1. CEE 262A H YDRODYNAMICS Lecture 18 Surface Ekman layer

2. Another exact solution: Coriolis   Viscous Stress: The Ekman Layer A rotational boundary layer in an infinite ocean: flow driven by a wind stress at the surface (x 3 = 0), acting in the x 1 direction x1x1 x2x2 x3x3 u1u1 u2u2 x 3 = 0 Assumptions: Steady flow, uniform density, constant viscosity, no pressure gradients

3. Steady Linear Constant density, only hydrostatic pressure with no motion Coriolis – Friction Balance

4. The governing equations reduce to (f-plane) With the BCs at x 3 = 0: Again, we wish to make the momentum equations dimensionless, and again, we look at the surface BC unknown velocity scale unknown length scale

5. These imply that Let’s define Thus if we choose the momentum equations are parameter free The Ekman layer thickness

6. These must be solved subject to the dimensionless BCs: Both at x 3 * =0 We also require that the flow decay to zero as This completes the physics part of the problem; what remains is the mathematical problem. To solve this system of two equations, we follow Ekman and define

8. In terms of , the two real o.d.e.s become one complex equation: which has the solution where A,B, 1, and 2 are all complex Using the ode itself, we see that for either i The two roots are given by

9. With the condition that the flow disappear at great depth implies that B=0 since The condition on the water surface Thus,

10. Taking the real part to find u 1 * (after using some trigonometric identities) Likewise, u 2 * is found from the imaginary part of  and a little more trigonometry to be Thus, the velocity vector decays and rotates with depth – aka the “Ekman Spiral”

11. The vertical structure of the Ekman spiral, as originally plotted by Ekman At the surface (x 3 =0): EE x2x2 x1x1

12. Friction + wind: Wind stress motion Friction (stress on bottom of parcel) Friction + wind + Coriolis: Wind stress motion Friction (stress on bottom of parcel) Coriolis In order to balance Coriolis and wind stress, motion must be at some angle to the wind

13. Finally, imagine we go deep enough into then ocean to have the stress be zero on the bottom of our slab Thus, we see that in an integral sense the motion we produce must be at right angles to the wind - this motion is known as Ekman drift. To see this more formally, we integrate the dimensional momentum equations: Wind stress motion Coriolis

14. Thus the net transports are found to be i.e., to the right of the wind in the Northern hemisphere!

15. California Wind from north Upwelling (downwelling): The effects of variations in Ekman transport Ekman drift Upwelling from depth California Wind from south Ekman drift Downwelling from surface

16. Why are California coastal waters so COLD? Upwelling! Upwelling-favorable winds during spring/summer.