Development of non-modal shear induced instabilities in atmospheric tornadoes
In this paper we consider the role of nonmodal instabilities in the dynamics of atmospheric tornadoes. For this purpose, we consider the Euler equation, continuity equation and the equation of state and linearize them. As an example we study several different velocity profiles: the so-called Rankine vortex model; the Burgers-Rott vortex model; Sullivan and modified Sullivan vortex models. It has been shown that in the two dimensional Rankine vortex model no instability appears in the inner region of a tornado. On the contrary, outside this area the physical system undergoes strong exponential instability. We have found that initially perturbed velocity components lead to amplified sound wave excitations. The similar results have been shown in Burgers-Rott vortex model as well. As it was numerically estimated, in this case, the unstable wave increases its energy by a factor of $400$ only in $\sim 0.5$min. According to the numerical study, in Sullivan and modified Sullivan models, the instability does not differ much by the growth. Despite the fact that in the inner area the exponential instability does not appear in a purely two dimensional case, we have found that in the modified Sullivan vortex even a small contribution from vertical velocities can drive unstable nonmodal waves.