Towards a theory for vortex filaments in stratified-rotating fluids
Résumé
In inviscid fluids with uniform density, it is common to idealize three-dimensional vortex tubes by filaments (i.e., single lines of an infinitesimal cross section). Thanks to the Kelvin and Helmholtz theorems, it is known that these vortex filaments are transported with the fluid and their circulation is conserved. The induced motions can be computed by the Biot–Savart law, with an appropriate cut off in the integral to avoid singularity. Hence, this approach allows one to model the linear or nonlinear dynamics of vortex flows. A priori, vortex filaments cannot be used in density-stratified and rotating fluids since the circulation is not conserved and the vortex lines are not material lines. However, in this paper we review a theory that is equivalent to vortex filaments. It is based on matched asymptotic expansions for small vortex-core size, weak curvature, and small vortex displacements. The resulting stability equations are formally identical to those of vortex filaments in homogeneous fluids. However, striking differences between homogeneous and stratified-rotating fluids exist, such as the reversal of the self-induced motion for strong stratification or complex self-induction for moderate stratification due to the presence of critical points. The three-dimensional linear stability of vertical vortex pairs and vortex arrays (Karman street, double symmetric row) in stratified and rotating fluids has been investigated using this analytical approach. The results are in very good agreement with the results of direct numerical stability analyses of smooth vortex configurations. Possible extensions to include nonlinear and baroclinic effects are briefly discussed.