How tidal waves interact with convective vortices in rapidly rotating planets and stars

¹ AIM, CEA, CNRS, Université Paris-Saclay, Université Paris Cité, Orme des Merisiers Bât 709, 91191 Gif-sur-Yvette Cedex, France
² Institute for Theoretical Astroparticle Physics, Karlsruhe Institute of Technology (KIT), Campus North, Bldg. 401, Postfach 3640 76021 Karlsruhe, Germany
³ Centre for Fluid and Complex Systems, Coventry University, Priory Street, Coventry CV1 5FB, UK
e-mail: junho.park@coventry.ac.uk
⁴ Department of Engineering Sciences and Applied Mathematics, Northwestern University, 2145 Sheridan Road, Evanston IL 60208, USA
⁵ CIERA, Northwestern University, 1800 Sherman Ave, Evanston IL 60201, USA
⁶ Department of Applied Mathematics, School of Mathematics, University of Leeds, Woodhouse Lane, Leeds, LS2 9JT, UK

Received: 18 March 2022
Accepted: 6 March 2023

Abstract

Context. The dissipation of tidal inertial waves in planetary and stellar convective regions is one of the key mechanisms that drive the evolution of star–planet and planet–moon systems. This dissipation is particularly efficient for young low-mass stars and gaseous giant planets, which are rapid rotators. In this context, the interaction between tidal inertial waves and turbulent convective flows must be modelled in a realistic and robust way. In the state-of-the-art simulations, the friction applied by convection on tidal waves is commonly modeled as an effective eddy viscosity. This approach may be valid when the characteristic length scales of convective eddies are smaller than those of the tidal waves. However, it becomes highly questionable in the case where tidal waves interact with potentially stable large-scale vortices such as those observed at the poles of Jupiter and Saturn. The large-scale vortices are potentially triggered by convection in rapidly-rotating bodies in which the Coriolis acceleration forms the flow in columnar vortical structures along the direction of the rotation axis.

Aims. We investigate the complex interactions between a tidal inertial wave and a columnar convective vortex.

Methods. We used a quasi-geostrophic semi-analytical model of a convective columnar vortex, which is validated by numerical simulations. First, we carried out linear stability analysis using both numerical and asymptotic Wentzel–Kramers–Brillouin–Jeffreys (WKBJ) methods. We then conducted linear numerical simulations of the interactions between a convective columnar vortex and an incoming tidal inertial wave.

Results. The vortex we consider is found to be centrifugally stable in the range –Ω_p ≤ Ω₀ ≤ 3.62Ω_p and unstable outside this range, where Ω₀ is the local rotation rate of the vortex at its center and Ω_p is the global planetary (stellar) rotation rate. From the linear stability analysis, we find that this vortex is prone to centrifugal instability with perturbations with azimuthal wavenumbers m = {0,1, 2}, which potentially correspond to eccentricity, obliquity, and asynchronous tides, respectively. The modes with m > 2 are found to be neutral or stable. The WKBJ analysis provides analytic expressions of the dispersion relations for neutral and unstable modes when the axial (vertical) wavenumber is sufficiently large. We verify that in the unstable regime, an incoming tidal inertial wave triggers the growth of the most unstable mode of the vortex. This would lead to turbulent dissipation. For stable convective columns, the wave-vortex interaction leads to the mixing of momentum for tidal inertial waves while it creates a low-velocity region around the vortex core and a new wave-like perturbation in the form of a progressive wave radiating in the far field. The emission of this secondary wave is the strongest when the wavelength of the incoming wave is close to the characteristic size (radius) of the vortex. Incoming tidal waves can also experience complex angular momentum exchanges locally at critical layers of stable vortices.

Conclusions. The interaction between tidal inertial waves and large-scale coherent convective vortices in rapidly-rotating planets (stars) leads to turbulent dissipation in the unstable regime and complex behaviors such as mixing of momentum and radiation of new waves in the far field or wave-vortex angular momentum exchanges in the stable regime. These phenomena cannot be modeled using a simple effective eddy viscosity.