Equatorial wave

An equatorial wave is trapped near the equator by the sign change of the Coriolis parameter. Its meridional structures are Gaussian-weighted Hermite polynomials.

An equivalent depth is the shallow-water depth whose gravity-wave speed matches one vertical mode of a continuously stratified fluid. Each vertical mode then obeys its own horizontal shallow-water wave equations.

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.

An equatorial Kelvin wave has zero meridional velocity, eastward phase propagation, and Gaussian meridional trapping.

An equatorial Rossby wave propagates westward and has a Gaussian-weighted Hermite-polynomial meridional structure.

Under meridional geostrophic balance, the mode with Hermite index $n\ge 1$ has $\omega =-kc/(2n+1)$. The full shallow-water equations bend this long-wave line into a dispersive Rossby branch.

An equatorial inertia--gravity wave is a high-frequency equatorially trapped mode restored jointly by buoyancy and the Coriolis force. Its dispersion relation has eastward and westward branches lying outside the low-frequency equatorial Rossby wave branch.

The Yanai wave is the $n=0$ branch of the equatorial shallow-water dispersion relation with positive frequency:Its group velocity is $d\omega /dk=c{\omega }^2/({\omega }^2+\beta c)>0$. At low frequency it has westward phase velocity but transports energy eastward; at high frequency its phase and group velocities approach $c$.

For dimensionless Hermite index zero, the equatorial shallow-water dispersion relation factorizes as $(\omega +k)({\omega }^2-k\omega -1)=0$. The root $\omega =-k$ is generally spurious: for $v=C{e}^{-y^2/2}$, the original equations require ${\delta }^{\prime }-y\delta =i(2k-1/k)v$, where $\delta =u-\eta$. Multiplication by $e^{-y^2/2}$ and integration over the real line shows that decay at both ends requires $C=0$ unless $k^2=1/2$. At that exception the root already belongs to the physical quadratic; $u=\eta =i\omega yv$ is regular and no extra wave branch is created. The assertion concerns nonzero $k$ and angular frequency.

With Rayleigh friction rate $\gamma$ and thermal damping rate $\alpha$, a steady localized equatorial forcing produces an eastward Kelvin tail of zonal scale $c/\sqrt{\alpha \gamma }$ and westward Rossby tails of scales $c/[(2n+1)\sqrt{\alpha \gamma }]$. Their common meridional scale is $[c\sqrt{\gamma /\alpha }/\beta {]}^{1/2}$.