Bistable flows in precessing spheroids

TL;DR

This paper derives analytical solutions for bistable precession-driven flows in spheroids, explores their stability and parameter ranges, and introduces the possibility of time-dependent multiple flows in triaxial ellipsoids.

Contribution

It provides the first analytical description of multiple coexisting solutions and their stability in precessing spheroids, extending previous models and comparing with classical theory.

Findings

Multiple flow solutions can coexist under certain parameters.

Analytical stability criteria for these solutions are established.

Time-dependent multiple flows are possible in triaxial ellipsoids.

Abstract

Precession driven flows are found in any rotating container filled with liquid, when the rotation axis itself rotates about a secondary axis that is fixed in an inertial frame of reference. Because of its relevance for planetary fluid layers, many works consider spheroidal containers, where the uniform vorticity component of the bulk flow is reliably given by the well-known equations obtained by Busse in 1968. So far however, no analytical result on the solutions is available. Moreover, the cases where multiple flows can coexist have not been investigated in details since their discovery by Noir et al. (2003). In this work, we aim at deriving analytical results on the solutions, aiming in particular at, first estimating the ranges of parameters where multiple solutions exist, and second studying quantitatively their stability. Using the models recently proposed by Noir \& C{\'e}bron (2013), which are more generic in the inviscid limit than the equations of Busse, we analytically describe these solutions, their conditions of existence, and their stability in a systematic manner. We then successfully compare these analytical results with the theory of Busse (1968). Dynamical model equations are finally proposed to investigate the stability of the solutions, which allows to describe the bifurcation of the unstable flow solution. We also report for the first time the possibility that time-dependent multiple flows can coexist in precessing triaxial ellipsoids. Numerical integrations of the algebraic and differential equations have been efficiently performed with the dedicated script FLIPPER (supplementary material).