Non-axisymmetric instabilities in discs with imposed zonal flows

TL;DR

This paper investigates non-axisymmetric instabilities in Keplerian discs with imposed zonal flows, revealing the conditions under which Kelvin-Helmholtz and gravitational instabilities occur, and how they depend on disc properties and self-gravity.

Contribution

It provides a linear stability analysis of zonal flows in discs, identifying the roles of Kelvin-Helmholtz and gravitational instabilities under various conditions, including self-gravity effects.

Findings

Kelvin-Helmholtz instability occurs at PV minima and is stable for wavelengths greater than 8H.

Self-gravity triggers a second instability around PV maxima, different from KH.

Gravitational instability can occur even with entropy structures alone, especially with increased amplitude.

Abstract

We conduct a linear stability calculation of an ideal Keplerian flow on which a sinusoidal zonal flow is imposed. The analysis uses the shearing sheet model and is carried out both in isothermal and adiabatic conditions, with and without self-gravity (SG). In the non-SG regime a structure in the potential vorticity (PV) leads to a non-axisymmetric Kelvin-Helmholtz (KH) instability; in the short-wavelength limit its growth rate agrees with the incompressible calculation by Lithwick (2007), which only considers perturbations elongated in the streamwise direction. The instability's strength is analysed as a function of the structure's properties, and zonal flows are found to be stable if their wavelength is $\gtrsim 8H$, where $H$ is the disc's scale height, regardless of the value of the adiabatic index $\gamma$. The non-axisymmetric KH instability can operate in Rayleigh-stable conditions, and it therefore represents the limiting factor to the structure's properties. Introducing SG triggers a second non-axisymmetric instability, which is found to be located around a PV maximum, while the KH instability is linked to a PV minimum, as expected. In the adiabatic regime, the same gravitational instability is detected even when the structure is present only in the entropy (not in the PV) and the instability spreads to weaker SG conditions as the entropy structure's amplitude is increased. This eventually yields a non-axisymmetric instability in the non-SG regime, albeit of weak strength, localised around an entropy maximum.