Torsional solutions of convection in rotating fluid spheres

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.Postprint (published version

Periodic orbits in tall laterally heated rectangular cavities

This study elucidates the origin of the multiplicity of stable oscillatory flows detected by time integration in tall rectangular cavities heated from the side. By using continuation techniques for periodic orbits, it is shown that initially unstable branches, arising at Hopf bifurcations of the basic steady flow, become stable after crossing Neimark-Sacker points. There are no saddle-node or pitchfork bifurcations of periodic orbits, which could have been alternative mechanisms of stabilization. According to the symmetries of the system, the orbits are either fixed cycles, which retain at any time the center symmetry of the steady flow, or symmetric cycles involving a time shift in the global invariance of the orbit. The bifurcation points along the branches of periodic flows are determined. By using time integrations, with unstable periodic solutions as initial conditions, we determine which of the bifurcations at the limits of the intervals of stable periodic orbits are sub- or supercritical.Postprint (author's final draft

Prandtl number dependence of convective fluids in tall laterally heated slots

The influence of the Prandtl number on the stable stationary and periodic flows of the fluids contained in laterally heated slots under realistic conditions is analyzed by using continuation methods. Namely, non-slip boundaries, insulated top and bottom horizontal limits and perfectly conducting lateral sides are considered. The branches of solutions are computed by decreasing the Prandtl number, for four Rayleigh numbers. The dynamical behavior depends strongly on both parameters. For a Rayleigh number, ${\rm Ra}=10^3$ the steady flow remains stable in the wide range of Prandtl numbers computed. At ${\rm Ra}=10^4$ and ${O(10^5)}$ the first bifurcations are of Hopf type giving rise to a type of oscillations that affect the bulk of the fluid, alternating from a general circulation to multi-vortex solutions, or the boundary layer, respectively. However, it is found that, in any case, the location of the shear determines the type of the oscillations. Moreover, at ${\rm Ra}=10^4$ the critical multipliers at the secondary bifurcations on the main branch of POs are real, giving rise to different kinds of very bounded stable periodic states of different symmetries and periods. At ${\rm Ra}$ of order $10^5$ the instability of the periodic orbits gives rise directly to quasi-periodic flows.Postprint (author's final draft

Effect of Robin boundary conditions on the onset of convective torsional flows in rotating fluid spheres

Torsional flows are preferred at the onset of thermal convection in fluid spheres with stress-free and perfectly conducting boundary conditions, in a narrow region of the parameter space for Prandtl numbers Pr¿0.9 and ratios Pr/Ek=O(10)¿, with Ek being the Ekman number. In this case, the transport of heat to the exterior is supposed instantaneous. When the thermal conductivity of the internal fluid is large, and the external convective heat transfer or radiative emissivity is low, the heat transmission is less efficient, and the thermal energy retained in the interior increases, enhancing the onset of convection. This study is devoted to analyzing the combined influence of the thermal conductivity and external conditions (temperature and resistance to heat transport) on the onset of the torsional convection by taking a Robin boundary condition for the temperature at the surface of the sphere. It is shown, by means of the numerical computation of the curves of simultaneous transitions to torsional flows and Rossby waves, that when the heat flux through the boundary decreases, the region where the axisymmetric flows are preferred shrinks, but it never strangles to an empty set. It has been found that with adequate scalings the curves delimiting the transition to torsional flows, and those of the critical Rayleigh number, Rac¿, and the frequencies of the modes vs Ek become almost independent of the parameter of the Robin boundary condition.Postprint (author's final draft

Three-dimensional quasiperiodic torsional flows in rotating spherical fluids at very low Prandtl numbers

The aim of this study is to determine through numerical simulations the extent and robustness of the three-dimensional torsional dynamics of the thermal convection in rotating spherical fluids at very low Prandtl numbers. It is known that the kinetic energy of the periodic axisymmetric flows propagates latitudinally on the surface of the sphere. Here, it is shown that when the axisymmetry is broken at a secondary Hopf bifurcation, the flow starts to drift in the azimuthal direction giving rise to a quasiperiodic motion that propagates the energy in latitude and longitude. The double direction of propagation gives rise to a meandering path of the kinetic energy, which is still concentrated on the surface, but highly localized. Several new stable states of convection with different symmetries have been identified in a large range of Rayleigh numbers, all of them retaining the torsional motion of the basic velocity field. Particular attention is paid to their dependence on the Rayleigh number and on the values of the frequencies, of the mean zonal flow, and of the kinetic energy of the fluid.Postprint (author's final draft

Generation of bursting magnetic fields by nonperiodic torsional flows

A mechanism for the cyclic generation of bursts of magnetic fields by nonlinear torsional flows of complex time dependence but very simple spatial structure is described. These flows were obtained numerically as axisymmetric solutions of convection in internally heated rotating fluid spheres in the Boussinesq approximation. They behave as repeated transients, which start with nearly periodic oscillations of the velocity field of slowly increasing amplitude. This regime is followed by a chaotic fast increase and a final decrease of the amplitude of, at least, one order of magnitude. The magnetic field decays due to the magnetic diffusion during the regular oscillations, but it grows in the form of bursts during the intervals of irregular time dependence of the velocity. The magnetic field is strongly localized in spirals, with spatial- and temporal-dependent intensity.Postprint (published version

Numerical study of the onset of thermosolutal convection in rotating spherical shells

The influence of an externally enforced compositional gradient on the onset of convection of a mixture of two components in a rotating fluid spherical shell is studied for Ekman numbers E = 10−3 and E = 10−6, Prandtl numbers σ = 0.1, 0.001, Lewis numbers τ = 0.01, 0.1, 0.8, and radius ratio η = 0.35. The Boussinesq approximation of the governing equations is derived by taking the denser component of the mixture for the equation of the concentration. Differential and internal heating, an external compositional gradient, and the Soret and Dufour effects are included in the model. By neglecting these two last effects, and by considering only differential heating, it is found that the critical thermal Rayleigh number Rec depends strongly on the direction of the compositional gradient. The results are compared with those obtained previously for pure fluids of the same σ. The influence of the mixture becomes significant when the compositional Rayleigh number Rc is at least of the same order of magnitude as the known Rec computed without mixture. For positive and sufficiently large compositional gradients, Rec decreases and changes sign, indicating that the compositional convection becomes the main source of instability. Then the critical wave number mc decreases, and the drifting waves slow down drastically giving rise to an almost stationary pattern of convection. Negative gradients delay the onset of convection and determine a substantial increase of mc and ωc for Rc sufficiently high. Potential laws are obtained numerically from the dependence of Rec and of the critical frequency ωc on Rc, for the moderate and small Ekman numbers explored.Postprint (published version

Stability analysis for the onset of convection in rotating fluid binary mixtures in spherical shells

The stability analysis of the basic conductive state of a rotating binary mixture of two fluids bounded by spherical shells is studied. The Boussinesq approximation of the mass conservation, Navier-Stokes, energy and concentration equations is used, and results for a moderate Ekman number E = 10-3 are presented for positive and negative external compositional gradients. Preliminary results are compared with those obtained for a pure fluid in the same range of parameters. They show an important influence of the presence of a mixture on the onset of the convection for solutal Rayleigh numbers Rc at least of the order of the thermal Rayleigh number Re. On the sphere the leading eigenvectors, which give the patterns of convection, have sectorial structure like those of a pure fluid, although slightly deformed due to the presence of two components.Postprint (author's final draft

Azimuthal waves and their stability in externally heated rotating spherical shells

Azimuthal waves appearing in the thermal convection of a pure fluid contained in a spherical shell with both boundaries at different temperatures are studied. They are computed by using continuation methods as steady solutions in the reference system of the wave. There stability is also studied, and the secondary bifurcations to modulated waves are detected.Postprint (author's final draft

Two computational approaches for the simulation of fluid problems in rotating spherical shells

Many geophysical and astrophysical phenomena such as magnetic fields generation, or the differential rotation observed in the atmospheres of the major planets are studied by means of numerical simulations of the Navier-Stokes equations in rotating spherical shells. Two different computational codes, spatially discretized using spherical harmonics in the angular variables, are presented. The first code, PARODY, solves the magneto-hydrodynamic anelastic convective equations with finite a difference discretization in the radial direction. This allows the parallelization on distributed memory computers to run massive numerical simulations of second order in time. It is mainly designed to perform direct numerical simulations. The second code, SPHO, solves the fully spectral Boussinesq convective equations, and its variationals, parallelized on shared memory architectures and it uses optimized linear algebra libraries. High-order time integration methods are implemented to allow the use of dynamical systems tools for the study of complex dynamics.Postprint (published version