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Mathematical study of degenerate boundary layers: a large scale ocean circulation problem. (English) Zbl 1423.35099

Memoirs of the American Mathematical Society 1206. Providence, RI: American Mathematical Society (AMS) (ISBN 978-1-4704-2835-8/print; 978-1-4704-4407-5/ebook). vi, 110 p. (2018).
This memoir is really a long research paper dealing with a simple model (the Munk equation) for the circulation of currents in closed basins, in terms of the variables \(x\) and \(y\) being respectively the longitude and the latitude. The paper improves the previous study made in [B. Desjardins and E. Grenier [SIAM J. Appl. Math. 60, No. 1, 43–60 (1999; Zbl 0958.76092)]. In some sense, the memoir culminates the series of papers on oceanography and meteorology initiated by J.-L. Lions, R. Temam, and S. Wang around 1993.
The analyzed stationary Munk equation [W. H. Munk and G. F. Carrier, “The wind-driven circulation in ocean basins of various shapes”, Tellus 2, 158–167 (1950)] can be written as \(\partial _{x}\psi -\varepsilon \Delta ^{2}\psi =\tau \), in a domain \(\Omega \subset \mathbb{R}^{2}\), supplemented with boundary conditions for \(\psi \) and \(\partial _{n}\psi \), where \(\psi \) represents the stream function of oceanic currents and \(\tau \) the wind forcing (Ekman pumping). A crude analysis shows that as \(\varepsilon\rightarrow 0\), the weak limit of \(\psi \) satisfies the so-called Sverdrup transport equation inside the domain, namely \(\partial _{x}\psi =\tau \), 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 behavior. 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 boundary, 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 behavior, previous studies imposed restrictive assumptions on the domain \(\Omega \) and on the forcing term \(\tau \). Here, the authors 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 (\(\varepsilon ^{1/4}\) vs. \(\varepsilon ^{1/3}\)). The explanation in detail of how the superposition takes place, depending on the geometry of the boundary, is offered in this paper. Moreover, when the domain \(\Omega \) is not connected 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 some kind of boundary layer separation appears. The authors prove a convergence theorem in this framework, so that the singular layers somehow remain stable, in spite of the separation.
It is also proved that 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.
For a general public presentation see the exposition made in [L. Saint-Raymond, in: Mathematical models and methods for planet Earth. Cham: Springer. 11–24 (2014; Zbl 1425.86008)].

MSC:

35J40 Boundary value problems for higher-order elliptic equations
35B25 Singular perturbations in context of PDEs
35B40 Asymptotic behavior of solutions to PDEs
35Q86 PDEs in connection with geophysics
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References:

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