Vortex Rossby Wave Theory and Literature
The term ‘vortex
Rossby wave’ (VRW) and a formal theory for the propagation and
interaction of the
waves with the mean flow was first presented by Montgomery and
(1997) and later generalized by McWilliams et al. (2003).
They showed that the axisymmetrization of potential vorticity
anomalies, such as those generated through moist-convective processes,
strong horizontal shear of the mean vortex was accompanied by outward
propagating VRWs that accelerated the tangential winds near the radius
of wave excitation. The VRWs propagated
the radial gradient of mean storm vorticity, analogous to Rossby waves
large-scale circulation (MacDonald 1968). Montgomery
and Enagonio (1998) and Möller and
Montgomery (1999, 2000) used a three-dimensional quasigeostrophic model
fully non-linear asymmetric balance model, respectively, to show
cyclogenesis and intensification could occur through the
ingestion of like-signed PV anomalies by a parent vortex with a
vorticity structure. Furthermore,
through the continual injection of PV pulses (to simulate the effects
of continuos convection), Möller and Montgomery (2000) showed that
tropical storm-strength vortex could develop a warm core and attain
strength on realistic time scales.
exhibiting monopoles of vorticity are, however, likely to occur only
tropical storm and weak hurricane stages of development, while rapidly
intensifying or mature hurricanes with well-developed eyewalls usually
rings of elevated PV on the inner edge of the eyewall (Kossin and
Mallen et al. 2005). Wang (2001,
2002a,b) showed that the asymmetric structure within 70 km of the
center of his
modeled vortex with an annular tower of PV
was dominated by wavenumber 1 and 2 VRWs. Similar
were found by Chen et al. (2003), who showed that the leading modes in
activity in the core were VRWs, generated in the lower eyewall through
heating. The waves were found to be well
coupled to convection as the enhanced vertical velocity associated with
wave led to the appearance of inner spiral rainbands (Chen and Yau
Figure 12 of Wang (2002b)
Figure 10 of Wang (2002a) (right): (Left) a) Radar reflectivity, b)
asymmetric PV and its c) wavenumber 1 and d) wavenumber 2 components
with positive values shaded, at 850 hPa after 84 h 30 min of
simulation, showing the relationship between inner and outer rainbands
and the activity of VRWs. (Right) The wavenumber 1 component of
a) PV and b)-h) PV tendencies due to terms on the right hand side of
the perturbation PV equation and the total local PV tendency (i)
estimated from all terms at 86 h 30 min of simulation at 850 hPa.
b) Horizontal and vertical advection of perturbation PV by the
symmetric cyclone flow (LADV); c) horizontal and vertical advection of
symmetric PV by the asymmetric flow (BETA); d) PV flux divergence due
to asymmetric diabatic heating (DIABE); e) PV flux divergence due to
symmetric diabatic heating (DIABM); f) non-conservative PV flux
divergence due to symmetric friction (FRICM); g) PV flux divergence due
asymmetric friction (FRICE); h) PV tendency due to all non-linear eddy
processes (NONL). The domain shown in each panel is 120 km x 120
km and circles are every 30 km from the center. Contour interval
is 1 PVU in a) and 1 PVU day-1 in the rest with positive
(anticyclonic) PV or PV tendency shaded.
Unlike their monopole vortex counterparts discussed
previously, numerical simulations of vortices with annular PV rings
to document a consistent influence of VRWs on the intensity of the mean
vortex. Chen and Yau (2001), Möller
(2002), and Chen et al. (2003) documented maximum VRW activity near the
of maximum wind (RMW), and the transport of high angular momentum and
PV by the
VRWs radially inward, from the eyewall towards the eye.
This inward transport of high vorticity by
VRWs was associated with intensification, as the maximum tangential
spin-up occurred just inside the RMW, causing the RMW to propagate
time and leading to contraction of the eyewall.
On the other hand, Wang (2002b) and Chen and Yau (2003) found
flow interactions tended to inhibit strengthening as the VRWs acted to
the tangential winds directly in the eye and decelerate the winds at
Further complicating the dynamics in storms with
elevated rings of vorticity in the eyewall is the possibility of
instability. Schubert et al. (1999) showed
that an annular vorticity structure contains
VRWs with respect to the flow on its inner and outer vorticity
gradients. If these waves became
phase-locked, they grew
in concert and led to the exponential instability of the ring, whereby
eyewall vorticity pooled into discrete areas, creating mesovortices. Depending
on the initial conditions of the PV
ring, the mesovortices either merged over time and relaxed to a
(Schubert et al. 1999; Chen and Yau 2003), or remained separate to form
quasi-steady rotating lattice of vortices that gave the appearance of
elliptical (two mesovortices) or polygonal (four or more mesovortices)
(Kossin and Schubert 2001). In either
case, the asymmetric mixing by the mesovortices between the eye and
brought high eyewall vorticity into the eye and low vorticity from the
outward. In order to conserve angular
momentum during such a rearrangement, some high eyewall vorticity
mixed outward, taking the form of vorticity filaments, or spiral
with VRW characteristics (Kuo et al. 1999; Schubert et al. 1999).
Left, Figure 3a of Schubert et al. (1999): Vorticity
contour plots for the representative experiment. The model domain
km x 600 km, but only the inner 200 km x 200 km is shown. The
begin at 0.0005 s-1. Low vorticity values are shaded
blue and high vorticity values
are shaded red. Vorticity from 0 h to 8 h is shown.
Given the copious
amount of numerical modeling studies dealing with VRWs in recent years,
somewhat surprising that only a handful of observational studies have
convective asymmetries in the core of tropical cyclones and evaluated
they exhibit the properties of the VRWs seen in numerical models. Muramatsu (1986) documented 15 h of
counterclockwise rotating eyewall shapes in Typhoon Wynne (1980). He made a fascinating analogy between
polygonal eyewalls and the multiple vortices sometimes seen rotating
inside of a parent tornado vortex, and noted barotropic instability as
possible cause of both phenomena. Kuo et
al. (1999) and Reasor et al. (2000) documented elliptical eyewalls in
Herb (1996) and Hurricane Olivia (1994), respectively.
Both studies noted elliptical eyewalls that
rotated at approximately half the maximum tangential wind speed (Vmax)
with deep convection located at the ends of the major axis of the
Other observational studies
have confirmed the barotropic modeling results of
Schubert et al. (1999) and Kossin and Schubert (2001) by documenting
existence of multiple mesovortices in the eye and substantial mixing
the eye and eyewall. Kossin et al.
(2002, 2004) and Kossin and Schubert (2004) showed spectacular
satellite imagery of low-level vortical swirls in the eyes of multiple Atlantic and Pacific basin tropical cyclones
resembled the long-lived mesovortices of Kossin and Schubert (2001). Using flight-level reconnaissance data,
Kossin and Eastin (2001) showed that the radial profiles of vorticity
equivalent potential temperature can undergo a
rapid transition from a
barotropically unstable regime with maximum values of vorticity and
equivalent potential temperature
in the eyewall, to a stable regime with both maxima in the eye. Knaff et al. (2003) found that the
development of annular hurricanes was systematically preceded by a
asymmetric mixing event between the eye and the eyewall involving one
The most complete observational
investigation of asymmetric vorticity dynamics and VRWs in the core of
tropical cyclone was carried out by Reasor et al. (2000) in Hurricane
(1994). The vorticity asymmetry in the
core was dominated by azimuthal wavenumber 2 below 3 km, and wavenumber
3 km, with maximum values of both found near the RMW.
The wavenumber 1 signature was hypothesized
to be the result of environmental vertical wind shear, as an
magnitude of the wavenumber 1 asymmetry coincided with an
magnitude of the vertical shear. The
wavenumber 2 asymmetry was found to be the most unstable wavenumber and
natural by-product of the breakdown of the unstable vorticity ring
(1994) possessed at the beginning of the period of study.
Reasor et al. (2000) also observed trailing
bands of vorticity, 5-10 km wide, associated with high reflectivity,
the eyewall of Olivia (1994). They
hypothesized that these bands were symmetrizing VRWs
resulting from the
expulsion of high vorticity air from the core during mixing.
Left, Figure 12 of Reasor et al. (2000): Radius-height
structure of the azimuthal variance of vorticity averaged over the
2027-2355 UTC. Also shown are the individual wavenumber
(1-3) to the azimuthal variance (i.e., wavenumber components of
vorticity squared and azimuthally averaged). Contour interval is
Above, Figures 15
16 (left) of Reasor et al. (2000): (Left) Hurricane Olivia inner core
lower fuselage reflectivity composites at 3 km height. Period
spanned is 2244-2251 UTC at 1 minute intervals. The contour
interval is 10 dBZ. (Right) Azimuthal wavenumber 2 component of
vorticity vertically averaged over the lowest 3 km for each
flight. Contour interval is 0.4x10-3 s-1.
Negative values are depicted by the dashed curves.
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