Nonlinear evolution of unstable solar inertial modes: The case of viscous modes on a differentially rotating sphere
Max-Planck-Institut für Sonnensystemforschung（马克斯·普朗克太阳系统研究所）；Institut für Astrophysik und Geophysik, Georg-August-Universität Göttingen（哥廷根大学天体物理与地球物理研究所）；Center for Astrophysics and Space Science, NYUAD Institute, New York University Abu Dhabi（纽约大学阿布扎比分校天体物理学与空间科学中心）；Department of Mechanical Engineering, King Abdullah University of Science and Technology（阿卜杜拉国王科技大学机械工程系）
Muneeb Mushtaq, Damien Fournier, Rama Ayoub, Peter J. Schmid, Laurent Gizon
在太阳上，观测到振幅最大的惯性模（均方根速度超过10米/秒）是高纬度模，其经向波数m=1。在二维情况下，在球体上，线性理论预测该模由于纬向差异旋转（快速赤道，较慢极区）相关的剪切不稳定性而不稳定。我们通过数值和理论方法研究了这种不稳定性的发展。通过直接数值模拟在时域中求解非线性涡度方程。唯一的控制参数是Ekman数E。对于10^{-3} ≲ E < E_c ≈ 1.5×10^{-3}，只有高纬度m=1模不稳定。我们提取其作为E函数的饱和振幅，并将其结果与非线性稳定性理论中的两种微扰方法的预测进行比较。模拟揭示了超临界Hopf分支点。接近起始点时，模振幅由Landau方程良好描述：d|A|/dt=σ_I |A|+β_I |A|^3，其中线性增长率σ_I为正，非线性系数β_I为负。系数β_I弱依赖于E，表明饱和振幅大致按|A|∝σ_I^{1/2}的比例变化。平衡模包含m=1基模和谐波m=2和m=3，其振幅按σ_I^{m/2}比例变化。饱和源于雷诺应力，这些应力平滑纬向差异旋转。对于E=4×10^{-4}，与太阳类湍流粘性一致，饱和速度达到28米/秒，与太阳观测相当。这些结果应谨慎解读，因为在三维情况下，不稳定性是Baroclinic的，并涉及不同的物理机制。
英文摘要：
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.
Effects of the radiative interior on solar inertial modes
Max-Planck-Institut für Sonnensystemforschung（马克斯·普朗克太阳系统研究所）；Institut für Astrophysik und Geophysik, Georg-August-Universität Göttingen（哥廷根大学天体物理与地球物理研究所）
