Western intensification
139
and the matching condition at λ →∞leads to
lim
λ→∞
ˆ
ψ
0 (λ, y)=C 2 (y) = lim
x↓x W
ψ
0 (x, y)=Ψ
0 (y).
(6.60)
This gives the Stommel boundary layer flow solution as
ˆ
ψ
0 (λ, y)=Ψ
0 (y)(1 − e
−λ ).
(6.61)
◮
Example 6.2: Stommel western boundary layer
Consider a square basin such as in Fig. 6.4 and take x W =0and x E =1 .A s
an example, the wind-stress field is given by
τ
x (x, y)=−
1
2π
cos 2πy,
(6.62a)
τ
y (x, y)=0 ,
(6.62b)
and the kinematic conditions on the boundaries are ψ =0 . The Sverdrup flow
(Fig. 6.4a), that satisfies the kinematic boundary conditions at the eastern boundary, was already determined in Example 6.1 and is given by
ψ
0 (x, y)=(1− x)sin2πy.
(6.63)
According to the boundary analysis, the boundary layer flow is
ˆ
ψ
0 (λ, y)=sin2πy(1 − e
−λ ),
(6.64)
and the total solution for δ S /L =0 .1 is plotted in Fig. 6.4b. Bottom friction, as
did lateral friction, can provide a boundary layer current at the western boundary
of the continent to balance the Sverdrup transport. The dimensional boundary
layer width is for this case 100 km and typical meridional horizontal velocities
V = UL/δ S in the boundary layer are about 10 cm/s.
The full solution to (6.49) can actually be analytically determined and with
η 1 = −L/(2δ S ) and η 2 =
η 2
1 +4π 2 it becomes
ψ(x, y)=
L
4π 2 δ S
1 − e
η1 e −η1 sinh η 2 x − sinh η 2 (x − 1)
sinh η 2
sin(2πy). (6.65)
Although this solution is useful to compare the validity of the asymptotic solutions, the boundary layer character of (6.65) is not so obvious. For δ S /L =0 .1