Wind-driven circulation
105
5.3.2. The bottom Ekman layer
For ease of understanding, we first consider the case of a flat bottom, with
h b (x, y) ≡ 0. The boundary layer variable ξ is introduced as
ξ =
z +1
ℓ
(5.38)
where ℓ is the scale of the (aprioriunknown) thickness of the bottom boundary
layer. The boundary layer (inner) solution is indicated by (˜ v, ˜
p) and it turns out to
be convenient to use ¯
E V =2E V and ¯
E H =2E H . The equations in the variables
(x, y, ξ) become
ǫ
D˜ u
dt
− ˜
v(1 + βǫy)=
−
∂ ˜
p
∂x
+
¯
E H
2
∂ 2 ˜
u
∂x 2 +
∂ 2 ˜
u
∂y 2
+
¯
E V
2ℓ 2
∂ 2 ˜
u
∂ξ 2 ,
(5.39a)
ǫ
D˜ v
dt
+˜ u(1 + βǫy)=
−
∂ ˜
p
∂y
+
¯
E H
2
∂ 2 ˜
v
∂x 2 +
∂ 2 ˜
v
∂y 2
+
¯
E V
2ℓ 2
∂ 2 ˜
v
∂ξ 2
(5.39b)
0=
∂ ˜
p
∂ξ
,
(5.39c)
ℓ
−1 ∂ ˜
w
∂ξ
+
∂˜ v
∂y
+
∂ ˜
u
∂x
=0,
(5.39d)
D
dt
=˜ u
∂
∂x
+˜ v
∂
∂y
+ ℓ
−1 ˜
w
∂
∂ξ
,
(5.39e)
and the boundary conditions become
ξ =0: ˜
u =˜ v =˜ w =0.
(5.40)
The condition that the vertical mixing of momentum should participate in the
O balance leads to
ℓ = ¯
E
1/2
V .
(5.41)
From the continuity equation (5.39d) it follows that
∂ ˜
w
∂ξ
= − ¯
E
1/2
V (
∂˜ v
∂y
+
∂ ˜
u
∂x
),
(5.42)
which indicates that ˜
w must be rescaled. The relevant scale of the vertical velocity in the boundary layer is not UD/L but ¯
E
1/2
V UD/L = Uδ E /L, with