Deformation radius ($R_d$)

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.

For an equatorial beta plane and a shallow water wave speed $c$, the natural meridional trapping scale is $R_{\mathrm{eq}}=\sqrt{c/\beta }$. A decaying equatorial Kelvin wave has $\eta \propto e^{-y^2/(2R_{\mathrm{eq}}^2)}$. An envelope written $e^{-y^2/L^2}$ instead uses $L=\sqrt{2}{R}_{\mathrm{eq}}$; stating the exponent convention prevents a factor-of-two error in the meridional Cauchy momentum equation.

The two-layer quasi-geostrophic potential vorticity coupling defines $R_d^{-2}=F_1+F_2$. With $F_i=f_0^2/(g\prime H_i)$ this gives the stated radius. The baroclinic energy ratio and deformation scale depends on $(l/R_d{)}^2$, so the thinner layer matters when the depths are strongly unequal.