The structure and spectrum of inertial waves of an incompressible viscous fluid inside a spherical shell are investigated numerically. These modes appear to be strongly featured by a web of rays which reflect on the boundaries. Kinetic energy and dissipation are indeed concentrated on thin conical sheets, the meridional cross-section of which forms the web of rays. The thickness of the rays is in general independent of the Ekman number E but a few cases show a scaling with E1/4 and statistical properties of eigenvalues indicate that high-wavenumber modes have rays of width O(E1/3). Such scalings are typical of Stewartson shear layers. It is also shown that the web of rays depends on the Ekman number and shows bifurcations as this number is decreased.This behaviour also implies that eigenvalues do not evolve smoothly with viscosity. We infer that only the statistical distribution of eigenvalues may follow some simple rules in the asymptotic limit of zero viscosity.

https://www.cambridge.org/core/services/aop-cambridge-core/content/view/B4D54DF6E30B5D3DA9046F1858F96800/S0022112097005491a.pdf/div-class-title-inertial-waves-in-a-rotating-spherical-shell-div.pdf

Inertial waves in a rotating spherical shell

Pattern selection with O(3) symmetry

Steady-State bifurcation with 0(3)-Symmetry

The analysis by Busse (1975) of preferred patterns of convection in spherical shells is extended to include the case of odd degrees l of spherical harmonics. In the general part of the paper only the property of spherical symmetry of the basic state is used. The results are thus applicable to all bifurcation problems with spherical symmetry. Except in the case 1 = 1 a pattern degeneracy of the linear problem exists, which is partly removed by the solvability conditions that are generated when nonlinear terms are taken into account as perturbations. In each of the cases 1 considered so far, at least 1 physically different solutions have been found. The preferred solution among I existing ones is determined for 1≥2 by a stability analysis. In the case l = 3 emphasized in this paper the axisymmetric solution is found to be always unstable, and the solution of tetrahedronal symmetry appears to be generally preferred. The latter result is rigorously established in the special case of a thin layer with nearly insulating boundaries treated in the second part of the paper.

https://www.cambridge.org/core/services/aop-cambridge-core/content/view/S0022112082003061

Patterns of convection in spherical shells. Part 2

Presents a broadly based discussion of 'catastrophe theory,' a mathematical discipline commonly associated with the names of Thom and Zeeman, placing emphasis on the development feedback between the mathematics and its applications, especially to the physical sciences. The author aims to present a typical selection of current work. Among the more prominent of the concepts that have emerged from this work are co-dimension, determinacy, unfoldings and organising centres. He shows, using specific applications as motivation, how these concepts may be used, and generalised to areas not obviously within the formal purview of 'catastrophe theory' as it is often presented. Structural stability, a concept from topological dynamics that provides a philosophical background for catastrophe theory, is also discussed. The mathematics of the subject has advanced considerably over the past decade, and in doing so has lost many of its original limitations. Some of these new directions are exhibited. On occasion, the catastrophe-theoretic methods are compared to more traditional ones.

https://iopscience.iop.org/article/10.1088/0034-4885/45/2/002/pdf

Catastrophe theory in physics

In this paper bifurcation theory and group theoretical methods are applied to the analysis of the stationary convection of a fluid filling a spherical shell which is rotating (or not rotating) about an axis with a constant angular speed. In the case with rotation, an analytical relation is found between the Rayleigh number and the Taylor number, for which a transcritical branch of stationary and axisymmetric (about the axis of rotation) solutions occur. At fixed Taylor number, these solutions are stable supercritically. When the shell doesn’t rotate, a two-parameter family of axisymmetric solutions is found to bifurcate supercritically, these solutions being deduced one from the other by a simple rotation. Under an assumption on the sign of a certain coefficient, these solutions are “orbitally stable.”

Pascal Chossat

Papers

Bifurcation and Stability of Convective Flows in a Rotating or not Rotating Spherical Shell