Linear Dynamics of the Multiple-Vortex Phenomenon in Tornadoes

Abstract

The reason for existence of two separate unstable modes, previously described by Gall for flows in vortex simulators, is explored. When the energy equation for an unstable disturbance is considered, it is clear that the most unstable wave must be centered inside the maximum in the vertical vorticity of the basic state if this vorticity has a radial distribution that is triangular-shaped and this triangle is near the center of the vortex. When this vorticity is at large radius, the most unstable wave can be centered near or even outside the basic-state vorticity maximum. This suggests different modes when the triangular profile of vorticity is near or far from the center and that the transition from one to another mode should be gradual. These notions are verified by a careful analysis of the stability properties of the triangular-shaped vorticity profile. It is shown that the triangular-shaped vorticity profile closely resembles the vorticity distribution in the vortex simulator after a downdraft has been established along the centerline of the vortex. In fact, the stability properties of these triangular profiles closely resemble the stability properties of the simulated vortex when the scale of the triangular profile is comparable to the vorticity distribution in the simulators. Square-shaped vorticity profiles, which have been considered in the past, have significantly different stability properties, as compared to the triangular profiles. In particular, there is only one unstable mode, and the instability is extinguished as the vorticity region approaches the center of the basic vortex. The reason for this is easily explained by considering the perturbation energy equation.