Chiral pattern in nonrotating spherical convection

When the Rayleigh number is low, Rayleigh-Bénard convection in a nonrotating spherical shell with central gravity has symmetric solutions in terms of three-dimensional discrete rotation. All the known patterns with the regular polyhedral symmetries accompany reflection symmetry. We found an alternative type of steady convection in a nonrotating spherical shell by computer simulations. The pattern has the discrete rotational symmetry of a regular tetrahedron with no reflection symmetry. The convection consists of six pairs of spiral rolls placed on 12 faces of a spherical dodecahedron. Depending on the winding of the spirals, there are two possible configurations that are mirror images of one another.

Figure 1

Time developments of convection flow energy for three different Rayleigh numbers $\overline{R}$. Time was normalized by the diffusion time ${\tau }_{\mathrm{d}}$. The dotted blue curve ($\overline{R}=1712$) and dashed green curve ($\overline{R}=1723$) show simulation results obtained starting from random perturbations. We assumed that the critical Rayleigh number ${\overline{R}}_{\mathrm{c}}$ was the midpoint of the two $\overline{R}$ values. The solid magenta curve ($\overline{R}=2209=1.29{\overline{R}}_{\mathrm{c}}$) shows the simulation result obtained starting from a controlled initial condition.

Figure 2

Time development of convection for $\overline{R}=1.29{\overline{R}}_{\mathrm{c}}$ when the initial condition is random. Time is presented on the diffusion timescale ${\tau }_{\mathrm{d}}$ in the upper-right corner of each panel. From panels (d) to (f), the pattern slowly changes in time.

Figure 3

Design of initial condition with chirality. (a) Six pairs of Archimedean spirals (blue) were placed on 12 faces of a regular dodecahedron (red). (b) Discrete rotational symmetry of the $T$ (tetrahedral) group, with four threefold rotations indicated by light blue triangles and three twofold rotations indicated by light blue ellipses.

Figure 4

Pressure perturbation profile $p_1(\vartheta ,\phi )$ used in the controlled initial condition in this study.

Figure 5

Time development of convection when $\overline{R}=1.29{\overline{R}}_{\mathrm{c}}$. Time is presented on the diffusion timescale ${\tau }_{\mathrm{d}}$ in the upper-right corner of each panel. The convection reached the steady state by $t=67.8{\tau }_{\mathrm{d}}$. No shift or rotation of spiral arms was observed. Panels (a) to (i) show the convection started by the pressure perturbation shown in Fig. 4. Panel (j) shows a snapshot of another simulation started by the mirrored perturbation of Fig. 4.

Figure 6

Dodecahedral convection pattern with discrete rotational symmetry of $T$ for $\overline{R}=1.29{\overline{R}}_{\mathrm{c}}$. Isosurfaces of positive $v_r$ were visualized using six different in situ visualization cameras. The viewing direction from each camera is indicated by arrows in the lower right corner of each panel.

Figure 7

Black or white circles with labels a–f in the upper panel show the range of Rayleigh numbers $\overline{R}$ for steady-state solutions; white circles indicate that steady, chiral, and symmetric solutions are obtained by simulations. The convection patterns are presented in the lower six panels with corresponding labels. Snapshots were taken at $t=113{\tau }_{\mathrm{d}}$ for panels (a)–(e), and at $t=12.2{\tau }_{\mathrm{d}}$ for panel (f).