Dynamics of Elevated Vortices

Abstract

Theoretical hydrodynamic models for the behavior of vortices with axially varying rotation rates are presented. The flows are inviscid, axisymmetric, and incompressible. Two flow classes are considered: (i) radially unbounded solid body?type vortices and (ii) vortex cores of finite radius embedded within radially decaying vortex profiles. For radially unbounded solid body?type vortices with axially varying rotation rates, the von Kármán?Bödewadt similarity principle is applicable and leads to exact nonlinear solutions of the Euler equations. A vortex overlying nonrotating fluid, a vortex overlying a vortex of different strength, and more generally, a vortex with N horizontal layers of different rotation rate are considered. These vortices cannot exist in a steady state because continuity of pressure across the horizontal interface between the vortex layers demands that a secondary (meridional) circulation be generated. These similarity solutions are characterized by radial and azimuthal velocity fields that increase with radius and a vertical velocity field that is independent of radius. These solutions describe nonlinear interactions between the vortex circulations and the vortex-induced secondary circulations, and may play a role in the dynamics of the interior regions of broad mesoscale vortices. Decaying, amplifying, and oscillatory solutions are found for different vertical boundary conditions and axial distributions of vorticity. The oscillatory solutions are characterized by pulsations of vortex strength in lower and upper levels associated with periodic reversals in the sense of the secondary circulation. These solutions provide simple illustrations of the ?vortex valve effect,? sometimes used to explain cyclic changes in updraft and rotation strength in tornadic storms. A linear analysis of the Euler equations is used to describe the short-time behavior of an elevated vortex of finite radius embedded within a radially decaying vortex profile (i.e., elevated Rankine-type vortices). The linear solution describes the formation of a central updraft (as in the similarity solution) and an annular downdraft ringing the periphery of the vortex core (not accounted for in the similarity solution). Downdraft strength is sensitive to both the vortex core aspect ratio and outer vortex decay rate, being stronger and narrower for broader vortices and larger decay rates. It is hypothesized that this dynamically induced downdraft may facilitate the transport of mesocyclone vorticity down to low levels in supercell thunderstorms.