Western intensification
137
From (6.45b) it follows that C 1 = C 2 =0and the conditions (6.45a) fix the
zeroth order boundary layer correction through C 3 and C 4 , with
C 3 (y)=−ψ
0 (x W ,y); C 4 (y)=
C 3
√
3
.
(6.46)
Finally, the boundary layer solution is given by
ψ(λ, y)=ψ
0 (x W ,y)(1 − e
−λ
2 cos
λ
√
3
2
−
1
√
3
e
−λ
2 sin
λ
√
3
2
),
(6.47)
and ψ(λ, y)/ψ 0 (x W ,y) is plotted as function of λ in Fig. 6.3. We see that indeed
-0.2
0
0.2
0.4
0.6
0.8
1
1.2
024681 0
λ λ
λ
λ
ψ ψ
ψ
ψ/ψ ψ
ψ
ψ
0
v > 0
v < 0
v > 0
Figure 6.3. Plot of the boundary layer solution ψ(λ, y)/ψ
0 (xW ,y) from (6.47).
ψ =0and ˆ
v(λ, y)=L/δ M ∂ ˆ
ψ 0 /∂λ =0at λ =0 . Note also that ψ oscillates
and hence there are intervals where ˆ
v is positive and negative.
6.2.2. The Stommel boundary layer
In case (ii), we consider the boundary layer structure due to bottom friction for
which
δ S ≫ max (δ I ,δ M ).
(6.48)
The potential vorticity equation reduces to
δ S
L
∇
2 ψ = −
∂ψ
∂x
+ ∇·(T ∧ e 3 ).
(6.49)