Hydrodynamical instability with noise in the Keplerian accretion discs: Modified Landau equation

TL;DR

This paper investigates how stochastic forces, representing noise, can induce hydrodynamical instability and turbulence in linearly stable Keplerian accretion disks by modifying the Landau equation to include external forcing.

Contribution

It introduces a modified Landau equation accounting for stochastic forces, demonstrating how noise can trigger instability and turbulence in otherwise stable shear flows.

Findings

External forcing causes linear modes to grow to large amplitudes.

Nonlinear perturbations diverge faster with forcing, leading to rapid turbulence.

Nonlinearity emergence depends only on force, not initial perturbation amplitude.

Abstract

Origin of hydrodynamical instability and turbulence in the Keplerian accretion disc as well as similar laboratory shear flows, e.g. plane Couette flow, is a long standing puzzle. These flows are linearly stable. Here we explore the evolution of perturbation in such flows in the presence of an additional force. Such a force, which is expected to be stochastic in nature hence behaving as noise, could be result of thermal fluctuations (however small be), Brownian ratchet, grain-fluid interactions and feedback from outflows in astrophysical discs etc. We essentially establish the evolution of nonlinear perturbation in the presence of Coriolis and external forces, which is modified Landau equation. We show that even in the linear regime, under suitable forcing and Reynolds number, the otherwise least stable perturbation evolves to a very large saturated amplitude, leading to nonlinearity and plausible turbulence. Hence, forcing essentially leads a linear stable mode to unstable. We further show that nonlinear perturbation diverges at a shorter timescale in the presence of force, leading to a fast transition to turbulence. Interestingly, emergence of nonlinearity depends only on the force but not on the initial amplitude of perturbation, unlike original Landau equation based solution.