Non-flat Ekman Boundary Layers: Topographic Lift, Generalized Ekman Pumping, and Anisotropic Asymptotic Behavior

Abstract

The Ekman boundary layer, a fundamental concept in geophysical fluid mechanics, describes the near-boundary fluid motion subject to rotation. Within the singular limit framework of rapid rotation and vanishing viscosity, classical studies of Ekman theory (e.g., Desjardins and Grenier (1999), Masmoudi (2000)) are predominantly restricted to flat or small-amplitude boundary assumptions. The conventional flat-boundary assumption obscures the complex mechanisms induced by topographic curvature; moreover, even small-amplitude perturbations reduce topographic effects to simple linear forcing terms. Consequently, this paper investigates the singular limit behavior of rotating fluids over a non-flat boundary $z=B(x,y)$ of $\mathcal{O}(1)$ amplitude with uniformly bounded slope and curvature. We elucidate how such topography modulates fluid dissipation through two distinct mechanisms: macroscopic topographic forcing and microscopic anisotropic pumping. First, using multi-scale asymptotic analysis, we construct a class of approximate solutions that explicitly depend on the boundary's geometric characteristics, yielding a two-dimensional limit system fundamentally distinct from classical models. A key innovation of this system is the introduction of a generalized velocity field defined via the topographic metric tensor. This formulation not only generalizes the traditional isotropic linear damping to anisotropic geometric damping but also couples rotational effects to macroscopic vertical acceleration. Furthermore, using energy methods, we establish the $L^2$ convergence of these variable-thickness approximate solutions to the weak solutions of the original three-dimensional system. Finally, we analyze the multiple mechanisms governing rotating fluid motion over large-amplitude topography using a representative class of boundary geometries.