302
DYNAMICAL OCEANOGRAPHY
1
cos θ
(
∂v 0
∂φ
−
∂(u 0 cos θ)
∂θ
)) +
1
4λ tan θ
(u
0 − v
0 ),
=˜ w
0 (φ, θ, 0) +
1
2λ
(∇·u
0 + ∇·(u
0 ∧ e 3 )+
1
2tanθ
(u
0 − v
0 )) =
=˜ w
0 (φ, θ, 0) +
1
2
(∇·(
u 0
λ
)+∇·(
u 0
λ
∧ e 3 )),
(13.17)
where u 0 =(u 0 ,v 0 , 0) T .
From the boundary conditions w =0at z = −1, it follows
¯
E
1/2
V ˜
w
0 (φ, θ, 0) + w
0 (φ, θ, −1) = 0,
(13.18)
and hence we finally find from (13.17) and (13.18) that
˜
w E (φ, θ) = lim
ξ→∞
¯
E
1/2
V ˜
w
0 (φ, θ, ξ) + lim
z→−1
w
0 (φ, θ, z)
=
1
2
¯
E
1/2
V (∇·(
u 0
λ
)+∇·(
u 0
λ
∧ e 3 )).
(13.19)
The dimensionless Ekman transport M E is determined through integration of
˜
u 0 − u 0 over the boundary layer and becomes
M E =
¯
E
1/2
V
2λ
(u
0 + u
0 ∧ e 3 ),
(13.20)
Note that this expression is very similar to that on the equatorial β-plane (section
11.2) where λ =
|y|.
Bottom Ekman layer
On the planetary scale the dimensional vertical Ekman pumping velocity (ms −1 ) and the Ekman volume transport (m 2 s −1 ) at the ocean (flat)
bottom are
˜
w E∗ =
1
2r 0
A V
Ω
(∇·(
u ∗
| sin θ|
)+∇·(
u ∗
| sin θ|
∧ e 3 )),
M E∗ =
1
2
A V
Ω
1
sin θ
(u ∗ + u ∗ ∧ e 3 ).
13.3.2. The free surface Ekman layer
In the boundary layer at the free surface, we introduce the boundary layer coordinate χ = −z/ ¯
E
1/2
V and the expansions (for u, v and p) become
ˆ
u(φ, θ, χ)=ˆ u
0 (φ, θ, χ)+ǫ p ˆ
u
1 (φ, θ, χ)+...
ˆ
w(φ, θ, χ)=−w
0 (φ, θ, z)+ ¯
E
1/2
V (ˆ w
0 (φ, θ, χ)+ǫ p ˜
w
1 (φ, θ, χ)+...