Nonlinear evolution of unstable solar inertial modes: The case of viscous modes on a differentially rotating sphere

Abstract

On the Sun, the inertial mode with the largest observed amplitude (rms velocity exceeding $10$ m/s) is the high-latitude mode with longitudinal wavenumber $m=1$. In two dimensions, on the sphere, linear theory predicts that this mode is unstable due to a shear instability associated with latitudinal differential rotation (fast equator, slower polar regions). We investigate the evolution of this instability numerically and theoretically. The nonlinear vorticity equation is solved using direct numerical simulations in the time domain. The only control parameter is the Ekman number $E$. For $10^{-3}\lesssim E< E_c \approx 1.5\times10^{-3}$, only the high-latitude $m=1$ mode is unstable. We extract its saturation amplitude as a function of $E$ and compare the results with predictions from two perturbative approaches in nonlinear stability theory. The simulations reveal a supercritical Hopf bifurcation. Near onset, the mode amplitude is well described by the Landau equation $d|A|/dt=σ_I |A|+β_I |A|^3$, with a positive linear growth rate $σ_I$ and a negative nonlinear coefficient $β_I$. The coefficient $β_I$ depends weakly on $E$, implying that the saturated amplitude scales approximately as $|A|\proptoσ_I^{1/2}$. The equilibrium mode contains the $m=1$ fundamental and harmonics $m=2$ and $m=3$, whose amplitudes scale as $σ_I^{m/2}$. Saturation results from Reynolds stresses that smooth the latitudinal differential rotation. For $E=4\times10^{-4}$, consistent with solar-like turbulent viscosity, the saturated velocity reaches $28$ m/s, comparable to solar observations. These results should be interpreted cautiously, since in three dimensions the instability is baroclinic and involves different physics.

Figures (13)

• Figure 1: Schematic diagram illustrating the two bifurcation types: supercritical (left) and subcritical (right). The thick blue curves show the stable equilibrium wave amplitudes as a function of Ekman number $E$, while the dashed red curves corresponds to the unstable solutions. The black arrows indicate the different possible evolutions of the system, which can be computed numerically. We will consider two perturbation methods to obtain the equilibrium amplitudes in the case of supercritical bifurcation, whereby the small parameter is either the distance to the critical Ekman number $E_{\rm c}$ or the amplitude itself.
• Figure 2: Initial rotation and associated $\Lambda$-effect. (a) The surface solar differential rotation profile $\Omega_0(\theta)$ inferred by helioseismology from six years of HMI data Larson2018. (b) The corresponding function $\Lambda_{\theta\phi}(\theta)$ computed using Eq. (\ref{['eqn:omega_ode']}). For comparison, we overplot the $\Lambda$-effect prescribed by Rempel2005, that is $\Lambda_{\theta\phi} = - 0.8 \sin^2 \theta \cos \theta \sin\left(\theta + 15^\circ \right)$ in the northern hemisphere.
• Figure 3: Time evolution of the radial vorticity $Z(t, \theta, \phi) - \langle Z(t,\theta,\phi) \rangle_\phi$ at a fixed point $(\theta, \phi) = (60^\circ, 30^\circ)$ from DNS for an Ekman number, $E = 10^{-3}$. The upper panel is a zoom between the two blue vertical lines on the bottom panel to show that the oscillations do not consist of a single harmonic due to nonlinearities in the system.
• Figure 4: Snapshots of the two-dimensional radial vorticity $Z(t, \theta, \phi) - \langle Z(t, \theta, \phi)\rangle_\phi$ at time $t =120$ year for three different Ekman numbers, $E = 1.4 \times 10^{-3}, 4 \times 10^{-4},$ and $10^{-5}$. The time variations can be seen on the https://doi.org/10.17617/3.U2TWXU.
• Figure 5: RMS velocity (Eq. \ref{['eqn:urms_amp']}) obtained using the unfiltered velocity fields, as a function of the Ekman number. The green shaded region corresponds to $E > E_{\rm c}$ where no modes are unstable. The white region corresponds to the region where only one mode is unstable, and hence the WNL theory can be applied, in contrast to the gray region. The points labeled as 1, 2, and 3, respectively, correspond to the Ekman numbers from the left, middle, and right subplots in Fig. \ref{['fig:snapshot_vorticity']}.