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Internal waves in a 2D subcritical channel

Zhenhao Li, Jian Wang, Jared Wunsch

TL;DR

This work analyzes low-frequency internal-wave scattering in a 2D unbounded channel with subcritical bottom. It develops a stationary scattering theory for the hyperbolic operator $P(\lambda)=-\lambda^2\partial_{x_1}^2+(1-\lambda^2)\partial_{x_2}^2$ under Dirichlet boundary conditions, showing that the scattering matrix $\mathbf{S}(\lambda)$ relates incoming to outgoing data up to a smoothing remainder, and is unitary on $\mathring{H}^{-1/2}$. By reducing the problem to boundary transport along characteristics and introducing the bounce map $b_{\lambda}$ (the chess-billiard map), the authors obtain an explicit structure for $\mathbf{S}(\lambda)$ as a pullback by $b_{\lambda}$, plus smoothing corrections. They further prove unique solvability of the inhomogeneous problem with an outgoing radiation condition, and construct the outgoing resolvent $\mathcal{R}(\lambda)$, providing a rigorous foundation for low-frequency ray-trace approximations in ocean-like waveguides. The results offer precise, operator-level insights into wave scattering in subcritical channels and quantify when ray-theory-based intuitions are valid at low wavenumbers.

Abstract

We analyze the scattering of linear internal waves in a two dimensional channel with subcritical bottom topography. We construct the scattering matrix for the internal wave problem in a channel with straight ends, mapping incoming data to outgoing data; this operator turns out to differ by a smoothing operator from the pullback by the ``bounce map'' for boundary data obtained by ray-tracing. As a consequence we obtain unique solvability of the inhomogeneous stationary scattering problem subject to an appropriate outgoing radiation condition.

Internal waves in a 2D subcritical channel

TL;DR

This work analyzes low-frequency internal-wave scattering in a 2D unbounded channel with subcritical bottom. It develops a stationary scattering theory for the hyperbolic operator under Dirichlet boundary conditions, showing that the scattering matrix relates incoming to outgoing data up to a smoothing remainder, and is unitary on . By reducing the problem to boundary transport along characteristics and introducing the bounce map (the chess-billiard map), the authors obtain an explicit structure for as a pullback by , plus smoothing corrections. They further prove unique solvability of the inhomogeneous problem with an outgoing radiation condition, and construct the outgoing resolvent , providing a rigorous foundation for low-frequency ray-trace approximations in ocean-like waveguides. The results offer precise, operator-level insights into wave scattering in subcritical channels and quantify when ray-theory-based intuitions are valid at low wavenumbers.

Abstract

We analyze the scattering of linear internal waves in a two dimensional channel with subcritical bottom topography. We construct the scattering matrix for the internal wave problem in a channel with straight ends, mapping incoming data to outgoing data; this operator turns out to differ by a smoothing operator from the pullback by the ``bounce map'' for boundary data obtained by ray-tracing. As a consequence we obtain unique solvability of the inhomogeneous stationary scattering problem subject to an appropriate outgoing radiation condition.

Paper Structure

This paper contains 8 sections, 9 theorems, 100 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Relation to the oceanographic literature
  3. Some Sobolev spaces
  4. Characteristics and Cauchy data
  5. Reduction to the boundary
  6. Scattering matrix
  7. Outgoing resolvent
  8. Outgoing solutions

Key Result

Theorem 1.1

Suppose $\Omega$ is subcritical for $\lambda\in (0,1)$. Then for any $q^i\in \mathring L^2(\mathbb T_{\lambda})$, there exist unique $q^o\in \mathring L^2(\mathbb T_{\lambda})$ and $u\in \dot H^1_{\mathrm{loc}}(\Omega)$ such that The resulting map is called the scattering matrix for $P(\lambda)$ in $\Omega$. Moreover, there exists a smoothing operator $R: \mathcal{D}'(\mathbb T_{\lambda}; T^*\ma

Figures (3)

  • Figure 1: Scattering of a low-frequency incoming wave (traveling from left to right in both figures) by smooth bottom bumps (black). Colors represent the velocity of the internal waves. Left: the topography is subcritical (see Definition \ref{['def:subcritical']} below). Right: the topography is supercritical. Figure from MaCaPe:14 (reproduced with permission).
  • Figure 2: Diagram of $\Omega$. Level lines of $\ell^+_\lambda$ are in red and level lines of $\ell^-_\lambda$ are in blue. For a point $\theta \in \partial \Omega_\uparrow$, the location of $\gamma^\pm(\theta)$ and $b_\lambda(\theta)$ are indicated. A choice of fundamental intervals $\mathcal{J}_{\mathrm{L}}$ and $\mathcal{J}_{\mathrm{R}}$ is also labeled.
  • Figure 3: Domain $\Omega$ in $y_\pm$ coordinates. The support of the forcing profile $f$ is shaded and labeled, and the rectangular shaded region is the integration kernel in \ref{['e-u0']}.

Theorems & Definitions (20)

  • Definition 1.1
  • Definition 1.2
  • Theorem 1.1
  • Theorem 1.2
  • Lemma 2.1
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • Remark 2.1
  • proof
  • ...and 10 more