A self-consistent numerical model of internal wave-induced mean flow oscillations in polar geometry

TL;DR

The paper develops a self-consistent two-zone polar model to study wave–mean flow interactions in star-like structures, focusing on internal gravity waves generated by a convective core and their ability to drive reversing azimuthal mean flows in the radiative envelope. By combining direct numerical simulations with a Plumb–McEwan–type reduced framework, it shows that a broad IGW spectrum can reproduce mean-flow reversals via a Hopf-bifurcation mechanism, with a critical parameter $\Lambda_1^c \approx 5.1 \times 10^{-4}$ and oscillation periods tied to the diffusive time scale. The results yield scaling relations for Reynolds stresses and wave spectra as functions of $Ra$, $Pr$, and $S$, demonstrating qualitative agreement with reduced models despite spectral complexity. The work highlights potential stellar implications for massive stars with convective cores and radiative envelopes and outlines future directions including rotation, 3D effects, and magnetic fields to better connect with realistic astrophysical contexts.

Abstract

The Earth's Quasi-Biennial Oscillation (QBO) is a natural example of wave-mean flow interaction and corresponds to the alternating directions of winds in the equatorial stratosphere. It is due to internal gravity waves (IGW) generated in the underlying convective troposphere. In stars, a similar situation is predicted to occur, with the interaction of a stably-stratified radiative zone and a convective zone. In this context, we investigate the dynamics of this reversing mean flow by modelling a stably-stratified envelope and a convectively unstable core in polar geometry. Here, the coupling between the two zones is achieved self-consistently, and IGW generated through convection lead to the formation of a reversing azimuthal mean flow in the upper layer. We characterise the mean-flow oscillations by their periods, velocity amplitudes, and regularity. Despite a continuous broad spectrum of IGW, our work show good qualitative agreement with the monochromatic model of Plumb and McEwan (1978). If the latter was originally developed in the context of the Earth's QBO, our study could prove relevant for its stellar counterpart in massive stars, which host convective cores and radiative envelopes.

Figures (10)

• Figure 1: Visualisations of convection, waves, and mean flows in the problem, in code units (see § \ref{['sec:method']}). a) Temperature field in a simulation, displaying an active core compared to the envelope. The contour value $T_i=0$ denotes the boundary between the convective (CZ) and radiative (RZ) zones. The colorbar is stretched, going from $-0.35$ (blue) to $0$ (white) in the RZ, and from $0$ to $0.05$ (red) in the CZ. b) Vorticity $\xi=\nabla \times \hbox{\boldmath $u$}$ of the flow, illustrating the typical turbulence generated in the core of the domain. c) Azimuthal flow developing in the stably-stratified layer, here showing a counter clockwise mean component whose reversal is at the core of the present study.
• Figure 2: Averaged Reynolds stress for different $Ra$ and $S$, in the CZ (a) and in the RZ (b). Note that contributions from the mean flows have been removed but keeping them gives the same results.
• Figure 3: Wave energy flux temporal spectra measurements for varying $Ra$, fixed $S$ (a-c) and fixed $Ra$, varying $S$ (d-f). Spectra are computed in the RZ $(r=0.7)$, and displayed for 3 values of horizontal wavenumbers. The dashed lines show comparisons with predictions by Lecoanet2013.
• Figure 4: Estimate of $\varLambda_1$ (\ref{['eq:paramPlumb']}) for the same points as in Figure \ref{['fig:3uphiur']}, showing how increasing $Ra$ tends to favour low $\varLambda_1$ values and therefore mean flow reversals when $\varLambda_1<\varLambda_1^c=5.10^{-4}$(see (\ref{['eq:lbda1DNS']}) for details).
• Figure 5: Mean flow visualisation for $S=1.98$. Hovmöller diagrams (Left, a-c-e, colours correspond to flow amplitudes) and phase portraits of local probes of the zonal velocity in the RZ (Right, b-d-f, colours correspond to time) for $Ra=10^{10}$ (a-b), $Ra=3.10^{10}$ (c-d) and $Ra=10^{11}$ (e-f).