Geometrical optics stability analysis of rotating visco-diffusive flows

The geometrical optics stability analysis has proved successful in solving problems of ideal hydrodynamics and magnetohydrodynamics related to the stability of 3D flows of both compressible and incompressible fluids [1-3]. The method is based on perturbations of a background flow in a small parameter, representing a short wavelength. These perturbations are localized wave envelopes moving along the trajectories of fluid elements. Thus, detecting instabilities localized near a particular fluid element location, moving with the flow, reduces to solving a system of ODEs for the wave vector and amplitude, along particle paths in the underlying flow, with coefficients depending on the unperturbed velocity field [1-3]. Inclusion of viscosity or diffusivity in the analysis was long-time believed to be only stabilizing. However, starting with the works [4,5] that considered both viscosity and magnetic diffusivity in the geometrical optics analysis of helical magnetorotational instability it became possible to extend the method to visco-diffusive [6] and multiple-diffusive [7] flows for a wide range of Prandtl, Schmidt and magnetic Prandtl numbers. In the talk we review these works and discuss new applications.

References:
[1] K.S. Eckhoff (1981) J. Differ. Equ. 40 (1), 94–115.
[2] A. Lifschitz, E. Hameiri (1991) Phys. Fluids 3, 2644–2651.
[3] S. Friedlander, M.M. Vishik (1991) Phys. Rev. Lett. 66, 2204–2206.
[4] O.N. Kirillov, F. Stefani, Y. Fukumoto (2014) JFM, 760: 591- 633.
[5] O.N. Kirillov (2017) Proc. A, 473(2205): 20170344.
[6] O.N. Kirillov, I. Mutabazi (2017) JFM, 818: 319-343.
[7] J. Labarbe, O.N. Kirillov (2021). Phys. Fluids, 33(10): 104108