On the three-dimensional instabilities of plane flows subjected to Coriolis force

Abstract

Linear stability of two-dimensional flows in a frame rotating with angular velocity vector Ω=Ωez perpendicular to their plane is considered. Sufficient conditions for instability have been derived for simple inviscid flows, namely parallel shear flows (characterized by the “Pedley” or “Bradshaw-Richardson” number), circular vortices (by the “generalized Rayleigh” discriminant) and unbounded flows having a quadratic streamfunction (with elliptical, rectilinear or hyperbolic streamlines). These exact criteria are reviewed and contrasted using stability analysis for both three-dimensional disturbances and oversimplified “pressureless” versions of the linear theory. These suggest that one defines a general inviscid criterion for rotation and curvature, based on the sign of the second invariant of the “inertial tensor,” and stating that, in a Cartesian coordinate frame: a sufficient condition for instability is thatΦ(x,y)=−12S:S+14Wt⋅Wt<0 somewhere in the flow domain. It involves the “tilting vorticity” Wt=W+4Ω [Cambon et al., J. Fluid Mech. 278, 175 (1994)] and the symmetric part S of the velocity gradient of the basic flow.