Excitation of Inertial Modes in 3D Simulations of Rotating Convection in Planets and Stars

TL;DR

The paper demonstrates that inertial modes can arise spontaneously in 3D rotating convection within spherical shells, without external forcing, when the flow is sufficiently rotation-dominated ($Ro_c\lesssim0.5$). Using a Boussinesq framework with fixed $Ra$ and varying $Ek$, the authors reveal discrete, retrograde, equatorially symmetric inertial modes with $| abla|<2\Omega_0$, and show that lowering the Prandtl number to $Pr=0.1$ enhances mode excitation and broadens the spectrum. They identify coherent mode structures linked to differential-rotation shear and mode attractors, including internal shear layers aligned with inertial-wave rays, and observe that low-$Pr$ regimes yield substantially richer inertial-mode spectra. The findings imply that similar inertial modes could operate in giant planets and stars, though their low frequencies pose detection challenges; they also discuss limitations of the current model and outline directions for incorporating stratification and magnetic fields in future work.

Abstract

Thermal convection in rotating stars and planets drives anisotropic turbulence and differential rotation, both capable of feeding energy into global oscillations. Using 3D simulations of rotating convection in spherical shells, we show that inertial modes--oscillations restored by the Coriolis force--emerge naturally in rotationally constrained turbulence, without imposing any external forcing other than thermal/buoyancy driving. By varying the rotation rate at fixed Rayleigh number, we find that coherent modes appear only when the convective Rossby number, the ratio of the rotation period to the convective turnover time, falls below about one-half, where rotation dominates the dynamics. These modes are mostly retrograde in the rotating frame, equatorially symmetric, and confined to mid and high latitudes, with discrete frequencies well below twice the background rotation rate. At lower viscosities, or smaller Prandtl number, mode excitation becomes more efficient and a broader spectrum of inertial modes emerges. While the precise excitation mechanism remains uncertain, our results suggest that the modes are driven by instabilities due to differential rotation rather than stochastic forcing by convection. We conclude that similar inertial modes are likely to exist in the interiors of giant planets and stars, though their low frequencies will make them difficult to detect.

Figures (10)

• Figure 1: 3D snapshots of the radial velocity $u_r$. The velocity in simulations of rotating flows is normalized by $\Omega_0 r_o$, while the non-rotating model is normalized by the free fall velocity $u_{\rm ff}$. Red and blue denote upflows and downflows, respectively. In the rotating cases, the flow exhibits markedly smaller, anisotropic spatial scales compared with the non-rotating case, where convection is more isotropic. All the simulations have the same Rayleigh number $\mathrm{Ra} = 5\times 10^6$ and Prandtl number $\mathrm{Pr} = 1$, while the Ekman number varies from $\mathrm{Ek}\sim$∼$7\times 10^{-5}$ to $\mathrm{Ek} = \infty$ (non-rotating case).
• Figure 2: 3D snapshots of the azimuthal velocity $u_\phi$ for all simulations. The velocity in simulations of rotating flows is normalized by $\Omega_0 r_o$, while the non-rotating model is normalized by free fall velocity $u_{\rm ff}$. Red and blue denote prograde (eastward) and retrograde (wesward) direction, respectively. Simulations of small convective Rossby number $\mathrm{Ro_c}$ produce prograde equatorial jets, while simulations of large $\mathrm{Ro_c}$ yield retrograde jets. No jets or differential rotation develop in the non-rotating case.
• Figure 3: Velocity power spectra for all simulations. Flows were sampled at a radius $r=0.85r_o$ to construct the spectra, with other radii yielding essentially identical results. Panels (a) and (b) show the power associated with all azimuthal wavenumbers $m$ and with only the non-axisymmetric components ($m > 0$), respectively. Panel (c) shows the power of the zonal flow, i.e., the power considering only the toroidal component with the $m=0$ contribution in Equation \ref{['eq:power']}. The dashed lines in panels (b) and (c) have a slope of $-2/3$ and $-5$ for comparison with Kolmogorov spectrum Kolmogorov1941 and with zonostropic turbulence Boning2023, respectively.
• Figure 4: Power spectra of the kinetic energy at $r=0.85r_o$, as a function of frequency for all rotating models. We show the frequency normalized to $2\Omega_0$, and to the convective frequency $\omega_{\rm ff}$. The power is normalized by the total over all frequencies, $\ell$, and $m$. Black curves show the sum over all $m$ and $\ell$, while colored curves indicate the contributions from all $\ell$ but individual $m$, allowing identification of the modes responsible for the observed peaks.
• Figure 5: Power spectra of the kinetic energy at $r=0.85r_o$, as a function of azimuthal order $m$ and dimensionless temporal frequency $\omega/2\Omega_0$ in the rotating frame. The left panels shows the antisymmetric contribution to the power (summing only over the signals with $\ell - m$ odd), while the right panels shows the symmetric contribution to the power (summing only the signals with $\ell - m$ even). The dashed lines indicate $\omega / 2\Omega_0 \simeq m \Delta\Omega / 2\Omega_0$, where $\Delta\Omega = |\min(\Omega)|$ represents the maximum retrograde (negative) shear of the mean flow in the rotating frame. Results are presented for simulations with $\mathrm{Ro_c} \leq 0.53$.