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DYNAMICAL OCEANOGRAPHY
c. Show that for v g∗ =0and constant u g∗ , the solution of the Ekman layer
velocities is given by
u E∗ = −u g∗ e
−z∗/δ E cos
z ∗
δ E
v E∗ = u g∗ e
−z∗/δ E sin
z ∗
δ E
where δ E =
2A V
f0 is the bottom Ekman layer thickness.
d. Check that the total velocity field (u, v) satisfies the boundary conditions
(z =0): u ∗ = v ∗ =0
(z ≫ 1) : u ∗ = u g∗ ; v ∗ =0
and make a sketch of the total velocity field in the Ekman layer.
(5.3) Ekman layer at the ocean-atmosphere interface
Suppose we have a basin of dimensions 1000 × 1000 × 1 km at 45 ◦ N
(domain [0, 1] × [−1, 1] × [−1, 0]) with the flow forced by the wind stress
τ x
∗ = τ 0 y, τ
y
∗ =0;takeτ 0 =10 −1 Pa and choose U =10 −2 ms −1 .
a. Write a computer program that calculates the dimensionless Ekman
velocities for this wind stress field for different values of A V . Make a plot
of the velocitities (ˆ u 0 , ˆ
v 0 ) versus χ ∈ [0, 10] for three different values of
A V =0.1, 0.01 and 0.0001 m 2 s −1 .
b. Calculate for each of the values of A V the dimensional Ekman transport
M E∗ .
(5.4) Ekman layer at an eastern boundary
Ekman layers can also occur near continental boundaries. Consider a flow
near an east coast x = x E driven by wind stress τ y = τ 0 , τ x =0, where τ 0 is
constant. Bottom friction can be neglected.
a. Write down the zonal momentum equations in this case (integrated over
depth) and determine the Ekman pumping at the surface.