A two-dimensional current (i.e. u=constant=0) conserves its potential vorticity. It consists of two layers that are separated by a pycnocline. The current has the following attributes:
The depth of the pycnocline H varies as a function of distance from the coast, i.e., H=H(x).
The pycnocline becomes flat at a distance L from the coast, i.e., H(x=L)=constant=H0.
The northward velocity vanishes at this location, i.e., v(x=L)=0.
(a) Utilize the statement of potential vorticity to find an expression for v(x) that may contain a single integral on the right hand side.
(b) Sketch the velocity profile v(x) assuming that the depth of the pycnocline increases monotonically from x=0 to its constant vakue at x=L.
(c) What kind of measurements would you need to verify the prediction of this theory?