour, previous studies imposed restrictive assumptions ÌĎon the domain and on the forcing term . Here, we prove ÌĎthat a superposition of two boundary layers occurs in the vicinity of ÌĎsuch points: the classical western or eastern boundary layers, and ÌĎsome northern or southern boundary layers, whose mathematical ÌĎderivation is completely new. The size of northern/southern boundary ÌĎlayers is much larger than the one of western boundary layers ÌĎ( vs. ). We explain in detail how the superposition ÌĎtakes place, depending on the geometry of the boundary. ÌĎ ÌĎMoreover, when the domain is not connex in the direction, ÌĎ is not continuous in , and singular layers appear in ÌĎorder to correct its discontinuities. These singular layers are ÌĎconcentrated in the vicinity of horizontal lines, and therefore ÌĎpenetrate the interior of the domain . Hence we exhibit some kind ÌĎof boundary layer separation. However, we emphasize that we remain ÌĎable to prove a convergence theorem, so that the singular layers ÌĎsomehow remain stable, in spite of the separation. ÌĎ ÌĎEventually, the effect of boundary layers is non-local in several ÌĎaspects. On the first hand, for algebraic reasons, the boundary layer ÌĎequation is radically different on the west and east parts of the ÌĎboundary. As a consequence, the Sverdrup equation is endowed with a ÌĎDirichlet condition on the East boundary, and no condition on the West ÌĎboundary. Therefore western and eastern boundary layers have in fact ÌĎan influence on the whole domain , and not only near the ÌĎboundary. On the second hand, the northern and southern boundary layer ÌĎprofiles obey a propagation equation, where the space variable ÌĎplays the role of time, and are therefore not local.

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