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The primitive equation

The primitive equation. (1). (2). (3). (4). Since the turbulent momentum transports are. ,. ,. 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.

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The primitive equation

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

  2. Assumption for the Ekman layer near the surface • Az=const • Steady state • 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

  3. Ekman layer • Near the surface, there is three-way force balance Coriolis force+vertical dissipation+pressure gradient force=0 Take and let ( , ageostrophic (Ekman) current, note that is not small in comparison to in this region) then

  4. The Ekman problem Boundary conditions At z=0, As z→-∞, ,. , . (complex variable) Let At z=0, As z→-∞,

  5. The solution Assuming f > 0, the general solution is Using the boundary conditions, we have Set , where and note that

  6. Define and we have • At the sea surface (z=0), the surface current flows at 45o to the right of the wind direction Depends on constant Az => • Current decreases exponentially with depth and. At the same time, its direction changes clockwise with depth (The Ekman spiral). • 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. , • At DE, the current magnitude is 4% of the surface current and its direction is opposite to that of the surface current. .

  7. Other properties (1) Relationship between surface wind speed W and (Vo, DE). Wind stress magnitude ) ( ,  , (2) Relationship between W and DE. Ekman’s empirical formula between W and Vo. , outside ±10o latitude (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. (4) has an error range of 2-5%.

  8. More comments (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.

  9. 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.)

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

  11. Ekman Transport Starting from a more general form of the Ekman equation (without assuming AZ or even a specific form for vertical turbulent flux Integrating from surface z=η to z=-2DE (e-2π=0.002), we have where and are the zonal and meridional mass transports by the by the Ekman current. Since , we have 

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

  13. Ekman pumping through the layer: through the layer: Integrating the continuity equation . Assume Where and are volume transports. Assume and let , we have and let , we have is transport into or out of the bottom of the Ekman layer to the ocean’s interior (Ekman pumping). , upwelling , downwelling Water pumped into the Ekman layer by the surface wind induced upwelling is from 200-300 meters, which is colder and reduces SST.

  14. Upwelling/downwelling are generated by curls of wind stress

  15. Coastal and equatorial upwelling Coastal 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).

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