Polar waves and chaotic flows in thin rotating spherical shells

Convection in rotating spherical geometries is an important physical process in planetary and stellar systems. Using continuation methods at a low Prandtl number, we find both strong equatorially asymmetric and symmetric polar nonlinear rotating waves in a model of thermal convection in thin rotating spherical shells with stress-free boundary conditions. For the symmetric waves, convection is confined to high latitude in both hemispheres but is only restricted to one hemisphere close to the pole in the case of asymmetric waves. This is in contrast to what is previously known from studies in the field. These periodic flows, in which the pattern is rotating steadily in the azimuthal direction, develop a strong axisymmetric component very close to onset. Using stability analysis of periodic orbits, the regions of stability are determined and the topology of the stable/unstable oscillatory flows bifurcated from the branches of rotating waves is described. By means of direct numerical simulations of these oscillatory chaotic flows, we show that these three-dimensional convective polar flows exhibit characteristics, such as force balance or mean physical properties, which are similar to flows occurring in planetary atmospheres. We show that these results may open a route to understanding unexplained features of gas giant atmospheres, particularly in the case of Jupiter. These include the observed equatorial asymmetry with a pronounced decrease at the equator (the so-called dimple), and the coherent vortices surrounding the poles recently observed by the Juno mission.

Figure 1

Bifurcation diagrams of time- and volume-averaged quantities for varying $\mathrm{Ra}$. Panels (a)–(c) are for the AP RW branch, and (d)–(f) also include the SP RW branch. (a) Kinetic energy density of the equatorially symmetric component of the flow $K^{\text{s}}$ and of the equatorially antisymmetric part $K^{\text{as}}=K-K^{\text{s}}$. (b) Ratio of the equatorially symmetric kinetic energy density over total kinetic energy density $K^{\text{s}}/K$. (c) Ratio of the equatorially symmetric kinetic energy density of the axisymmetric ($m=0$) flow over total axisymmetric kinetic energy density $K_a^{\text{s}}/K_a$ and the same ratio $K_{na}^{\text{s}}/K_{na}$ for the nonaxisymmetric ($m>0$) flow. (d) Total kinetic energy density $K$. (e) Ratios of the axisymmetric kinetic energy over the total kinetic energy density $K_a/K$. (f) Rotation frequency. Solid (dashed) lines mean stable (unstable) RWs. The points (circles) of panel (f) correspond to RWs shown in Figs. 2 and 3.

Figure 2

First and second rows: Contour plots of $m=19$ RW on the AP branch at $\mathrm{Ra}=3.057\times {10}^2, 3.1\times {10}^2,3.25\times {10}^2,3.56\times {10}^2,3.63\times {10}^2,3.99\times {10}^2$ (from left to right). First row: Spherical sections of $v_{\varphi }$ at $r_o$. Second row: Spherical sections of ${\mathbf{v}}^2/2$ at $r_o$. Third and fourth rows: Same as the previous rows but for RW on the SP branch at $\mathrm{Ra}=3.27\times {10}^2,3.33\times {10}^2,3.49\times {10}^2,3.78\times {10}^2,3.64\times {10}^2,3.56\times {10}^2$ (from left to right). All waves are marked with a circle in Fig. 1.

Figure 3

Meridional sections of $v_{\varphi }$ through a relative maximum. First row: Contour plots of $m=19$ RW on the AP branch at $\mathrm{Ra}=3.057\times {10}^2,3.1\times {10}^2,3.25\times {10}^2,3.56\times {10}^2,3.63\times {10}^2,3.99\times {10}^2$ (from left to right). Second row: Same as the previous row but for RW on the SP branch at $\mathrm{Ra}=3.27\times {10}^2,3.33\times {10}^2,3.49\times {10}^2,3.78\times {10}^2,3.64\times {10}^2,3.56\times {10}^2$ (from left to right). All waves are marked with a circle in Fig. 1.

Figure 4

First row: Contour plots of the 1st, 4th, 6th, 7th, 9th, and 13th (from left to right) dominant eigenfunctions on the AP RW branch at $\mathrm{Ra}=3.2\times {10}^2$. Spherical section of $v_{\varphi }$ at $r_o$. Second row: Same as the first row but for the 1st, 2nd, 7th, 8th, 9th, and 10th dominant eigenfunctions on the SP RW branch at $\mathrm{Ra}=3.78\times {10}^2$.

Figure 5

Instantaneous contour plots of DNS at $\mathrm{Ra}=3.5\times {10}^2$ (left group of three plots) and $\mathrm{Ra}=5\times {10}^2$ (right group). For each group the spherical sections (from left to right) are of $\mathrm{\Theta }$ (at $r\approx r_i+0.5d$), $v_{\phi }$, and $\langle v_{\phi }\rangle$ (at $r=r_o$), respectively.

Figure 6

(a) Time series of convective heat transfer at the outer surface $\mathrm{Nu}-1$ and (b) a detail of (a) including the time span of the supplementary movies [72]. (c) The kinetic energy spectrum $K_m$ vs the azimuthal wave number $m$ and the theoretical [73] Rhines' scaling (dashed line).

Figure 7

${\mathrm{Ro}}_v=\langle v_{\phi }\rangle /\mathrm{\Omega }{r}_o$ vs latitude in degrees. (a) Asymmetric polar rotating wave $u_{\text{as}}$ at $\mathrm{Ra}=3.98\times {10}^2$ (solid line), its eighth dominant eigenfunction (dotted-dashed line), symmetric polar rotating wave $u_{\text{s}}$ at $\mathrm{Ra}=3.78\times {10}^2$ (dashed line), and its $u_{\text{2nd}}$ second dominant eigenfunction (dotted line). The eigenfunctions are scaled for comparative purposes. (b) DNS at $\mathrm{Ra}=5\times {10}^2$ (solid line) and the linear superposition $u_{\text{as}}+4u_{\text{s}}+1.5\times {10}^4{u}_{\text{2nd}}$ of the curves of ${\mathrm{Ro}}_v$ shown in (b) (dashed line). (c) Hubble data for Jupiter [76] (solid line) and DNS of (b) scaled by a factor of 6.3 (dotted line). (d) Hubble data for the antisymmetric Jupiter zonal flow and the eighth dominant eigenfunction of the asymmetric polar rotating wave scaled by a factor of 6.5 (dotted line).

Figure 8

Radial vorticity at $r=r_o$ (viewed from the north pole, the equator, and the south pole) and axial vorticity on the equatorial plane (from left to right). Cyclonic (anticyclonic) radial vorticity is red (blue) on the northern hemisphere and blue (red) on the southern hemisphere. A supplementary movie [72] for each panel is included.