 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2019/ib/paper-4/18c/c/solution

Codex  0 Created 2026-09-24 Updated 2026-10-06

For a steady flow independent of

y

u = 0, v = \frac{g}{f} \frac{d η}{d x} .

Hence

ζ = v_{x} = (g / f) η^{''}

, and the definition of

q

becomes

q = \frac{g}{f} η^{''} - \frac{f}{h} η .

Equivalently,

η^{''} - \frac{f ^{2}}{g h} η = \frac{f}{g} q .

R = g h / f

, the bounded solution for the stated potential-vorticity step is

η (x) = ⎩ ⎨ ⎧ - \frac{h q _{1}}{f} + A e^{x / R}, - \frac{h q _{2}}{f} + B e^{- x / R}, x < 0, x > 0.

Continuity of

η

and

v = (g / f) η^{'}

x = 0

gives

A = \frac{h ( q _{1} - q _{2} )}{2 f}, B = - \frac{h ( q _{1} - q _{2} )}{2 f} .

Therefore

η (x) = ⎩ ⎨ ⎧ - \frac{h q _{1}}{f} + \frac{h ( q _{1} - q _{2} )}{2 f} e^{x / R}, - \frac{h q _{2}}{f} - \frac{h ( q _{1} - q _{2} )}{2 f} e^{- x / R}, x < 0, x > 0,

and

v (x) = \frac{R ( q _{1} - q _{2} )}{2} e^{- ∣ x ∣/ R} .

When

q_{1} > q_{2}

, the surface height increases smoothly from

- h q_{1} / f

- h q_{2} / f

, while

v

is a positive eastward jet with its maximum at the discontinuity and exponential decay on the Rossby-radius scale:

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