Torsional solutions of convection in rotating fluid spheres

Abstract

A numerical study of the nonlinear torsional solutions of convection in rotating, internally heated, self-gravitating fluid spheres is presented. Their dependence on the Rayleigh number has been found for two pairs of Ekman (E) and small Prandtl (Pr) numbers in the region of parameters where, according to Zhang et al. [J. Fluid Mech. 813, R2 (2017)], the linear stability of the conduction state predicts that they can be preferred at the onset of convection. The bifurcation to periodic torsional solutions is supercritical for sufficiently small Pr. They are not rotating waves, unlike the nonaxisymmetric case. Therefore they have been computed by using continuation methods for periodic orbits. Their stability with respect to axisymmetric perturbations and physical characteristics have been analyzed. It was found that the time-and space-averaged equatorially antisymmetric part of the kinetic energy of the stable orbits splits into equal poloidal and toroidal parts, while the symmetric part is much smaller. Direct numerical simulations for E = 10 −4 at higher Rayleigh numbers (Ra) show that this trend is also valid for the nonperiodic flows and that the mean values of the energies remain almost constant with Ra. However, the modulated oscillations bifurcated from the quasiperiodic torsional solutions reach a high amplitude, compared with that of the periodic, increasing slowly and decaying very fast. This repeated behavior is interpreted as trajectories near heteroclinic chains connecting unstable periodic solutions. The torsional flows give rise to a meridional propagation of the kinetic energy near the outer surface and an axial oscillation of the hot nucleus of the metallic fluid sphere.

Generation of bursting magnetic fields by nonperiodic torsional flows

“…The magnetic field is split into toroidal, G, and poloidal, H, components, and treated numerically as the velocity field v(r, θ, t ) = ∇ × ( (r, θ, t )r) + ∇ × ∇ × ( (r, θ, t )r) in Ref. [36], where the physical and dynamical properties of v were studied. Then…”

“…Since stress-free boundary conditions are applied, the conservation of the z component of the angular momentum is achieved in the way explained in Sánchez Umbría and Net [36]. At r = 0 only regularity conditions are required.…”

“…This paper is mainly devoted to solving a fundamental problem of magnetohydrodynamics (MHD): to find out if nearly heteroclinic cycles with these latter symmetry properties are able to generate and sustain magnetic fields by a purely kinematic action. The flows were found by DNSs as solutions of the axisymmetric Navier-Stokes equation for an internally heated rotating fluid sphere with stress-free conditions at the boundary [36].…”

Weak branch and multimodal convection in rapidly rotating spheres at low Prandtl number

“…However, in this regime weakly energetic nonlinear flows (weak branch of [32]), which include nonaxisymmetric RWs (steadily drifting flows in the azimuthal direction), have not been found although they were predicted by the linear theory [39]. The situation is different in the case of low Pr and stress-free boundaries [40], because the first convective instability is axisymmetric, i.e periodic torsional oscillations develop at the onset. That study revealed a rich dynamical regime including bifurcations to quasiperiodic flows and solutions in which the amplitude is slowly increasing and rapidly decaying, repeatedly.…”

The life and fate of a bubble in a geometrically perturbed Hele-Shaw channel

“…Recent examples include studies of turbulence (Budanur et al. 2017), convection (Sánchez Umbría & Net 2019) and droplet breakup (Gallino, Schneider & Gallaire 2018).…”