Continuation and stability of convective modulated rotating waves in spherical shells

Abstract

Modulated rotating waves (MRW), bifurcated from the thermal-Rossby waves that arise at the onset of convection of a fluid contained in a rotating spherical shell, and their stability, are studied. For this purpose, Newton-Krylov continuation techniques are applied. Nonslip boundary conditions, an Ekman number E = 10 −4 , and a low Prandtl number fluid Pr = 0.1 in a moderately thick shell of radius ratio η = 0.35, differentially heated, are considered. The MRW are obtained as periodic orbits by rewriting the equations of motion in the rotating frame of reference where the rotating waves become steady states. Newton-Krylov continuation allows us to obtain unstable MRW that cannot be found by using only time integrations, and identify regions of multistability. For instance, unstable MRW without any azimuthal symmetry have been computed. It is shown how they become stable in a small Rayleigh-number interval, in which two branches of traveling waves are also stable. The study of the stability of the MRW helps to locate and classify the large sequence of bifurcations, which takes place in the range analyzed. In particular, tertiary Hopf bifurcations giving rise to three-frequency stable solutions are accurately determined.

Continuation and stability of rotating waves in the magnetized spherical Couette system: secondary transitions and multistability

“…Bifurcation and continuation methods have been successfully applied during the last years to a great variety of problems in the fluid dynamics context [27,28]. Computations based on continuation of periodic orbits of nontrivial time dependence [29,30] and even tori [31] or other invariant objects [32] have provided useful information to clarify the dynamics.…”

Radial collocation methods for the onset of convection in rotating spheres

“…Most sources of instability of the time stepping algorithms can be detected and corrected by looking at the linearized problem, which separates into one for each azimuthal wave number. It is known that collocation methods usually give rise to worst conditioned systems than Galerkin or Petrov-Galerkin methods [11] but, if one has in mind fully tridimensional time evolutions or to apply continuation techniques like in [39,40], it is not expected that more than a few hundreds of radial collocation points will be used. Otherwise the total size of the discretization would be prohibitive.…”

Modulated Rotating Waves in the Magnetised Spherical Couette System

“…This sequence of bifurcations -basic steady state, RW 3 3 , quasiperiodic with two frequencies MRW 3 1 and quasiperiodic with three frequencies MRW 3 1 -seen at low Ha for the radial jet instability with m max = 3 corresponds to the Ruelle-Takens scenario [35,8] which is typical for systems with symmetry [33,16]. Figure 4 We note that an accurate computation of bifurcation points to MRW with 3 frequencies in SO(2) systems could be done efficiently as in [12] by considering MRW as periodic orbits.…”