On Mesoscale Vortex Rossby Wave in Zonal Shear Flow

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On Mesoscale Vortex Rossby Wave in Zonal Shear
Flow Xinyong Shen Department of Atmospheric
Sciences, Nanjing University of Information
Science and Technology, Nanjing, China Yunqi
Ni Chinese Academy of Meteorological Sciences,
Beijing, China Tongli Shen Department of
Atmospheric Sciences, Nanjing University of
Information Science and Technology, Nanjing,
China Deying Wang Chinese Academy of
Meteorological Sciences, Beijing, China
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• Introduction
• Ougra et al., 1982 Field observations show that
  there often range meso-scale surface convergence
  systems and temperature waves with wavelengths
  approximately 400 km long in the direction of a
  cold front and thermal winds and the vorticity
  fields show a similar pattern, the train of
  disturbances ranging along a southwest jet axis
  in front of a 500-hPa cold vortex
• Yang et al., 1994 observational studies based on
  surface and upper-air conventional records as
  well as satellite and radar measurements reveal
  that a number of aligned meso-ß convective cloud
  nuclei (or rainstorm masses) exist in the meso-a
  system of a Meiyu front, and move eastward so
  fast as to make it difficult for surface stations
  to forecast

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Charney (1947) and Eady (1949) were the first to
discover the
instability of quasi
geostrophic synoptic-scale disturbance happening
in baroclinic flows, i.e., baroclinic instability
Kuo (1978) and Kuo and Seitter (1985)
addressed the structure of TTD instability in a
neutral and a partly unsteady stratified
atmosphere and layered it vertically through
numerical differencing calculation, thereby
obtaining more than one spectrum of developing
extra-long waves, in addition to the long wave
interception of Charney modes. Tokioka (1971)
made layering of 4-km-deep shear flow of the
Meiyu front, leading to a most unsteady
wavelength.
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Zhang (1988) By extending the Eady model into a
non-geostrophic domain so as to get numerically
the phase velocity, growth rate and
characteristic flow pattern of disturbance under
steady stratification, she obtained ageostrophic
baroclinic meso modes, tens to hundreds of
kilometers in scale in addition to the Eady
equivalents at synoptic and sub-synoptic scales
and she also showed the growth rate of her modes
to be about 4 times that of the Eady modes,
which, as she indicated, maybe serves as a
dynamic mechanism for initiating and organizing
deep convective cloud cluster. Additionally, she
got the meso TTD characteristic wave structure by
means of a matrix method and the targeting,
discovering an asymmetric cats eye pattern of
the mode on a vertical section.

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Because of more difficulties solving
theoretically an ordinary differential equation
with variable coefficients, the solution to the
problem of meso TTD instability depends on a
numerical differencing scheme in most cases. The
present work is an attempt to mathematically
prove the characteristics of propagation when TTD
experiences unstable development, leading to an
expression of phase velocity of vortex Rossby
wave, followed by exploring the physical origin
of the waves genesis and its properties.
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2. Equations of Traversal Type Disturbance
With no account of external forcings, i.e.,
surface friction, topography, external heat
transport and latent heat effect, the equations
of the atmospheric motion take the form
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All physical variables are discomposed, leading
to
and
We let the stratification and baroclinic
stabilities be
and
meso-ß weather systems are set to be arranged E
W , all disturbance variables are assumed to be
independent of y, Therefore, we find the
following equation
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and
Streamfunctions
are introduced because
disturbance velocity is non-divergent on the
plane,
we get a partial differential equation containing
only disturbance streamfunction as the variable,
viz.,
In finding the characteristic wave solution to
the streamfunction by means of
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we arrive at an ordinary differential equation
0
3. Phase velocity and energy frequency-dispersion
relation of vortical Rossby wave properties of
TTD instability
we set basic flow
to be z 0 at surface and the flow in linear
distribution at arbitrary height z follows
with
ltlt1 where
is a constant, indicating that the
second-order vertical background wind shear is
very small.
for convenience, when we discuss non-vortical
fluid, i.e.,
(21) has the form
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where we set
which is made to satisfy
we obtain the following frequency dispersion
relation
a) For
but
We have
which indicates that with no second
-order shear in the flow available, disturbance
is an internal gravity wave propagating both
east- and westward with respect to basic flow.
b) For
but
We have phase velocity
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(33)
which actually denotes the expression for phase
velocity of vortex
Rossby wave.
From (33) we see that the wave propagates
unidirectionally, i.e., east- (westward) with
respect to basic flow
As a result, if, compared to the middle
troposphere, basic flow has its greater
horizontal velocity in the lower and higher
troposphere (related to the existence of low- and
high-level jet streams), then
gt0 is met for vortical Rossby wave propagating
eastward,
and even at greater velocity, i.e., cgt
and if, compared to the
lower and higher troposphere, basic flow has its
greater velocity in the middle troposphere, then
is satisfied for the wave
propagating westward with respect to the basic
flow.
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In addition, because the phase velocity of the
wave bears a relation to zonal wavenumber k, its
energy is frequency dispersive and its zonal
group velocity takes the form
c) For
and
Therefore, (32)-denoted TTD is mixed vortex
Rossby gravity wave. In this sense, the
instability of TTD is that of the mixed vortex
Rossby gravity wave.
In fact, for
we get an ordinary differential equation
the Weber equation of first kind
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the frequency dispersion relation for meso TTD in
the vertically second shear flow
(38) is a cubic equation and has three
characteristic roots, two of which denote
internal inertial gravity wave, one propagating
east- and the other westward with respect to
basic flow, and the third of which denotes
vortical Rossby wave.
4. Analysis of vorticity and divergence
equations for TTD
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Fig.1. The structures of meso-ßweather systems
with n 1 disturbance propagating eastward under
more weakly unsteady stratification (N20).
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we observe no disturbance of vertical velocity
and potential temperature on surface (z0)
synoptic maps that the center of the pressure
field coincides with that of the vorticity field,
with the high (low) pressure core corresponding
to that of anticyclonic (cyclonic) vorticity the
center of surface convergence (divergence) is
ahead of that of surface low (high) pressure
system, i.e., ahead of (behind) the surface
trough. It is true of the situation at zH
(tropopause) except for the opposite distribution
of pressure fields (only in the sense of n1
mode). That the core of divergence (convergence)
does not coincide with that of the high (low)
pressure system displays stronger ageostrophy of
meso motions at middle troposphere (), a warm
(cold) core is over the center of a surface low
(high) pressure system, with the maximum rising
(sinking) occurring over the surface convergence
(divergence) center, i.e., above ahead of
(behind) the surface trough.
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5. Analysis of energy equation for TTD
development
the total energy for local disturbance
development comes from mean effective potential
energy, mean kinetic energy and the advection of
the basic flow upon the total energy.
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6. Equation for TTD conservation and physical
mechanism for generating vortex Rossby wave
Yu (2002) made an excellent overview on the
formation of vortical Rossby wave in typhoon
spiral rainbands from the contributions of
MacDonald (1968), Guinn et al. (1993), Smith and
Montgomery (1995) and Montgomery et al. (1997a,
b) and Wang (2002). We make attempt to explain
the physical processes of the formation of
vortical Rossby wave in meso TTD in order to
understand possible mechanisms of the genesis and
development of meso-ß rainstorm masses in the
Meiyu front.
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(No Transcript)
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Fig.3. Possible mechanism for the genesis and
propagation of the rainstorm mass in the Meiyu
front.
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7. Numerical experiment by means of the MM5
we undertake numerical simulation of one
torrential rain event occurring in the mid-lower
basins of the Changjiang from 0000 UTC, July 29
to 0000 of 30, 1998 by means of the meso-scale
non-hydrostatic model MM5 developed by US
PSU/NCAR
Fig.4 Rainfall distribution for 0500 0600 UTC,
July 29, 1998. Units mm.
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Fig.5. Divergence (10-5) pattern at 700 hPa at
0600 UTC, July 29, 1998.
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Fig.6. as in Fig.5 but for vorticity (10-5).
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Fig.10. A U-wind profile in 29N, 116E at 0600
UTC, July 29, 1998.
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Fig.11. the same as in Fig.10 but for at 1000
UTC. Otherwise as in Fig.10.
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Averaging U windspeeds over the study area with a
vigorous rain mass results in z-dependent EW
curves of U at 0600 and 1000 UTC (Figs.10-11).
Investigation shows that in the free atmosphere
(850 hPa), by and large, U decreases linearly
versus height on account of low-level jets around
850-800 hPa. It is seen from the theoretical
conclusions presented in Part I that with the
linearly reduced windspeed profile available the
TWT disturbances (mainly inertia-gravity waves
rather than VRoW) experience instability, thereby
giving rise to the development and enhancement of
meso-beta rainstorm masses anterior to 1400 UTC,
July 29, and are kept in a quasi-stationary
state.
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Fig.14. The U-wind profile (29N, 119E) at 1000
UTC, July 22, 1998.
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Fig.15. as in Fig.14 but at 1500 UTC.
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Height-dependent U-wind curves in the E W
direction are presented for 1000 and 1500 UTC,
July 22, 1998, as shown in Figs.14 and 15,
respectively. Their analysis indicates that in
the free atmosphere (850-200 hPa) U changes in a
way that satisfies the condition of the second
derivative of it with respect to height, with
greater velocity around 850 and 200 hPa (in
relation to the low- and high-level jets,
respectively) and low easterly and westerly
speeds in the neighborhood of 400 hPa. As shown
by the theoretical findings given in Part I, with
the effect of Uzz ? 0, TWT disturbance
(dominantly VRoW) will display instability,
thereby driving the meso- rain mass to travel
eastward, as portrayed in Figs.16-19 where the
isopluvials 5 mm are plotted, indicating that
the meso-ß-scale rain mass shifts at the rates of
more than 50 km an hour between 0900 to 1200 UTC,
July 22.
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8. Concluding remarks
To investigate the physical mechanism of the
genesis/development, and the structure, of a
meso-ß rainstorm mass (convective nucleus) in the
Meiyu front and explain the physics of TTD
vortical Rossby wave (VRoW), the approximate
Boussinesq equations for 2D meso TTD in basic
flow are used to make theoretical study of the
physical representation of VRoW propagating in a
zonal shear flow, thus revealing the fields of
disturbance physical variables and energy
sources. The main conclusions are as follows 1)
Theoretical analysis shows that vertical wind
shear in basic flow causes the unsteady
development of meso TTD and the consideration of
second-order shear in the flow would lead to the
expression of phase velocity of VRoW indicating
that the wave is unidirectional in propagation
with respect to the basic flow.
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2) The physical origin of the vortex Rossby wave
lies in the second shear of the basic flow
plays a role similar to the effect of ß component
of
midlatitude planetary Rossby wave. The waves
phase velocity is functionally related to zonal
wavenumber k, with the energy being frequency
dispersive and group velocity available in the x
direction.
3) For the second-order shear of wind , the TTD
instability is the instability of mixed vortex
Rossby gravity wave. With wind subject to linear
shear rather than a second shear , the
instability of TTD is that of internal inertial
gravity wave.
4) With constant flow and (weaker unsteady
stratification) available, we see on the surface
synoptic chart that the center of the pressure
coincides with that of the vorticity field, the
core of
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the high (low) field corresponds to that of the
anticyclonic (cyclonic) vorticity field the
center of surface convergence (divergence) is
ahead of that of the surface low (high), i.e., in
front (at the back) of the trough in the mid
troposphere, a warm (cold) core is over the
surface low (high) pressure core and the maximum
rising (sinking) motion occurs over the center of
surface convergence (divergence), i.e., above
ahead of (behind) the surface trough.
5) The total energy for disturbance development
on a local basis originates from mean effective
potential energy, mean kinetic energy and
advection of basic wind upon disturbance total
energy. Under no account taken of basic flow the
total energy is conserved, and the disturbance
kinetic energy comes only from disturbance
effective potential energy, with no energy
provided by the background field for disturbance
development.
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6) The environmental field has its mean vorticity
changing with z, viz., , leading to the fact
that during the vertical down- and upward
movement, to keep constant, the air parcel is
bound to exhibit vibrations in vertical,
resulting in the formation of a VRoW that
propagates in a zonal direction. When displaying
unsteady development (TTD instability), the wave
takes energy from the shear flow, resulting in
amplitude that gets progressively bigger to such
an extent as to produce a weather system of
meso-ß rainstorm mass in the Meiyu front.
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