Completeness of Inertial Modes of an Incompressible Non-Viscous Fluid in a Corotating Ellipsoid

TL;DR

This paper mathematically analyzes inertial modes in a rotating incompressible fluid within a rigid ellipsoid, proving their completeness and spectral properties, and clarifying the structure of solutions in this hyperbolic boundary-value problem.

Contribution

It formulates the Poincaré problem in a Hilbert space, showing the boundedness and self-adjointness of the Poincaré operator, and proves the completeness of inertial modes in an ellipsoidal domain.

Findings

The Poincaré operator is bounded and self-adjoint in the Hilbert space of square-integrable functions.

Inertial modes form a complete basis of polynomial velocity fields in an ellipsoid.

For axisymmetric ellipsoids, the inertial modes include Bryan's modes and geostrophic modes.

Abstract

Inertial modes are the eigenmodes of contained rotating fluids restored by the Coriolis force. When the fluid is incompressible, inviscid and contained in a rigid container, these modes satisfy Poincar\'e's equation that has the peculiarity of being hyperbolic with boundary conditions. Inertial modes are therefore solutions of an ill-posed boundary-value problem. In this paper we investigate the mathematical side of this problem. We first show that the Poincar\'e problem can be formulated in the Hilbert space of square-integrable functions, with no hypothesis on the continuity or the differentiability of velocity fields. We observe that with this formulation, the Poincar\'e operator is bounded and self-adjoint and as such, its spectrum is the union of the point spectrum (the set of eigenvalues) and the continuous spectrum only. When the fluid volume is an ellipsoid, we show that the inertial modes form a complete base of polynomial velocity fields for the square-integrable velocity fields defined over the ellipsoid and meeting the boundary conditions. If the ellipsoid is axisymmetric then the base can be identified with the set of Poincar\'e modes, first obtained by Bryan (1889), and completed with the geostrophic modes.