Geostrophic & gradient-wind adjustmentm-mak/Adjustment-chapter.pdf · 2007. 2. 19. · Mankin Mak...

Mankin Mak 2/19/2007, Adjustment , version 1.0, p. 1

Geostrophic & gradient-wind adjustment

1. The problem of rotational adjustment …….……….…..…….….…… 2

2.a An exact analysis of Rossby’s problem of geostrophic adjustment… 5

.b Quantitative properties of the solution ……………………………… 12

.c Essence of geostrophic adjustment …………….. ……………………. 14

.d Energetics of geostrophic adjustment ……………………………….. 16

3.a An exact analysis of the complimentary Rossby problem ……….. 18

.b Comparison of the approximate and exact solutions ……………… 21

4.a Gradient-wind adjustment of an axisymmetric vortex…………….. 25

.b Formulation ………………………………………………………… 26

.c Analysis ……………………………………………………………… 27

.d Result ……………………………………………………………….. 29

5. Concluding remarks …………………………….…………….……. 35

6. Reference …………………………………………………………….. 38


1. The problem of rotational adjustment

The pressure field of a large-scale flow stems from the distribution of mass of the

air in the atmosphere by virtue of hydrostatic balance. Under the influence of rapid

rotation of the earth, the large-scale mass field (pressure field) and velocity field in the

atmosphere tend to mutually adjust towards a quasi-balance state. A manifestation of

this intrinsic tendency is that the observed large-scale wind vectors everywhere on an

upper level isobaric surface in mid-latitude are always approximately parallel to the

height contours. An example is shown in Fig. 1 for the circulation at 500 mb level over

North America on a summer day. It means that such wind field is approximately in

geostrophic balance with the pressure field everywhere. It is actually more precise to say

that they are approximately in gradient-wind balance because the curvature effect can be

significant in the neighborhood of troughs and ridges.


Fig. 1 Distribution of the actual wind vectors (green arrows) , the geostrophic vectors (red arrows) and the contours of the height fields at 500 mb on 7 August 2006.

We should distinguish the intrinsic evolution of a quasi-balance flow in a closed

system from the adjustment of a flow in response to a significant local and transient

external influence. The former can be adequately described from the perspective of

quasi-geostrophic theory . From the perspective of that theory, a large-scale wind field

consists of a primary component and a secondary component. The former is horizontally

non-divergent whereas the latter is divergent. The former may remain in quasi-balance

all the time due to the effectively instantaneous feedback influence of the much weaker

divergent component. Internal gravity waves need not play any role in such evolution.

The situation is totally different when the balance between the pressure field and

wind field is significantly disrupted by a strong transient local external forcing. For

example, heavy precipitation in a locale would release a large amount of heat over a short

time interval. It would momentarily change the local values of pressure and temperature

but not the wind. Nevertheless, a large-scale flow appears to be able to quickly adjust

itself to a new balanced state as suggested by the fact that quasi-balance prevails almost

all the times. In this case, gravity waves play an indispensable role. The imbalance

would be removed by a redistribution of the mass and/or velocity fields. The gravity

waves would quickly propagate to distant locations leaving behind a new balanced flow

in the original location of imbalance.

The question then is: what would the adjusted state be like for a given initial

unbalanced state in a geophysical fluid? Rossby (1938) hypothesized that it is possible

to deduce the adjusted state solely on the basis of the initial unbalanced state without


having to work out the intermediate states. This is the classical problem of geostrophic

adjustment. We will elucidate the nature of geostrophic adjustment for two types of

diametrically different initial disturbances. An initial disturbance could be entirely

associated with the velocity field in the presence of a uniform pressure field (Rossby’s

problem). Alternatively, an initial disturbance could be entirely associated with the

pressure field in the absence of a wind field (complimentary Rossby problem). There

are subtle differences in the two problems. Ironically, existing textbooks only discuss

the complimentary Rossby problem.

A more general form of adjustment is gradient-wind adjustment. It is a more

difficult problem because of the explicit nonlinear relationship between the centrifugal

force and velocity. An analysis of such adjustment in the context of an unbalanced

axisymmetric vortex will be discussed. Rotational adjustment is used as a generic term

for the process of establishing a quasi-balance state in a rotating fluid.

The transient aspect of the adjustment problem entails inertial oscillation as well

as gravity wave excitation/propagation. There may even be turbulence. An investigation

of such detailed motions would require a technically much more involved analysis.

There is however reason to believe that the transient stage of adjustment is relatively

brief. As such, it is a secondary aspect of the adjustment process. On the other hand, we

should not dismiss the transient part of the adjustment process altogether. Significant

meso-scale disturbances could be triggered during geostrophic adjustment according to

some investigations. Such discussion is beyond the scope of this chapter on the

adjustment problem.


2.a An exact analysis of Rossby’s problem of geostrophic adjustment

Rossby (1938) posed the geostrophic adjustment problem in the context of an

idealized ocean current in an unbounded shallow-water model. The fluid in his initial

state is moving uniformly in a certain direction designated here as the y-direction

( ) in an infinite strip of width and is at rest elsewhere (Vvo = a2 0vo = ). Suppose that

the transfer of momentum from the wind to the central strip of this model ocean has

rapidly taken place without affecting the pressure field. The initial depth in Rossby’s

problem is uniform everywhere, Hho = . This initial state is clearly not in balance and

therefore necessarily evolves in time. Friction may be assumed to have negligible

influence on the subsequent development of the flow if the adjustment time is not very

long. The upper panel of Fig. 2 is a schematic of the initial state. The valuable insight in

Rossby’s contribution is that it is possible to deduce the adjusted state solely on the basis

of the unbalanced initial state without having to work out the intermediate states because

of a fundamental dynamical constraint.

It is instructive to begin by applying rudimentary physical reasoning for the

purpose of deducing how this fluid system might initially evolve. The northward

moving fluid in the central strip would be deflected eastward by the Coriolis force

immediately after . This would create convergence of mass next to its eastern

boundary and divergence of mass next to its western boundary. The elevation of the fluid

surface should then rise to the east and would sink to the west. This change in pressure

field would tend to oppose the initial deflection in accordance with the old saying, “to

every action there is a reaction”. The resulting pressure gradient force would also set

the fluid to motion in the domain further east and west of the boundaries. In other words,

0t =


the initial action would tend to induce oscillatory motion of the fluid in the central strip

and would also excite gravity waves. The velocity and height of the central strip would

soon change. The gravity waves would begin to propagate away to far distance. There

would be mutual adjustment of the mass field and velocity field in the whole system.

The adjustment would continue for some time until a new steady balanced state is

eventually established. For the geometry of the system under consideration, the adjusted

state must be in geostrophic balance. We have reduced the question to: how can one

quantify the adjusted state?

The initial state has symmetry in the x-direction with respect to the center of

moving fluid (upper panel of Fig. 2). Coriolis force and gravity do not alter the intrinsic

symmetry of the system, even though the central fluid column in the adjusted state does

not correspond to the central fluid column in the initial state. Thus, we may expect odd

symmetry in the surface elevation of the adjusted state with respect to a certain point and

even symmetry in the corresponding velocity field. That point of symmetry is designated

without loss of generality. The lower panel of Fig. 2 indicates the general

character of the surface elevation in the adjusted state. The surface elevation at in

the adjusted state is

0x =

0x =

H . The domain of the adjusted state has three distinct regions: I, II

and III. The symmetry consideration and the incompressibility of the fluid suggest that

the width of region I also has a width of . a2


a2

II I III

-a 0 a x

Fig. 2. A sketch of the initial state (upper panel) and adjusted state ( lower panel) of a shallow-water model. The moving fluid of the initial state is located in the shaded area, ( ) ( )11 xaxxa −≤≤−− . See text for details.

Each fluid column in a balanced state of a shallow-water model moves as a

column. The fluid columns at the two boundaries of the central strip are unique markers

because their initial velocity has a discontinuity. In light of the earlier discussion, they

can be identified as having moved to the location ax ±= in the adjusted state. We

indicate the left boundary in the two states by an arrow linking the two panels. By the

same token, an arrow links a certain fluid column in the initial state to the column at

( )0X o( )aX oHho = −

0vo = Vvo = 0vo =


0x = in the adjusted state. The red rectangles in the two panels indicate what should

happen to a particular fluid column of finite width. It would be displaced eastward by a

certain distance. It becomes shorter and thereby wider as required by mass conservation.

The task is to determine the zonal displacement of all fluid columns as well as the height

and velocity fields at large time. Although Rossby briefly mentioned how an exact

solution could be obtained in a footnote, he did not carry it out but only presented an

approximate solution. What he actually did was to obtain the solution in the two regions

outside the central strip and assumed a uniform velocity in the central strip. He closed

the problem by imposing a requirement that the absolute momentum of the fluid in the

central strip conserves. The exact solution was first reported by Mihaljan (1963).

Now let us perform a complete analysis of this geostrophic adjustment problem.

The equation of motion in the y-direction for a fluid column of unit mass may be used as

a starting point of an analysis, viz.

fudtdv

−= (1)

Integrating (1) from t =0 to a certain large time when a balanced state has been essentially

established would yield

( )oo XXfvv −−=− (2)

where is the original position of a fluid column which is located at in the

adjusted state. is the initial velocity of the fluid column and should be thought of as a

function of . Eq. (2) is a statement of conservation of absolute momentum in the y-

direction of a fluid parcel of unit mass.

( )xX o xX =

ov

oX


By mass conservation, the length ( h ) and width ( xΔ ) of a fluid column in the

adjusted state are related to its counterparts at t=0 by

oo Xhxh ΔΔ =

dxdX

hh oo=

(3)

Since the flow in the adjusted state is by definition steady and unidirectional, it must be

in geostrophic balance. Hence, we have

2o

2o

dxXd

fgh

dxdh

fgv ==

(4)

Writing oXX −≡ξ as the displacement of a fluid column in the x-direction, we treat it.

as a dependent variable. We obtain from (2) and (4)

o2

2o vf

dxd

fgh

−=− ξξ (5)

It is convenient to measure x and ξ in unit of , in unit of V and h in

unit of

a v

H . We thus define non-dimensional variables: axx~ = ,

a~ ξξ = ,

Vvv~ = and

Hhh~ = . Two dimensionless parameters are particularly pertinent: the ratio of to

radius of deformation

a

gHfa

=λ and the Rossby number faVR = . The non-

dimensional form of (5) is (after dropping tilda)

In region I , Rdxd 22

2

2

λξλξ−=−

In regions II and III, 0dxd 2

2

2

=− ξλξ

(6)


ξ is to be finite as ±∞→x . The general solution is then

For I, RBeAe xx ++= −λλξ

For II, xCeλξ =

For III, xDe λξ −=

(7)

On the basis of (4), the corresponding non-dimensional solution of the surface elevation

is

1dxdh +−=ξ

For I, ( ) 1BeAeh xx +−−= −λλλ

For II, 1Ceh x +−= λλ

For III, 1Deh x += −λλ

(8)

Since at , we must have 1h = 0x = BA = . Symmetry consideration also implies

that . Furthermore, we have matching conditions that DC = ξ and must be

continuous at and thereby get

h

1x ±=

λ−−== e2RBA

( )λλ −−== ee2RDC

(9)

The non-dimensional form of (4) is

dxdh

R1v 2λ

= (10)


So the explicit form of the complete solution is

For I, 1x1 ≤≤− ( ) 1ee2

Rh )1x()1x( +−= +−− λλλ

( ) Ree2R )1x()1x( ++−= +−− λλξ

( ))1x()1x( ee21v +−− += λλ

For II, 1x −≤<∞− ( ) 1eee2

Rh x +−−= − λλλλ

( ) xeee2R λλλξ −−=

( ) xeee21v λλλ −−−=

For III, ∞≤< x1 ( ) 1eee2

Rh x +−= −− λλλλ

( ) xeee2R λλλξ −−−=

( ) xeee21v λλλ −−−−=

(11)

The eastward displacement of the eastern boundary of the central strip of moving fluid is

( ) ( ) ( )λξ 2o e1

2R1X11 −−=−=

(12)

The dimensional form of (12) is ( ) ( )gH/fa2dim e1

f2Va −−=ξ which shows that the shift

is proportional to the velocity of the initial disturbance and is larger at lower latitude.

The shift has an additional dependence on the scale of the disturbance. The larger the

parameter gHfa

=λ is, the larger would be the displacement. In other words, a


disturbance of larger scale (larger ) would be displaced more. For a small disturbance, a

fgH

a << , the shift becomes ( )gH

Vaadim ≈ξ which is independent of the rotation rate

and proportional to the scale of the initial disturbance. For a large disturbance,

fgH

a >> , the shift would approach the maximum possible value, ( )f2

Vadim ≈ξ . It has

been verified that the error in Rossby’s approximate solution is quite substantial for

3≥λ .

2.b Quantitative properties of the solution

To visualize what the results look like, we plot the solutions of and for h v

95.0=λ and in Fig. 3 . 0.1R =

-0.5

0

0.5

1

1.5

-10 -5 0 5 10

x

h

v

Fig. 3. Distribution of the non-dimensional meridional velocity and height of the free surface for .1R = and 95.0=λ


It is confirmed that the elevation has odd symmetry and the velocity has even symmetry

with respect to . The plot reveals that the elevation of the free surface increases

from a minimum value at to a maximum value at

0x =

1x −= 1x = . It tapers from 1x ±=

towards zero at a rate equal to λ . A southerly flow can be therefore geostrophically

supported in the central region and a northerly flow can be supported on each side. The

southerly flow is reduced to about half of its initial value. It should be noted according

to the solutions (11) that while h depends on both R and λ , whereas v only depends

on λ because v is scaled in terms of V . The pressure field changes relatively little for

a small R with a given value of λ . This is illustrated in Fig. 4.

-0.5

0

0.5

1

1.5

-10 -5 0 5 10

x

h

v

Fig. 4 Distribution of the meridional velocity and height of free surface for and 1.0R = 95.0=λ

On the other hand, if the radius of deformation is short compared to the length

scale of the initial disturbance, as exemplified by .1R = and 0.2=λ , a larger change of

the height field takes place in geostrophic adjustment (Fig. 5). The corresponding change


in the velocity field would be larger. Rossby’s approximation is less justifiable in this

case.

-0.5

0

0.5

1

1.5

2

-10 -5 0 5 10

x

h

v

Fig. 5. Distribution of the meridional velocity and height of free surface for and

.1R =.2=λ

2.c Essence of geostrophic adjustment

The essence of geostrophic adjustment can be most succinctly understood from

the potential vorticity point of view. In 1938, the concept of potential vorticity was not

yet firmly established. What Rossby referred to as “absolute vorticity” in his article is

equivalent to what we nowadays call “potential vorticity”. The potential vorticity of the

adjusted state is ⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜

⎝

⎛∂∂

+

hxvR1

in unit of Hf . According to the solution (11), it is equal to

1.0 at all but two points. The exceptions are the two boundaries of the central strip,


1x ±= . The non-dimensional velocity of the initial state and of the adjusted state have a

discontinuity equal to 1 at the boundaries of the central strip. The PV is therefore infinite

at those two points. The distribution of PV can be represented with the Dirac delta

function, ( )xδ , as

( ) (( )1x1xR1h

xvR1

−−++=∂∂

+δδ )

(13)

It is noteworthy that by taking a derivative of (2) with respect to x and making

use of (3) and (4), we would get an equation in dimensional form as

oo

oo

o

o

hhf

Xv

dxdXf

Xvf

xv

⎟⎟⎠

⎞⎜⎜⎝

⎛+

∂∂

=⎟⎟⎠

⎞⎜⎜⎝

⎛+

∂∂

=+∂∂

o

o

o2

2

h

fXv

h

fdx

hdfg +

∂∂

=+

(14)

Eq. (14) is a statement of conservation of potential vorticity of a fluid column in this

system. Indeed, one could alternatively use (14) as the starting point of an analysis. The

same solution (11) can be obtained by solving (14) instead of (6). In doing so, special

care must be taken to impose proper matching conditions at ax ±= because of the

discontinuity in . The matching conditions atov ax ±= are: (i) is continuous and (ii) h

dxdh has a jump related to the discontinuity of the initial velocity.

In light of the discussion above, we may conclude that the mutual adjustment of

the velocity and height fields must be such that the PV of every fluid column conserves.

It is this fundamental dynamical constraint of inviscid incompressible fluid that makes it

possible to deduce the adjusted state directly from an initial state.


2.d Energetics of geostrophic adjustment

The initial amount of kinetic energy of the fluid in a section of length is

. The amount of kinetic energy of the adjusted state in this section is

l

2o VHaK lρ=

∫∞

∞−

= hdxvVHa21K 22lρ

(15)

Using the solution (11), we obtain

⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎠⎞

⎜⎝⎛ −+= − λ

λλρ 22 e

211

21VHa

21K l

(16)

The amount of kinetic energy is therefore reduced during the adjustment process by

⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎠⎞

⎜⎝⎛ −−−=−= − λ

λλρΔ 22

o e211

212VHa

21KKK l

(17)

In the limit of 1<<λ (small disturbance), we get oK21K =Δ . In the limit of 1>>λ ,

oKK ≈Δ .

The amount of potential energy in a section of length in the initial state and

adjusted state are

l

( ) dx1Hag21P 22

o ∫∞

∞−

= lρ

dxhHag21P 22 ∫

∞

∞−

= lρ

(18)

According to the solution (11), the potential energy is increased by


{ }

⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎠⎞

⎜⎝⎛ +−=

−=−=

−

∞

∞−∫

λ

λλρ

ρΔ

22

22o

e211

21VaH

21

dx1hHag21PPP

l

l

(19)

The net loss of energy in the fluid is equal to ( )PK ΔΔ − . This amount of energy must

have been propagated to far distance from the initial disturbance by the gravity waves

during adjustment. The ratio of KE reduction to PE increase is

λ

λ

λλ

λλΔΔ

2

2

e211

21

e211

212

PK

−

−

⎟⎠⎞

⎜⎝⎛ +−

⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎠⎞

⎜⎝⎛ −−−

=

(20)

This value is equal to ~3.8 for 1=λ . In the limit of 1<<λ (small disturbance), we

get ( 1)PK

−→ΔΔ . In the limit of 1>>λ , we get ( 14

PK

−→ λΔ

)Δ . This amounts to

saying PK ΔΔ >> .


3.a An exact analysis of the complimentary Rossby problem

By the “complimentary Rossby problem”, we mean the adjustment of an initial

disturbance in a shallow-water model which is structurally opposite of that considered in

the Rossby’s problem. Specifically, the initial velocity is ( ) 0xvo = and the initial

surface elevation has a top-hat configuration, namely ( )xho has a larger constant value in

ax ≤ than in ax ≥ . The task is to directly deduce the adjusted state solely on the

basis of this initial state. The discussion of geostrophic adjustment in textbooks is

typically given only for a perturbation analysis of the complimentary Rossby problem.

Since this problem is analytically tractable even without restricting the magnitude of the

initial disturbance to be weak, we will go over such an analysis.

It is again helpful to anticipate what should happen to the fluid by rudimentary

physical reasoning before undertaking a mathematical analysis. Right after , some

fluid is expected to flow outward across

0t =

ax ±= because of the pressure differential.

The elevation of the fluid in the central strip would naturally become lower. The Coriolis

force acting on the eastward moving fluid in ax > would deflect it southward.

Likewise, the westward moving fluid in ax −< would be deflected northward. The

subsequent evolution may be quite complicated in details possibly involving turbulent

motions as well as generation and propagation of gravity waves. But the motions in the

region near the central strip may be expected to become steady at large time. The initial

changes alluded to above might be indicative of what we could expect to see in the

adjusted state at large time. How much the depth of the layer in the central strip would

decrease and how strong the northerly and southerly flows on the two side regions would


be are obviously functions of the parameters. They can only be deduced by a

mathematical analysis.

Now let us perform the mathematical analysis. Let us use ( ) HAxho += in

ax ≤ and in ( ) Hxho = ax ≥ as the initial state. It is convenient to introduce the

following non-dimensional variables and parameters:

Ahh~′

= ,

HgA

vv~ = , axx~ = , ε = A

H,

gHfa

=λ , f

QHQ~ = (21)

ε is a measure of the strength of the initial imbalance. is the potential vorticity (PV)

of the initial state,

Q

ε+=

11Q in 1x ≤ and 1Q = in 1x ≥ . The surface elevation and

the velocity in the adjusted state are ( )h~1h~total ε+= and xh~1v~λ

= . The non-dimensional

form of conservation of PV is (after dropping the tilda)

hxx −Qλ2h =λ2

εQ−1( )

(22)

When we consider a strong disturbance ( )1O~HA

=ε , it would be necessary to

take into account of the displacement of fluid columns from their original positions.

Because of the innate symmetry of the set-up with respect to 0x = , we may simply

consider the positive half of the infinite domain. That half domain consists of two

regions, which are designated as (i) and (ii) in a description of the adjusted state. They

are separated at x = x1. In other words, ( )1x1 − is the displacement of the PV boundary

originally at . The fluid columns have a constant PV, 1x =ε+

=1

1Q , in region (i) and


a different constant PV, , in region (ii). The value of is itself to be determined

as part of the solution. We write the governing equations in the following explicit form

1Q = 1x

In (i), , with 1xx0 ≤≤ 22xx bhbh −=−

ελ+

=1

b2

2

In (ii), , 1xx > 0hh 2xx =− λ

(23)

The proper boundary conditions are : 0hx = at 0x = and 0h = as ∞→x .

The general solution is then

In (i) ( )bxbx eeA1h −++=

In (ii) xCeh λ−=

(24)

The integration constants and can be determined on the basis of the matching

conditions at which are continuity in both h and . The complete solution is

A C

1xx = xh

In , 1xx0 ≤≤( )

( ) ( )aeaeee1h

11 axax

axax

−+++

−= −

−

λλλ ,

( )( ) ( )aeae

eeav11 axax

axax

−++−

−= −

−

λλ,

In , 1xx > ( )( ) ( )

( )xxaxax

axax1

11

11

eaeae

eeah −−

−

−++−

= λ

λλ,

( )( ) ( )

( )xxaxax

axax1

11

11

eaeae

eeav −−

−

−++−

−= λ

λλ

(25)

The condition of mass conservation can be used to determine . We thereby obtain 1x


( ) ( )dx1dxh11

0

x

0

1

∫∫ +=+ εε

( ) ( )( ) ( )( ) ε

λλελε +=

−++−

−+ −

−

1aeaea

eex111

11

axax

axax

1

(26)

1x can be readily evaluated as the intersection of two curves that represent the LHS

and RHS of (26).

It is instructive to compare this exact solution with the approximate solution for a

perturbation. The latter is obtained by using conservation of the linearized form of PV

and neglecting the displacement of fluid columns (i.e. setting .1x1 = ). It can be shown

that such solution in the positive half of the domain is

In , 1x0 ≤≤( )

λ

λλ

e2ee1h

xx −+−= , ( )

λ

λλ

e2eev

xx −−−= ,

In , 1x > ( ) xeee21h λλλ −−−= , ( ) xeee

21v λλλ −−−−=

(27)

3.b Comparison of the approximate and exact solutions

To visualize what the results look like, we plot the approximate and exact

solutions of h1htotal ε+= for 1=ε , 5.0=λ in Fig. 6. The length scale of disturbance

in this case is equal to the radius of deformation. It is found that the PV boundary in the

exact solution is displaced to 5.1x1 ≈ and ( )( ) 34.10h1 ≈+ ε . The new boundaries of

PV discontinuity are indicated by two vertical dash lines at 1xx ±= in Fig. 6. The

decrease of mass between and 0x = 1x = from the initial state to the adjusted state is

about . It is indeed equal to the mass in the column between 65.0 1x = and . 1xx =


That is why the linear solution of near totalh 1x = is higher than that of the exact

solution.

0

0.5

1

1.5

2

2.5

-10 -5 0 5 10

FGI

x

Fig. 6 A comparison of the distributions of h1htotal ε+= , the non-dimensional height of the surface in the initial state (F), the linear solution (G), and the exact solution (I) for the case of 5.0=λ , 1=ε : .

The corresponding distributions of the velocity are shown in Fig. 7 v


-0.4

-0.2

0

0.2

0.4

-10 -5 0 5 10

JKM

x Fig. 7 A comparison of the non-dimensional velocity of the initial state (J), the linear solution (K), and the exact solution (M) for the case of 5.0=λ , 1=ε .

Both solutions reveal that a shear flow is induced in this problem by geostrophic

adjustment. They confirm the expection from rudimentary physical reasoning that there

should be a northerly (southerly) jet in the eastern (western) side of the initial

disturbance. The displacement of the PV boundaries is more evident in Fig. 7. The

position of each jet is collocated with the PV boundary in each solution. The strength of

the jet is considerably overestimated by the linear solution.

Let us next examine the solutions for a case of larger disturbance, .2=λ This is

a case where the length scale of disturbance is equal to four times the radius of

deformation. It is found that 2.1x1 = for this case. The counterpart solutions of

h1htotal ε+= and v are shown in Fig. 8 and 9 respectively.


0

0.5

1

1.5

2

2.5

-10 -5 0 5 10

FGI

x

Fig.8 A comparison of the distributions of the non-dimensional height of the surface of the initial state (F), the linear solution (G), and the exact solution (I) for the case of .2=λ , 1=ε .

-0.4

-0.2

0

0.2

0.4

-10 -5 0 5 10

JKM

x

Fig. 9. A comparison of the non-dimensional velocity of the initial state (J), the linear solution (K), and the exact solution (M) for the case of .2=λ , 1=ε .


One key difference between the two adjusted states in Fig. 6 and Fig. 8 is that the

elevation changes by a much smaller extent from the initial configuration in the case of a

larger disturbance. The pressure gradient at the PV boundary in the adjusted state is

correspondingly much larger. That in turn can support a stronger jet on each side (0.4 vs

0.25).

The comparison above reveals that geostrophic adjustment is achieved mostly by

adjusting the velocity field if the length scale of the disturbance is long compared to the

radius of deformation. Conversely, geostrophic adjustment is achieved mostly by

adjusting the pressure field if the length scale is short compared to the radius of

deformation.

4.a Gradient-wind adjustment problem of an axisymmetric vortex

A generalization of the classical geostrophic adjustment problem is the

adjustment of an axisymmetric vortex that has significant curvature in gradient wind

balance. It may be regarded as a prototype large-scale atmospheric disturbance,

exemplified by hurricanes. When such a disturbance is significantly disturbed to an

unbalanced state by transient external influence, it is expected to eventually adjust to a

new steady balanced state. We again seek to directly deduce the structure and intensity

of the adjusted state of the vortex in a simple setting without having to work out the

intermediate states.

The gradient-wind adjustment problem even for the case of an axisymmetric

vortex is analytically intractable because of the nonlinear functional relationship between


the pressure field and velocity field. It is beyond the scope of the previous analyses of

adjustment of weak axisymmetric perturbations in the presence of a background vortex

(e.g. Schubert et al, 1980).

4.b Formulation

Let us consider an axisymmetric vortex in a shallow-water model without

friction. Gradient wind balance initially prevails everywhere except in a certain radial

band. As the basis for deducing the adjusted state at large time, we invoke the principle

of conservation of potential vorticity of each fluid column. For a cyclonic axisymmetric

flow, the conservation of PV can be written as

HQhQfrv

drdv

++−−= (28)

where r is the radial distance, the Coriolis parameter, the departure from a constant

reference depth

f h

H , v the azimuthal velocity of a fluid column and o

oo

hH

fr

vdr

dv

Q+

++= is the

initial known PV of the same fluid column. The adjusted vortex is assumed to be in gradient

wind balance, namely

grvv

gf

drdh 2

+= (29)

g is reduced gravity. The values of v and are expected very close to and

respectively at a sufficiently large distance

h ov oh

maxrr = from the center of the vortex.

Let us first consider a balanced Rankin vortex in a shallow-water model. Its

vorticity is equal to orV2 in an inner core, and is zero in the outer region. The distribution of

its azimuthal velocity is then


orVr*v = in orr ≤ ,

rVr

*v o= in orr > (30)

By (29), we obtain the corresponding as *h

2

o

o2

rr

Vfr

1g2

V*h ⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎠

⎞⎜⎝

⎛+= in orr ≤

2o

2

o

oo2

rr

g2V

rrn

gfVr

Vfr

2g2

V*h ⎟⎟⎠

⎞⎜⎜⎝

⎛−⎟⎟

⎠

⎞⎜⎜⎝

⎛+⎟

⎠

⎞⎜⎝

⎛ += l in orr >

(31)

It is convenient to measure r in unit of , v in unit of V , in unit of or h H and

in unit of QHf . The dynamics depends on two non-dimensional parameters, the Rossby

number ofr

VR = and the ratio of to the radius of deformation,orgHfro=λ . Then the non-

dimensional properties of a balanced Rankin vortex are

222

rR11

2R*h ⎟

⎠⎞

⎜⎝⎛ +=

λ , r*v = , ( )*h11R2*Q

++

= in 1r ≤ ;

( )⎟⎠⎞

⎜⎝⎛ ++

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛⎟⎠⎞

⎜⎝⎛−= rn

21R

r12

2R*h 2

222lλλ ,

r1*v = , ( )*h1

1*Q+

= in 1r ≥

(32)

4.c Analysis

The initial vortex is prescribed as an unbalanced Rankin vortex

*vvo = , ⎟⎟

⎠

⎞

⎜⎜

⎝

⎛⎟⎠⎞

⎜⎝⎛ −

−+=2

o barexp*hh ε ,

( )oh11R2Q

++

= in 1r ≤ ; ( )oh11Q+

= in 1r ≥

(33)


where and are given in (32). The non-dimensional governing equations take the form

of

*v *h

( )1QR1h

RQ

rv

drdv

−++−= (34)

2222

rvRvR

drdh λλ +=

(35)

The task is to determine the structure of and solely on the basis of the given information

about which is a function of

v h

Q ( )rvo and ( )rho . A solution must be also compatible with the

kinematic constraint that the fluid at the center of the adjusted vortex is at rest.

Eqs. (34) and (35) could be combined to one single but highly nonlinear ordinary

differential equation (ODE) that governs either or . We however choose to directly solve

(34) and (35) as a pair of coupled 1

v h

st order ODE for two unknowns simultaneously. Therefore,

we may pose the problem of solving (34) and (35) as an initial value problem in space. The

solution is to satisfy the following far field condition and kinematic constraint at the center,

@ maxrr = , *vv = and *hh =

@ , 0r → 0v→

(36)

Since we need to consider a domain much larger than the inner core of a vortex, it would be

advantageous to introduce a stretch-coordinate so that the inner core can be depicted in high

resolution whereas the outer region can be depicted with progressively coarser resolution.

Thus, we introduce a new radial coordinate

( )rns l= (37)

The computational domain is maxmin sss ≤≤ corresponding to a range of radial distance

. The computation is done using maxmin rrr ≤≤ 2smin −= and 2smax = corresponding to 13.0rmin =


and . The domain is depicted with (J+2) uniform grids in the stretch-coordinate. We

integrate (34) and (35) radially inward.

4.7rmax =

The imbalance in the initial state under consideration solely stems from the

departure of from . It is strongest at oh *h ar = exponentially decreasing away from it. We

will consider two distinctly different locations of the initial imbalance; one centering at the

edge of the inner core and the other centering at twice the distance further out from the center.

It suffices to present the results for one pertinent combination of the parameter values, namely

and 1R = 1=λ . For this condition the Coriolis force and centrifugal force are equally strong

at the edge of the inner core of the initial vortex.

4.d Result

4.d.1 Case (1)

A fairly strong localized imbalance is prescribed with the use of 0.1=ε , 0.1a = and

. The maximum initial unbalanced depth is then located at the edge of the inner core of

the vortex, . Rudimentary reasoning suggests that some fluid would move inward leading

to an increase in the depth and also possibly in the azimuthal velocity. The PV boundary

would be displaced inward as well. We denote the location of the displaced PV boundary by

which is unknown a priori. The properties of the initial state of the vortex under

consideration are shown in Fig. 10.

6.0b =

.1r =

1r


0

1

2

3

4

5

0 1 2 3 4 5 6 7 8

HIJK

r

Fig. 10 Radial distribution of the non-dimensional depth ,velocity and potential vorticity of an initial unbalanced vortex and the depth of a corresponding balanced vortex for case 1 ( ,

)J( )I()K(

)H( .1R = .1=λ , .1=ε , .1a = and 6.0b = ) .

The solution is obtained with an iterative algorithm. 225 uniform grids in the s

coordinate are used in this computation. We begin by making a guess of which allows us to

compute a corresponding distribution of PV,

1r

( )rQ , according to (33). We solve (34) and (35)

with a 4th order Runge-Kutter scheme subject to boundary conditions (36) for that specific

distribution of Q . The provisional solution of ( )rh is then checked to see if it satisfies the

mass conservation requirement for that particular value of , namely

. If not, the guess for is modified and the computation is repeated.

It is found that this algorithm quickly yields to convergence of the result. It is found that this

algorithm quickly yields convergence of the result. It is found that

1r

( ) ( +1 )⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=+ ∫∫ rdrhrdrh1

1

0o

r

0

1

1r

81.0r1 = is a self-consistent

value for this case. The distributions of the total depth of the adjusted vortex, that of the initial


vortex and that of a counterpart Rankin vortex are shown in Fig. 11. The corresponding

velocity field is shown in Fig. 12.

0

1

2

3

4

5

0 1 2 3 4 5 6 7

r

8

Fig. 11. Radial distribution of the non-dimensional total depth of an initial unbalanced vortex (dash), of the adjusted vortex (solid) and of the corresponding Rankin vortex for comparison (dot) for case 1.

It is seen that the height field has been considerably modified due to redistribution of

mass. The change of potential energy of the vortex in unit of is found to be πρ 2o

2rgH

( ) ( )( )∫ +−+=max

min

r

r

2o

2 h1h1PΔ 1.5rdr = (38)

The mass redistribution leads to an increase of pressure gradient inward of , but a

decrease outward of . The adjusted velocity field inward of

81.0r1 =

81.0r1 = 81.0r1 = is enhanced,

but the velocity in most of the vortex is substantially weakened. The maximum velocity

is now located at . 81.0rr 1 ==


0

0.2

0.4

0.6

0.8

1

1.2

0 1 2 3 4 5 6 7

r

8

Fig. 12 Radial distribution of the non-dimensional velocity field of the initial unbalanced vortex (dash) and of the adjusted vortex (solid) for case 1.

The change of kinetic energy of the vortex in unit of is found to be πρ 2o

2rgH

( ) ( )( ) .125rdrvh1vh1RKmax

min

r

r

2oo

222 −=+−+= ∫λΔ (39)

While the potential energy increases, the kinetic energy decreases as a result of the

adjustment process. 120 units of energy must have been transmitted to far distances by

gravity waves as a result of the adjustment process. We check to see if the distribution of

potential vorticity of the adjusted state ,h1

1rv

drdvR

+

+⎟⎠⎞

⎜⎝⎛ +

agrees with that of the initial state (Fig.

13). It is found that except for the displacement of the PV boundary, the two distributions

are indeed very close to one another.


0

0.5

1

1.5

2

2.5

3

0 1 2 3 4 5 6 7

r

8

Fig. 13. Radial distribution of potential vorticity of the initial state (dash) and of the adjusted state (solid) for case 1.

4.d.2 Case (2)

It is also of interest to consider a case in which the initial imbalance is located

further outside of the inner core. The parameter values are 2a = , 6.0b = , and .1=ε . The

center of the initial imbalance is now located at 2r = instead of 1r = . The PV boundary

is found to shift outward to associated with an outward mass flux. The results for

the velocity and height fields of the adjusted vortex are shown in Figs. 14 and 15.

35.1r1 =

0

1

2

3

4

5

0 1 2 3 4 5 6 7

r

8


Fig. 14 Radial distribution of the non-dimensional total depth of the initial vortex (dash), of the adjusted vortex (solid) and of a corresponding undisturbed vortex (dot) for case 2 ( , 1R = 1=λ . , 2a = , 6.0b = and .1=ε ).

0

0.2

0.4

0.6

0.8

1

0 1 2 3 4 5 6 7 8

r

Fig. 15 Radial distribution of the velocity field of an initial vortex (dot) and that of the velocity field of the adjusted vortex (solid) for case 2.

The change of potential energy of the vortex in unit of is found to be πρ 2o

2rgH 3.5P −=Δ .

A comparison of Fig. 15 with Fig. 12 reveals the main difference between the adjusted

state in case-2 versus that in case-1. It is found that when the initial imbalance is located

outside its inner core (i.e. at . instead of .2r = 1r = ), the wind in the inner core changes

very little, but the maximum velocity shifts outward and is enhanced by about 10%. The

velocity extending out to about .2r = is enhanced. The velocity further out is weaker

than that in the initial state. Consequently, the zone of relatively strong wind of the

adjusted vortex becomes narrower. This is an interesting consequence of the

redistribution of mass and related PV. There is a compensating decrease of velocity

farther out from the center of the vortex. The change of kinetic energy of the vortex in

unit of is found to be πρ 2o

2rgH 7.57K −=Δ This change is much smaller than in case-1. It


is noteworthy that both the potential energy and kinetic energy of the vortex become

smaller after adjustment in this case. Sixty-three units of energy must have been

transmitted to far distance by gravity waves.

The characteristics of the response in both cases of initial states suggest one

overall conclusion, namely the azimuthal velocity of the adjusted state on the radially

inward (outward) side of the maximum initial unbalanced depth is enhanced (weakened)

with respect to that of the initial vortex. The results for different combinations of the

values of R and λ are qualitatively similar to those of the two cases reported above.

The algorithm works less well in cases where the centrifugal force is much stronger.

This approach of directly deducing the adjusted state from a given unbalanced

initial state is not only more efficient but also more illuminating than the alternative

approach of explicit simulation of a flow. While the transient development of a vortex is

an interesting issue in its own right, it is a secondary aspect of the adjustment process.

There is however one caveat. If the adjusted state takes a very long time to establish, the

neglected effect of friction would be eventually important and the actual adjusted state

could be somewhat different.

5. Concluding remarks

We have seen that it is feasible to get exact solution of the geostrophic adjustment

problem for relatively simple initially unbalanced disturbances of any intensity in the

context of a shallow-water model. It is important to take into account of the displacement

of fluid columns from their original locations as a result of the adjustment process. It

follows that the spatial distribution of their potential vorticity in the adjusted state would


differ from that in the initial state. The mutual adjustment between the mass and velocity

fields is such that the potential vorticity of each fluid column conserves. It is this

dynamical constraint that makes it possible to deduce the adjusted state solely on the

basis of the information about the initial state without having to know the intermediate

states.

Symmetry consideration is helpful for a physical system that has innate

symmetry. For example, the depth of the fluid in the adjusted state has odd symmetry

and the velocity field has even symmetry with respect to the center of the domain in

Rossby’s problem. In the complimentary Rossby problem , the elevation of the fluid in

the adjusted state has even symmetry, whereas the velocity field has odd symmetry with

respect to the center of the domain. Most of the adjustment involves the velocity (mass)

field for the case of a large (small) disturbance relative to the radius of deformation.

Gradient-wind adjustment problem is a generalization of geostrophic adjustment

problem. The former is a more difficult problem because the centrifugal force is a

quadratic function of velocity. For the case of a relatively simple initial unbalanced

Rankin vortex, the analysis can be done numerically in an iterative manner. It is found

that the adjusted state depends very much upon the location of the initial imbalance

relative to the inner core of the Rankin vortex. There are mass fluxes in both inward and

outward directions from the location of maximum initial imbalance. The azimuthal

velocity of the adjusted state on the radially inward (outward) side of the maximum initial

unbalanced depth is found to be greater (weaker) than that of the initial vortex.

Consequently, the belt of strong wind becomes narrower.


We have not discussed the transient aspect of geostrophic adjustment. Broadly

speaking, the transient evolution of an initial unbalanced state in a shallow-water model

involves excitation and propagation of inertio-gravity wave away from the initial region

of imbalance. The time it takes for geostrophic adjustment to complete is relatively short.

For this reason, it is generally regarded as a secondary aspect of the adjustment process.

Details about the transient process can only be delineated with a high resolution

numerical simulation.


6. Reference

Rossby, C.-G, 1938: On the mutual adjustment of pressure and velocity distributions in

certain simple current systems, II. J. Marine Research, 239-263.

Mihaljan, J.M., 1963: The exact solution of the Rossby adjustment problem. Tellus XV,

150-

W. H. Schubert, J. J. Hack, P. L. Silva Dias and S. R. Fulton. 1980: Geostrophic

Adjustment in an Axisymmetric Vortex. J. Atmos. Sci.: 37, 1464–1484.

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