Equatorial shallow-water dispersion relation

With $c=\sqrt{gH}$, beta plane parameter $\beta >0$, and Hermite polynomial index $n\ge 0$, meridional trapping of the nonzero meridional-velocity branch requiresIts meridional velocity is proportional to $H_n(Y)e^{-Y^2/2}$, with $Y=y\sqrt{\beta /c}$. Exceptional roots with ${\omega }^2=c^2{k}^2$ must be checked in the original equations, since eliminating velocity and height divides by this factor.

In dimensionless equatorial wave units with gravity wave speed and beta plane coefficient one, a Fourier mode $e^{i(kx-\omega t)}$ obeys $\omega u-k\eta =iyv$ and $\omega \eta -ku=-i{v}^{\prime }$. Inverting this two-by-two linear system gives the displayed wave polarization, provided ${\omega }^2\ne k^2$. Inserting it in the meridional momentum equation gives the Hermite differential equation after a Gaussian factor is removed. At singular values the original equations, rather than these quotient formulae, determine whether a trapped mode exists.

For fixed positive frequency and $n\ge 1$, the equatorial shallow-water dispersion relation has real zonal wavenumbers only ifBetween these values the zonal roots are complex, giving an evanescent response rather than freely propagating modes.