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.

Suction–shear–Coriolis instability in a flow between parallel plates

A rotating fluid flow between differentially translating parallel plates, which induce uniform suction and injection, is studied as a canonical model of swirling flow where suction, shear and Coriolis effects compete. This relatively simple modelling yields several reduced equations that are valid for asymptotically large suction, shear and/or rotation rates. The linear stability problems derived from the full Navier–Stokes and reduced problems are numerically solved and compared. In addition to Taylor-vortex modes, transverse-roll-type instabilities are found in Rayleigh-stable and -unstable parameter regions when weak suction is applied. These instabilities, separated by the so-called Rayleigh line, are characterised by vortices attached to the suction wall. Another type of instability, which exists beyond the Rayleigh line and shows inviscid motion in the fluid core, is found when suction is sufficiently strong. The relation of this instability to the stability results by Gallet, Doering & Spiegel (Phys. Fluids, vol. 22, 2010, 034105) is discussed. Our nonlinear analyses indicate subcritical and supercritical bifurcations of finite-amplitude solutions for the near-wall and fluid-core instabilities, respectively.