= Solution
The equations describe the linear <reduced gravity> <shallow water equations> for a uniform upper layer over a deep, resting lower layer, or equivalently one separated internal vertical mode. The layer is shallow compared with its horizontal scales, permitting <hydrostatic balance>, and its horizontal <velocity> is independent of depth. The <Boussinesq approximation> retains the small <density> contrast through $g\prime=g(\rho_2-\rho_1)/\rho_0>0$. We neglect <friction>, external forcing, background currents, and quadratic perturbation terms. On an equatorial <beta plane>, $f=\beta y$ with $\beta>0$, and the <equatorial Kelvin wave> branch imposes $v=0$. The meridional <Cauchy momentum equation> then reduces exactly within this linear model to a balance between the <Coriolis force> and <pressure gradient force>.
Strictly, the printed $h$ must denote the \b[thickness perturbation] $\eta$, with total thickness $h_0+\eta$ and $|\eta|\ll h_0$. A signed oscillation around zero cannot represent the total positive layer thickness. Define $c=\sqrt{g\prime h_0}$. For a nonzero zonal <wavenumber> $k$, the zonal <Cauchy momentum equation> and <conservation of mass> give
$$
\omega\widehat u=kg\prime\widehat\eta,\qquad
\omega\widehat\eta=kh_0\widehat u,
\qquad \omega^2=c^2k^2.
$$
On the eastward branch $\omega=ck$, the <wave polarization> is $u=(c/h_0)\eta$. Meridional <geostrophic balance> gives
$$
g\prime\eta_y=-\beta y\frac c{h_0}\eta,
\qquad \frac{\eta_y}{\eta}=-\frac{\beta y}{c},
\qquad \eta\propto\exp\left(-\frac{\beta y^2}{2c}\right).
$$
The branch $\omega=-ck$ instead grows as $\exp(+\beta y^2/(2c))$ and is excluded by decay away from the equator. Matching the paper's <Gaussian function> therefore gives
$$
\boxed{L_{\rm eq}^2=\frac{2c}{\beta},\qquad
\eta=A_0e^{-y^2/L_{\rm eq}^2}\cos(kx-ckt),\qquad
u=\frac c{h_0}\eta,\quad v=0.}
$$
The usual <equatorial deformation radius> is $R_{\rm eq}=\sqrt{c/\beta}$, with the envelope written $e^{-y^2/(2R_{\rm eq}^2)}$. Thus the printed exponent uses $L_{\rm eq}=\sqrt2R_{\rm eq}$. Calling that width a <deformation radius> is a convention; using the standard radius without the factor $1/2$ would fail the printed meridional equation.
\Image[/past-exam-of-the-geophysical-fluid-dynamics-course-of-the-university-of-oxford/2024-equatorial-kelvin-wave.png]
{title=Equatorial Kelvin-wave thickness anomaly and zonal velocity at one instant, using the paper's Gaussian-width convention}
At an instant, alternating positive and negative thickness anomalies have their largest magnitudes on the equator and decay symmetrically away from it. Eastward arrows occupy the positive anomalies, westward arrows the negative anomalies; the <velocity> vanishes at the nodes and weakens away from the equator. The whole pattern travels eastward at speed $c$. \b[Meridionally], $fu=-g\prime\eta_y$ is <geostrophic balance>. For a positive anomaly north of the equator, the northward <pressure gradient force> balances the southward <Coriolis force>; both directions reverse south of the equator. \b[Zonally], $fv=0$ and $u_t=-g\prime\eta_x$: the zonal <pressure gradient force> accelerates the flow, so it is not in zonal <geostrophic balance>. On the equator the meridional terms both vanish smoothly.
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