Western intensification
135
6.2.1. The Munk boundary layer
Case (i) of the previous section is characterized by
δ M ≫ max (δ I ,δ S ),
(6.29)
and hence lateral friction dominates over inertia and bottom friction. We start
from the Sverdrup solution ψ 0 in (6.6), with
ψ
0 (x, y)=
x
x W
∇.(T ∧ e 3 )(s, y)ds +Ψ
0 (y),
(6.30)
where Ψ 0 (y) is still to be determined. For constant x W and x E , the kinematic
(6.10) and no-slip (6.11) boundary conditions are
x = x W ,x E : ψ =0,
∂ψ
∂x
=0.
(6.31)
Consider first the boundary layer at the eastern boundary. With ℓ ∗ = δ M and
with (6.29), (6.26) becomes
∂ 2
∂μ 2 + ℓ
2 ∂ 2
∂y 2
2
ψ =
∂ψ
∂μ
+ ℓ∇·(T ∧ e 3 ).
(6.32)
It appears convenient to introduce the boundary layer correction φ B by
ψ(μ, y)=ψ
0 (x, y)+φ B (μ, y).
(6.33)
Substitution of (6.33) into (6.32) and use of
∂ψ 0
∂μ
= −ℓ
∂ψ 0
∂x
,
(6.34)
gives, with ℓ = δ M /L ≪ 1, the dominant balance
∂ 4 φ B
∂μ 4 +
∂φ B
∂μ
=0.
(6.35)
The characteristic polynomial is found through substitution of e μz into (6.35)
which gives
z
4 + z =0⇒ z =0∨ z = −1 ∨ z =
1
2
(1 + i
√
3) ∨ z =
1
2
(1 − i
√
3). (6.36)
The general solution of (6.35) is
φ B (μ, y)=C 1 (y)+C 2 (y)e
−μ +C 3 (y)e
µ
2 cos
μ
√
3
2
+C 4 (y)e
µ
2 sin
μ
√
3
2
. (6.37)