 Solution

ID: past-exam-of-the-physics-of-atmospheres-and-oceans-course-of-the-university-of-oxford/2018/5/d/solution

Per unit horizontal area, the linear shallow-water kinetic and gravitational potential energies are

K = \frac{1}{2} ρ H (u^{2} + v^{2}), P = \frac{1}{2} ρ g η^{2} .

For the Kelvin mode,

u = 0

and

v = (c / H) η

with

c^{2} = g H

. Therefore

H v^{2} = g η^{2}

, and the two energies are actually equal pointwise at this order. For a harmonic wave

η = A_{0} e^{- r / L_{R}} cos (k y - ω t)

\overline{K} = \overline{P} = \frac{ρ g A _{0}^{2}}{4} e^{- 2 r / L_{R}} .

The averaging supplies

\overline{cos^{2}} = 1/2

. These energies refer to the perturbation relative to the flat resting state; background ocean potential energy is not included.

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