Weak branch and multimodal convection in rapidly rotating spheres at low Prandtl number

The focus of this study is to investigate primary and secondary bifurcations to weakly nonlinear flows (weak branch) in convective rotating spheres in a regime where only strongly nonlinear oscillatory sub- and supercritical flows (strong branch) were previously found [E. J. Kaplan, N. Schaeffer, J. Vidal, and P. Cardin, Phys. Rev. Lett. 119, 094501 (2017)]. The relevant regime corresponds to low Prandtl and Ekman numbers, indicating a predominance of Coriolis forces and thermal diffusion in the system. We provide the bifurcation diagrams for rotating waves (RWs) computed by means of continuation methods and the corresponding stability analysis of these periodic flows to detect secondary bifurcations giving rise to quasiperiodic modulated rotating waves (MRWs). Additional direct numerical simulations (DNS) are performed for the analysis of these quasiperiodic flows for which Poincaré sections and kinetic energy spectra are presented. The diffusion timescales are investigated as well. Our study reveals very large initial transients (more than 30 diffusion time units) for the nonlinear saturation of solutions on the weak branch, either RWs or MRWs, when DNS are employed. In addition, we demonstrate that MRWs have multimodal nature involving resonant triads. The modes can be located in the bulk of the fluid or attached to the outer sphere and exhibit multicellular structures. The different resonant modes forming the nonlinear quasiperiodic flows can be predicted with the stability analysis of RWs, close to the Hopf bifurcation point, by analyzing the leading unstable Floquet eigenmode.

Figure 1

Bifurcation diagrams of rotating waves for the 3 sets of parameters: $P_1$ with $(Pr,\mathrm{E})=(0.03,3\times {10}^{-6})$, $P_2$ with $(Pr,\mathrm{E})=(0.01,{10}^{-6})$, and $P_3$ with $(Pr,\mathrm{E})=(0.003,3\times {10}^{-7})$. For the set $P_3$ two different branches with $m_0=11$ (violet) and with $m_0=12$ (magenta) are shown. (a) Péclet number $\mathrm{Pe}$ versus $\widetilde{\mathrm{Ra}}=\mathrm{Ra}/{\mathrm{Ra}}_{\mathrm{c}}-1$. The dashed line indicates the $\sqrt{\mathrm{Ra}-{\mathrm{Ra}}_{\mathrm{c}}}$ scaling predicted in Ref. [39]. (b) Scaled rotating frequency $\omega \mathrm{E}$ versus $\widetilde{\mathrm{Ra}}$. Solid/dashed lines mark stable/unstable rotating waves.

Figure 2

Rotating wave with $m_0=12$, in the case of the set $P_3$ ($\mathrm{E}=3\times {10}^{-7}$, $\mathrm{Pr}=0.003$) and $\mathrm{Ra}=1.2634\times {10}^8$ ($\widetilde{\mathrm{Ra}}=7.9\times {10}^{-3}$). Top row: Contour plots for the temperature perturbation $\mathrm{\Theta }$ on an equatorial and meridional section. Bottom row: Contour plots for the vertical vorticity ${\widehat{\omega }}_z$ on an equatorial section and for the azimuthal velocity $v_{\phi }$ on a meridional section.

Figure 3

Rotating waves with $m_0=12$, in the case of the set $P_3$ ($\mathrm{E}=3\times {10}^{-7}$, $\mathrm{Pr}=0.003$). Contour plots for the temperature perturbation $\mathrm{\Theta }$ (top row) and for the kinetic energy density $K$ (bottom row) on an equatorial section. The Rayleigh numbers are $\mathrm{Ra}=1.3030\times {10}^8$ (a, f), $\mathrm{Ra}=1.4511\times {10}^8$ (b, g), $\mathrm{Ra}=1.5932\times {10}^8$ (c, h), $\mathrm{Ra}=1.9650\times {10}^8$ (d, i), and $\mathrm{Ra}=2.5067\times {10}^8$ (e, j). This corresponds to $\widetilde{\mathrm{Ra}}=3.9\times {10}^{-2}$, $\widetilde{\mathrm{Ra}}=1.6\times {10}^{-1}$, $\widetilde{\mathrm{Ra}}=2.7\times {10}^{-1}$, $\widetilde{\mathrm{Ra}}=5.7\times {10}^{-1}$, and $\widetilde{\mathrm{Ra}}={10}^0$.

Figure 4

(a) Real and imaginary part of the leading Floquet multipliers $\lambda$ corresponding to the first unstable RWs with azimuthal symmetry $m_0=12$ for the set $P_1=(Pr,\mathrm{E})=(0.03,3\times {10}^{-6})$ (squares, orange online) and for the set $P_3=(Pr,\mathrm{E})=(0.003,3\times {10}^{-7})$ (circles, blue online). The Rayleigh numbers are $\mathrm{Ra}=2.43043\times {10}^7$ ($\widetilde{\mathrm{Ra}}=4\times {10}^{-2}$) and $\mathrm{Ra}=1.27467\times {10}^8$ ($\widetilde{\mathrm{Ra}}=1.7\times {10}^{-2}$), respectively. (b) Leading Floquet multipliers for the last stable (triangles down, red online) and first (circles, blue online) and second unstable (triangles up, dark-green online) RWs with azimuthal symmetry $m_0=12$ for the set $P_3$ at ${\mathrm{Ra}}_1=1.26344\times {10}^8$, ${\mathrm{Ra}}_2=1.27467\times {10}^8$, and ${\mathrm{Ra}}_3=1.3030\times {10}^8$, respectively. The first unstable RW with azimuthal symmetry $m_0=11$ at ${\mathrm{Ra}}_3^{11}=1.3011\times {10}^8$ is also shown (diamonds, green online). The conjugate Floquet multipliers are not shown in panel (b) and the solid line marks the unit circle.

Figure 5

Leading Floquet multipliers for unstable RWs with azimuthal symmetry $m_0=12$ and the set $P_3$ ($\mathrm{E}=3\times {10}^{-7},\mathrm{Pr}=0.003$) at ${\mathrm{Ra}}_3=1.3030\times {10}^8$ (triangles, dark-green online), at ${\mathrm{Ra}}_4=1.3228\times {10}^8$ (circles, orange online), ${\mathrm{Ra}}_5=1.3919\times {10}^8$ (asterisk, violet online), and ${\mathrm{Ra}}_6=1.4511\times {10}^8$ (pentagon, yellow online). The labels indicate the azimuthal symmetry $m_1$ of the eigenfunction and its most energetic wave number $m_{\text{max}}$. The conjugate Floquet multipliers are not shown and the solid line marks the unit circle.

Figure 6

Leading eigenfunction of a RW with $m_0=11$, in the case of the set $P_3$ ($\mathrm{E}=3\times {10}^{-7}$, $\mathrm{Pr}=0.003$), and $\mathrm{Ra}=1.3011\times {10}^8$. Top row: Contour plots for the temperature perturbation $\mathrm{\Theta }$ on an equatorial and meridional sections. Middle row: Contour plots for the vertical vorticity ${\widehat{\omega }}_z$ on an equatorial section and for the azimuthal velocity $v_{\phi }$ on a meridional section. Bottom row: Contour plots for the kinetic energy density $K$ on an equatorial and meridional sections. The azimuthal symmetry and most energetic wave number are $m_1=1$ and $m_{\text{max}}=10$, respectively.

Figure 7

Leading eigenfunctions of rotating waves with $m_0=12$, in the case of the set $P_3$ ($\mathrm{E}=3\times {10}^{-7}$, $\mathrm{Pr}=0.003$). Contour plots for the temperature perturbation $\mathrm{\Theta }$ (top row) and for the kinetic energy density $K$ (bottom row) on an equatorial section. The Rayleigh numbers are $\mathrm{Ra}=1.3030\times {10}^8$ (a, f), $\mathrm{Ra}=1.3228\times {10}^8$ (b, g), $\mathrm{Ra}=1.3919\times {10}^8$ (c, h), and $\mathrm{Ra}=1.4511\times {10}^8$ (d, e, i, j). For the latter $\mathrm{Ra}$, panels (d, i) and (e, j) correspond to the first and third leading eigenfunctions, respectively. Their respective azimuthal symmetry and most energetic wave number are $m_1=2$ and $m_{\text{max}}=10$, $m_1=3$ and $m_{\text{max}}=9$, $m_1=4$ and $m_{\text{max}}=8$, $m_1=2$ and $m_{\text{max}}=10$, and $m_1=1$ and $m_{\text{max}}=7$.

Figure 8

Time integration with initial conditions obtained by adding a random perturbation to the RWs for the set $P_3$ ($\mathrm{E}=3\times {10}^{-7},\mathrm{Pr}=0.003$). Norm of the amplitudes of the potential scalars and the temperature perturbation $\left|\right|u\left|\right|$, and the norm $\left|\right|u_{\text{nd}}\left|\right|$ for only $m\ne 12k,k\in \mathbb{Z}$ [in panels (a, c, d, e, f)] and $m\ne 11k,k\in \mathbb{Z}$ [in panel (b)], versus diffusion time (also rotation time on top horizontal axis). The volume averaged kinetic energies for each wave number $m=1,10,11,12,13,14$ are displayed as well. The Rayleigh numbers are (a) $\mathrm{Ra}=1.2634\times {10}^8$, (b) $\mathrm{Ra}=1.2747\times {10}^8$, (c) $\mathrm{Ra}=1.3030\times {10}^8$, (d) $\mathrm{Ra}=1.3228\times {10}^8$, (e) $\mathrm{Ra}=1.3919\times {10}^8$, and (f) $\mathrm{Ra}=1.4511\times {10}^8$.

Figure 9

(a) Peclet number versus diffusion time (also rotation time on top horizontal axis), in the case of the set $P_3$ ($\mathrm{E}=3\times {10}^{-7}$, $\mathrm{Pr}=0.003$) for the same solutions as shown in Fig. 8. The Rayleigh numbers, increasing from bottom to top, are $\mathrm{Ra}=1.2634\times {10}^8$, $\mathrm{Ra}=1.2747\times {10}^8$, $\mathrm{Ra}=1.3030\times {10}^8$, $\mathrm{Ra}=1.3228\times {10}^8$, $\mathrm{Ra}=1.3919\times {10}^8$, and $\mathrm{Ra}=1.4511\times {10}^8$. (b) Bifurcation diagrams of the time averaged Peclet number corresponding to the branches of RWs with azimuthal symmetry $m_0=12$ and $m_0=11$ and of MRWs with azimuthal symmetry $m_2=1$. The points correspond to the curves shown in panel (a).

Figure 10

(a) Poincaré section defined by $0=\mathrm{\Theta }(r,\phi ,\theta )$ with $(r,\phi ,\theta )=(0.51,0,5\pi /8)$. The temperatures ${\mathrm{\Theta }}_1=\mathrm{\Theta }(0.16,0,5\pi /8)$ and ${\mathrm{\Theta }}_2=\mathrm{\Theta }(0.86,0,5\pi /8)$ are displayed on the horizontal and vertical axis, respectively. We recall that $\eta =0.01$ implies $r_i=0.0101$ and $r_o=1.0101$ and that solutions belong to the set $P_3$ ($\mathrm{E}=3\times {10}^{-7},\mathrm{Pr}=0.003$). The Rayleigh numbers of each section increase from right to left in the figures (see the arrow). They are $\mathrm{Ra}=1.2634\times {10}^8$, $\mathrm{Ra}=1.2747\times {10}^8$, $\mathrm{Ra}=1.3030\times {10}^8$, $\mathrm{Ra}=1.3228\times {10}^8$, $\mathrm{Ra}=1.3919\times {10}^8$, and $\mathrm{Ra}=1.4511\times {10}^8$, corresponding to panels (a–f), respectively, of Fig. 8.

Figure 11

(a) Temperature ${\mathrm{\Theta }}_3=\mathrm{\Theta }(r,\phi ,\theta )$, with $(r,\phi ,\theta )=(0.51,0,5\pi /8)$, versus diffusion time. (b) Same as panel (a) but for the temperature ${\mathrm{\Theta }}_1=\mathrm{\Theta }(0.16,0,5\pi /8)$. The Rayleigh number is $\mathrm{Ra}=1.4511\times {10}^8$ corresponding to panel (f) of Fig. 8. The solution belongs to the set $P_3$ ($\mathrm{E}=3\times {10}^{-7},\mathrm{Pr}=0.003$). The time averages of the kinetic energy spectra of Figs. 12 and 12 are taken over the time intervals $I_1$ and $I_2$, respectively. Panels (c) and (e) correspond to details of panel (a) in the intervals $I_1$ and $I_2$, respectively. Panels (d) and (f) correspond to details of panel (b) in the intervals $I_1$ and $I_2$, respectively.

Figure 12

Time and volume averaged kinetic energy spectra $K_m$ versus the azimuthal wave number $m$. In panels (a) and (b) the Rayleigh number is $\mathrm{Ra}=1.4511\times {10}^8$ and the time average is taken over the interval $I_1$ and $I_2$, respectively, which are shown in Fig. 11. In panel (c) several Rayleigh numbers are shown which increase from right to left in the figure (see the arrow). They are $\mathrm{Ra}=1.2634\times {10}^8$, $\mathrm{Ra}=1.2747\times {10}^8$, $\mathrm{Ra}=1.3030\times {10}^8$, $\mathrm{Ra}=1.3228\times {10}^8$, $\mathrm{Ra}=1.3919\times {10}^8$, and $\mathrm{Ra}=1.4511\times {10}^8$, corresponding to panels (a–f), respectively, of Fig. 8. All these solutions belong to the set $P_3$ ($\mathrm{E}=3\times {10}^{-7},\mathrm{Pr}=0.003$). For each $m$, the maximum and minimum values of $K_m$ over the time interval are shown with error bars. The ime average, maximum and minimum values are taken over the last 5 diffusion time units of each time series.

Figure 13

Solution bifurcating from rotating waves with $m_0=12$, in the case of the set $P_3$ ($\mathrm{E}=3\times {10}^{-7},\mathrm{Pr}=0.003$) at $\mathrm{Ra}=1.4511\times {10}^8$. The snapshot is taken in the transient phase, at the end of the time interval $I_1$ shown in Fig. 11. Top row: Contour plots for the temperature perturbation $\mathrm{\Theta }$ on an equatorial and meridional section. Bottom row: Contour plots for the vertical vorticity ${\widehat{\omega }}_z$ on an equatorial section and for the azimuthal velocity $v_{\phi }$ on a meridional section.

Figure 14

As Fig. 13 but with the snapshot taken in the saturated phase, at the end of the time interval $I_2$ shown in Fig. 11.

Figure 15

Frequency analysis for two solutions at $\mathrm{Ra}=1.3030\times {10}^8$ (squares) and at $\mathrm{Ra}=1.3919\times {10}^8$ (circles), in the case of the set $P_3$ ($\mathrm{E}=3\times {10}^{-7},\mathrm{Pr}=0.003$). (a) Leading frequencies, $f_m$, and amplitudes, $A_m$, of the time series of poloidal component, $\Re {\mathrm{\Phi }}_m^m\left(r_d\right)$, of the different modes $(m,m)$, $1\le m\le 20$. (b) as in panel (a) but for the second leading frequencies $f_m^2$.

Figure 16

Modulated rotating wave with azimuthal symmetry $m_2=1$ in the case of the set $P_3$ ($\mathrm{E}=3\times {10}^{-7},\mathrm{Pr}=0.003$) at $\mathrm{Ra}=1.3030\times {10}^8$. The contour plots of the temperature perturbation $\mathrm{\Theta }$ on an equatorial (a, f, k) and a meridional (b, g, l) section, and of the radial velocity $v_r$ (c, h, m), azimuthal velocity $v_{\phi }$ (d, i, n), and vertical vorticity ${\widehat{w}}_z$ (e, j, o), normalized by the planetary vorticity ${\widehat{w}}_z=w_z\mathrm{E}/2$, on equatorial sections, are displayed from left to right in each row. From top to bottom only the $m=1$ (a–e), $m=10$ (f–j), and $m=21$ (k–o), respectively, azimuthal wave numbers, rather than all $m$'s, are considered for the contour plots.

Figure 17

Leading eigenfunction, with azimuthal symmetry $m_1=1$, of a rotating wave with $m_0=11$, in the case of the set $P_3$ ($\mathrm{E}=3\times {10}^{-7},\mathrm{Pr}=0.003$) at $\mathrm{Ra}=1.3011\times {10}^8$. The contour plots of the temperature perturbation $\mathrm{\Theta }$ on an equatorial (a, f, k) and a meridional (b, g, l) section, and of the radial velocity $v_r$ (c, h, m), azimuthal velocity $v_{\phi }$ (d, i, n), and vertical vorticity ${\widehat{w}}_z$ (e, j, o), normalized by the planetary vorticity ${\widehat{w}}_z=w_z\mathrm{E}/2$, on equatorial sections, are displayed from left to right in each row. From top to bottom only the $m=1$ (a–e), $m=10$ (f–j), and $m=21$ (k–o), respectively, azimuthal wave numbers, rather than all $m$'s, are considered for the contour plots.

Figure 18

Modulated rotating wave with azimuthal symmetry $m_2=1$ at in the case of the set $P_3$ ($\mathrm{E}=3\times {10}^{-7},\mathrm{Pr}=0.003$) at $\mathrm{Ra}=1.3030\times {10}^8$. The contour plots of the temperature perturbation $\mathrm{\Theta }$ (a–e), of the radial velocity $v_r$ (f–j), and of the azimuthal velocity $v_{\phi }$ (k–o) on an equatorial section are displayed from top to bottom rows. The $m=2,...,6$ azimuthal components of the solution are displayed in panels (a, f, k), (b, g, l), (c, h, m), (d, i, n), and (e, j, o), respectively.

Figure 19

Modulated rotating wave with azimuthal symmetry $m_2=1$ in the case of the set $P_3$ ($\mathrm{E}=3\times {10}^{-7},\mathrm{Pr}=0.003$) at $\mathrm{Ra}=1.3030\times {10}^8$. The contour plots for kinetic energy $K$ on a spherical section at $r\approx 0.99r_o$ (a–e), on an equatorial section (f– j), and on a colatitudinal section at $\theta ={75}^{\circ }$ (k–o). The $m=1,2,3,10,21$ azimuthal components of the solution are displayed in panels (a, f, k), (b, g, l), (c, h, m), (d, i, n), and (e, j, o), respectively.