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DYNAMICAL OCEANOGRAPHY
15.5. Exercises on chapter 15
(15.1) Limits
There are two interesting limits in the problem discussed in Example 15.2.
a. First consider the limit ǫ 0 H 1 /DA f →∞ . Determine the resulting
streamfunction solutions and provide a physical interpretation for the result.
b. Second, consider the limit ǫ 0 → 0 as was used in the Figs. 15.8. Again,
determine the resulting solutions and provide a physical interpretation of the
result.
(15.2) Homogenization of PV
The solution of the problem in Example 15.3 has an interesting potential
vorticity distribution.
a. Determine the potential vorticity q 2 (of the bottom layer) and in particular
consider the limit ǫ 0 → 0.
Assume more generally that the dissipation term in the lower layer vorticity
equation can be written as ∇·(κ∇q 2 ) instead of using the interfacial friction
term. Instead of (15.19) we then obtain with ǫ 0 =0,
J(ψ 2 , ˆ
q 2 )=∇·(κ∇q 2 )
b. Show that
C
κ∇q 2 · n ds =0
where the integral is taken over a closed streamline in the lower layer.
We can also use the constraint of potential vorticity conservation in the frictionless limit, i.e., q 2 =Ψ(ψ 2 ).
c. Show that for all closed streamlines
dΨ
dψ 2
=0
and provide a physical interpretation of the result.