Ekman Flow September 27, 2006

Remember from last time…

Define c (current speed) as: u and v change, but c stays constant: Coriolis force does no work!

Flow is in a circle: Inertial or Centripetal force = Coriolis force where Inertial radius If c ~ 0.1 m/s f ~ 10-4 then r ~ 1 km

Inertial Period is given by T where: If f ~ 10-4 then T ~ 6.28x104sec ~ 17.4 hrs

r fc c c2/r

Inertial Oscillations Latitude (j) Ti (hr) D (km) for V = 20 cm/s 90° 11.97 2.7 35° 20.87 4.8 10° 68.93 15.8 Table 9.1 in Stewart Note: V is equivalent to c from previous slides, D is equal to the diameter

Inertial currents in the North Pacific in October 1987 Figure 9.1 in Stewart

Ekman flow Fridtjof Nansen noticed that wind tended to blow ice at an angle of 20°-40° to the right of the wind in the Artic Nansen hired Ekman (Bjerknes graduate student) to study the influence of the Earth’s rotation on wind-driven currents Ekman presented the results in his thesis and later expanded the study to include the influence of continents and differences of density of water (Ekman, 1905)

Balances of forces acting on an iceberg on a rotating earth Figure 9.2 in Stewart

So again…

So again…

Balance in the surface boundary layer is between vertical friction and Coriolis – all other terms are neglected

At the surface (z=0) friction c.f. 45°

If we assume the wind is blowing in the x-direction only, we can show:

Wind Stress Frictional force acting on the surface skin ρa: density of air cD: Drag coefficient – depends on atmospheric conditions, may depend on wind speed itself W: usually measured at “standard Anemometer height” ~ 10m above the sea surface

Ekman Depth (thickness of Ekman layer) For Mid-latitudes: Av = 10 ρ = 103 f = 10-4 Plug these into the D equation and: meters

Figure 9.3 in Stewart

Typical Ekman Depths Table 9.3 in Stewart U10(m/s) Latitude 15° 45° 5 20 180m 110m Table 9.3 in Stewart

At z=0 y x v u 45°

At z=-D y,v v x,u u -π -π/4

Ekman Number The depth of the Ekman layer is closely related to the depth at which frictional force is equal to the Coriolis force in the momentum equation The ratio of the forces is known as Ekman depth Solving for d:

Ekman Transport Transport in the Ekman Layer is 90° to the right of the wind stress in Northern Hemisphere

Ekman Pumping

By definition, the vertical velocity at the sea surface w(0)=0, and the vertical velocity at the base of the layer wE (-D) is due to divergence of the Ekman flow: horizontal divergence operator vector mass transport due to Ekman flow

If we use the Ekman mass transports in we can relate Ekman pumping to the wind stress.

ME Ek pile up of water anticyclonic wE Hi P Lo P

Hydrostatic Equilibrium Typical Scales: L » 106m f» 10-4s-1 U» 10-1m/s g» 10 m/s2 H» 103m r» 103kg/m3 with these we can “scale” the equations of motion

The momentum equation for vertical velocity is therefore: and the only important balance in the vertical is hydrostatic: Correct to 1:106

The momentum equation for horizontal velocity in the x direction is The Coriolis balances the pressure gradient, known as the geostrophic balance