Subcritical transition to turbulence in accretion disc boundary layer

TL;DR

This paper investigates the subcritical transition to turbulence in accretion disc boundary layers, proposing a mechanism involving three-dimensional vortices generated by swing amplification, supported by analytical and numerical studies.

Contribution

It introduces a new hydrodynamical instability mechanism involving cross-roll vortices driven by transient growth, explaining turbulence onset in accretion disc boundary layers.

Findings

Transition Reynolds number $R_T$ is about 50,000 in tall box simulations.

Transition Reynolds number could be as high as $10^8$ in Keplerian flows.

Sub-Keplerian shear rate (~1/2) is most favorable for turbulence testing.

Abstract

Enhanced angular momentum transfer through the boundary layer near the surface of weakly magnetised accreting star is required in order to explain the observed accretion timescales in low-mass X-ray binaries, cataclysmic variables or young stars with massive protoplanetary discs. Accretion disc boundary layer is locally represented by incompressible homogeneous and boundless flow of the cyclonic type, which is linearly stable. Its non-linear instability at the shear rates of order of the rotational frequency remains an issue. We argue that hydrodynamical subcritical turbulence in such a flow is sustained by the non-linear feedback from essentially three-dimensional vortices, which are generated by quasi-two-dimensional trailing shearing spirals grown to high amplitude via the swing amplification. We refer to those three-dimensional vortices as cross-rolls, since they are aligned in the shearwise direction in contrast to streamwise rolls generated by the anti-lift-up mechanism in rotating shear flow on the Rayleigh line. Transient growth of cross-rolls is studied analytically and further confronted with direct numerical simulations (DNS) of dynamics of non-linear erturbations. DNS performed in a tall box show that transition Reynolds number $R_T$ as function of shear rate accords with the line of constant maximum transient growth of cross-rolls. The transition in the tall box has been observed until the shear rate three times higher than the rotational frequency, when $R_T\sim 50000$. Assuming that the cross-rolls are also responsible for turbulence in the Keplerian flow, we estimate $R_T\lesssim 10^8$ in this case. The most favourable shear rate to test the existence of turbulence in the quasi-Keplerian regime may be sub-Keplerian and equal approximately to $1/2$.