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

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 $d|A|/dt=\sigma_I |A|+\beta_I |A|^3$, with a positive linear growth rate $\sigma_I$ and a negative nonlinear coefficient $\beta_I$. The coefficient $\beta_I$ depends weakly on $E$, implying that the saturated amplitude scales approximately as $|A|\propto\sigma_I^{1/2}$. The equilibrium mode contains the $m=1$ fundamental and harmonics $m=2$ and $m=3$, whose amplitudes scale as $\sigma_I^{m/2}$. 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.