On self-sustaining processes in Rayleigh-stable rotating plane Couette flows and subcritical transition to turbulence in accretion disks
Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Centre for Mathematical Sciences, Wilberforce Road, Cambridge CB3 0WA, UK e-mail: F.Rincon@damtp.cam.ac.uk
2 Laboratoire d'hydrodynamique (LadHyX), CNRS – École polytechnique, 91128 Palaiseau Cedex, France
Accepted: 6 December 2006
Context.Subcritical transition to turbulence in Keplerian accretion disks is still a controversial issue and some theoretical progress is required in order to determine whether or not this scenario provides a plausible explanation for the origin of angular momentum transport in non-magnetized accretion disks.
Aims.Motivated by the recent discoveries of exact nonlinear steady self-sustaining solutions in linearly stable non-rotating shear flows, we attempt to compute similar solutions in Rayleigh-stable rotating plane Couette flows and to identify transition mechanisms in such flows.
Methods.Numerical simulations (time-stepping approaches) have been employed in most recent studies of the problem. Here, we instead combine nonlinear continuation methods and asymptotic theory to study the nonlinear dynamics of plane Couette flows in Rayleigh-stable regimes.
Results.We obtain exact nonlinear solutions for Rayleigh-stable cyclonic regimes using continuation techniques proposed by Waleffe and Nagata but show that it is not possible to compute solutions for Rayleigh-stable anticyclonic regimes, including Keplerian flow, using similar techniques. We also present asymptotic descriptions of these various problems at large Reynolds numbers that provide some insight into the differences between the non-rotating and Rayleigh-stable anticyclonic regimes and derive some necessary conditions for mechanisms analogous to the non-rotating self-sustaining process to be present in (constant angular momentum) flows on the Rayleigh line.
Conclusions.Our results demonstrate that subcritical transition mechanisms cannot be identified in wall-bounded Rayleigh-stable anticyclonic shear flows by transposing directly the phenomenology of subcritical transition in cyclonic and non-rotating wall-bounded shear flows. Asymptotic developments, however, leave open the possibility that nonlinear self-sustaining solutions may exist in unbounded or periodic flows on the Rayleigh line. These could serve as a starting point to discover solutions in Rayleigh-stable flows, but the nonlinear stability of Keplerian accretion disks remains to be determined.
Key words: accretion, accretion disks / hydrodynamics / instabilities / turbulence
© ESO, 2007