Some studies on vortices with density stratification

Abstract

The effects of density stratification on the fie stability and evolution of vortices is investigated. Baroclinic vorticity generated due to density inhomogeneities can have important implications for the behavior of vortices. In atmospheres and oceans, due to strong effects of density stratification and rotation, the flow can be considered to be in a quasi two-dimensional state. Typically, large scale vortical structures are accompanied by an inverse cascade of energy owing to the two-dimensional nature of the flow field. In these situations, density stratification occurs along the vortex axis. We have analysed the effect of density variations on a single vortex, and then considered the interaction of two such vortices in a general stratified medium. A combination of linear stability analysis and direct numerical simulation of the nonlinear Navier-Stokes equations have been carried out. When density variations occur in the plane of the vortex, it is shown in this thesis that small scale instabilities arise in the flow resulting in a strong direct cascade of energy. Such variations occur in a variety of situations, example in aircraft trailing vortices in a stratified medium, in cyclones/hurricanes travelling across regions of strong density gradients as would be encountered when we move in meridional directions or across the ocean-land interface, and also in polar vortices. In all the single vortex instabilities, gravity is completely neglected, and density effects arise from the inertial terms in the governing equations. The results are also valid for sharp density interfaces in the presence of weak gravity. We begin with the linear stability of the classical piece-wise continuous mixing layer profile. Though a great deal of work has been undertaken using the 'Rayleigh' method in normal mode analysis, often, the physical mechanisms of these instabilities is not made clear. An exception is the kinematic mechanism for the instability of a vortex sheet dealt with by Batchelor in his classic text. A similar understanding in more general barotropic and baroclinic instabilities is still lacking. The linear wave interaction mechanism attempts to fill this gap in our understanding. We solve an initial value problem analytically to obtain further insight into the instabihty mechanism. Specifically, the temporal evolution of the initial disturbance into a normal mode is analysed. We then extend these ideas to the stability of a simple stratified shear flow problem with two interfaces, one with a vorticity jump and the other with a density jump. The phase relationship between the waves at these interfaces is explicitly calculated. We extend the above analysis to a cylindrical geometry. The problem we investigate is the stability of a vortex with radial density distribution. Baroclinic vorticity is generated in this geometry due to the presence of centrifugal forces. A heavy-cored vortex, where density decreases raonotonically away from the vortex axis is a potentially unstable configuration. Similarly, a light-cored vortex is expected to be stable due to the stabilizing effect of a centrifugal buoyancy force. But it is shown that even a light-cored vortex can be unstable, contrary to common intuition. The entire range of vortex profiles of smoothness varying from a Rankine vortex to a Gaussian vortex with analogous density profiles are studied. This is carried out by defining a single parameter family of vorticity and density profiles, all having the same circulation. In the case of a Rankine vortex with a step density jump, we again interpret instabilties in terms of wave-interactions between Kelvin modes of the vortex and internal waves due to density jump. For smooth prcvfiles which do not possess discrete Kelvin modes, we resort to the ideas of quasi-modes. Quasi-modes are exponentially decaying eigenmodes of the inviscid stability problem with a wave-like response, and are a manifestation of a pecuhar cooperative effect of the continuous spectrum modes localized at the critical layer. It is suggested in this thesis that quasi-modes of the vortex interact with internal waves leading to a linearly unstable flow. A detailed analysis of quasi-modes of all the vortex profiles is carried out.