11

Ekman FlowEkman Flow

September 27, 2006September 27, 2006

22

Remember from last time…Remember from last time…

2

2

2

22

2

22

2

22

1

1

yx

z

vAvAfu

y

p

dt

dv

z

uAuAfv

x

p

dt

du

H

zHH

zHH

33

fvdt

duu

fudt

dvv

0)(2

1 22 uvuvfvudt

d 0

22

dt

dc

vuc

Define c (current speed) as:

u and v change, but c stays constant: Coriolis force does no work!

002

12

2

dt

dc

td

dc

44

Flow is in a circle: Inertial or Centripetal force = Coriolis force

fcr

c

2where

f

cr

Inertial radius

If

c ~ 0.1 m/s

f ~ 10-4

then

r ~ 1 km

55

Inertial Period is given by T where:

fT f

2

If

f ~ 10-4

then

T ~ 6.28x104sec ~ 17.4 hrs

66

r

fc

fc

fc

fc

cc

cc

c2/r

c2/r

c2/r

c2/r

77

Latitude () Ti (hr) D (km)

for V = 20 cm/s

90° 11.97 2.7

35° 20.87 4.8

10° 68.93 15.8

Table 9.1 in Stewart

Inertial Oscillations

Note: V is equivalent to c from previous slides, D is equal to the diameter

88

Figure 9.1 in Stewart

Inertial currents in the North Pacific in October 1987

99

Ekman flowEkman flow

Fridtjof Nansen noticed that wind tended to blow Fridtjof Nansen noticed that wind tended to blow ice at an angle of 20ice at an angle of 20°-40° to the right of the wind °-40° to the right of the wind in the Articin the Artic

Nansen hired Ekman (Bjerknes graduate Nansen hired Ekman (Bjerknes graduate student) to study the influence of the Earth’s student) to study the influence of the Earth’s rotation on wind-driven currentsrotation on wind-driven currents

Ekman presented the results in his thesis and Ekman presented the results in his thesis and later expanded the study to include the influence later expanded the study to include the influence of continents and differences of density of water of continents and differences of density of water (Ekman, 1905)(Ekman, 1905)

1010

Figure 9.2 in Stewart

Balances of forces acting on an iceberg on a rotating earth

1111

So again…So again…

2

2

2

22

2

22

2

22

1

1

yx

z

vAvAfu

y

p

dt

dv

z

uAuAfv

x

p

dt

du

H

zHH

zHH

1212

So again…So again…

2

2

2

22

2

22

2

22

1

1

yx

z

vAvAfu

y

p

dt

dv

z

uAuAfv

x

p

dt

du

H

zHH

zHH

1313

Balance in the surface boundary layer is between vertical Balance in the surface boundary layer is between vertical friction and Coriolis – all other terms are neglectedfriction and Coriolis – all other terms are neglected

0

0

2

2

2

2

z

vAfu

z

uAfv

z

z

1414

friction

u

c.f.

45°

At the surface (z=0)

1515

If we assume the wind is blowing in the x-direction only, we can show:

4sin

4cos

)(

)(

D

ze

fAv

D

ze

fAu

Dz

z

x

Dz

z

x

1616

Wind StressWind Stress Frictional force acting on the surface skinFrictional force acting on the surface skin

2WcDa ρa: density of air

cD: Drag coefficient – depends on atmospheric conditions, may depend on wind speed itself

W: usually measured at “standard Anemometer height” ~ 10m above the sea surface

2

3

3

/5.01.0

/1

1021

mNt

mkg

c

a

D

1717

Ekman Depth (thickness of Ekman layer)Ekman Depth (thickness of Ekman layer)

f

AD z

2

For Mid-latitudes:

Av = 10

ρ = 103

f = 10-4

Plug these into the D equation and:

4520031010

1023

43

D meters

1818

Figure 9.3 in Stewart

1919

U10(m/s)Latitude

15° 45°

540m 30m

10 90m 50m

20 180m 110m

Typical Ekman Depths

Table 9.3 in Stewart

2020

4sin

4cos

)(

)(

z

x

z

x

fAv

fAu

At z=0

y

x

v

u45°u

)(x

2121

4sin

4cos

)(

)(

efA

v

efA

u

z

x

z

x

At z=-D

y,v

x,u

v

u -π/4

u

)(x

-π

2222

Ekman NumberEkman Number

The depth of the Ekman layer is closely related to the The depth of the Ekman layer is closely related to the depth at which frictional force is equal to the Coriolis depth at which frictional force is equal to the Coriolis force in the momentum equation force in the momentum equation

The ratio of the forces is known as Ekman depthThe ratio of the forces is known as Ekman depth

2fD

AE zz

Solving for d:

z

z

fE

AD

2323

Ekman TransportEkman Transport

fvdzM

fudzM

xyE

yxE

)(0)(

)(0)(

Transport in the Ekman Layer is 90° to the right of the wind stress in Northern Hemisphere

2424

Ekman PumpingEkman Pumping

)()0(

0

0 00

0

Dwwy

M

x

M

dzz

wvdz

yudz

x

dzz

w

y

v

x

u

yxEE

D DD

D

2525

By definition, the vertical velocity at the sea surface w(0)=0, and the vertical velocity at the base of the layer wE (-D) is due to divergence of the Ekman flow:

)(

)(

DwM

Dwy

M

x

M

EEH

E

EE yx

vector mass transport due to Ekman flow

horizontal divergence operator

2626

If we use the Ekman mass transports in we can relate Ekman pumping to the wind stress.

yxcurl

fcurlDw

fyfxDw

xy

E

xyE

)(

1)(

wind stress

2727

ME Ek

pile up

of water

wE wE

Hi P

Lo P Lo P

anticyclonic

2828

Hydrostatic EquilibriumHydrostatic Equilibrium

L 106m f 10-4s-1

U 10-1m/s g 10 m/s2

H 103m 103kg/m3

Typical Scales:

with these we can “scale” the equations of motion

2929

sU

LT

PagzP

smsmL

UHW

L

U

H

W

y

v

x

U

z

W

7

7313

46

31

10

10101010

/10/10

1010;

3030

The momentum equation for vertical velocity is therefore:

10101010101010

cos21

514111111

2

gfUH

P

L

W

L

UW

L

UW

T

W

guz

p

z

ww

y

wv

x

wu

t

w

and the only important balance in the vertical is hydrostatic:

gz

p

Correct to 1:106

3131

The momentum equation for horizontal velocity in the x direction is

558888 101010101010

1

fvx

P

z

uw

y

uv

x

uu

t

u

The Coriolis balances the pressure gradient, known as the geostrophic balance

gz

pfu

y

pfv

x

p

1

;1

;1