Define fAD z

E2π=

• Current decreases exponentially with depth and. At the same time, its direction changes clockwise with depth (The Ekman spiral).

Ez

o fDfAV ρ

τπρ

τ 2== we have

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛−+=

4cos πφπ

π

SE

zED

oE zD

eVu

,

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛−+=

4sin πφπ

π

SE

zED

oE zD

eVv

.

and

• At the sea surface (z=0), the surface current flows at 45o to the right of the wind direction

⎟⎟⎠

⎞⎜⎜⎝

⎛ −=4

cos πφSoE Vu ⎟⎟⎠

⎞⎜⎜⎝

⎛ −=4

sin πφSoE Vv

• At DE, the current magnitude is 4% of the surface current and its direction is opposite to that of the surface current.

• DE (≈100 m in mid-latitude) is regarded as the depth of the Ekman layer. DE is not the mixed layer depth (hm). The latter also depends on past history, surface heat flux (heat balance) and the stability of the underlying water. In reality, DE < hm because hm can be affected by strong wind burst of short period.

Depen

ds o

n co

nsta

nt A z

=>

2WCDa

ρτ = ( 33.1 mkga =ρ , 3104.1 −×≈DC , 23108.1 W−×=τ )

smfD

W

fD

WfD

VEE

Eo

25

23

1079.01025

108.122 −−

×=××== πρ

τπ

(2) Relationship between W and DE.

Ekman’s empirical formula between W and Vo.

( )ϕsin0127.0=

WVo , outside ±10o latitude ( )

mWDEϕsin

3.4=

(3) There is large uncertainty in CD (1.3 to 1.5 x 10-3 ±20% for wind speed

up to about 15 m/s). CD itself is actually a function of W.

(1) Relationship between surface wind speed W and (Vo, DE).

Wind stress magnitude

( )ϕsin0127.0=

WVo has an error range of 2-5%.(4)

Other properties

(1) DE is not the mixed layer depth (hm). The latter also

depends on past history, surface heat flux (heat balance) and the stability of the underlying water. In reality, DE < hm

because hm can be affected by strong wind burst of short

period.

(2) Az = const and steady state assumptions are questionable.

(3) Lack of data to test the theory. (The Ekman spiral has been observed in laboratory but difficult to observe in fields).

(4) Vertically integrated Ekman transport does not strongly depend on the specific form of Az.

More comments

Progressive vector diagram, using daily averaged currents relative to the flow at 48 m, at a subset of depths from a moored ADCP at 37.1°N, 127.6°W in the California Current, deployed as part of the Eastern Boundary Currents experiment. Daily averaged wind vectors are plotted at midnight UT along the 8-m relative to 48-m displacement curve. Wind velocity scale is shown at bottom left. (Chereskin, T. K., 1995: Evidence for an Ekman balance in the California Current. J. Geophys. Res., 100, 12727-12748.)

Surface Drifter Current Measurements a platform designed to move with the ocean current

Ekman Transport

0=∂∂+

zfv x

Eτρ

0=∂∂+−

zfu y

Eτρ

Integrating from surface z= to z=-2DE (e-2DE=0.002), we have

( ) ( )∫ ∫ +−=∂∂−==

− −−

τττρ

ED EDEDxx

xE

yE dz

zdzvffM

2 22

( ) ( )∫ ∫ −=∂∂

==− −

−

τττ

ρED ED

EDyy

yE

xE dz

zdzuffM

2 22

by the Ekman current. Since ( ) ( ) 022 ≈≈ −−ED

yEDx ττ , we have

τ ⎟

⎠⎞

⎜⎝⎛−=x

yEfM

Starting from a more general form of the Ekman equation

(without assuming AZ or even a specific form for vertical turbulent flux)

where xEM and y

EM are the zonal and meridional mass transports by the

τ ⎟⎟

⎠

⎞⎜⎜⎝

⎛=y

xEfM

( ) kff

MMM xyyE

xEE

rrr×=−== ⎟

⎠⎞⎜

⎝⎛ τττ 1,1,

Ekman transport is to the right of the direction of the surface winds

Ekman pumping

yEV

xEV

Integrating the continuity equation

020 =−=−=+∂

∂+∂∂

⎟⎠⎞

⎜⎝⎛⎟

⎠⎞⎜

⎝⎛

EEE

yE

xE Dzwzw

yV

xV

is transport into or out of the bottom of the Ekman layer to the ocean’s interior (Ekman pumping).

0=∂∂+

∂∂+

∂∂

zw

yv

xu EEE through the layer:

Where and are volume transports. Assume and let , we have00 ≈= ⎟

⎠⎞⎜

⎝⎛zwE

DEEE wDzw =−= ⎟

⎠⎞⎜

⎝⎛ 2

kf

kffyfxy

VxVw xy

yE

xED

E

rrrr⋅×∇≈⋅×∇=

∂∂−

∂∂=

∂∂+

∂∂= ⎟

⎠⎞⎜

⎝⎛

⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜

⎝

⎛

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

τρρ

τρτ

ρ

τ 1

DEw

0>DEw , upwelling

Water pumped into the Ekman layer by the surface wind induced upwelling is from 200-300 meters, which is colder and reduces SST.

0<DEw , downwelling

Upwelling/downwelling are generated by curls of wind stress

Coastal and equatorial upwellingCoastal upwelling: Along the eastern coasts of the Pacific and Atlantic Oceans the Trade Winds blow nearly parallel to the coast towards the Doldrums. The Ekman transport is therefore directed offshore, forcing water up from below (usually from 200 - 400 m depth).

Equatorial Upwelling: In the Pacific and Atlantic Oceans the Doldrums are located at 5°N, so the southern hemisphere Trade Winds are present on either side of the equator. The Ekman layer transport is directed to the south in the southern hemisphere, to the north in the northern hemisphere. This causes a surface divergence at the equator and forces water to upwell (from about 150 - 200 m).

An example of coastal upwelling

Water property sections in a coastal upwelling region, indicating upward water movement within about 200 km from the coast. (This particular example comes from the Benguela Current upwelling region, off the coast of Namibia.) The coast is on the right, outside the graphs; the edge of the shelf can just be seen rising to about 200 m depth at the right of each graph.

Note how all contours rise towards the surface as the coast is approached; they rise steeply in the last 200 km. On the shelf the water is colder, less saline and richer in nutrients as a result of upwelling.

Cold SST associated

with the coastal and equatorial upwelling