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