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The Burger number compares the square of a relevant deformation radius with the square of the horizontal flow scale: $\mathrm{Bu}=R_d^2/l^2$. In a two-layer geostrophic model $R_d^2=g\prime H_1{H}_2/[f_0^2(H_1+H_2)]$. Its inverse measures the scale ratio of baroclinic available potential energy to kinetic energy. Keeping it of order unity retains both interface and relative-vorticity contributions in QG scaling.

A horizontal scale comparing gravitational or buoyancy restoring forces with rotation. For a shallow-water gravity mode with propagation speed $c$, the deformation radius is $c/|f|$. The barotropic deformation radius uses $c=\sqrt{gH}$; internal layered modes use a speed set by reduced gravity and layer depths.

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 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 giveOn the eastward branch $\omega =ck$, the wave polarization is $u=(c/h_0)\eta$. Meridional geostrophic balance givesThe 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 givesThe usual equatorial deformation radius is $R_{\mathrm{eq}}=\sqrt{c/\beta }$, with the envelope written $e^{-y^2/(2R_{\mathrm{eq}}^2)}$. Thus the printed exponent uses $L_{\mathrm{eq}}=\sqrt{2}{R}_{\mathrm{eq}}$. Calling that width a deformation radius is a convention; using the standard radius without the factor $1/2$ would fail the printed meridional equation.

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 $c$. 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. 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.

Write $h=h_0+\eta$, with small interface displacement $\eta$ and horizontal variation scale $L$. The geostrophic pressure-Coriolis balance givesSubtracting the uniform reference state and the mass-conserving linear potential-energy term, the perturbation energy densities per unit horizontal area areThereforeThe internal deformation radius $L_R$ is the distance an internal gravity wave travels in an inverse-Coriolis time. Balanced broad disturbances $L\gg L_R$ mainly store available potential energy; balanced fine structures $L\ll L_R$ require relatively large kinetic energy. This scale result presupposes small Rossby number $U/(|f|L)$; reducing the length at fixed displacement eventually violates geostrophy. It does not say an initially resting small-scale height perturbation already has that large kinetic energy: such a perturbation undergoes geostrophic adjustment.

Homogenizing the potential vorticity over a finite constant bottom slope creates the odd anomaly displayed here while leaving the exterior unchanged. Inverting $({\partial }_y^2-L_R^{-2})\psi =\Delta Q$ with decay at both infinities and matching $\psi ,{\psi }_y$ gives a central return current and flanking jets. The deformation radius sets the smoothing length. For $L_R\ll a$, the central surface follows the bottom slope to leading order, but edge corrections and decaying exterior flow are essential.