Critical torsional modes of convection in rotating fluid spheres at high Taylor numbers

Sánchez, J.; García, F.; Net, Marta

doi:10.1017/jfm.2016.52

Cited by 24 publications

(32 citation statements)

References 24 publications

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“…Then the behavior is very close to that of the eigenfunction at the onset of convection (see Ref. [17]), with an almost complete exchange of kinetic energy between the toroidal and poloidal components because they are a quarter period out of phase, their mean value and amplitude are the same, and the minimum is almost zero. This is why the two curves represented in each plot of Fig.…”

Section: Results: Periodic Solutionssupporting

confidence: 62%

“…In the case of shells, it was found that at low Ekman numbers (E) it is possible to find nonaxisymmetric and equatorially antisymmetric modes at the beginning of convection for the very small Prandtl numbers (Pr < 0.01) of the liquid metals [16]. Moreover, preferred torsional modes, i.e., axisymmetric and equatorially antisymmetric modes, were found numerically in rotating fluid spheres for Pr 0.01 at low E [17]. This latter result was confirmed analytically by using asymptotic methods in Ref.…”

Section: Introductionmentioning

confidence: 96%

“…The pairs of parameters used fulfill the relation Pr/E = O (10) that, according to Refs. [17] and [18], ensures having periodic torsional solutions at the onset of convection with stress-free boundary conditions. The latter authors also proved that, for this ratio, the torsional modes are never preferred with nonslip boundary conditions.…”

Section: Introductionmentioning

confidence: 99%

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Torsional solutions of convection in rotating fluid spheres

Umbría

Net

2019

Phys. Rev. Fluids

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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.

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Section: Results: Periodic Solutionssupporting

confidence: 62%

Section: Introductionmentioning

confidence: 96%

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Torsional solutions of convection in rotating fluid spheres

Umbría

Net

2019

Phys. Rev. Fluids

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“…The main characteristic is that the patterns of convection depend on the value of σ , and the preferred eigenfunctions are even axisymmetric in the σ → 0 limit [60].…”

Section: Results On the Stability Analysismentioning

confidence: 99%

Radial collocation methods for the onset of convection in rotating spheres

Sánchez

García

Net

2016

Journal of Computational Physics

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The viability of using collocation methods in radius and spherical harmonics in the angular variables to calculate convective flows in full spherical geometry is examined. As a test problem the stability of the conductive state of a self-gravitating fluid sphere subject to rotation and internal heating is considered. A study of the behavior of different radial meshes previously used by several authors in polar coordinates, including or not the origin, is first performed. The presence of spurious modes due to the treatment of the singularity at the origin, to the spherical harmonics truncation, and to the initialization of the eigenvalue solver is shown, and ways to eliminate them are presented. Finally, to show the usefulness of the method, the neutral stability curves at very high Taylor and moderate and small Prandtl numbers are calculated and shown.

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“…In this paper, we vary Pr between [10 −2 , 10 −1 ]. Note that for small Prandtl numbers, convection cells attached to the equator of the outer sphere are preferred to thermal Rossby waves at the linear onset of convection when the Ekman number is moderately small (Zhang & Busse, 1987;Busse & Simitev, 2004;Plaut & Busse, 2005;Sánchez et al, 2016). In the limit Ek 1 and Ek /Pr 1, the critical Rayleigh number at the onset of these equatorially-attached modes scales as Ek −2 , while the critical Rayleigh number at the onset of the thermal Rossby waves scales as Ek −4/3 (Busse & Simitev, 2004).…”

Section: Introductionmentioning

confidence: 99%

Subcritical convection of liquid metals in a rotating sphere using a quasi-geostrophic model

Guervilly

Cardin

2016

J. Fluid Mech.

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We study nonlinear convection in a rapidly rotating sphere with internal heating for values of the Prandtl number relevant for liquid metals (Pr ∈ [10 −2 , 10 −1 ]). We use a numerical model based on the quasi-geostrophic approximation, in which variations of the axial vorticity along the rotation axis are neglected, whereas the temperature field is fully three-dimensional. We identify two separate branches of convection close to onset: (i) a well-known weak branch for Ekman numbers greater than 10 −6 , which is continuous at the onset (supercritical bifurcation) and consists of thermal Rossby waves, and (ii) a novel strong branch at lower Ekman numbers, which is discontinuous at the onset. The strong branch becomes subcritical for Ekman numbers of the order of 10 −8 . On the strong branch, the Reynolds number of the flow is greater than 10 3 , and a strong zonal flow with multiple jets develops, even close to the nonlinear onset of convection. We find that the subcriticality is amplified by decreasing the Prandtl number. The two branches can co-exist for intermediate Ekman numbers, leading to hysteresis (Ek = 10 −6 , Pr = 10 −2 ). Non-linear oscillations are observed near the onset of convection for Ek = 10 −7 and Pr = 10 −1 .

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Critical torsional modes of convection in rotating fluid spheres at high Taylor numbers

Cited by 24 publications

References 24 publications

Torsional solutions of convection in rotating fluid spheres

Torsional solutions of convection in rotating fluid spheres

Radial collocation methods for the onset of convection in rotating spheres

Subcritical convection of liquid metals in a rotating sphere using a quasi-geostrophic model

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