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 areFor the Kelvin mode, $u=0$ and $v=(c/H)\eta$ with $c^2=gH$. Therefore $H{v}^2=g{\eta }^2$, and the two energies are actually equal pointwise at this order. For a harmonic wave $\eta =A_0{e}^{-r/L_R}\cos (ky-\omega t)$,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.