Geometric constraints on Ekman boundary layer solutions in non-flat regions with well-prepared data

TL;DR

This paper analyzes how the geometric shape of non-flat boundaries influences Ekman boundary layer solutions, deriving approximate solutions under certain curvature conditions without smallness restrictions, and validating their convergence.

Contribution

It introduces a new approach to construct approximate Ekman boundary layer solutions considering boundary curvature, extending previous work to more general geometries.

Findings

Derived approximate boundary layer solutions under geometric constraints.

Established convergence of the approximate solutions.

No smallness condition needed on boundary amplitude.

Abstract

The construction of Ekman boundary layer solutions near the non-flat boundaries presents a complex challenge, with limited research on this issue. In Masmoudi's pioneering work [Comm. Pure Appl. Math. 53 (2000), 432--483], the Ekman boundary layer solution was investigated on the domain $\mathbb{T}^2\times [\varepsilon f(x,y), 1]$, where $\varepsilon$ is a small constant and $f(x,y)$ denotes a periodic smooth function. This study investigates the influence of the geometric structure of the boundary $B(x,y)$ within the boundary layer. Specifically, for well-prepared initial data in the domain $\mathbb{R}^2\times[B(x,y), B(x,y)+2]$, if the boundary surface $B(x,y)$ is smooth and satisfies certain geometric constraints concerning its Gaussian and mean curvatures, then we derive an approximate boundary layer solution. Additionally, according to the curvature and incompressible conditions, the limit system we constructed is a 2D primitive system with damping and rotational effects. From the model's background, it reflects the characteristics arising from rotational effects. Finally, we validate the convergence of this approximate solution. No smallness condition on the amplitude of boundary $B(x, y)$ is required.