Mathematical study of degenerate boundary layers : a large scale ocean circulation problem This paper is concerned with a complete asymptotic analysis as E \\to 0 of the Munk equation \\partial _x\\psi 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 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