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EVAT 554OCEAN-ATMOSPHERE
DYNAMICS
SVERDRUP TRANSPORT
LECTURE 15
(Reference: Peixoto & Oort, Chapter 8,10)
Sverdrup Transport
)/v(/ˆ1u z
zp
af V
)/(/cosˆ1v zu
zp
af V
Let us consider again the approximate form of the governing equations for the horizontal circulation where we ignore horizontal,
but not vertical friction:
zxp
af
/
cos1v
zyp
af
/1u
Let us rewrite the friction terms in terms of stress, and multiply through by ,
Sverdrup Transport
zxp
af
/
cos1v
zyp
af
/1u
Integrate these equations vertically, from some depth h well below the Ekman depth:
dz0
dz/cos1
0vdz
0
zx
h
pah
f
h
dz0
dz/10
udz0
zy
h
pah
f
h
Sverdrup Transport
zxp
af
/
cos1v
zyp
af
/1u
Now re-arrange the equations
dz0
vdz0
dz/cos1
0
zx
h
f
h
pah
dz0
udz0
dz/10
zy
h
f
h
pah
Evaluate the right hand side…
dz0
vdz0
dz/cos1
0
zx
h
f
h
pah
Sverdrup Transport
zxp
af
/
cos1v
zyp
af
/1u
dz0
udz0
dz/10
zy
h
f
h
pah
yfM
xfM
dz0
vdz0
dz/cos1
0
zx
h
f
h
pah
Sverdrup Transport
zxp
af
/
cos1v
zyp
af
/1u
dz0
udz0
dz/10
zy
h
f
h
pah
yfM
xfM
0x
0y
Sverdrup Transport
dz0
vdz0
dz/cos1
0
zx
h
f
h
pah
yfM0x
dz0
udz0
dz/10
zy
h
f
h
pah
xfM
0y
a
xyfM
ap
ah
a
)cos(cosdz/1
00
a
yxfM
ap
ah
a0dz/1
0
Multiply first equation by cos and differentiate the two equations with respect to latitude and longitude respectively
Sverdrup Transport
a
xyfM
ap
ah
a
)cos(cosdz/1
00
a
yxfM
ap
ah
a0dz/1
0
Sverdrup Transport
a
xyfM
ap
ah
a
)cos(cosdz/1
0021
a
yxfM
ap
ah
a0dz/1
01 2
Sverdrup Transport
cos
)cos(cos
coscos0
cos0
ayMf
aa
yxfM
ax
a
xyfM
ap
ah
a
)cos(cosdz/1
0021
a
yxfM
ap
ah
a0dz/1
01 2
coscoscos0
af
aMxMa
fy
x
cos
)cos(cos
cos0
aaf
yMyMaf x
coscos0
axMa
fy
cos
)cos(cos
cos0
aaf
yMyMaf x
Sverdrup Transport
Re-arrange, and collect like terms,
coscos
)cos(cos
coscos00
aaaf
yMyMaxMaf
yx
cos
)cos(cos
coscos0
cos0
ayMf
aa
yxfM
ax
coscoscos0
af
aMxMa
fy
x
cos
)cos(cos
cos0
aaf
yMyMaf x
coscos0
axMa
fy
cos
)cos(cos
cos0
aaf
yMyMaf x
Since there can be no vertically-integrated mass convergence,
coscos
)cos(0 00
aaaf
yMyx
Sverdrup Transport
cos/)cos(1/
cos1
00x
ayaaf
yM
Re-arrange this equation,
We can then write ,
kyM )(
Sverdrup Equation
af
And define,
coscos
)cos(0 00
aaaf
yMyx
Sverdrup Transport
kyM )(1
Sverdrup Transport
Sverdrup Transport
kyM )(1
Sverdrup Transport
Sverdrup Transport represents the total mass transport in the wind-influenced layer, including both Ekman and Geostrophic
transport, each of which can be written separately,
dz/cos11dzgv
0 0
paf
hyM
hG
dzvE
0
h
EyM fx /
0 (from previous lecture)
EyMyM fxk
/0
)(1
0x
Sverdrup Transport
)( /1/cos11
00
xayayMSverdrup Transport
Lets estimate the Ekman and Sverdrup Transports at 35N
Pa05.00x
Ekman Transport fyM xE
/0
14o15 s01)35)(sins10x27.7(2sin2 f
/10xa
3Nm8105
14
0s01/Pa05.0/ fM xEy
11116o15 sm01m1037.6/)35)(coss10x27.7(2/cos2 a
/10xay
M 111138 sm01/Nm105
11smkg500 11smkg5000
0x
Sverdrup TransportExpress these as Volume Transports
Pa05.00x
/10xa
3Nm8105
14
0s01/Pa05.0/ fM xEy
/10xay
M 111138 sm01/Nm105
11smkg500
11smkg5000
/EE yy
MQ s/m5.0)kg/m1000/skgm500( 2311
/yy
MQ s/m5)kg/m1000/skgm5000( 2311
0x
Sverdrup Transport
Pa05.00x
/10xa
3Nm8105
14
0s01/Pa05.0/ fM xEy
/10xay
M 111138 sm01/Nm105
11smkg500
11smkg5000
/EE yy
MQ s/m5.0)kg/m1000/skgm500( 2311
/yy
MQ s/m5)kg/m1000/skgm5000( 2311
Express these as Basin-Integrated (Atlantic) Volume Transports
)cos)(( aQVEE yy
)35)(cosm0x137.6)(3/)(s/m5.0( o62 s/m01 36
)cos)(( aQVyy
)35)(cosm0x137.6)(3/)(s/m5( o62 s/m01 37
0x
Sverdrup Transport
Pa05.00x
/10xa
3Nm8105
Express these as Basin-Integrated (Atlantic) Volume Transports
)cos)(( aQVEE yy
)30)(cosm0x137.6)(3/)(s/m5.0( o62 s/m01 36
)cos)(( aQVyy
)30)(cosm0x137.6)(3/)(s/m5( o62 s/m01 37
=-1 “Sverdrups”
=-10 Sverdrups
Windstress and Circulation in the Upper Ocean
0x
Ekman circulation contributes no depth-integrated flow (a
geostrophic return flow balances near-surface Ekman
flow)
By contrast, Sverdrup transport contributes a non-zero depth-integrated flow
fxM
Ey
0
0x
By contrast, Sverdrup transport contributes a non-zero depth-integrated flow
Sverdrup Transport
ax
/0
0
cos2/
/ 00
x
yM
yMaxMa cosBy continuity, we require
2
/cos21
0
2
xa
2
/
cos0
x
a
xMa cos
Sverdrup Transport
yMaxMa cosBy continuity, we require
0x
ax
/0
0
cos2/
/ 00
x
yM
2
/cos21
0
2
xa
2
/
cos0
x
a
xMa cos
Sverdrup Transport
yMaxMa cosBy continuity, we require
This equation can be integrated from the eastern boundary...
2/
2 0
20
xaxM
ax
/0
0
cos2/
/ 00
x
yM
2
/cos21
0
2
xa
2
/
cos0
x
a
xMa cos
Sverdrup Transport
cos2/
/ 0
x
yM
What about the western boundary???
2/
2 0
20
xaxM
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