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On gravito-inertial surface waves

Yves Colin de Verdìère, Jérémie Vidal

Abstract

In geophysical environments, wave motions that are shaped by the action of gravity and global rotation bear the name of gravito-inertial waves. We present a geometrical description of gravito-inertial surface waves, which are low-frequency waves existing in the presence of a solid boundary. We consider an idealized fluid model for an incompressible fluid enclosed in a smooth compact three-dimensional domain, subject to a constant rotation vector. The fluid is also stratified in density under a constant Brunt-V{ä}is{ä}l{ä} frequency. The spectral problem is formulated in terms of the pressure, which satisfies a Poincaré equation within the domain, and a Kelvin equation on the boundary. The Poincaré equation is elliptic when the wave frequency is small enough, such that we can use the Dirichlet-to-Neumann operator to reduce the Kelvin equation to a pseudo-differential equation on the boundary. We find that the wave energy is concentrated on the boundary for large covectors, and can exhibit surface wave attractors for generic domains. In an ellipsoid, we show that these waves are square-integrable and reduce to spherical harmonics on the boundary.

On gravito-inertial surface waves

Abstract

In geophysical environments, wave motions that are shaped by the action of gravity and global rotation bear the name of gravito-inertial waves. We present a geometrical description of gravito-inertial surface waves, which are low-frequency waves existing in the presence of a solid boundary. We consider an idealized fluid model for an incompressible fluid enclosed in a smooth compact three-dimensional domain, subject to a constant rotation vector. The fluid is also stratified in density under a constant Brunt-V{ä}is{ä}l{ä} frequency. The spectral problem is formulated in terms of the pressure, which satisfies a Poincaré equation within the domain, and a Kelvin equation on the boundary. The Poincaré equation is elliptic when the wave frequency is small enough, such that we can use the Dirichlet-to-Neumann operator to reduce the Kelvin equation to a pseudo-differential equation on the boundary. We find that the wave energy is concentrated on the boundary for large covectors, and can exhibit surface wave attractors for generic domains. In an ellipsoid, we show that these waves are square-integrable and reduce to spherical harmonics on the boundary.
Paper Structure (18 sections, 8 theorems, 50 equations, 3 figures)

This paper contains 18 sections, 8 theorems, 50 equations, 3 figures.

Table of Contents

  1. Introduction
  2. The Poincaré operator
  3. Equations for the boundary value of the pressure
  4. Poincaré equation
  5. Kelvin equation
  6. Computing the Kelvin equation
  7. Explicit computations
  8. Vertical stratification and rotation
  9. Stratification parallel to the boundary
  10. Stratification orthogonal to the boundary
  11. Dynamics of waves
  12. The case of ellipsoids
  13. Spherical harmonics
  14. Liouville integrability
  15. Perspectives for future work
  16. ...and 3 more sections

Key Result

Theorem 2.1

The Poincaré operator $\mathcal{P}$ is a bounded self-adjoint operator in $L^2 (D, \mathbb{C}^4)$, and its spectrum $\sigma(\mathcal{P})$ is given by

Figures (3)

  • Figure 1: Non-ellipticity of the Kelvin equation for different cases. Left: Aligned case with $\overrightarrow{\Omega} \propto \overrightarrow{g}$. The Kelvin equation is non-elliptic near the equator when $\cos \phi \leq \omega/N$ (see §\ref{['ss:vertrot']}). Right: The Kelvin equation is elliptic when the tangent plane (gray) is parallel to gravity (see §\ref{['ss:strat-parallel']}), whereas it is non-elliptic when the tangent plane is orthogonal to gravity (see §\ref{['ss:strat-orth']}).
  • Figure 2: Number of eigenvalues in the interval $0<|\omega|< \omega_-$ for polynomial eigenvectors of degree less than $n$. Aligned rotation and gravity (as considered in §\ref{['ss:vertrot']}) when $\partial D$ is a sphere. For every degree $n$, the number of eigenvalues is bounded by the number $2n+3$ of spherical harmonics of degree $n+1$. Numerical calculations following the method presented in vidal2024igw.
  • Figure 3: Discrete and essential spectra defined in Theorem \ref{['theo:specelip']}.

Theorems & Definitions (17)

  • Theorem 2.1
  • proof
  • Theorem 3.1
  • proof
  • Lemma 7.1
  • proof
  • Lemma 7.2
  • proof
  • Theorem 7.3
  • proof
  • ...and 7 more