Wind Driven Circulation I Planetary boundary Layer near

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

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 midlatitudes), 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 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.

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/m 2, from Surface Marine Data (NODC)

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

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

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 • • 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

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

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

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

Define and we have • At the sea surface (z=0), the surface current flows at 45 o to the right of the wind direction • Current decreases exponentially with depth and. At the same time, , its direction changes clockwise with depth (The Ekman spiral). • 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.

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 ± 10 o 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%.

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 theory. (The Ekman spiral has been observed in laboratory but yet to be found in field observations). (4) Vertically integrated Ekman transport does not strongly depend on the specific form of Az.

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=-2 DE (e-2π=0. 002), we have where and are the zonal and meridional mass transports by the Ekman current. Since , we have

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

Ekman pumping through the layer: Integrating the continuity equation Where and are volume transports. Assume 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.

Upwelling/downwelling are generated by curls of wind stress

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).

An example of coastal upwelling 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. 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.

Cold SST associated with the coastal and equatorial upwelling

Properties of Sea Water • • What is the pressure at the bottom of the ocean relative to sea surface pressure? What unit of pressure is very similar to 1 meter? What is salinity and why do we use a single chemical constituent (which one? ) to determine it? What other physical property of seawater is used to determine salinity? What are the problems with both of these methods? What properties of seawater determine its density? What is an equation of state? What happens to the temperature of a parcel of water (or any fluid or gas) when it is compressed adiabatically? What quantity describes the effect of compression on temperature? How does this quantity differ from the measured temperature? (Is it larger or smaller at depth? ) What are the two effects of adiabatic compression on density? What are σt and σθ? How do they different from the in situ density? Why do we use different reference pressure levels for potential density? What are the significant differences between freezing pure water and freezing seawater?

Basic Dynamics • • What are the differences between the centrifugal force and the Coriolis force? Why do we treat them differently in the primitive equation? What is the definition of dynamic height? In geostrophic flow, what direction is the Coriolis force in relation to the pressure gradient force? What direction is it in relation to the velocity? Why do we use a method to get current based on temperature and salinity instead of direct current measurements for most of the ocean? How are temperature and salinity information used to calculate currents? What are the drawbacks to this method? What is a "level of no motion"? Why do we need a "level of known motion" for the calculation of the geostrophic current? (What can we actually compute about the velocity structure given the density distribution and an assumption of geostrophy? ) What are the barotropic and baroclinic flows? Is there a “thermal wind” in a barotropic flow? What can you expect about the relation between the slopes of thermocline depth and the sea surface height?