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 $\mathrm{Pr}$. 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 $\mathrm{E}={10}^{-4}$ 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 $\mathrm{Ra}$. 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.

Figure 1

Period along the branches of periodic orbits versus the Rayleigh number. Solid lines mean stable solutions and dashed lines unstable.

Figure 2

(a, b) Decomposition of the kinetic energy into its toroidal and poloidal components versus the Rayleigh number. Solid lines mean stable solutions and dashed lines unstable. The lines following the circles correspond to stable nonperiodic solutions. Full circles indicate ${\overline{K}}_T$, and empty ${\overline{K}}_P$.

Figure 3

(a, b) Decomposition of the kinetic energy into its symmetric and antisymmetric parts versus the Rayleigh number. The symbols and type of lines are the same as in Fig. 2.

Figure 4

(a, b) Maximum Nusselt (left vertical axis) and helicity (right vertical axis) numbers along the branches of solutions versus the Rayleigh number. The symbols and type of lines are the same as in Fig. 2.

Figure 5

Snapshots of the velocity field (arrows) superposed to the contour plots of the temperature for a period of oscillation of a stable solution of $\mathrm{E}={10}^{-3}$, $\mathrm{Pr}={10}^{-2}$, and $\mathrm{Ra}=12209$. At left from top to bottom $t=0$, $T/8$, $T/4$, $3T/8$, and at right from top to bottom $t=T/2$, $5T/8$, $3T/4$, $7T/8$. The radial, equatorial, and meridional lines indicate the place where the sections are taken.

Figure 6

(a, c) Trajectories of the Floquet multipliers spanning along $\mathrm{Ra}\in [12192,12243]$ for $\mathrm{E}={10}^{-3}$ (17 multipliers), and $\mathrm{Ra}\in [7806,8330]$ for $\mathrm{E}={10}^{-4}$ (31 multipliers). (b, d) Floquet multipliers at the bifurcation points. The phases on the unit circle are signaled every $\pi /6$, with the origin in the horizontal axis.

Figure 7

Spectra of frequencies of the solution found at $\mathrm{Ra}=12500$. (a) The colatitudinal component of the velocity field $v_{\theta }(r_o,\pi /6,\phi )$, (b) ${\mathrm{\Psi }}_1^0\left(r_m\right)$, with $r_m=r_o/2$, and (c) ${\mathrm{\Phi }}_1^0\left(r_m\right)$.

Figure 8

Evolution of the spectrum of frequencies of ${\mathrm{\Phi }}_1^0\left(r_m\right)$, with $r_m=r_o/2$, from a periodic orbit to temporal chaos.

Figure 9

(a) Poincaré sections taken at ${\mathrm{\Phi }}_1^0\left(r_m\right)=0$, for $\mathrm{E}={10}^{-3}$, $\mathrm{Pr}={10}^{-2}$, showing the evolution of the shape of the two-tori up to the appearance of the third frequency. The Rayleigh numbers are 12 200, 12 300, 12 500, 12 700, 13 000, 13 400, 13 700, 14 000, 14 400, 15 200, 16 000, 17 000, 18 500, 20 000, 20 500, 21 000, 21 150; (b) evolution of the three-tori up to the appearance of temporal chaotic solutions for $\mathrm{Ra}=21150$, 21 170, 21 200, and 21 250.

Figure 10

(a) Poincaré sections taken at ${\mathrm{\Phi }}_1^0\left(r_m\right)=0$, for $\mathrm{E}={10}^{-4}$, $\mathrm{Pr}={10}^{-3}$, showing the evolution of the shape of the two-tori up to the appearance of the third frequency. The parameters are $\mathrm{Ra}=7921$, 7930, 7950, 8000, 8100, 8200, and 8250. (b) Enlargement of the three-tori. The parameters are $\mathrm{Ra}=8250$, 8300, 8325, and 8350. (c) Transition to temporal chaos showing the three-tori found at $\mathrm{Ra}=8350$ and at $\mathrm{Ra}=8400$, and the chaotic attractors at $\mathrm{Ra}=8500$ and $\mathrm{Ra}=10000$. (d) Detail of the resonant torus at $\mathrm{Ra}=8400$.

Figure 11

(a–e) Evolution of the spectrum of frequencies of ${\mathrm{\Psi }}_1^0\left(r_m\right)$ (upper spectra) and ${\mathrm{\Phi }}_1^0\left(r_m\right)$ (lower spectra), with $r_m=r_o/2$, and (f) of ${\mathrm{\Psi }}_1^0\left(r_m\right)$, from a periodic orbit to temporal chaos. The parameters are $\mathrm{E}={10}^{-4}$, $\mathrm{Pr}={10}^{-3}$.

Figure 12

(a, c, e) Temporal evolution of the coefficient of the perturbation of the temperature ${\mathrm{\Theta }}_0^0\left(r_m\right)$, and (b, d, f) of the azimuthal component of the velocity field $v_{\phi }(r_o,\pi /3,\phi )$. In panel (e) the thick curve (red online) corresponds to the Poincaré sections taken at ${\mathrm{\Phi }}_1^0\left(r_m\right)=0$. The parameters are $\mathrm{E}={10}^{-4}$, $\mathrm{Pr}={10}^{-3}$.

Figure 13

(a, b) Surface corresponding to the unstable manifold of a cycle and contained in the stable manifold of another, at $\mathrm{Ra}=10000$ (in magenta online), and the Poincaré section cutting the ${\mathrm{\Phi }}_1^0\left(r_m\right)=0$ hyperplane (in blue online). (c) Same surface shown with a different variable in the $z$ axis, including the unstable periodic orbit of the main branch. (d) Same representation as in panel (b) for the resonant three-torus at $\mathrm{Ra}=8400$. The other parameters are $\mathrm{E}={10}^{-4}$, $\mathrm{Pr}={10}^{-3}$.