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DYNAMICAL OCEANOGRAPHY
where y 0 is an arbitrary north-south location for which ψ =0for all x.F r o m
now, take y 0 =0.
Consider now the new situation in which there is also a continental boundary
at x =1. Between the continents, the flow is still purely zonal.
d. Show that the total solution (with western and eastern boundary layers) can
be written as
ψ WE (x, y)=y(1 − e
−xL/δ I − e
−(1−x)L/δ I )
Finally, we consider a closed basin with a southern boundary at y = −1 and a
northern boundary at y =1(where ψ =0).
e. Why are there boundary layers at these northern and southern boundary?
Show that the solution is given by
ψ B (x, y)=ψ WE (x, y)+c 1 e
−(1−y)L/δ I + c 2 e
−(y+1)L/δ I
and determine the constants c 1 and c 2 .
f. Make a sketch (or plot) of the resulting flow under e. This is the Fofonoff
inertial flow.
(6.5) Weakly nonlinear Stommel model
Consider the barotropic vorticity equation (6.22)
(
δ I
L
)
2
∂ψ
∂x
∂
∂y
−
∂ψ
∂y
∂
∂x
∇
2 ψ +
δ S
L
∇
2 ψ = −
∂ψ
∂x
−
∂τ x
∂y
for the dimensionless wind-stress field
τ
x = −
1
π
cos πy
in an ocean basin [0, 1] × [0, 1].L e tδ S /L = ε (where ε ≪ 1 is not the Rossby
number but just a small parameter) and scale δ I /L = Rε p for certain p and
R = O(1) with respect to ε.
For this case, the Sverdrup (ψ 0 ) and Stommel solution were determined in
section 6.4. Introduce now a western boundary layer coordinate λ, with
λ =
x
ε q