In the most cases of thermally driven flows the inner vortex moves upward because of the buoyancy force, one of the effects of this motion is the characteristic folding behind the raising air mass. We can often observe also the Kelvin-Helmholtz instability on the edge of a rising thermal. I have attached a simple computer simulation depicting solution of the inviscid Euler equations for such a case; the initial condition assumed fluid at the state of rest in the external gravity field and a small initial perturbation of the temperature.
Creation of the outer downward vortex is often due to shedding of vortices in the wake of a thermal or in result of additional thermal forcing.