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DYNAMICAL OCEANOGRAPHY
Assume now that ψ b is a function only of q.
b. Consider the linear friction law τ b = RC D v b and integrate the vorticity
equation (15.44) along a closed contour C of constant f/H. Show that the
result can be written as
RC D
ρ 0 f
v b · ds =
A
(e 3 ·∇∧
τ s
ρ 0 f
−
1
f
V s ·∇f ) dxdy
where A is the area enclosed by C.
c. Now use the results under a. in that of b. to derive an expression for ∂ψ b /∂q.
d. Show that
v b = α b
A
(e 3 ·∇∧
τ s
ρ 0 f
−
1
f
V s ·∇f ) dxdy
t
where t is the tangent vector to C and determine α b .
(15.5) Explicit bottom velocities in Example 15.3
As an application of the results in exercise (15.4), it is possible to determine
explicit expressions for the bottom velocities in the case of Example 15.3.
a. Show that for this case the expression for v b in exercise 15.4 (part d.)
reduces to
v b = |∇H|
f
R
Fdxdy
|∇H|f 2 ds
t
b. Use the expression for F in Example 15.3 to derive
v b =
fF 0
8Rx 2
0
(r
3 +4x
2
0 r) t
where r =
x 2 + y 2 and interpret the result.