On the role of sheared waves on instabilities and turbulence in rotating hydrodynamics and magneto--hydrodynamics

In this paper we make a survey of some of our previous works devoted to the studies of sheared flows with emphasis put on their applications to geophysics and astrophysics. Some relevant related topics are geophysical fluids, with possible involved dynamo effects in magnetized cases, fluid behavior in stellar interiors and the study of stability of astrophysical accretion disks. These works include the findings of new invariants and the generalization of previous results of the current literature in terms of dispersion relations, wave-vortex decomposition and so on, for various systems which all share the common feature of being under differential rotation (i .e with both pure rotation and shear) but with also possible additional effects like precession, medium stratification, buoyancy, gravity and magnetic fields. Our main tool relies on the use of rapid distorsion theory for analysis of perturbations and interactions with the mean flow in terms of sheared waves with time dependent wave vectors with further linear theory performed in the spectral space. This method allows in particular the description of sub-critical instabilities which can exhibit finite time transient growth phenomena and non linear transverse cascades, in otherwise time asymptotically linearly spectral stable flows. We stress also on the general comment/observation that for these systems the linear perturbations can be regenerated by suitable phase tuning feedbacks between them and their triggered non linear perturbations, which make possible the comparison at least asymptotically in time of the predictions of the linear spectral analysis 1 with the results obtained with full DNS (direct numerical simulations) performed on the same total systems. (This paper may be considered as a 'complement' to the paper subcritical instabilities in neutral fluids and plasmas , to appear in the same special issue)