# Wind Driven Circulation I: Planetary boundary Layer near the sea surface.

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Wind Driven Circulation I: Planetary boundary Layer near the sea surface

Monin-Obukhoff Similarity Theory Surface layer (several tens of meters above surface, 10-15% of the planetary boundary layer) in nearly steady condition 1.Vertical turbulent flux is nearly constant 2.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 constant Momentum flux  (  ) is a universal function of L is the Monin-Obukhoff length u * is frictional velocity At altitudes below L, shear production of turbulent kinetic energy dominates over buoyant production of turbulence.

In neutral condition,  (  )=1 von Karman logarithmic law of wall The surface momentum flux is If we choose wind measurement at a certain height, e.g., 10m above the sea surface, the bulk formula is is 10m neutral drag coefficient z o 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

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 Where is air density (about 1.2 kg/m 3 at mid- latitudes), V (m/s), the wind speed at 10 meters above the sea surface, C d, the empirical determined drag coefficient

Drag Coefficient C d C d 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. C d Dependence on wind speed. C d 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.

C d 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/m 2, from Surface Marine Data (NODC)

December-January-February mean wind stress Unit: N/m 2, from Surface Marine Data (NODC)

December-January-February mean wind stress Unit: N/m 2, from Scatterometer data from ERS1 and 2

June-July-August mean wind stress Unit: N/m 2, from Surface Marine Data (NODC)

Unit: N/m 2, Scatterometers from ERS1 and 2 June-July-August mean wind stress

The primitive equation Since the turbulent momentum transports are (1) (2) (3) (4),,etc We can also write the momentum equations in more general forms At the sea surface (z=0), turbulent transport is wind stress.,

Assumption for the Ekman layer near the surface A z =const Steady state (steady wind forcing for long time) Small Rossby number Large vertical Ekman Number Homogeneous water (  =const) f-plane (f=const) no lateral boundaries (1-d problem) infinitely deep water below the sea surface

Ekman layer Near the surface, there is a three-way force balance Coriolis force+vertical dissipation+pressure gradient force=0 Take and let (note that V E is not small in comparison to V g in this region) then Geostrophic current Ageostrophic (Ekman) current

The Ekman problem Boundary conditions At z=0, As z  - ,.,. Let (complex variable), take (1) + i(2), we have,.,. (1) (2) (3) (4) (5) (6) Since (7)

z=0, As z  - , Group equations (7), (8), and (9) together, we have At z=0, As z  -  (3) (4) (5) (6) Take (3) + i (4), we have Define Take (5) + i (6), we have (8) (9) (7) (8) (9)

Assume the solution for (7) has the following form Take into We have If f > 0, If f < 0, In above derivations, we have used the following equality: For f > 0, the general solution of (7) can be written as

At z=0, (8) As z  -  (9) Therefore, B=0 because grow exponentially as z  -  Then and then and The final solution to (7), (8), (9) is

Set, where and Also note that Given We have Current Speed: Phase (direction): (  =0, eastward)

If f < 0,

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