For a rotating solid object, the vorticity is two times of its angular velocity Vortici ty In physical oceanography, we deal mostly with the vertical component of vorticity, which is notated as k y u x v j x w z u i z v y w k j i V z y x r r r r r r r r ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ∂ ∂ − ∂ ∂ + ∂ ∂ − ∂ ∂ + ∂ ∂ − ∂ ∂ = + + = × ∇ = ω ω ω ω y u x v ∂ ∂ − ∂ ∂ = ς Relative vorticity is vorticity relative to rotating earth Absolute vorticity is the vorticity relative to an inertia frame of reference (e.g., the sun) Planetary vorticity is the part of absolute vorticty associated with Earth rotation f=2sin, which is only dependent on latitude. Absolute vorticity =Relative vorticity + Planetary
Transcript
For a rotating solid object, the vorticity is two times of its angular velocity
Vorticity
In physical oceanography, we deal mostly with the vertical component of vorticity, which is notated as
kyu
xvj
xw
zui
zv
ywkjiV zyx
rrrrrrrr⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛
∂∂−
∂∂+
∂∂−
∂∂+
∂∂−
∂∂=++=×∇= ωωωω
yu
xv
∂∂−
∂∂=ς
Relative vorticity is vorticity relative to rotating earth
Absolute vorticity is the vorticity relative to an inertia frame of reference (e.g., the sun)
Planetary vorticity is the part of absolute vorticty associated with Earth rotation f=2sin, which is only dependent on latitude.
Considering the case of constant ρ. For a shallow layer of water (depth H<<L), u and v are not function of z because the horizontal pressure gradient is not a function of z. (In general, the vortex tilting term, is small.
Then we have the simplified vorticity equation
⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛⎟⎠⎞⎜
⎝⎛
∂∂+
∂∂=
∂∂+−+
∂∂+
∂∂+
∂∂
2
2
2
2
yxA
zwfv
yv
xu
t Hςςςβςςς
yw
xw
zu
yw
zv
xw
yx ∂∂−
∂∂−=
∂∂
∂∂−
∂∂
∂∂ ωω
Since vdy
dfv
dt
df β==
the vorticity equation can be written as (ignoring friction)
( ) ( )z
wf
dt
fd
∂
∂+=
+ς
ς
+f is the absolute vorticity
0=∂∂
+∂∂
+∂∂
zw
yv
xu
Using the Continuity Equation
( ) ( ) ⎟⎟⎠
⎞⎜⎜⎝
⎛∂∂
+∂∂
+−=+
y
v
x
uf
dt
fdζ
ζ
For a layer of thickness H, consider a material column
⎟⎟⎠
⎞⎜⎜⎝
⎛∂∂
+∂∂
−=y
v
x
u
dt
dH
H
1
( )( )
dt
dH
Hdt
fd
f
11=
+
+
ζ
ς
0=⎟⎠⎞
⎜⎝⎛ +
Hf
dtd
We get
or Potential Vorticity Equation
Alternative derivation of Sverdrup Relation
xp
gfv ∂∂=
ρ1
ypfu
g ∂∂−=
ρ1
Construct vorticity equation from geostrophic balance
In the ocean’s interior, for large-scale movement, we have the differential form of the Sverdrup relation
zwfv∂∂=β
z
wf
dt
df
∂∂
=
i.e., <<f
If f is not constant, then
⎟⎟⎠
⎞⎜⎜⎝
⎛∂∂
−∂∂
−⎟⎟⎠
⎞⎜⎜⎝
⎛
∂∂
−∂∂
+∂∂
−≈y
u
x
vK
yxy
fv xy ττ
ρ
10
€ βv+Kς=1ρ∂τy∂x−∂τx∂y ⎛ ⎝ ⎜ ⎞ ⎠ ⎟
Assume geostrophic balance on -plane approximation, i.e.,
yff o 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 Ar
yxDxJ
tWe 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,
Posing the gyre problem
ψψττρ
ψβψψψ 4222 1, ∇+∇−∂∂−
∂∂=
∂∂+∇+∇
∂∂
⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛⎟⎠⎞⎜
⎝⎛
Hxy Ar
yxDxJ
t
( ) ( ) ( )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-dimensional Equation, An Example
2
2
2
2
2
21
z
uA
y
u
x
uA
x
pyvvf
z
uw
y
uv
x
uu
t
uzho ∂∂
+⎟⎟⎠
⎞⎜⎜⎝
⎛
∂∂
+∂∂
+∂∂
−=−−∂∂
+∂∂
+∂∂
+∂∂
ρβ
Consider a homogeneous fluid on a β-plane
0=∂∂
+∂∂
+∂∂
zw
yv
xu
1<<=LHδ
Define the following non-dimensional variables:
( ) ( )vuUvu ′′= ,,
( ) ( )yxLyx ′′= ,, zHz ′=wUwWw ′=′= δ
pULfpPp o ′=′= ρ
x
u
L
U
x
u′∂′∂
=∂∂
( ) ( )1~,,, Opwvu ′′′′
By definition
(geostrophy)
( )1~ Ox
u′∂
′∂
tU
LtTt ′=′=
( ) ( )1~,,, Otzyx ′′′′
1<<=Lf
U
o
ε ( ) 1~ <<= εβ
β Of
L
oo
Define the non-dimensional parameters
Rossby Number
2Lf
AE
o
hh =
2
2
2⎟⎠
⎞⎜⎝
⎛==HD
HfA
E E
o
zz
Horizontal Ekman Number
Vertical Ekman Numbero
zE f
AD
2=
Ekman depth
2
2
2
2
2
2
z
uE
y
u
x
uE
x
pyvv
z
uw
y
uv
x
uu
t
uvho ∂∂
+⎟⎟⎠
⎞⎜⎜⎝
⎛
∂∂
+∂∂
+∂∂
−=−−⎟⎟⎠
⎞⎜⎜⎝
⎛∂∂
+∂∂
+∂∂
+∂∂ βε
0=∂∂
+∂∂
+∂∂
zw
yv
xu
Then, we have (with prime dropped)
( )zho EEzyxuu ,,,,,, βε=The solution
In the interior of the ocean, Eh<<1 and Ez<<1
x
pv
∂∂
= (geostrophy)
Near the bottom or surface, Ez≈O(1)
2
2
z
u
x
pv
∂∂
+∂∂
−=−
( )1~ OE z EDH ~
In the surface and bottom boundary layers, the vertical scales are redefined(shortened, a general character of a boundary layer)
Taking into the equations, we have
2
2
22
2
2
2
2
2
z
u
H
UA
y
u
x
u
L
UA
x
pUf
vyLUvUfz
uw
y
uv
x
uu
t
u
L
U
zho
o
′∂′∂
+⎟⎟⎠
⎞⎜⎜⎝
⎛′∂′∂
+′∂′∂
+′∂′∂
−=
′′−′−⎟⎟⎠
⎞⎜⎜⎝
⎛′∂′∂′+
′∂′∂′+
′∂′∂′+
′∂′∂ β
0=⎟⎟⎠
⎞⎜⎜⎝
⎛′∂′∂
+′∂′∂
+′∂′∂
zw
yv
xu
LU
2
2
22
2
2
2
2 z
u
Hf
A
y
u
x
u
Lf
A
x
p
vyf
Lv
z
uw
y
uv
x
uu
t
u
Lf
U
o
z
o
h
oo
′∂′∂
+⎟⎟⎠
⎞⎜⎜⎝
⎛′∂′∂
+′∂′∂
+′∂′∂
−=
′′−′−⎟⎟⎠
⎞⎜⎜⎝
⎛′∂′∂′+
′∂′∂′+
′∂′∂′+
′∂′∂ β
0=′∂′∂
+′∂′∂
+′∂′∂
zw
yv
xu
Non-dimensional vorticity equationNon-dimensionalize all the dependent and independent variables in the quasi-geostrophic equation as
where UL=Ψ ULT = 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.,
€ x,y()=L′ x ,′ y ()€ t=T′ t € ψ=Ψ′ ψ € τ=Θ′ τ
Interior (Sverdrup) solutionIf <<1, S<<1, and M<<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 −=
∂∂+∇ , S<<1.
0=ψ
(Since the horizontal friction is neglected, the no-slip condition can not be enforced. No-normal flow condition is used).
( )yx
I ππψ
sin−=∂
∂
( )yxI ππψ sin1 ⎟⎠⎞⎜
⎝⎛ −=
Interior solution
at x=0, 1; y=0, 1. No-normal flow boundary condition
Let S
S
xxδεξ
*
==, we
have
( )ySyySS ππψεψεψε ξξξ sin11 −=++ −−
( ) 0sin2 ==−=+ ⎟⎠⎞
⎜⎝⎛
SSyySOy εππεψεψψ ξξξ
Re-scaling in the boundary layer:
€ ∂ψ∂x=∂ψ∂ξ∂ξ∂x=1εS∂ψ∂ξ
€ ∂2ψ∂x2=1εS∂∂ξ∂ψ∂x ⎛ ⎝ ⎜ ⎞ ⎠ ⎟=1εS2∂2ψ∂ξ2
€ ∇2ψ=∂2ψ∂x2+∂2ψ∂y2=1εS∂2ψ∂ξ2+∂2ψ∂y2
( )yxS ππψψε sin2 −=
∂∂+∇Take into
As ξ=0, ψ=0. As ξ,ψψI
The solution for 0=+ ξξξ ψψ is
( ) ( ) S
x
BeAeyxByxA εξψ−− +=+= ,,
0=ξ , 0=ψ . A=-B
( ) ⎟⎟
⎠
⎞
⎜⎜
⎝
⎛ −−= S
x
eyxA εψ 1,
, ( ) ( ) ( )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
curlvoρ
τβ = ( )D
curldt
dfoρ
τ=
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?
