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Characterization of the spectra of rotating truncated gas planets and inertia-gravity modes

Maarten V. de Hoop, Sean Holman, Alexei Iantchenko

Abstract

We study the essential spectrum, which corresponds to inertia-gravity modes, of the system of equations governing a rotating and self-gravitating gas planet. With certain boundary conditions, we rigorously and precisely characterize the essential spectrum and show how it splits from the portion of the spectrum corresponding to the acoustic modes. The fundamental mathematical tools in our analysis are a generalization of the Helmholtz decomposition and the Lopantinskii conditions.

Characterization of the spectra of rotating truncated gas planets and inertia-gravity modes

Abstract

We study the essential spectrum, which corresponds to inertia-gravity modes, of the system of equations governing a rotating and self-gravitating gas planet. With certain boundary conditions, we rigorously and precisely characterize the essential spectrum and show how it splits from the portion of the spectrum corresponding to the acoustic modes. The fundamental mathematical tools in our analysis are a generalization of the Helmholtz decomposition and the Lopantinskii conditions.

Paper Structure

This paper contains 11 sections, 14 theorems, 230 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Acousto-gravitational system of equations and well-posedness
  3. Decompositions of Hilbert space and spectrum
  4. Riesz projectors and acoustic mode decomposition
  5. Inertia-gravity modes and essential spectrum
  6. Full spectrum bound
  7. Discussion
  8. Acknowledgements
  9. A study of $\operatorname{Ker}(A_2)$
  10. Non-rotating planet
  11. Rotating planet

Key Result

Lemma 1

Suppose that $M$ is compact with smooth boundary $\partial M$, $c$, $g_0' \in C(M)$, $\rho_0 \in C^1(M)$, $c^2> 0$ on $M$, $g_0'$ and $\nabla \rho_0$ are parallel in $M$, $g_0'$ and $n$ are parallel on $\partial M$, and $g_0'\cdot n < 0$ on $\partial M$. Then $a_2$ defined by eq:a2 is a continuous s for all $u \in H_{\mathop{\mathrm{Div}}\nolimits}(M,L^2(\partial M))$.

Figures (2)

  • Figure 1: The solid black regions give the set \ref{['eq:spt']} for fixed $x \in M$; in (a) on the left we see the case when $N^2\geq 0$ (note that this region includes the origin as indicated by the black dot), while in (b) on the right we see the case $N^2<0$. In reference to Theorem \ref{['thm:pp']}, the solid black regions are the areas where ellipticity fails at the point $x$, and appear for a single $x \in M$ in the union on the top line of \ref{['eq: SpectrumCondition']}. The dashed black region on the left is the set where the system \ref{['eq:bigsys']} fails the Lopatinskii condition at the boundary but not in the interior, and appears for a single $x \in \partial M$ in the union on the second line of \ref{['eq: SpectrumCondition']}. Note that in (a) it is possible for the dashed set to intersect the solid set.
  • Figure 2: An illustration of the spectrum $\sigma(L)$ after Rogister and Valette RogisterValette:2009. The dark cross is the set $\mathfrak{S}_1$ which by Theorem \ref{['thm:pp']}, Lemma \ref{['lem:pt']} and the discussion after the proof of that lemma, we have shown must contain the essential spectrum $\sigma_{ess}(L)$, but may in general be larger than the essential spectrum. By Proposition \ref{['prop:DS']}, the full spectrum $\sigma(L)$ is contained in the union of the imaginary axis and region surrounded by the dashed curve. The crosses on the imaginary axis are included to indicate eigenvalues, which could also occur within the dashed curve. The crosses which appear outside of the essential spectrum are red, indicating that they are part of $\sigma(S_1)$ which is the acoustic part of the spectrum.

Theorems & Definitions (32)

  • Lemma 1
  • Remark 1
  • proof
  • Corollary 1
  • Corollary 2
  • Definition 1
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • ...and 22 more