This paper further examines the rate at which potential vorticity in the core of a monotonic cyclone becomes vertically aligned and horizontally axisymmetric. We consider the case in which symmetrization occurs by the damping of a discrete vortex Rossby (VR) wave. The damping of the VR wave is caused by its stirring of potential vorticity at a critical radius r*, outside the core of the cyclone. The decay rate generally increases with the radial gradient of potential vorticity at r*. Previous theories for the decay rate were based on “balance models” of the vortex dynamics. Such models filter out inertia–buoyancy (IB) oscillations, i.e., gravity waves. However, if the Rossby number is greater than unity, the core VR wave can excite a frequency-matched outward propagating IB wave, which has positive feedback. To accurately account for this radiation, we here develop a theory for the decay rate that is based on the hydrostatic primitive equations. Starting from conservation of wave activity (angular pseudomomentum), an expression for the decay rate is derived. This expression explicitly demonstrates a competition between the destabilizing influence of IB wave emission, and the stabilizing influence of potential vorticity stirring at r*. Moreover, it shows that if the radial gradient of potential vorticity at r* exceeds a small threshold, the VR wave will decay, and the vortex will symmetrize, even at large Rossby numbers.

1.
J. C.
McWilliams
,
J. B.
Weiss
, and
I.
Yavneh
, “
The vortices of homogeneous geostrophic turbulence
,”
J. Fluid Mech.
401
,
1
(
1999
).
2.
J.
Enagonio
and
M. T.
Montgomery
, “
Tropical cyclogenesis via convectively forced Rossby waves in a shallow water primitive equation model
,”
J. Atmos. Sci.
44
,
685
(
2001
).
3.
J. D.
Moller
and
M. T.
Montgomery
, “
Vortex Rossby waves and hurricane intensification in a barotropic model
,”
J. Atmos. Sci.
56
,
1674
(
1999
).
4.
M. T.
Montgomery
and
J.
Enagonio
, “
Tropical cyclogenesis via convectively forced vortex rossby waves in a three-dimensional quasigeostrophic model
,”
J. Atmos. Sci.
55
,
3176
(
1998
).
5.
L. M.
Polvani
, “
Two-layer geostrophic vortex dynamics. Part 2: Alignment and two-layer V-states
,”
J. Fluid Mech.
225
,
241
(
1991
).
6.
F.
Viera
, “
On the alignment and axisymmetrization of a vertically tilted geostrophic vortex
,”
J. Fluid Mech.
289
,
29
(
1995
).
7.
G. G.
Sutyrin
,
J. C.
McWilliams
, and
R.
Saravanan
, “
Co-rotating stationary states and vertical alignment of geostrophic vortices with thin cores
,”
J. Fluid Mech.
357
,
321
(
1998
).
8.
P. D.
Reasor
and
M. T.
Montgomery
, “
Three-dimensional alignment and co-rotation of weak, TC-like vortices via linear vortex-Rossby-waves
,”
J. Atmos. Sci.
58
,
2306
(
2001
).
9.
D. A.
Schecter
,
M. T.
Montgomery
, and
P. D.
Reasor
, “
A theory for the vertical alignment of a quasigeostrophic vortex
,”
J. Atmos. Sci.
59
,
150
(
2002
).
10.
For a related discussion of 2D symmetrization, the reader may consult the following:
M. V.
Melander
,
J. C.
McWilliams
, and
N. J.
Zabusky
, “
Axisymmetrization and vorticity gradient intensification of an isolated two-dimensional vortex through filamentation
,”
J. Fluid Mech.
178
,
137
(
1987
).
11.
J. Pedlosky, Geophysical Fluid Dynamics (Springer, New York, 1987), Chap. 6.
12.
J. C.
McWilliams
, “
A uniformly valid model spanning the regimes of geostrophic and isotropic, stratified turbulence: Balanced turbulence
,”
J. Atmos. Sci.
42
,
1773
(
1985
).
13.
Vortex Rossby waves are analogous to planetary Rossby waves; however, they were first studied in a simplified context by Kelvin. See
Lord
Kelvin
, “
On the vibrations of a columnar vortex
,”
Philos. Mag.
10
,
155
(
1880
).
14.
R. J.
Briggs
,
J. D.
Daugherty
, and
R. H.
Levy
, “
Role of Landau damping in crossed-field electron beams and inviscid shear flow
,”
Phys. Fluids
13
,
421
(
1970
).
15.
S.
Pillai
and
R. W.
Gould
, “
Damping and trapping in 2D inviscid fluids
,”
Phys. Rev. Lett.
73
,
2849
(
1994
).
16.
N. R.
Corngold
, “
Linear response of the two-dimensional pure electron plasma: Quasimodes for some model profiles
,”
Phys. Plasmas
2
,
620
(
1995
).
17.
R. L.
Spencer
and
S. N.
Rasband
, “
Damped diocotron quasi-modes of nonneutral plasmas and inviscid fluids
,”
Phys. Plasmas
4
,
53
(
1997
).
18.
D. A.
Schecter
,
D. H. E.
Dubin
,
A. C.
Cass
,
C. F.
Driscoll
,
I. M.
Lansky
, and
T. M.
O’Neil
, “
Inviscid damping of asymmetries on a two-dimensional vortex
,”
Phys. Fluids
12
,
2397
(
2000
).
19.
N. J.
Balmforth
,
S. G.
Llewellyn Smith
, and
W. R.
Young
, “
Disturbing vortices
,”
J. Fluid Mech.
426
,
95
(
2001
).
20.
D. A.
Schecter
and
M. T.
Montgomery
, “
On the symmetrization rate of an intense geophysical vortex
,”
Dyn. Atmos. Oceans
37
,
55
(
2003
).
21.
L. J.
Shapiro
and
M. T.
Montgomery
, “
A three-dimensional balance theory for rapidly rotating vortices
,”
J. Atmos. Sci.
50
,
3322
(
1993
).
22.
S.
Ren
, “
Further results on the stability of rapidly rotating vortices in the asymmetric balance formulation
,”
J. Atmos. Sci.
56
,
475
(
1999
).
23.
J. C.
McWilliams
,
L. P.
Graves
, and
M. T.
Montgomery
, “
A formal theory for vortex Rossby waves and vortex evolution
,”
Geophys. Astrophys. Fluid Dyn.
97
,
275
(
2003
).
24.
R.
Plougonven
and
V.
Zeitlin
, “
Internal gravity wave emission from a pancake vortex: An example of wave–vortex interaction in strongly stratified flows
,”
Phys. Fluids
14
,
1259
(
2002
).
25.
R.
Ford
, “
The instability of an axisymmetric vortex with monotonic potential vorticity in rotating shallow water
,”
J. Fluid Mech.
280
,
303
(
1994
).
26.
R.
Ford
, “
The response of a rotating ellipse of uniform potential vorticity to gravity wave radiation
,”
Phys. Fluids
6
,
3694
(
1994
).
27.
K. C.
Chow
and
K. L.
Chan
, “
Angular momentum transports by moving spiral waves
,”
J. Atmos. Sci.
60
,
2004
(
2003
).
28.
R.
Ford
,
M. E.
McIntyre
, and
W. A.
Norton
, “
Balance and the slow quasi-manifold: Some explicit results
,”
J. Atmos. Sci.
57
,
1236
(
2000
).
29.
S.
Saujani
and
T. G.
Shepherd
, “
Comments on ‘Balance and the slow quasimanifold: Some explicit results,’ 
J. Atmos. Sci.
59
,
2874
(
2002
).
30.
R.
Ford
,
M. E.
McIntyre
, and
W. A.
Norton
, “
Reply to Comment by S. Saujani and T. G. Shepherd on ‘Balance and the slow quasimanifold: Some explicit results,’ 
J. Atmos. Sci.
59
,
2878
(
2002
).
31.
E. G.
Broadbent
and
D. W.
Moore
, “
Acoustic destabilization of vortices
,”
Philos. Trans. R. Soc. London, Ser. A
290
,
353
(
1979
).
32.
W. M.
Chan
,
K.
Shariff
, and
T. H.
Pulliam
, “
Instabilities of two-dimensional inviscid compressible vortices
,”
J. Fluid Mech.
253
,
173
(
1993
).
33.
J. C.
McWilliams
,
I.
Yavneh
,
M. J. P.
Cullen
, and
P. R.
Gent
, “
The breakdown of large-scale flows in rotating, stratified fluids
,”
Phys. Fluids
10
,
3178
(
1998
).
34.
M. T.
Montgomery
and
R. J.
Kallenbach
, “
A theory of vortex Rossby-waves and its application to spiral bands and intensity changes in hurricanes
,”
Q. J. R. Meteorol. Soc.
123
,
435
(
1997
).
35.
A. P.
Bassom
and
A. D.
Gilbert
, “
The spiral wind-up of vorticity in an inviscid planar vortex
,”
J. Fluid Mech.
371
,
109
(
1998
).
36.
G.
Brunet
and
M. T.
Montgomery
, “
Vortex Rossby waves on smooth circular vortices: I. Theory
,”
Dyn. Atmos. Oceans
35
,
153
(
2002
).
37.
M. T.
Montgomery
and
G.
Brunet
, “
Vortex Rossby waves on smooth circular vortices: II. Idealized numerical experiments for tropical cyclone and polar vortex interiors
,”
Dyn. Atmos. Oceans
35
,
179
(
2002
).
38.
Y.
Chen
and
M. K.
Yau
, “
Spiral bands in a simulated hurricane. Part I: Vortex Rossby wave verification
,”
J. Atmos. Sci.
58
,
2128
(
2001
).
39.
Y.
Chen
,
G.
Brunet
, and
M. K.
Tau
, “
Spiral bands in a simulated hurricane. Part II: Wave activity diagnostics
,”
J. Atmos. Sci.
60
,
1239
(
2003
).
40.
Y.
Wang
, “
Vortex Rossby waves in a numerically simulated tropical cyclone. Part I: Overall structure, potential vorticity, and kinetic energy budgets
,”
J. Atmos. Sci.
59
,
1213
(
2002
).
41.
Y.
Wang
, “
Vortex Rossby waves in a numerically simulated tropical cyclone. Part II: The role in tropical cyclone structure and intensity changes
,”
J. Atmos. Sci.
59
,
1239
(
2002
).
42.
P. D. Reasor, M. T. Montgomery, and L. D. Grasso, “A new look at the problem of tropical cyclones in shear flow: Vortex resiliency,” J. Atmos. Sci. (in press).
43.
D. A. Schecter and M. T. Montgomery, “The symmetrization rate of a geophysical vortex: Extension of theory to large Rossby numbers,” Proceedings of the 14th Conference on Atmospheric and Oceanic Fluid Dynamics (American Meteorological Society, San Antonio, TX, 2003), p. 48.
44.
W. D.
Smyth
and
J. C.
McWilliams
, “
Instability of an axisymmetric vortex in a stably stratified rotating environment
,”
Theor. Comput. Fluid Dyn.
11
,
305
(
1998
).
45.
The eventual destabilizing effect of nonlinear critical layer stirring has been observed in other systems. See, for example,
N. J.
Balmforth
and
Y.-N.
Young
, “
Stratified Kolmogorov flow
,”
J. Fluid Mech.
450
,
131
(
2002
).
46.
T. B.
Mitchell
and
C. F.
Driscoll
, “
Symmetrization of 2D vortices by beat-wave damping
,”
Phys. Rev. Lett.
73
,
2196
(
1994
).
47.
A.
Kubokawa
, “
Instability of a geostrophic front and its energetics
,”
Geophys. Astrophys. Fluid Dyn.
33
,
223
(
1985
).
48.
N. J.
Balmforth
, “
Shear instability in shallow water
,”
J. Fluid Mech.
387
,
97
(
1998
).
49.
E. S.
Benilov
, “
Short-wave, localized disturbances in jets, with applications to flows on a beta plane with topography
,”
Phys. Fluids
15
,
718
(
2003
).
50.
J. C. B.
Papaloizou
and
J. E.
Pringle
, “
The dynamical stability of differentially rotating discs—III
,”
Mon. Not. R. Astron. Soc.
225
,
267
(
1987
).
51.
J. C. B.
Papaloizou
and
D. N. C.
Lin
, “
Theory of accretion disks I: Angular momentum transport processes
,”
Annu. Rev. Astron. Astrophys.
33
,
505
(
1995
).
52.
I. G.
Shukhman
, “
Nonlinear evolution of spiral density waves generated by the instability of the shear layer in a rotating compressible fluid
,”
J. Fluid Mech.
233
,
587
(
1990
).
53.
B. J.
Hoskins
and
F. P.
Bretherton
, “
Atmospheric frontogenesis models: Mathematical formulation and solution
,”
J. Atmos. Sci.
29
,
11
(
1972
).
54.
M. E.
McIntyre
, “
On the ‘wave-momentum’ myth
,”
J. Fluid Mech.
106
,
331
(
1981
).
55.
I. M.
Held
, “
Pseudomomentum and orthogonality of modes in shear flows
,”
J. Atmos. Sci.
42
,
2280
(
1985
).
56.
I. M.
Held
and
P. J.
Phillips
, “
Linear and nonlinear barotropic decay on the sphere
,”
J. Atmos. Sci.
42
,
200
(
1987
).
57.
M. E.
McIntyre
and
T. G.
Shepherd
, “
An exact local conservation theorem for finite amplitude disturbances to non-parallel shear-flows, with remarks on Hamiltonian structure and on Arnol’d’s stability theorems
,”
J. Fluid Mech.
181
,
527
(
1987
).
58.
P. H.
Haynes
, “
Forced, dissipative generalizations of finite amplitude wave-activity conservation relations for zonal and nonzonal basic flows
,”
J. Atmos. Sci.
45
,
2352
(
1988
).
59.
T. A.
Guinn
and
W. H.
Schubert
, “
Hurricane spiral bands
,”
J. Atmos. Sci.
50
,
3380
(
1993
).
60.
The minimum frequency of an environmental IB wave is the Coriolis parameter f (see Ref. 61). In general, the VR wave frequency satisfies ωR∼nZ0. Frequency matching can occur only if ωR⩾f. For a Rossby wave with n of order unity, this condition becomes Ro≳1.
61.
A. E. Gill, Atmosphere-Ocean Dynamics (Academic, San Diego, CA, 1982), p. 258.
62.
M. T.
Montgomery
and
L. J.
Shapiro
, “
Generalized Charney–Stern and Fjortoft theorems for rapidly rotating vortices
,”
J. Atmos. Sci.
52
,
1829
(
1995
).
63.
G. Birkhoff and G. C. Rota, Ordinary Differential Equations (Wiley, New York, 1989), p. 44.
64.
As noted in Sec. I, Papaloizou and Pringle (Ref. 50) derived an analogous formula for the growth rate of an accretion disk wave. Such waves are affected by both critical layer stirring and acoustic radiation.
65.
M. T.
Montgomery
and
C.
Lu
, “
Free waves on barotropic vortices. I. Eigenmode structure
,”
J. Atmos. Sci.
54
,
1868
(
1997
).
66.
D. S.
Nolan
and
M. T.
Montgomery
, “
Nonhydrostatic, three-dimensional perturbations to balanced, hurricane-like vortices. Part I: Linearized formulation, stability and evolution
,”
J. Atmos. Sci.
59
,
2989
(
2002
).
67.
K. C.
Chow
,
K. L.
Chan
, and
A. K. H.
Lau
, “
Generation of moving spiral bands in tropical cyclones
,”
J. Atmos. Sci.
59
,
2930
(
2002
).
68.
J. Mathews and R. L. Walker, Mathematical Methods of Physics (Addison-Wesley, Redwood City, CA, 1970), p. 481.
This content is only available via PDF.
You do not currently have access to this content.