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Geometry-dependent Ekman layer approximations on curved domains: L^{\infty} convergence

Yifei Jia, Yi Du, Lihui Guo

TL;DR

This work advances the theory of Ekman boundary layers by treating non-flat, curved boundary geometries and proving $L^\infty$ convergence of a carefully constructed multi-scale approximate solution to the full rotating Navier–Stokes system in the vanishing-viscosity limit. The authors develop a geometry-aware expansion that separates interior dynamics from bottom and top boundary layers, uses a cut-off-based coupling, and introduces a divergence-free correction to satisfy incompressibility. Key contributions include establishing how boundary geometry, quantified through Gaussian curvature bounds and curvature-related matrices, shapes the near-boundary flow and yields a limiting 2D damped Euler-type system for the horizontal velocity, while providing explicit, exponentially decaying boundary-layer profiles. The analysis removes the small-boundary-amplitude restriction and recovers classical flat-boundary results in the planar limit, with potential implications for atmospheric and oceanic geophysical flows over complex terrains. Overall, the paper delivers rigorous, high-precision $L^\infty$ error control for Ekman layers on curved domains, under explicit geometric conditions and well-prepared initial data, broadening the applicability of boundary-layer theory to realistic geophysical geometries.

Abstract

The Ekman boundary layer is a fundamental concept in fluid dynamics that describes fluid motion near boundaries affected by Earth's rotation. Most theoretical studies have simplified their analysis by assuming a planar boundary surface, resulting in limited exploration of structures with general smooth boundary conditions. Investigating the impact of boundary geometry in the Ekman boundary layer is essential, as initially suggested by J.L. Lions and further examined in Masmoudi's study [Comm. Pure Appl. Math. 53 (2000), 432-483] under small amplitude periodic boundary conditions. This paper clarifies how boundary geometry influences flow fields and characterizes its effects on near-boundary layer flow. We construct a class of multi-scale approximate solutions based on the boundary's geometric features and establish their convergence in the L^{\infty} framework. Our findings do not require a small-amplitude assumption, only an upper bound on the Gaussian curvature of the boundary surface. Notably, when the boundary is planar, our approach aligns with existing studies. Additionally, in the vanishing-viscosity limit, we derive a limiting-state system dependent on boundary geometric parameters. These contributions extend the theoretical understanding of boundary-layer interactions to general curved geometries and have possible applications in atmospheric, oceanic, and other geophysical flow contexts.

Geometry-dependent Ekman layer approximations on curved domains: L^{\infty} convergence

TL;DR

This work advances the theory of Ekman boundary layers by treating non-flat, curved boundary geometries and proving convergence of a carefully constructed multi-scale approximate solution to the full rotating Navier–Stokes system in the vanishing-viscosity limit. The authors develop a geometry-aware expansion that separates interior dynamics from bottom and top boundary layers, uses a cut-off-based coupling, and introduces a divergence-free correction to satisfy incompressibility. Key contributions include establishing how boundary geometry, quantified through Gaussian curvature bounds and curvature-related matrices, shapes the near-boundary flow and yields a limiting 2D damped Euler-type system for the horizontal velocity, while providing explicit, exponentially decaying boundary-layer profiles. The analysis removes the small-boundary-amplitude restriction and recovers classical flat-boundary results in the planar limit, with potential implications for atmospheric and oceanic geophysical flows over complex terrains. Overall, the paper delivers rigorous, high-precision error control for Ekman layers on curved domains, under explicit geometric conditions and well-prepared initial data, broadening the applicability of boundary-layer theory to realistic geophysical geometries.

Abstract

The Ekman boundary layer is a fundamental concept in fluid dynamics that describes fluid motion near boundaries affected by Earth's rotation. Most theoretical studies have simplified their analysis by assuming a planar boundary surface, resulting in limited exploration of structures with general smooth boundary conditions. Investigating the impact of boundary geometry in the Ekman boundary layer is essential, as initially suggested by J.L. Lions and further examined in Masmoudi's study [Comm. Pure Appl. Math. 53 (2000), 432-483] under small amplitude periodic boundary conditions. This paper clarifies how boundary geometry influences flow fields and characterizes its effects on near-boundary layer flow. We construct a class of multi-scale approximate solutions based on the boundary's geometric features and establish their convergence in the L^{\infty} framework. Our findings do not require a small-amplitude assumption, only an upper bound on the Gaussian curvature of the boundary surface. Notably, when the boundary is planar, our approach aligns with existing studies. Additionally, in the vanishing-viscosity limit, we derive a limiting-state system dependent on boundary geometric parameters. These contributions extend the theoretical understanding of boundary-layer interactions to general curved geometries and have possible applications in atmospheric, oceanic, and other geophysical flow contexts.

Paper Structure

This paper contains 17 sections, 15 theorems, 171 equations.

Table of Contents

  1. Introduction
  2. Notations
  3. Main results
  4. Preliminaries
  5. Construction of the approximate solution
  6. Calculation of boundary coordinates
  7. Asymptotic analysis
  8. Construction of ($\boldsymbol{u}^{I,0}, \boldsymbol{u}^{B,0}, \boldsymbol{u}^{T,0}$) and (${p}^{I,0}, {p}^{B,0}, {p}^{T,0},{p}^{B,1},{p}^{T,1}$)
  9. Construction of ($\boldsymbol{u}^{I,1}, \boldsymbol{u}^{B,1}, \boldsymbol{u}^{T,1}$) and (${p}^{I,1}, {p}^{B,2}, {p}^{T,2}$)
  10. Construction of ($\boldsymbol{u}^{I,2}, \boldsymbol{u}^{B,2}, \boldsymbol{u}^{T,2}$) and (${p}^{I,2}, {p}^{B,3}, {p}^{T,3}$)
  11. Modification of incompressible condition
  12. Estimates of the approximate solution \ref{['2.42']}
  13. $L^\infty$ Error estimation
  14. Estimations of approximate solution $\boldsymbol{u}_{app}^\varepsilon$
  15. $L^2$ estimation for $\boldsymbol{v}^\varepsilon$
  16. ...and 2 more sections

Key Result

Theorem 1

Let $\Omega=\mathbb{T}^2\times[0,1]$, $(\boldsymbol{u}^\varepsilon,{p}^\varepsilon)$ be a pair of global weak solutions of system 1.1-1.3 with initial data $\boldsymbol{u}_0^\varepsilon$. Assume $\boldsymbol{u}_0^\varepsilon$ strongly converges to ${\boldsymbol{u}}_0 (x,y)= ({\boldsymbol{u}}_{0,h},

Theorems & Definitions (31)

  • Theorem 1: Grenier and Masmoudi Grenier1997
  • Remark 1
  • Theorem 2: Gong, Guo, and Wang Gong2015
  • Remark 2
  • Theorem 3
  • Remark 3
  • Theorem 4
  • Remark 4
  • Lemma 1: Grenier and Masmoudi Grenier1997
  • Lemma 2
  • ...and 21 more