= Solution
Write $h=h_0+\eta$, with small interface displacement $\eta$ and horizontal variation scale $L$. The geostrophic pressure-Coriolis balance gives
$$
fU\sim g'\eta/L,\qquad U\sim\frac{g'\eta}{|f|L}.
$$
Subtracting the uniform reference state and the mass-conserving linear potential-energy term, the perturbation <energy> densities per unit horizontal area are
$$
\mathrm{KE}\simeq\tfrac12\rho_1h_0U^2,\qquad
\mathrm{PE}\simeq\tfrac12\rho_1g'\eta^2.
$$
Therefore
$$
\boxed{\frac{\mathrm{KE}}{\mathrm{PE}}\sim
\frac{g'h_0}{f^2L^2}=\left(\frac{L_R}{L}\right)^2,
\qquad L_R=\frac{\sqrt{g'h_0}}{|f|}.}
$$
The 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 <Geostrophic flow>[geostrophy]. It does not say an initially resting small-scale height perturbation already has that large <kinetic energy>: such a perturbation undergoes <geostrophic adjustment>.
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