Mathematical Study of Degenerate Boundary Layers: A Large Scale Ocean Circulation Problem
Anne-Laure Dalibard, Laure Saint‐Raymond · Memoirs of the American Mathematical Society · 2018
This paper is concerned with a complete asymptotic analysis as E → 0 \mathfrak {E} \to 0 of the stationary Munk equation ∂ x ψ − E Δ 2 ψ = τ \partial _x\psi -\mathfrak {E} \Delta ^2 \psi = \tau in a domain Ω ⊂ R 2 \Omega \subset \mathbf {R}^2 , supplemented with boundary conditions for ψ \psi and ∂ n ψ \partial _n \psi . This equation is a simple model for the circulation of currents in closed basins, the variables x x and y y being respectively the longitude and the latitude. A crude analysis shows that as E → 0 \mathfrak {E} \to 0 , the weak limit of ψ \psi satisfies the so-called Sverdrup transport equation inside the domain, namely ∂ x ψ 0 = τ \partial _x \psi ^0=\tau , while boundary layers appear in the vicinity of the boundary. These boundary layers,