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).