Non-linear simulations of the onset and non-linear dynamics of inertial waves in solar and stellar interiors

TL;DR

This work develops and validates an anelastic hydrodynamics framework to study non-linear inertial waves in solar-like interiors using EULAG-MHD. It shows that differential rotation combined with subadiabatic stratification triggers baroclinic instabilities that generate polar vortices and high-latitude inertial modes, which transport angular momentum poleward as shear increases. Rossby-like inertial waves emerge under both solid-body and differential rotation, with energy cascading to multiple scales and frequencies consistent with the Rossby dispersion relation; equatorial modes are weak unless externally seeded. The findings illuminate how inertial waves interact with differential rotation and stratification to shape angular-momentum transport, though magnetic fields and more realistic forcing remain important topics for future work.

Abstract

Inertial modes have been recently detected in the Sun via helioseismology, yet their origin, evolution, and role in the dynamics of the solar plasma and magnetic field remain poorly understood. In this study, we employ global numerical simulations to investigate the excitation mechanisms and dynamical consequences of inertial modes in the Sun and stellar interiors. We validate first our numerical setup by analyzing the evolution of sectoral and tesseral perturbations imposed on a rigidly rotating sphere. The results confirm that a perturbation of a given mode can excite neighboring modes with both smaller and larger wavenumbers along the dispersion relation of Rossby waves. Subsequently, we use a physically motivated forcing to impose differential rotation with varying shear amplitudes, and examine the spontaneous onset and nonlinear evolution of inertial modes. The simulations reveal that the growth of velocity perturbations is primarily driven by baroclinic instability. It gives rise to high-latitude inertial modes in the form of retrograde polar vortices whose properties depend on the imposed shear. Equatorial Rossby modes are also excited, albeit with lower intensity than their high-latitude counterpart. Perturbations with arbitrary azimuthal wavenumbers lead to the excitation of Rossby modes for all available wave numbers, sustained by both direct and inverse energy cascades. In simulations with stronger shear, the high latitude modes produce Reynolds stresses able to modify the imposed differential rotation and accelerate the rotation of the poles.

Figures (17)

• Figure 1: Thermodynamic structure of the simulations including shear. The black and blue lines depict the density and temperature of the ambient state as a function of radius. The thin red dashed line shows the same quantities for the solar model of christensen1996current. The red line correspond to the adiabaticity at different layers(see the text). The vertical dashed and dotted lines correspond to the transition between a highly sub-adiabatic and a slightly sub-adiabatic layers, and the location of the tachocline, respectively.
• Figure 2: Mollweide projection of the velocity components $u$ (a) and $v$ (b) corresponding to the initial condition of the simulation with the sectoral mode $(m,l)=(4,4)$. Note that $u$ is anti-symmetric whereas $v$ is symmetric across the equator.
• Figure 3: Longitude-time, Hovmöller, diagrams showing the propagation of the longitudinal velocity, $u$ for simulation with sectoral initial condition $m=l=2,4,6,8$, panels (a) to (d), respectively, at a radius $r=0.64R_{\odot}$ and $45^{\circ}$ latitude. The tilt of the propagating pattern depicts the phase speed of the wave. The thickness of the bands with alternating colors depicts the frequency of the mode.
• Figure 4: Same as Fig. \ref{['fig:hovmoller_sectoral']} but for tesseral initial perturbations with (a) $m=2$, $l=3$, (b) $m=2$, $l=4$, (c) $m=2$, $l=5$, and (d) $m=2$, $l=8$.
• Figure 5: Wave frequency with highest power as a function of the longitudinal wave number, $m$ imposed as IC. The left (rigth) panels correspond to sectoral (tesseral) modes. The black lines show the theoretical dispersion relation, Eq. (\ref{['eq:tdr']})