Schedule for final exam

• Last day of class for CLIM712 (Dec. 6, 06)

• Exam period: Dec., 12 - Dec., 19

• Two choices:

a) close-book exam (12/11 or 12/13)

b) take home

Kufvx

g x −++∂∂

−=ρτη

0

Consider the vorticity balance of an homogeneous fluid (ρ=constant) on an f-plane

Kvfux

g y −+−∂∂

−=ρτη

0

⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

−∂∂

−⎟⎟⎠

⎞⎜⎜⎝

⎛

∂∂

−∂∂

+⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

+∂∂

−=y

u

x

vK

yxy

v

x

uf xy ττ

ρ

10

⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

−∂∂

−⎟⎟⎠

⎞⎜⎜⎝

⎛

∂∂

−∂

∂≈

y

u

x

vK

yxxy ττ

ρ

10

If f is not constant, then

⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

−∂∂

−⎟⎟⎠

⎞⎜⎜⎝

⎛

∂∂

−∂∂

+∂∂

−≈y

u

x

vK

yxy

fv xy 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<<Lf

U

o

, we have

of

P

ρψ =

( ) ( ) ( )xyyx

J∂

∇∂

∂

∂−

∂

∇∂

∂

∂=∇

ψψψψψψ

222,

Quasi-Geostrophic Vorticity Equation

ψψττρ

ψβψψψ 4222 1, ∇+∇−∂∂−

∂∂=

∂∂+∇+∇

∂∂

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛⎟⎠⎞⎜

⎝⎛

Hxy AryxDx

Jt

( ) ( ) ( )xyyx

J∂

∇∂

∂

∂−

∂

∇∂

∂

∂=∇

ψψψψψψ

222,

4

4

22

4

4

44 2

yyxx ∂∂+∂∂

∂+∂∂=∇ ψψψψ

( ) 0=∇×⋅=⋅ ψknVnrrrr

0=∂∂

=∇⋅l

lψψ

r ( )0== constψ

( ) 0=∂

∂=∇⋅=∇×⋅=⋅

nnklVl

ψψψ

rrrrr

0=Vr

Boundary conditions on a solid boundary L

(1) No penetration through the wall

(2) No slip at the wall

( )0== constψ

Non-dmensional vorticity 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 ππ

ψsin−=

∂∂ ( )yxI ππψ sin1 ⎟

⎠⎞⎜

⎝⎛ −=

In the boundary layer, let S

S

xxδεξ*

== ( ∞→0~ξ), we

have

( )ySyySS ππψεψεψε ξξξ sin11 −=++ −−

( ) 0sin2 ==−=+ ⎟⎠⎞

⎜⎝⎛

SSyySOy 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?