Linear evolution of the vortex induced by localized temperature disturbance in stratified shear flow

We study the combined effect of the shear flow velocity and its (stable) vertical stratification on the evolution of the three-dimensionally localized vortical disturbance induced by the initial temperature perturbation embedded at the initial time into a local region of the flow. Small geometric scales of perturbations compared to the characteristic scales of velocity and temperature variation of the background flow allow to consider vertical gradients of the horizontal velocity and temperature to be not dependent on the coordinates. Assuming a disturbance sufficiently weak, we use linear theory to calculate fields of vorticity and temperature. The problem is solved analytically using a three-dimensional Fourier transform of the basic set of equations and further transition to a Lagrange variables in the Fourier space. It is shown that the growth of the intensity of the vortex (a measure of which are enstrophy and circulation) is obliged to both stratification and shear. However, the character of this growth (monotonous or oscillating) depends on what of two factors dominates. In the case where the dissipation effects are negligible, enstrophy grows indefinitely (in the framework of the linear theory), but dissipative factors (viscosity and thermal diffusivity) modified this growth and make it only transient, so that eventually the perturbation decays. Perturbation domain stretches along the direction of flow, but its vertical and horizontal movement as a whole in the framework of the linear theory doesn’t occur, since it is the nonlinear effect. Nonlinear evolution of the vortex induced by temperature disturbance is considered in a separate paper.