Wind Driven Circulation III

• Closed Gyre Circulation• Quasi-Geostrophic Vorticity Equation• Westward intensification• Stommel Model• Munk Model• Inertia boundary layer• Numerical results• Observations

Kux

fvx

p x −∂∂++

∂∂−= τ

ρρ110

Kvy

fux

p y −∂∂

+−∂∂−=

τρρ110

Consider the balance on an f-plane

⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂−

∂∂−⎟⎟

⎠

⎞⎜⎜⎝

⎛∂∂

−∂∂

∂∂+⎟⎟

⎠

⎞⎜⎜⎝

⎛∂∂+

∂∂=

x

v

y

uK

xyzy

v

x

uf yx

ττ

ρ

10

⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

−∂∂

−⎟⎟⎠

⎞⎜⎜⎝

⎛

∂∂

−∂∂

∂∂

≈x

v

y

uK

xyzyxττ

ρ

10

If f is not constant, then

⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

−∂∂

−⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

−∂∂

∂∂

+∂∂

≈x

v

y

uK

xyzy

fv yx

ττ

ρ

10

Assume geostrophic balance on β-plane approximation, i.e.,

yff o β+= (β is a constant)

Vertically integrating the vorticity equation

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

∂∂+

∂∂=

∂∂−+

∂∂+

∂∂+

∂∂

2

2

2

2

yxA

zwfv

yv

xu

t Hoςςβςςς

we have

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

⎟⎠⎞

⎜⎝⎛

∂∂+∂

∂=

−−+∂∂+∂

∂+∂∂

2

2

2

2

yxA

wwD

fv

yv

xu

t

H

BEo

ςς

βςςς

The entrainment from bottom boundary layer

ρβτττ

ρ 21

o

xxy

oE

fyxfw +

∂∂−

∂∂=

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

ςπ2E

BDw =

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜

⎝

⎛

∂∂+∂

∂+−+∂∂−∂

∂=+∂∂+∂

∂+∂∂

2

2

2

21yx

ArfyxD

vy

vx

ut H

o

xxy

o

ςςςβτττ

ρβςςς

The entrainment from surface boundary layer

We have

where DDfr E

π2=

barotropic

xxP

fv

o∂∂=∂

∂= ψρ1

1<<ofLβ

1~ <<=⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜

⎝

⎛

∂∂

⎟⎟⎠

⎞⎜⎜⎝

⎛

o

o

x

o

x

fL

L

fO

x

f βτ

τβ

τ

βτ

For and ψς 2∇=

where

and

Moreover, (Ekman transport is negligible)

ψψττρ

ψβψψψ 4222 1, ∇+∇−∂∂−

∂∂=

∂∂+∇+∇

∂∂

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛⎟⎠⎞⎜

⎝⎛

Hxy AryxDx

Jt

We have

Quasi-geostrophic vorticity equation

where

4

4

22

4

4

44 2

yyxx ∂∂+

∂∂∂+

∂∂=∇ ψψψψ

yyP

fu

o∂∂−=∂

∂−= ψρ1

1<<LfU

o

, we have

of

P

ρψ =

( ) ( ) ( )xyyx

J∂

∇∂

∂

∂−

∂

∇∂

∂

∂=∇

ψψψψψψ

222,

Non-dmensional equationNon-dimensionalize all the dependent and independent variables in the quasi-geostrophic equation as

( )yxLyx ,),( =′′ Ttt =′ ψψ Ψ=′ ττ Θ=′where UL=Ψ U

LT= UDLρβ=Θ

xU

xL

UL

x ′∂′∂

=′∂′∂

=∂∂ ψβψβψβ

For example,

( ) ψεψεττψψψψε 4222 , ∇+∇−∂∂−

∂∂=

∂∂+∇+∇

∂

∂⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

MSxy

yxxJ

t

The non-dmensional equation

where 2

2 ⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

==LL

U Iδ

βε , βδ UI = ,

nonlinearity.

LLr S

S

δβε == βδ r

S = , bottom friction.

3

3 ⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

==LL

A MHM

δβε

,

3βδ H

MA= , lateral friction.,

Interior (Sverdrup) solutionIf ε<<1, εS<<1, and εM<<1, we have the interior (Sverdrup) equation:

yxxxyI

∂∂−

∂∂=

∂∂ ττψ

∫ ∂

∂−∂∂−=

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛Ex

x

xyEI

dxyxττψ

(satistfying eastern boundary condition)

∫ ∂∂−∂

∂=⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛x

Wx

xyWI

dxyxττψ

Example:Let ( )yx πτ cos−= ,

0=yτOver a rectangular

basin (x=0,1; y=0,1)

( )yxEI ππψ sin1⎟

⎠⎞⎜

⎝⎛ −−=

( )yxWI ππψ sin−=

(satistfying western boundary condition)

.

Westward IntensificationIt is apparent that the Sverdrup balance can not satisfy the mass conservation and vorticity balance for a closed basin. Therefore, it is expected that there exists a “boundary layer” where other terms in the quasi-geostrophic vorticity is important. This layer is located near the western boundary of the basin. Within the western boundary layer (WBL),

IB ψψ ~ , for mass balance

δξ x=

In dimensional terms,

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

∂∂−

∂∂>>

∂∂−

∂∂=

=∂∂

xyDxyDLO

ULOOx

yx

o

yx

o

BB

ττρ

ττρδ

δβ

δψβψβ

11

~

The Sverdrup relation is broken down.

, the length of the layer δ <<L The non-dimensionalized distance is

The Stommel modelBottom Ekman friction becomes important in WBL.

( )yxS ππψψε sin2 −=

∂∂+∇ , εS<<1.

0=ψ at x=0, 1; y=0, 1. free-slip boundary condition(Since the horizontal friction is neglected, the no-slip condition

can not be enforced. No-normal flow condition is used).

( )yxI ππψ

sin−=∂

∂ ( )yxI ππψ sin1 ⎟⎠⎞⎜

⎝⎛ −=

In the boundary layer, let S

S

xxδεξ

*

== ( ∞→0~ξ), we

have

( )ySyySS ππψεψεψε ξξξ sin11 −=++ −−

( ) 0sin2 ==−=+ ⎟⎠⎞

⎜⎝⎛SSyyS

Oy εππεψεψψ ξξξ

Interior solution

Re-scaling:

The solution for 0=+ ξξξ ψψ is

( ) ( ) S

x

BeAeyxByxA εξψ−− +=+= ,,

0=ξ , 0=ψ . A=-B

( ) ⎟⎟

⎠

⎞

⎜⎜

⎝

⎛ −−= S

x

eyxA εψ 1,

ξ→∞, ( ) ( ) ( )yxyxyxA I ππψψ sin1,, ⎟⎠⎞⎜

⎝⎛ −==→

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛ −−= S

x

Ie εψψ 1 ( Iψ can be the interior solution under different winds)

For ( )SOx ε<

( )ye S

xB ππψ ε sin1 ⎟

⎟

⎠

⎞

⎜⎜

⎝

⎛ −−=

( )yevS

x

B S

ππεε

sin−

=

For ( ) 1≤≤ xO Sε ,

( )yxI ππψ sin1 ⎟⎠⎞⎜

⎝⎛ −= ,

( )yv I ππ sin−= .

,

.

,

The dynamical balance in the Stommel model

In the interior,Dx

pfvo

x

o ρτ

ρ +∂∂−=− 1

Dypfu

o

y

o ρτ

ρ +∂∂−= 1

( )D

curlvoρτβ = ( )

Dcurl

dt

dfoρτ=

Vorticity input by wind stress curl is balanced by a change in the planetary vorticity f of a fluid column.(In the northern hemisphere, clockwise wind stress curl induces equatorward flow).

In WBL,xpfv

o ∂∂=ρ

1

rvypfu

o−

∂∂−= ρ

10=+

∂∂ vxvr β x

vrdtdf

∂∂−=

Since v>0 and is maximum at the western boundary, 0<∂∂xv

the bottom friction damps out the clockwise vorticity.

,

Question: Does this mechanism work in a eastern boundary layer?

Munk modelLateral friction becomes important in

WBL.( )yxM ππψψε sin4 −=+∇−

Within the boundary layer, let MM

xxδε

ξ *

31== , we have

( )τεξψψεξ

ψεξψ curl

yy MMM31

4

434

22

432

4

42 =

∂∂+

∂∂+

∂∂∂+

∂∂−

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

Wind stress curl is the same as in the interior, becomes negligible in the boundary layer.

For the lowest order, 04

4=

∂∂−

∂∂

ξψ

ξψ . If we let ( ) ⎟

⎠⎞⎜

⎝⎛+= yyxI ,, ξφψψ , we have

04

4=

∂∂−

∂∂

ξφ

ξφ

. And for +∞→ξ , 0→φ .

The general solution is⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛−

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛− +++= ξξφξξ

ξ23sin

23cos 2

42

321 eCeCeCC

Since +∞→ξ

,

0→φ , C1=C2=0.

.

( )⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛−

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛− ++= ξξψψξξ

23sin

23cos, 2

42

3 eCeCyxI

Using the no-slip boundary condition at x=0,

03 =∂∂−

∂∂−=

∂∂−=

yC

yyu Iψψ KyC I +−= ⎟

⎠⎞⎜

⎝⎛ ,03 ψ (K is a constant).

023

21

43

31=+−+

∂∂=

∂∂=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡CC

xxv

M

Iε

ψψ .to ⎟⎟

⎠

⎞

⎜⎜

⎝

⎛3

1

MO ε , 33

4

CC =

Total solution

( )⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛−

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛− +++−= ξξξξψψξξ

23sin

31

23cos

23sin

31

23cos1, 22 KeeyxI

Considering mass conservation 0),1(,01

0=−=∫ ⎟

⎠⎞⎜

⎝⎛ yyvdx ψψ

0),1(,0 ==⎟⎠⎞⎜

⎝⎛ yy ψψ K=0 ( )

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛− +−= ξξψψξ

23sin

31

23cos1, 2eyxI

Western boundary current

ξε

ψξ

23sin

32,0

31

2

M

Ieyv

−

⎟⎠⎞⎜

⎝⎛=B

Scaling

ξε

ψξ

23sin

32,0

31

2

M

Ieyv

−

⎟⎠⎞⎜

⎝⎛=B

The cross-stream distance from boundary to maximum velocity is

Given

31

* 2.1 ⎟⎟⎠

⎞⎜⎜⎝

⎛=

βδ HA

The ratio between the nonlinear and dominant viscous terms is

( )2

*

3**

2*

*

2.1)(

)(⎟⎠⎞

⎜⎝⎛=⎟⎟

⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛===

+=

δ

δβδ

β

δ

δ

δ I

HHHxxH

yx

A

U

A

UVA

UV

vAO

vvuvOR βδ U

I =where

The continuity relation is also used:L

VU

*δ=

Using U=O(2 cm/s), ß=O(10-13 cm-1 s-1), AH=4×106 cm2/s, we have R=4.i.e., the nonlinear terms neglected are larger than the retained viscous terms,which causes an internal inconsistency within the frictional boundary layer.

Inertial Boundary Layer

( ) yxxJ

txy

∂∂−

∂∂=

∂∂+∇+∇

∂

∂⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛ ττψψψψε 22 ,If ε>>εI and εM,

Given SMI δδδ ,>>

I

xδξ =

a boundary layer exists in the west where

03

3

2

2=

∂∂+

∂∂

∂∂−

∂∂

∂∂

∂∂

ξψ

ξψψ

ξψ

ξψ BBBBB

yy Re-scaling with

0,2

2=+

∂∂

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛yJ B

B ξψψ Conservation of potential vorticity.

,BB uv >> ,yx ∂∂>>∂

∂2

2

xxv

yu

xv

∂∂=

∂∂≈

∂∂−

∂∂= ψς

, we have

or

The conservation equation may be integrated to yield

( )BB Gy ψξψ =+

∂∂

2

2

where ( )BGψ is an arbitrary function of Bψ

This equation states that the total vorticity is constant following a specific streamline.

Let ( ) ⎟⎠⎞⎜

⎝⎛+= yyxBIB ,, ξφψψ

(interior stream function plus a boundary layer correction),

Bφ must satisfy 0,lim =⎟⎠⎞⎜

⎝⎛

∞→yB ξφ

ξ

Now consider the region of large ξ, where IB ψφ <<

Take BIB φψψ += into equation

03

3

2

2=

∂∂+

∂∂

∂∂−

∂∂

∂∂

∂∂

ξψ

ξψψ

ξψ

ξψ BBBBB

yy

Retain only linear term in Bφ (and neglect some other small terms), we have

0,3

3=

∂∂+

∂∂

∂∂− ⎟

⎠⎞⎜

⎝⎛

ξφ

ξφψ BB

WI yXy

Integrate once and use the boundary condition

0,lim =⎟⎠⎞⎜

⎝⎛

∞→yB ξφ

ξ, we have

0,2

2=+

∂∂⎟⎠⎞⎜

⎝⎛

BB

WI yXu φξφ

If 0, >⎟⎠⎞⎜

⎝⎛ yXu WI ,

BφA necessary condition for the existence of a

pure inertial boundary current is

0, <⎟⎠⎞⎜

⎝⎛ yXu WI

The decaying solution is of the form

( )⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

⎟⎠⎞⎜

⎝⎛−

−=yXu

yCWI

B,

exp ξφ

will be oscillatory and not satisfy the boundary condition.

The dimensional width of the inertial boundary layer is

( )βδ yU−=

At those y’s where U is on shore and small, the width of the inertial current is small. As the point y0 is approached where U=0, δ will

shrink and finally be swallowed up within the thickness of a frictional layer.

Since equation

0,2

2=+

∂∂⎟⎠⎞⎜

⎝⎛

BB

WI yXu φξφ

is symmetric under transformation

ξξ −→ A similar inertial boundary layer can exist at the eastern boundary.

Inertial Currents with Small FrictionIn the presence of a small lateral friction, we can derive the perturbation equation as

which makes the boundary layer possible only in the western ocean. Moreover, it can be shown that a inertial-vicious boundary layer can be generated in the northern part of the basin where

0, >⎟⎠⎞⎜

⎝⎛ yXu WI

characterized by a standing Rossby wave.

3

33

2

2,

ξφ

δδ

φξφ

∂∂=+∂

∂⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜

⎝

⎛

⎟⎠⎞⎜

⎝⎛ B

I

MB

BWI yXu

ζβζ Kvv −=+∇⋅r

Assume the simple balance

A parcel coming into the boundary layer has 0<∇⋅= ζζv

dtd r 0>= v

dtdf β

The effect of friction is reduced and the boundary layer is broadened.

Bryan (1963) integrates the vorticity equation with nonlinear term and lateral friction. The Reynolds number is define as

HA

UL=Re

And δI/δM ranges from 0.56 for Re=5 to 1.29 for Re=60.

Veronis (1966), nonlinear Stommel Model

Western Boundary current: Gulf Stream

Gulf Stream Transport