Asymptotic analysis of the motion and structure of sheared mesoscale vortices

Abstract:

We present matched asymptotic expansions for concentrated atmospheric vortices in the gradient wind regime using the mathematical framework proposed recently in [Klein, 2004] and utilized in the construction of multiscale models in [Majda and Klein, 2003], [Klein and Majda, 2006]. Our presentation extends related two-dimensional theories by Ting and Ling [Ling and Ting, 1988] and subsequent work by Reznik and co-workers [Reznik, 1992] to three dimensions, allowing for weak vertical shear and diabatic source terms. Preliminary results were summarized in [Mikusky, 2007].

Focusing on dry flows with explicitly prescribed diabatic forcings, we consider a nearly axisymmetric vortex and allow for strong vertical tilt comparable in magnitude to the vortex diameter. The vortex is embedded, in the sense of matched asymptotics, in a quasi-geostrophic background flow. We obtain a closed asymptotic theory that describes

1. the motion of the vortex center, which turns out to be advected by the large-scale leading-order barotropic mean flow
2. a vertical, shear-induced vortex tilt and an associated precession reminiscent of that found by Reasor and Montgomery [Reasor and Montgomery, 2004]
3. the vortex core evolution, which is governed by momentum drag and radial advection of total angular momentum.

Importantly, the radial advection velocities can be positive or negative depending on the relative arrangement of vortex tilt and diabatic heating. As a consequence, diabatic heating and tilt can amplify or weaken the vortex depending on that same arrangement.

Bibliography

Klein, 2004

Klein, R. (2004). An Applied Mathematical View of Theoretical Meteorology (invited lecture ICIAM 2003), volume 116 of SIAM Proceedings in Applied Mathematics. SIAM.

Klein and Majda, 2006

Klein, R. and Majda, A. J. (2006). Systematic multiscale models for deep convection on mesoscales. Theoretical and Computational Fluid Dynamics, 20:525-552.

Ling and Ting, 1988

Ling, G. and Ting, L. (1988). Two-time scales inner solutions and motion of a geostrophic vortex. Scienta Sinica, XXXI(7).

Majda and Klein, 2003

Majda, A. J. and Klein, R. (2003). Systematic multi-scale models for the tropics. Journal of the Atmospheric Sciences, 60:393-408.

Mikusky, 2007

Mikusky, E. (2007). On the structure of concentrated atmospheric vortices in a gradient wind regime and its motion on synoptic scales. PhD thesis, Universität Hamburg, Fachbereich Geowissenschaften.

Reasor and Montgomery, 2004

Reasor, P. D. and Montgomery, M. T. (2004). A new look an the problem of tropical cyclones in vertical shear flow. Journal of the Atmospheric Sciences, 61(1).

Reznik, 1992

Reznik, G. M. (1992). Dynamics of singular vortices on a beta-plane. Journal of Fluid Mechanics, 240:405-432.