A study of transient dynamics of perturbations in Keplerian discs using a variational approach

TL;DR

This paper investigates the transient growth of perturbations in Keplerian discs using a variational approach, revealing conditions under which large-scale vortices can be significantly amplified, especially in non-Keplerian regions.

Contribution

It introduces a variational method to analyze transient dynamics in Keplerian discs, highlighting the potential for significant vortex amplification in certain conditions.

Findings

Optimal growth scales with $(\,\Omega/\kappa)^4$ for large azimuthal wavelengths.

Large-scale vortices can be transiently amplified by dozens of times within a few Keplerian periods.

Global perturbations with wavelengths much larger than the disc thickness can still experience substantial growth.

Abstract

We study linear transient dynamics in a thin Keplerian disc employing a method based on variational formulation of optimisation problem. It is shown that in a shearing sheet approximation due to a prominent excitation of density waves by vortices the most rapidly growing shearing harmonic has azimuthal wavelength, $\lambda_y$, of order of the disc thickness, $H$, and its initial shape is always nearly identical to a vortex having the same potential vorticity. Also, in the limit $\lambda_y\gg H$ the optimal growth $G\propto (\Omega/\kappa)^4$, where $\Omega$ and $\kappa$ stand for local rotational and epicyclic frequencies, respectively, what suggests that transient growth of large scale vortices can be much stronger in areas with non-Keplerian rotation, e.g. in the inner parts of relativistic discs around the black holes. We estimate that if disc is already in a turbulent state with effective viscosity given by the Shakura parameter $\alpha<1$, the considered large scale vortices with wavelengths $H/\alpha>\lambda_y>H$ have the most favourable conditions to be transiently amplified before they are damped. At the same time, turbulence is a natural source of the potential vorticity for this transient activity. We extend our study to a global spatial scale showing that global perturbations with azimuthal wavelengths more than an order of magnitude greater than the disc thickness still are able to attain the growth of dozens of times in a few Keplerian periods at the inner boundary of disc.