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Journal ArticleDOI

Geostrophic Adjustment in an Axisymmetric Vortex

01 Jul 1980-Journal of the Atmospheric Sciences (American Meteorological Society)-Vol. 37, Iss: 7, pp 1464-1484
TL;DR: In this paper, a linearized system of equations for the atmosphere's first internal mode in the vertical is derived, which governs small-amplitude, forced, axisymmetric perturbations on a basic-state tangential flow which is independent of height.
Abstract: A linearized system of equations for the atmosphere's first internal mode in the vertical is derived. The system governs small-amplitude, forced, axisymmetric perturbations on a basic-state tangential flow which is independent of height. When the basic flow is at rest, solutions for the transient and final adjusted state are found by the method of Hankel transforms. Two examples are considered, one with an initial top hat potential vorticity and one with an initial Gaussian-type potential vorticity. These two examples, which extend the work of Fischer (1963) and Obukhov (1949), indicate that the energetical efficiency of cloud-cluster-scale heating in producing balanced vortex flow is very low, on the order of a few percent. The vast majority of the energy is simply partitioned to gravity-inertia waves. In contrast the efficiency of cloud-cluster-scale vorticity transport is very high. When the basic state possesses positive relative vorticity in an inner region, the energy partition can be subst...

Summary (2 min read)

INTRODUCTION

  • The problem of geostrophic adjustment is to determine the final adjusted state and the transient states which occur when atmospheric or oceanic flows mutually adjust the pressure field and the momentum field to a state of geostrophic balance.
  • Since gravity-inertia waves are then filtered, the transient aspects of the adjustment problem are not simulated.
  • Since a circular domain of radius 1200-1300 km occupies only about 1% of the total surface area of the globe, the region surrounding a tropical cyclone can, for practical purposes, be consi~ered infinite in extent.

METHOD OF SOLUTION

  • If the flow initially deviates from this balanced state a transient adjustment process occurs and ultimately a balanced flow results.
  • The authors have chosen the form given by (3.6) and (3.7) since it makes the following analysis somewhat simpler.
  • If the authors assume that there is no radial flow initially all the kinetic energy of the initial and final states is associated with v. Then (i) For small scale initial disturbances in the tangential wind field most of the initial energy ends up in the geostrophic flow.
  • Thus, for small scale momentum forcing the efficiency of geostrophic energy generation is very high.

5.2 Discontinuous initial tangential wind

  • In general the initial wind case shows just the opposite energetic .characteristics as the initial geopotential case.
  • In the initial wind case for a = 0.2, about 99% of the initial energy remains in the final balanced flow while about 1% is partitioned to gravity inertia waves.

25 8.2 Primitive Equation Model

  • When at = 5 ninety-six percent of the eventual total forcing has already occurred.
  • Although the final state will be the same for the use of balanced models when the time scale of the forcing is large compared to l/f.
  • A criticisn, of the use of balanced models for tropical studies is that the vertical motion patterns associated with gravity-inertia waves can interact nonlinearly with the moisture field, a pr)cess which cannot be simulated with a filtered model.

NON-RESTING BASIC STATE

  • Equation (3.4) can be regarded as the differential equation for ¢oo when the basic state, the initial conditions and the forcing are all known.
  • 00 00 The energetics for the non-resting basic state case are somewhat more complicated than the resting basic state case since energy can be extracted from the basic flow.
  • For any given horizontal scale of the initial perturbation, the final balanced state is obtained from the numerical solution of (3.4) subject to the boundary conditions (9.1) and (9.3), followed by numerical evaluation of (3.3).
  • But for a < 1 there are some significant changes in the quantitative character of the energy curves.
  • Thus, it would appear that, o when a tropical disturbance acquires a significant relative vorticity field, convective heating within the region of positive relative vorticity can become much more efficient at producing balanced flow.

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Geostrophic Adjustment
in
an Axisymmetric Vortex
by
Wayne H. Schubert
James
J.
Hack
Pedro L. Silva Dias
Scott R. Fulton
Department
of
Atmospheric Science
Colorado State University
Fort Collins, Colorado

GEOSTROPHIC
ADJUSTMENT
IN
AN
AXISYMMETRIC
VORTEX
by
Wayne
H.
Schubert
James
J.
Hack
Pedro
L.
Silva
Dias
Scott
R.
Fulton
This research
was
supported
by
the
Global
Atmospheric Research Program,
National Science Foundation,
and
the
GATE
Project Office,
NOAA
under
Grant
No.
ATM-7808125.
Department of Atmospheric Science
Colorado
State
University
Fort
Collins,
Colorado
80523
November,
1979
Atmospheric Science Paper
No.
317

CONTENTS
~~
ABSTRACT.
. . . . . . . . . . . . . . . . . . . . . . . . • . . . . . . . . . . . . . . . . . . . .
..
iii
1.
INTRODUCTION
............................................
.
2.
GOVERNING
EqUATIONS......................................
5
3.
METHO
0
OF
SOLUTI
ON.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
3.1
Final
Adjusted
State...
...............
...
.......
....
8
3.2
Transient
State.....................................
9
4.
GENERAL
PROPERTIES
OF
THE
SOLUTIONS
......................
12
5.
INITIAL
TOP-HAT
POTENTIAL
VORTICITy
......................
15
5.1 Continuous
Initial
Tangential
Wind
..................
15
5.2 Discontinuous
Initial
Tangential
Wind
...............
17
5.3
Transient
Solution
..................................
18
6.
INITIAL
GAUSSIAN-TYPE
POTENTIAL
VORTICITY
................
21
7.
INITIAL
RADIAL
WIND......................................
23
8.
THE
FORCED
CASE..........................................
25
8.1 Balanced
Model......................................
25
8.2
Primitive
Equation
Model
............................
26
9.
NON-RESTING
BASIC
STATE..................................
28
10.
IMPLICATIONS
FOR
BOUNDARy-CONDITIONS
.....................
33
11.
CONCLUDING
REMARKS
.......................................
39
ACKNOWLEDGEMENTS.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
41
REFERENCES.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
42
LIST
OF
FIGURES..........................................
47
i ;

ABSTRACT
A
linearized
system
of
equations
for
the atmosphere's
first
inter-
nal
mode
in the
vertical
is
derived.
The
system governs small amplitude,
forced, axisymmetric perturbations
on
a basic
state
tangential flow
which
is
independent
of
height.
When
the basic flow
is
at
rest,
solutions for
the
transient
and
final
adjusted
state
are
found
by
the
method
of
Hankel
transforms.
Two
examples are considered,
one
with
an
initial
top-hat
potential
vorticity
and
one
with
an
initial
Gaussian-type
potential
vorticity.
These
two
examples,
which
extend the
work
of
Fischer
(1963)
and
Obukhov
(1949),
indicate
that
the energetical
efficiency
of cloud
cluster
scale
heating in producing balanced vortex flow
is
very low,
on
the order of a
few
percent.
The
vast
majority
of
the energy
is
simply
partitioned
to
gravity-inertia
waves.
In
contrast
the
efficiency
of
cloud
cluster
scale
vorticity
transport
is
very high.
When
the basic
state
possesses
positive
relative
vorticity
in
an
.
inner region, the energy
partition
can
be
substantially
modified,
and
cloud
cluster
scale
heating
can
become
considerably
more
efficient.
The
energy
partition
result~
have
important implications
for
the
lateral
boundary condition
used
in
tropical
cyclone models.
Faced
with
the
fact
that
a
perfect
non-reflecting
condition
is
possible but imprac-
tical
to implement,
one
is
forced to use
an
approximate condition
which
causes
some
reflection
of
gravity-inertia
waves
and
hence
some
distor-
tion
of the geostrophic adjustment process.
The
distortion
can
be
kept
small
by
the use
of
a
suitable
radiation
condition.
iii

1.
INTRODUCTION
The
problem of geostrophic adjustment
is
to determine the final
adjusted
state
and
the
transient
states
which
occur
when
atmospheric or
oceanic flows mutually
adjust
the pressure
field
and
the
momentum
field
to a
state
of geostrophic balance. This
problem
was
first
studied
by
Rossby
(1938),
Cahn
(1945),
and
Obukhov
(1949).
Rossby
studied only the
relationship
between
the
initial
unbalanced
state
and
the final geo-
strophica11y balanced
state.
The
linear
transient
adjustment
was
studied
for the one-dimensional case
by
Cahn
and
for
the two-dimensional case
by
Obukhov.
Since these
classical
studies (primarily barotropic) there
have
been
many
contributions to
this
problem,
e.g.
the
effect
of
strati-
fication
(Bolin, 1953; Kibel, 1955, 1957, 1963;
Fjelsted,
1958;
Monin,
1958, Fischer, 1963), the
effect
of horizontal shear of the basic
flow
(31umen
and
Washington, 1969), the
effect
of nonlinear terms
(Blumen,
1967), the
effect
of a variable
coriolis
parameter (Dobrischman, 1964;
Geisler
and
Dickinson, 1972), the
effect
of a
transient
(rather
than
implusive) forcing of the
momentum
field
(Veronis,
1956)
and
of the
mass
field
(Paegle, 1978). Geisler
(1970)
has
also
shown
that
the
linear
response of the ocean to a
moving
hurricane
is
similar
in
many
respects
to the
problem
of geostrophic adjustment. Analytic solutions to the
adjustment
problem
also
serveas
useful guides in the design of
finite
differencing
schemes
for
more
complicated
models
(Arakawa
and
Lamb,
1977;
Schoenstadt, 1977, 1979, 1980). A review of the early Soviet
literature
on
geostrophic adjustment
(and
numerical weather prediction)
can
be
found
in
Phillips
et
a1. (1960).
An
excellent recent
and
comprehensive review
of the adjustment
problem
can
be
found
in the paper by
Blumen
(1972).

Citations
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Journal ArticleDOI
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Abstract: Enhanced infrared satellite imagery and conventional surface and sounding data are used to