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On the boundary layer arising from fast internal waves dynamics

Rupert Klein, Xin Liu

TL;DR

This work analyzes boundary-layer dynamics arising from fast internal waves in the Boussinesq system with a Brunt–Vaisala frequency scaled as $1/\varepsilon$. By performing a boundary-layer expansion and employing an $\iota$-approximation with frozen transport, the authors derive an inviscid boundary-layer model near $z=0$ and prove a local-in-time well-posedness result for the nonlinear boundary-layer system in spaces of analytic functions, with exponential decay in the stretched vertical coordinate $\eta$. The analysis combines uniform-in-$\iota$ $H^s$ estimates for the approximating problems and a contraction mapping in analytic spaces to obtain existence and uniqueness, while clarifying the absence of feedback from the boundary layer to the bulk in this framework. The results provide a rigorous foundation for near-boundary fast internal-wave dynamics and set the stage for future multi-scale couplings between bulk and boundary-layer processes in stratified geophysical flows.

Abstract

In this paper, we investigate the boundary layer arising from the fast internal waves in the Boussinesq equations with the Brunt-Vaisälä frequency of order $ \mathcal O(1/\varepsilon) $. For the homogeneous-in-height stratification, previous work by \emph{Desjardins, Lannes, Saut, 3(1):153--192, Water Waves, 2021} establishes uniform-in-$ε$ estimates locally in time, with additional constraints on the boundary data initially, which essentially restricts the dynamics in the spatially periodic domain. Removing such constraints, our goal is to investigate the general near-boundary behavior. We observe that the fast internal waves will give rise to large growth of the spatial derivatives in the normal direction of the solutions in the vicinity of the boundary. To capture this phenomenon, we introduce an inviscid boundary layer using a natural scaling. In addition, we investigate the well-posedness of such a boundary layer system in the space of analytic functions.

On the boundary layer arising from fast internal waves dynamics

TL;DR

This work analyzes boundary-layer dynamics arising from fast internal waves in the Boussinesq system with a Brunt–Vaisala frequency scaled as . By performing a boundary-layer expansion and employing an -approximation with frozen transport, the authors derive an inviscid boundary-layer model near and prove a local-in-time well-posedness result for the nonlinear boundary-layer system in spaces of analytic functions, with exponential decay in the stretched vertical coordinate . The analysis combines uniform-in- estimates for the approximating problems and a contraction mapping in analytic spaces to obtain existence and uniqueness, while clarifying the absence of feedback from the boundary layer to the bulk in this framework. The results provide a rigorous foundation for near-boundary fast internal-wave dynamics and set the stage for future multi-scale couplings between bulk and boundary-layer processes in stratified geophysical flows.

Abstract

In this paper, we investigate the boundary layer arising from the fast internal waves in the Boussinesq equations with the Brunt-Vaisälä frequency of order . For the homogeneous-in-height stratification, previous work by \emph{Desjardins, Lannes, Saut, 3(1):153--192, Water Waves, 2021} establishes uniform-in- estimates locally in time, with additional constraints on the boundary data initially, which essentially restricts the dynamics in the spatially periodic domain. Removing such constraints, our goal is to investigate the general near-boundary behavior. We observe that the fast internal waves will give rise to large growth of the spatial derivatives in the normal direction of the solutions in the vicinity of the boundary. To capture this phenomenon, we introduce an inviscid boundary layer using a natural scaling. In addition, we investigate the well-posedness of such a boundary layer system in the space of analytic functions.
Paper Structure (12 sections, 2 theorems, 123 equations)

This paper contains 12 sections, 2 theorems, 123 equations.

Table of Contents

  1. Internal waves and the Boussinesq equations
  2. Motivation and linear analysis
  3. Linear system and motivation
  4. Linear boundary layer equations
  5. Some physical remarks
  6. Boundary layer expansion: Formulation of nonlinear equations
  7. Well-posedness theory of the boundary layer equations, i.e., \ref{['sys:boundary_eqs']}
  8. The $\iota$-approximating system with frozen transport speed and uniform-in-$\iota$ estimates
  9. The contraction mapping
  10. Preliminaries and the main theorem
  11. Boundedness estimate
  12. Contracting and the fixed point

Key Result

Theorem 1.1

As $\varepsilon \rightarrow 0^+$, there exists a boundary layer dynamics of system sys:boussinesq for general initial data without the assumption of hp:lannes on the slow time scale, i.e., for $t = O(1)$. Under the requirement that the boundary layer flow has no feedback to the internal flow, the bo

Theorems & Definitions (3)

  • Theorem 1.1
  • Remark 1
  • Theorem 4.1