An optimal transient growth of small perturbations in thin gaseous discs

A thin gaseous disc with an almost keplerian angular velocity profile, bounded by a free surface and rotating around point-mass gravitating object is nearly spectrally stable. Despite that the substantial transient growth of linear perturbations measured by the evolution of their acoustic energy is possible. This fact is demonstrated for the simple model of a non-viscous polytropic thin disc of a finite radial size where the small adiabatic perturbations are considered as a linear combination of neutral modes with a corotational radius located beyond the outer boundary of the flow.Comment: 15 pages, 5 figures, accepted for publication in Ast

Subcritical transition to turbulence in accretion disc boundary layer

Context. 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. The 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 the order of the rotational frequency remains an issue. Aims. We put forward a conjecture 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. Methods. Transient growth of cross-rolls is studied analytically and further confronted with direct numerical simulations (DNS) of the dynamics of non-linear perturbations in the shearing box approximation. Results. A substantial decrease of transition Reynolds number RT is revealed as one changes a cubic box to a tall box. DNS performed in a tall box show that RT as a 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 is three times higher than the rotational frequency, when RT ∼ 50 000. Conclusions. Assuming that the cross-rolls are also responsible for turbulence in the Keplerian flow, we estimate R T ≲ 108 in this case. Our results imply that non-linear stability of Keplerian flow should be verified by extending turbulent solutions found in the cyclonic regime across the solid-body line rather than entering a quasi-Keplerian regime from the side of the Rayleigh line. 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