Centrifugal Waves in Tornado-Like Vortices: Kelvin’s Solutions and Their Applications to Multiple-Vortex Development and Vortex Breakdown

Abstract

AbstractAbout 140 years ago, Lord Kelvin derived the equations describing waves that travel along the axis of concentrated vortices such as tornadoes. Although Kelvin’s vortex waves, also known as centrifugal waves, feature prominently in the engineering and fluid dynamics literature, they have not attracted as much attention in the field of atmospheric science. To remedy this circumstance, Kelvin’s elegant derivation is retraced, and slightly generalized, to obtain solutions for a hierarchy of vortex flows that model basic features of tornado-like vortices. This treatment seeks to draw attention to the important work that Lord Kelvin did in this field, and reveal the remarkably rich structure and dynamics of these waves. Kelvin’s solutions help explain the vortex breakdown phenomenon routinely observed in modeled tornadoes, and it is shown that his work is compatible with the widely used criticality condition put forth by Benjamin in 1962. Moreover, it is demonstrated that Kelvin’s treatment, with the slight generalization, includes unstable wave solutions that have been invoked to explain some aspects of the formation of multiple-vortex tornadoes. The analysis of the unstable solutions also forms the basis for determining whether, for example, an axisymmetric or a spiral vortex breakdown occurs. Kelvin’s work thus helps explain some of the visible features of tornado-like vortices.