Bifurcations of rotating waves in rotating spherical shell convection

The dynamics and bifurcations of convective waves in rotating and buoyancy-driven spherical Rayleigh-Bénard convection are investigated numerically. The solution branches that arise as rotating waves (RWs) are traced by means of path-following methods, by varying the Rayleigh number as a control parameter for different rotation rates. The dependence of the azimuthal drift frequency of the RWs on the Ekman and Rayleigh numbers is determined and discussed. The influence of the rotation rate on the generation and stability of secondary branches is demonstrated. Multistability is typical in the parameter range considered.

Figure 1

Critical Rayleigh numbers of the $m=2,3,4,5$ modes as a function of the Ekman number plotted with a logarithmic scale for $\mathrm{Ek}$. Crosses label the points at which bifurcations have been calculated; the solid lines are drawn to guide the eye.

Figure 2

Radial velocity normalized to its maximum modulus in the equatorial plane for RW2 (top left), RW3 (top right), RW4 (bottom left), and RW5 (bottom right) at $\mathrm{Ek}=0.001$ and $\mathrm{Ra}=100$. Positive values (red) correspond to radial outflow and negative values (blue) to radial inflow.

Figure 3

Radial velocity in the meridional plane at $\mathrm{Ek}=0.001$ and $\mathrm{Ra}=100$. RW2 (top left), RW3 (top right), RW4 (bottom left), and RW5 (bottom right).

Figure 4

Kinetic energy of RW3 as function of the Ekman and Rayleigh numbers. Thin (thick) lines correspond to unstable (stable) solutions and the lower dashed line shows the critical Rayleigh number as function of the Ekman number.

Figure 5

Same as described in the caption of Fig. 4, but for RW4.

Figure 6

Same as described in the caption of Fig. 4, but for RW2.

Figure 7

Same as described in the caption of Fig. 4, but for RW5.

Figure 8

Nusselt-number dependence of the rotating waves. For each $m$, curves from left to right correspond to $\mathrm{Ek}=0.002,0.003,0.004,0.005,0.006,0.007,0.008,0.009,0.01$.

Figure 9

Drift frequency $\omega$ as function of $\mathrm{Ra}$ for RW2 (dashed line), RW3 (solid line), RW4 (dotted line), and RW5 (dash-dotted line).

Figure 10

Bifurcations breaking the $Z_4$ symmetry of the RW4 branch and generating a new branch, RW4-2, with $Z_2$ symmetry are drawn in the plane spanned by Ra (abscissa) and the kinetic energy contained in the $m=4$ mode (ordinate). The RW4 (RW4-2) branches are shown in blue (red). From top to bottom, $\mathrm{Ek}=0.003,\mathrm{Ek}=0.004$, and $\mathrm{Ek}=0.007$. Solid (dashed) lines mark the stable (unstable) parts of the RWs.

Figure 11

Radial velocity as in Fig. 2, but for the RW4-2 solution at $\mathrm{Ek}=0.007$ and $\mathrm{Ra}=120$ at midgap (left) and in the equatorial plane (right).