Wind Driven Circulation I:Planetary boundary Layer

near the sea surface

Monin-Obukhoff Similarity TheorySurface layer (several tens of meters above surface, 10-15% of

the planetary boundary layer) in nearly steady condition 1. Vertical turbulent flux is nearly constant2. horizontal homogeneity (the scale of vertical variation is

much smaller than horizontal) 3. The turbulent mixing length l=z

=0.4±0.01 is von Karman constantMomentum flux

() is a universal function of€ ς=zL

L is the Monin-Obukhoff length€ zu*∂u∂z=φς() € τ=′ u ′ w =ρu*2

u* is frictional velocity

At altitudes below L, shear production of turbulent kinetic energy

dominates over buoyant production of turbulence. € L=−Tvu*3gκ′ w ′ T v

In neutral condition, ()=1

€ ∂u∂z=u*κz

von Karman logarithmic law of wall

€ u=u*κlnzzo

€ τ=ρu*2=ρκ2lnzzou2The surface momentum flux is

If we choose wind measurement at a certain height, e.g., 10m above the sea surface, the bulk formula is

€ τ=ρC10Du102€

C10D=κ2lnz10zois 10m neutral drag coefficient

zo is aerodynamic roughness length

Surface wind stress• Approaching sea surface, the geostrophic

balance is broken, even for large scales. • The major reason is the influences of the winds

blowing over the sea surface, which causes the transfer of momentum (and energy) into the ocean through turbulent processes.

• The surface momentum flux into ocean is called the surface wind stress ( ), which is the tangential force (in the direction of the wind) exerting on the ocean per unit area (Unit: Newton per square meter)

• The wind stress effect can be constructed as a boundary condition to the equation of motion as

τr

τρ rr

==

∂∂

0zz

VA Hz

Wind stress Calculation• Direct measurement of wind stress is difficult.• Wind stress is mostly derived from meteorological

observations near the sea surface using the bulk formula with empirical parameters.

• The bulk formula for wind stress has the form

VVC ad

rr ρτ =

aρWhere is air density (about 1.2 kg/m3 at mid-latitudes), V (m/s), the wind speed at 10 meters above the sea surface, Cd, the empirical determined drag coefficient

Drag Coefficient Cd

• Cd is dimensionless, ranging from 0.001 to 0.0025 (A median value is about 0.0013). Its magnitude mainly depends on local wind stress and local stability.

• Cd Dependence on wind speed.

• Cd Dependence on stability (air-sea temperature difference).

More important for light wind situation For mid-latitude, the stability effect is usually small but in tropical and subtropical regions, it should be included.

Cd dependence on wind speed in neutral

condition

Large uncertainty between estimates(especially in low wind speed).

Lack data in high wind

Annual Mean surface wind stress

Unit: N/m2, from Surface Marine Data (NODC)

December-January-February mean wind stress

Unit: N/m2, from Surface Marine Data (NODC)

December-January-February mean wind stress

Unit: N/m2, from Scatterometer data from ERS1 and 2

June-July-August mean wind stress

Unit: N/m2, from Surface Marine Data (NODC)

Unit: N/m2, Scatterometers from ERS1 and 2

June-July-August mean wind stress

The primitive equation

gzp ρ−=∂∂

xuAxxx ∂∂=ρτ y

uAyxy ∂∂=ρτ

2

2

2

2

2

21zuA

yuA

xuAfv

xp

zuw

yuv

xuu

tu

zyx ∂∂+∂

∂+∂∂++∂

∂−=∂∂+∂

∂+∂∂+∂

∂ρ

2

2

2

2

2

21zvA

yvA

xvAfu

yp

zvw

yvv

xvu

tv

zyx ∂∂+∂

∂+∂∂+−∂

∂−=∂∂+∂

∂+∂∂+∂

∂ρ

0=∂∂+∂

∂+∂∂

zw

yv

xu

Since the turbulent momentum transports are

(1)

(2)

(3)

(4)

zuAzxz ∂∂=ρτ, , etc

⎟⎟⎠

⎞⎜⎜⎝

⎛

∂∂

+∂

∂+

∂∂

++∂∂

−=∂∂

+∂∂

+∂∂

+∂∂

yyxfv

x

p

z

uw

y

uv

x

uu

t

u xzxyxx τττ

ρρ

11

⎟⎟⎠

⎞⎜⎜⎝

⎛

∂

∂+

∂

∂+

∂

∂+−

∂∂

−=∂∂

+∂∂

+∂∂

+∂∂

yyxfu

y

p

z

vw

y

vv

x

vu

t

v yzyyyx τττ

ρρ

11

We can also write the momentum equations in more general forms

zuAzxz ∂∂=ρτ 0

zvAzyz ∂∂=ρτ 0At the sea surface (z=0),

turbulent transport is wind stress. ,

Assumption for the Ekman layer near the surface

• Az=const• Steady state (steady wind forcing for long time)• Small Rossby number

• Large vertical Ekman Number

• Homogeneous water (ρ=const)• f-plane (f=const)• no lateral boundaries (1-d problem)• infinitely deep water below the sea surface

€ Ro=Ou∂u∂x ⎛ ⎝ ⎜ ⎞ ⎠ ⎟Ofv()=U2LfoU=UfoL<<1€

Ez=OAz∂2u∂z2 ⎛ ⎝ ⎜ ⎞ ⎠ ⎟Ofv()=AzUH2foU=AzfoH2

Ekman layer

• Near the surface, there is a three-way force balance

Coriolis force+vertical dissipation+pressure gradient force=0

011 =∂∂−

∂∂+

xp

zfv x

ρτ

ρ

011 =∂∂−

∂∂+−

yp

zfu y

ρτ

ρ

zuAzx ∂∂=ρτ

Take

zvAzy ∂∂=ρτ

and let

(note that VE is not small in comparison to Vg in this region)

02

2

2

2

≈∂∂

−=∂∂

+zu

AzuAfv g

zE

zE

02

2

2

2

≈∂∂

−=∂∂− +

zv

AzvAfu g

zE

zE

then € u,v()=ug,vg()+uE,vE()Geostrophic current

Ageostrophic (Ekman) current

The Ekman problem0

2

2

=∂∂

+zuAfv E

zE

02

2

=∂∂− +

zvAfu E

zE

Boundary conditionsAt z=0,

xE

z zuA τρ =∂∂

yE

z zvA τρ =∂∂

As z -0→Eu,

.0→Ev

Let EEE ivuV += (complex variable), take (1) + i(2), we have

02

2=−

∂∂

EE

z ifVzVA

,.

(1)

(2)

(3)

(4)

(5)

(6)

€ fvE+Az∂2uE∂z2+i−fuE+Az∂2vE∂z2 ⎛ ⎝ ⎜ ⎞ ⎠ ⎟=0€

Az∂2uE+ivE( )∂z2−ifuE−vEi ⎛ ⎝ ⎜ ⎞ ⎠ ⎟=0€ i2=i⋅i=−1Since

€ i=−1i

€ Az∂2uE+ivE( )∂z2−ifuE+ivE( )=0 (7)

z=0,x

Ez z

uA τρ =∂∂

yE

z zvA τρ =∂∂

As z - ,0→Eu

0→Ev

Group equations (7), (8), and (9) together, we have

At z=0,

As z - 0→EV

(3)

(4)

(5)

(6)

Take (3) + i (4), we have

€ ρAz∂uE+ivE( )∂z=τx+iτy

€ τ=τx+iτyDefine

€ ρAz∂VE∂z=τ

Take (5) + i (6), we have

€ uE+ivE=VE→0(8)

(9)

€ ∂2VE∂z2−ifAzVE=0€ ∂VE∂z=τρAz (7)

(8)

(9)

Assume the solution for (7) has the following form

€ ∂2VE∂z2−ifAzVE=0€ VE=eαzTake into

€ α2eαz−ifAzeαz=0We have

€ α2=ifAz

If f > 0,

€ i=eiπ2=eiπ4=cosπ4 ⎛ ⎝ ⎜ ⎞ ⎠ ⎟+isinπ4 ⎛ ⎝ ⎜ ⎞ ⎠ ⎟=12+i12=1+i2€ α=±ifAz=±1+i2fAz

If f < 0,

€ α=±ifAz=±1+i2−fAz=±1+i2ifAz=±1−i2fAz

In above derivations, we have used the following equality:

For f > 0, the general solution of (7) can be written as

⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜

⎝

⎛

⎟⎠⎞⎜

⎝⎛

⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜

⎝

⎛

⎟⎠⎞⎜

⎝⎛

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

+−++=−+= ziA

fBziA

fAzAifBz

AifAV

zzzz

E 12

exp12

expexpexp

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

⎟⎠⎞⎜

⎝⎛

⎟⎠⎞⎜

⎝⎛

+−

−= ziAf

fA

iV

zzE 1

2exp

2

1

ρ

τ

At z=0,

€ ∂VE∂z=τρAz

(8)As z - 0→EV

(9)

⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜

⎝

⎛

⎟⎠⎞⎜

⎝⎛

⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜

⎝

⎛

⎟⎠⎞⎜

⎝⎛

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

+−++=−+= ziA

fBziA

fAzAifBz

AifAV

zzzz

E 12

exp12

expexpexp

Therefore, B=0 because

€ exp−f2Azi+1()z ⎡ ⎣ ⎢ ⎤ ⎦ ⎥=exp−if2Azz ⎡ ⎣ ⎢ ⎤ ⎦ ⎥⋅exp−f2Azz ⎡ ⎣ ⎢ ⎤ ⎦ ⎥

grow exponentially as z-

€ VE=Aexpf2Azi+1()z ⎡ ⎣ ⎢ ⎤ ⎦ ⎥Then and

€ ∂VE∂z=Af2Azi+1()expf2Azi+1()z ⎡ ⎣ ⎢ ⎤ ⎦ ⎥

€ Af2Azi+1()=τρAzthen

€ A=τρAzf2Azi+1()=2τi−1()ρfAzi+1()i−1()=i−1()τρ2fAzand

The final solution to (7), (8), (9) is

Set , where Si

eφττ = and ⎟

⎟⎠

⎞⎜⎜⎝

⎛−=x

yS τ

τφ 1tan

⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜

⎝

⎛

−+= 42

2 πφ

ρτ Sz

zAfi

z

zzA

f

E efA

eV

Also note that iei 4

21 π−=−−

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

⎟⎠⎞⎜

⎝⎛

⎟⎠⎞⎜

⎝⎛

+−

−= ziAf

fA

iV

zzE 1

2exp

2

1

ρ

τGiven

€ τ=τx2+τy2We have

€ VE=τef2AzzρfAzCurrent Speed:

Phase (direction):€ φ=f2Azz+φS−π4(=0, eastward)

€ VE=τef2AzzρfAzeif2Azz+φS−π4 ⎛ ⎝ ⎜ ⎜ ⎞ ⎠ ⎟ ⎟=τef2AzzρfAzcosf2Azz+φS−π4 ⎛ ⎝ ⎜ ⎜ ⎞ ⎠ ⎟ ⎟+isinf2Azz+φS−π4 ⎛ ⎝ ⎜ ⎜ ⎞ ⎠ ⎟ ⎟ ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥

€ uE=τef2AzzρfAzcosf2Azz+φS−π4 ⎛ ⎝ ⎜ ⎜ ⎞ ⎠ ⎟ ⎟

€ vE=τef2AzzρfAzsinf2Azz+φS−π4 ⎛ ⎝ ⎜ ⎜ ⎞ ⎠ ⎟ ⎟€

uE=τef2AzzρfAzcos−f2Azz+φS+π4 ⎛ ⎝ ⎜ ⎜ ⎞ ⎠ ⎟ ⎟€

vE=τef2AzzρfAzsin−f2Azz+φS+π4 ⎛ ⎝ ⎜ ⎜ ⎞ ⎠ ⎟ ⎟If f < 0,