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
- the motion of the vortex center, which turns out to
be advected by the large-scale leading-order barotropic
mean flow
- a vertical, shear-induced vortex tilt and
an associated precession reminiscent of that found by
Reasor and Montgomery [Reasor and Montgomery, 2004]
- 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.