Muneeb Mushtaq

Context . On the Sun, the inertial mode with the largest observed amplitude (rms velocity exceeding 10 m/s) is the high-latitude mode with a longitudinal wavenumber of m = 1. In two dimensions, linear theory predicts this mode to be unstable due to a shear instability associated with latitudinal differential rotation (i.e., fast equator, slower polar regions). Aims . In this work, we investigated the evolution of this instability numerically and theoretically. Methods . The nonlinear vorticity equation was solved using direct numerical simulations in the time domain. The only control parameter was set as the Ekman number, E . For 10 ⁻³ ≲ E < E _c ≈ 1.5×10 ⁻³ , only the high-latitude m = 1 mode was unstable. We extracted its saturation amplitude as a function of E and compared the results with predictions from two perturbative approaches in nonlinear stability theory. Results . The simulations reveal a supercritical Hopf bifurcation. Near the onset, the mode amplitude is aptly described by the Stuart-Landau equation d | A |/ dt = σ _I | A |+ β _I | A | ³ , derived from perturbation theories, with a positive linear growth rate, σ _I , and a negative nonlinear coefficient, β _I . This description remains valid throughout the range of 10 ⁻³ ≲ E < E _c . The coefficient β _I depends weakly on E , implying that the saturated amplitude scales approximately as | A |∝ σ _I ^1/2 . The equilibrium mode shape consists of the m = 1 fundamental mode and higher harmonics, m = 2 and m = 3, whose amplitudes scale approximately as σ _I ^{m /2} . The saturation mechanism arises from the Reynolds stress that feeds back on and smooths the latitudinal differential rotation. At E = E _⊙ = 4×10 ⁻⁴ , consistent with solar-like values inferred from supergranular turbulent viscosity, the saturated velocity reaches 28 m/s, comparable to the observed value on the Sun. Conclusions . In two dimensions, the onset and saturation of the solar m = 1 inertial mode are governed by a supercritical Hopf bifurcation associated with a shear instability that is aptly described by weakly nonlinear theory. Future works will benefit from extending the model to include solar-cycle variations in the differential rotation and moving toward a 3D setup that incorporates baroclinic effects into the dynamics.