Three-dimensional centrifugal-type instabilities of two-dimensional flows in rotating systems

This paper deals with the stability of incompressible inviscid planar basic flows in a rotating frame. We give a sufficient condition for such flows t o undergo three-dimensional shortwave centrifugal-type instabilities. This criterion reduces to the Bradshaw-Richardson (1969) or Pedley (1969) criter ion in the specific case of parallel shear flows subject to rotation, to Ra yleigh's centrifugal criterion (1916) in the case of axisymmetric vortices in inertial frames, to the Kloosterziel and van Heijst (1991) criterion in the case of axisymmetric vortices subject to rotation and to Bayly's criter ion (1988) in the case of general two-dimensional flows in inertial frames. The criterion states that a steady 2D basic flow subject to rotation Omega is unstable if there exists a streamline for which at each point 2(V/R+Ome ga)(W+2 Omega)< 0 where W is the vorticity of the streamline, R is the loca l algebraic radius of curvature of the streamline and V is the local norm o f the velocity. If this condition is satisfied then the flow is unstable ac cording to the geometrical optics method introduced by Lifschitz and Hameir i (1991), which consists in following wave packets along the flow trajector ies using a Wentzel-Kramers-Brillouin formalism. When the streamlines are c losed, it is further shown that a localized unstable normal mode can be con structed in the vicinity of a streamline. As an application, this new crite rion is used to study the centrifugal-type instabilities in the Stuart vort ices, which is a family of exact solutions describing a row of periodic co- rotating eddies. For each solution of that family and for each rotation par ameter f=2 Omega, we give the unstable streamline interval, according to th e criterion of instability. This criterion gives only a sufficient conditio n of centrifugal instability. The equations of the geometrical optics metho d are therefore numerically solved to obtain the true centrifugally unstabl e streamline intervals. It turns out that our criterion gives excellent res ults for highly concentrated vortices, i.e., the two approaches yield the s ame unstable streamline intervals. In less concentrated vortices, some stre amlines undergo centrifugal instability although our criterion is not fulfi lled. From these numerical results, another criterion of centrifugal instab ility for a flow with closed streamlines is conjectured which reduces to th e change of sign of the absolute vorticity W+2 Omega somewhere in the flow. (C) 2000 American Institute of Physics. [S1070-6631(00)00107-0].