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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
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