Vortical disturbances in a linearly stratified linear shear flow. I. Linear theory

Abstract

The purpose of this theoretical study is to explore the combined effect of shear and stratification on the evolution of a three-dimensional localized vortical disturbance induced by an initial temperature perturbation embedded in linearly (stable) stratified linear shear flow. The base flow velocity includes only the horizontal component which together with the temperature field varies in the vertical direction. The geometrically small confinement of the disturbance relative to the characteristic scales of variation of the base flow velocity and temperature fields justifies the assumption that the vertical gradients of base flow velocity and temperature are uniform. Assuming a sufficiently weak disturbance, linear theory is applied for calculating the disturbed vorticity and temperature fields. The solution is carried out by first transforming the set of equations to Fourier space and a subsequent transition into Lagrangian variables. It is shown that the growth of the strength of the vortex (measured by its integrated volumetric enstrophy and its circulation) is caused by both shear and stratification while the characteristic curve of amplification (monotonic or oscillatory) depends on which one of the two factors dominates (shear or stratification). In the non-dissipative case, the enstrophy growth continues indefinitely (within the framework of linear theory); however, dissipative effects (viscosity and thermal diffusion) modify this growth to become transient. The disturbance is stretched in the streamwise direction by the shear. As long as the initial temperature disturbance is spherically localized and sufficiently weak, the centers of the induced vortical and temperature disturbances remain in their original position where the streamwise base flow velocity is zero. The rise of the disturbance and its consequence streamwise movement as a whole does not occur as it is a nonlinear effect. This and other nonlinear effects are considered in our complementary paper [Weiss Tewner et al., “Vortical disturbances in a linearly stratified linear shear flow. II. Nonlinear evolution,” Phys. Fluids 27, 024104 (2015)].