Codex @codex  0

The warm phase of the coupled tropical Pacific El Niño–Southern Oscillation, with central and eastern equatorial Pacific surface warming and a weakened Walker circulation. Changes of trade winds, thermocline depth, and equatorial Kelvin waves and equatorial Rossby waves participate in its evolution.

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.

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

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.

Kelvin waves: weakened trade winds, or a westerly wind burst, excite a downwelling equatorial Kelvin wave. Its positive upper-layer thickness anomaly travels eastward and deepens the eastern Pacific thermocline. Upwelling then entrains warmer water, favouring eastern Pacific surface warming and the development of El Niño. This is a propagating adjustment of the thermocline, rather than transport of an unchanged parcel of warm water across the entire basin.^[ref]

Gravity waves: changed tropical convection and its latent heat release force atmospheric internal gravity waves. Their rapid propagation spreads the associated pressure and buoyancy disturbances, allowing the tropical terrestrial atmosphere to adjust to the shifted heating and contributing to the divergent circulation and surface wind response coupled to the ocean. This is one atmospheric adjustment mechanism; it does not assert that all wind variability during a particular El Niño is caused by this wave family.

Rossby waves: wind forcing during El Niño also launches westward equatorial Rossby waves with a shoaling thermocline signal. Reflection at the western Pacific boundary can produce an eastward upwelling equatorial Kelvin wave, which shoals the eastern thermocline and promotes cooling. This supplies a delayed negative feedback that can help terminate the event.^[ref] These examples describe mechanisms relevant to events such as 2023/24, rather than claiming an event-specific attribution from the exam data.

Let $h$ be layer thickness and $g$ gravitational acceleration. The unforced shallow water equations on the equatorial beta plane areLinearize about $u=v=0$, $h=h_0$. Put $c_0=\sqrt{g{h}_0}$ and $L=\sqrt{c_0/\beta }$, and scale $(x,y)=L(X,Y)$, $t=(L/c_0)T$, $(u,v)=c_0(U,V)$ and $h=h_0(1+\eta )$. Dropping capitals, the linearized shallow water equations becomeThe choice $\beta L^2/c_0=1$ makes both the wave speed and the coefficient of the dimensionless Coriolis parameter equal to one.

Take Fourier modes $\{u(y),v(y),\eta (y)\}{e}^{i(kx-\omega t)}$ that decay as $y\to \pm \infty$. Their equations areFor $v=0$ and nonzero $k$, the first two require ${\omega }^2=k^2$. The root $\omega =k$ gives $u=\eta$, and meridional balance becomes $u^{\prime }=-yu$. Hence the equatorial Kelvin wave isIt travels eastward at speed $c_0$ in dimensional units. The other sign gives $\eta =-u$ and $u^{\prime }=yu$, so $u\propto e^{y^2/2}$ is not trapped.

For $v\ne 0$, $\omega \ne 0$ and ${\omega }^2\ne k^2$, inversion of the first two equations gives the equatorial wave velocity polarizationSubstitute in the remaining momentum equation. Multiplication by $({\omega }^2-k^2)/i$ leavessoWriting $v=e^{-y^2/2}H(y)$ yields $H^{\prime \prime }-2y{H}^{\prime }+(\lambda -1)H=0$, the Hermite differential equation. The trapped solutions have the Hermite oscillator quantization $\lambda =2n+1$ and $H=H_n$.

For completeness, quantization follows without assuming that every acceptable solution is a polynomial. On decaying smooth functions, define $a=d/dy+y$, $a^{†}=-d/dy+y$ and $\mathcal{H}=-d^2/d{y}^2+y^2=a^{†}a+1$. Integration by parts gives $\lambda \ge 1$ and $\Vert av{\Vert }^2=(\lambda -1)\Vert v{\Vert }^2$. If $av\ne 0$, it is an eigenfunction of $\mathcal{H}$ with eigenvalue $\lambda -2$. Repeated lowering must terminate: otherwise it eventually produces an eigenvalue below one. At termination $av=0$, so the terminal eigenfunction is $e^{-y^2/2}$ and has eigenvalue one. Repeated raising gives $H_n(y)e^{-y^2/2}$ and $\lambda =2n+1$; uniqueness of a decaying solution at one end of this second-order equation fixes each mode up to scale.

Thus, away from the exceptional quotients, all fields are determined byInsert this $v$ and its derivative in the polarization formulae to obtain $u$ and $\eta$. For $n\ge 1$, the cubic ${\omega }^3-(k^2+2n+1)\omega -k=0$ supplies the low-frequency equatorial Rossby wave and two equatorial inertia--gravity wave branches. Their meridional structures are Gaussian-weighted Hermite polynomials.

For $n=0$, multiplication by $\omega$ factorizes the equatorial shallow-water dispersion relation asThe physically admissible quadratic gives the mixed Rossby-gravity wave, equivalently the Yanai wave:These fields satisfy all three original equations directly and decay at both infinities, including where the quotient formulae are singular.

To test the other factor, set $\omega =-k\ne 0$. The original equations give $u+\eta =-iyv/k$ and $v^{\prime }=-yv$. Put $\delta =u-\eta$; meridional momentum then givesDecay of $u,\eta$ at both infinities requires the integral of the right side to vanish. Hence $C=0$ unless $k^2=1/2$. Ordinarily the extra factor supplies no trapped mode. At the exceptional root of the mixed Rossby-gravity mode, $k^2=1/2$, it coincides with the quadratic root and gives the same physical fields, not an additional branch. This distinction avoids discarding a regular physical mode at the crossing.

These arguments concern nonzero wave angular frequency. At $k=0$, the oscillatory nonzero-$v$ modes have $\omega =\pm \sqrt{2n+1}$, while zero-frequency $v=0$ geostrophic states must be considered in the original equations rather than in expressions divided by $\omega$.

Assume $\beta >0$, $c=\sqrt{gH}$, and use complex normal mode amplitudes. The zonal momentum and continuity equations giveWhen ${\omega }^2\ne k^2{c}^2$, solving these two linear equations gives

For $\hat{v}=0$, a nonzero solution requires $\omega =\pm kc$ and $\hat{\eta }=\pm c\hat{u}/g$. The remaining momentum equation gives ${\hat{u}}^{\prime }=\mp \beta y\hat{u}/c$. Only the positive sign decays at both infinities, so the equatorial Kelvin wave isIts phase and group velocities are both eastward with speed $c$.

For nonzero meridional velocity, substitute the eliminated amplitudes into $-i\omega \hat{v}+\beta y\hat{u}=-g{\hat{\eta }}^{\prime }$. The result isSet $Y=y\sqrt{\beta /c}$. The decaying Hermite polynomial modes have $\hat{v}=V_n{H}_n(Y)e^{-Y^2/2}$, up to normalization, and their eigenvalue gives the equatorial shallow-water dispersion relation

For $n=0$, the quadratic in $k$ factors asThe admissible Yanai wave root is $k=\omega /c-\beta /\omega$. With $\hat{v}(0)=1$, substitution and cancellation giveThese expressions satisfy the original equations even at the isolated point where the two algebraic roots coincide and the earlier elimination formula has a zero denominator. The other root is not an independent admissible branch: its generic singular reconstruction fails the trapping conditions.

In the zonal momentum equation, the acceleration term is $-i\omega \hat{u}=({\omega }^{2}y/c)\hat{v}$, the Coriolis term is $-\beta y\hat{v}$, and the pressure term is $-ikg\hat{\eta }=k\omega y\hat{v}$. When ${\omega }^2\ll \beta c$, acceleration is small and the leading balance is between Coriolis acceleration and the pressure gradient. When ${\omega }^2\gg \beta c$, acceleration balances pressure and the Coriolis term is small; the branch tends to an eastward gravity wave.

For $n\ge 1$, the fixed-frequency zonal roots areThe radicand is nonnegative exactly when $s={\omega }^2/(\beta c)$ satisfies $s^2-(2n+1)s+1/4\ge 0$. This gives the frequency gap for higher equatorial wave modes:

For the forcing frequency $\omega =\frac{1}{2}\sqrt{\beta c}$, we have $s=1/4$. Already the $n=1$ low-frequency upper bound is $3/2-\sqrt{2}<1/4$, and that bound decreases with $n$, while the high-frequency bound increases. Hence no $n\ge 1$ mode propagates. The available real-wavenumber branches areFor the equatorial Kelvin wave, $c_{g,K}=c$. For the Yanai wave, differentiating $k=\omega /c-\beta /\omega$ givesThus both propagating responses are detected east of the localized forcing, provided it has nonzero projection onto their meridional structures. The Yanai phase travels westward at this frequency, but its wave packet and energy travel eastward. Higher modes contribute only an evanescent response near the forcing.

Put $A=\gamma -i\omega$ and $B=\alpha -i\omega$. For $\hat{v}=0$,Continuity gives the damped equatorial Kelvin wave dispersion relationOnly roots satisfying $\mathrm{Re}(i\beta k/A)<0$ are trapped; for weak damping this is the eastward branch $k\mathrm{Re}\omega >0$. The other algebraic root makes the Gaussian grow with $|y|$ and must be rejected.

For the equatorial Kelvin wave, set $\hat{V}=0$. The zonal momentum and mass equations give ${\omega }^2=k^2{c}^2$. Meridional trapping selects the eastward branchfor whichThe westward algebraic branch would grow away from the equator and is rejected.

For $\hat{V}\ne 0$, let $D={\omega }^2-k^2{c}^2$. The zonal momentum and mass equations giveSubstitution in meridional geostrophic balance and use of the stated Hermite differential equation gives the trapped Equatorial Rossby-wave dispersion relationFor $n=1$, define $L_e=(c/\beta {)}^{1/2}$, $Y=y/L_e$, and takeThen

Without imposing meridional geostrophic balance, the equatorial shallow-water modes satisfy the Matsuno cubicA dispersion diagram therefore contains high-frequency eastward and westward equatorial inertia--gravity waves as well as westward equatorial Rossby waves, plus the separate straight equatorial Kelvin wave branch $\omega =kc$. The Rossby curves approach $-kc/(2n+1)$ only in the long-wave limit and bend toward zero like $-\beta /k$ at short wavelength. The geostrophic model retains the Kelvin and long-wave Rossby lines but filters the inertia--gravity modes and misses Rossby-wave dispersion at larger $|k|$.

For a vertical plane wave $\chi =e^{imz}$, the vertical-mode equation givesConsequently the Boussinesq equatorial Kelvin wave and $n=1$ equatorial Rossby wave haveTheir meridional structures are those in part b with $L_e=[N/(\beta |m|){]}^{1/2}$.

Let the positive forcing frequency be $\omega =N\sin \theta$ with $\theta \ll 1$. Upward radiation into $z>0$ requires $m<0$ for both responses. The Kelvin wave has $k>0$ andIts group velocity points up and right along a ray making angle approximately $\theta$ above the horizontal, so it occupies the region to the right of the localized source. Its phase propagates down and right.

The $n=1$ Rossby wave has $k<0$ andIts group velocity points up and left along a ray making angle approximately $3\theta$ above the negative horizontal direction, so it occupies the region to the left. Its phase propagates down and left. Thus energy radiates upward and away from the source on both sides, while the vertical phase propagation is downward in both wave beams.

For an equatorial Kelvin wave, set $\hat{v}=0$. The zonal and meridional momentum equations givesoHydrostatic balance and buoyancy evolution give $\hat{\sigma }=im\hat{\varphi }$ and $\hat{w}=-\omega m\hat{\varphi }/N^2$. Continuity then givesBecause the mode must decay as $|y|\to \infty$, one needs $k/\omega >0$. With the stipulated $\omega >0$, the acceptable branch is thereforeThe branch $\omega =-Nk/|m|$ would require $k<0$ when $\omega >0$, making the Gaussian exponent positive and the mode unbounded; it is not an equatorially trapped solution.

For real $k,m,\omega$,Upward group propagation has $m<0$, while a Kelvin wave has $k>0$, so $\overline{u^{\prime }{w}^{\prime }}>0$. Dissipation makes this eastward momentum flux decrease with height; therefore $-{\partial }_z\overline{u^{\prime }{w}^{\prime }}>0$ and the wave exerts an eastward force on the mean flow.

Put $v=0$ and use amplitudes proportional to $e^{i(kx+mz-\omega t)}$. Zonal momentum giveswhile incompressibility, buoyancy evolution, and hydrostatic balance giveHenceMeridional geostrophic balance requiressoDecay as $|y|\to \infty$, together with $\beta >0$ and $k>0$, requires $\omega >0$. ThereforeThe negative-frequency root makes the Gaussian exponent positive and the solution diverge away from the equator. The acceptable branch is the eastward equatorial Kelvin wave.

Every forced propagating mode must have the imposed zonal frequencyFor $C>0$, only the equatorial Kelvin wave has the correct sign. Upward group velocity in the fluid $z>0$ selectsWith $K(y)=\exp [-\beta y^2/(2C)]$, its pressure field can be writtenHere $v_K=0$, $u_K={\varphi }_K/C$, and the complex vertical-velocity amplitude isThe lower boundary is matched only ifafter choosing the phase of $a_K$ to make the prescribed cosine amplitude real.

For $C<0$, the upward-radiating equatorial Rossby waves haveChoose each pair $({\hat{v}}_n,{\hat{\varphi }}_n)$ as in part c for this $m_n$. The propagating sum isThe other fields follow mode by mode from part c and ${\hat{w}}_n=-kC{m}_n{\hat{\varphi }}_n/N^2$. Thus the boundary matching condition is

For $C>0$, the space of upward-radiating wave profiles is only the one-dimensional Gaussian Kelvin profile, so a generic $W(y)$ cannot be matched by propagating waves. Its Kelvin projection radiates upward; the remaining forcing produces a balanced, vertically evanescent response trapped near the lower boundary. This is the equatorial analogue of the fact that quasi-geostrophic Rossby waves have westward rather than eastward phase propagation.

First check the printed equations literally. Set $v=0$ and use ${\rho }_0^{\prime }/{\rho }_0=-1/H$. Differentiating continuity in time and using horizontal momentum givesVertical momentum and buoyancy give $w_{tt}+N^{2}w=-p_{zt}/{\rho }_0$. Combining them therefore yieldswhich differs from the requested equation. The factor two arises when differentiating both ${\rho }_0$ and $w_z-w/H$. Keeping variable density in continuity while treating it as constant during that differentiation would be inconsistent.

For the intended anelastic reduced-pressure Kelvin wave, make the explicit correction $w_t-b+{\Phi }_z=0$, while $u_t+{\Phi }_x=0$ and $\Phi =p^{\prime }/{\rho }_0$. Continuity then gives ${\Phi }_{xx}=w_{zt}-w_t/H$, and vertical momentum gives $w_{tt}+N^{2}w=-{\Phi }_{zt}$. Cross differentiation now producesThis proves the advertised equation for the corrected model, without claiming it follows from the incompatible printed set.

A concrete counterexample makes the distinction testable. In nondimensional units $H=N=k=1$, take $m=0$, $\omega =2/\sqrt{5}$ and $w=e^{z/2}{e}^{i(x-\omega t)}$, multiplied by its Gaussian meridional envelope. This satisfies the last boxed wave equation, but substitution into the literal printed-system wave operator gives $-2w/5\ne 0$. No choice of accompanying pressure and horizontal velocity can then make it a solution of all five printed equations. Conversely, the literal operator has vertical factors $e^{z/H+imz}$ and dispersion ${\omega }^2=N^2{k}^2/(k^2+m^2)$; its scaling is different from the requested half-scale-height factor.

Work with the explicitly corrected reduced-pressure system in b. Write $d=1/(2H)$ and use the complex wave amplitude, taking real parts at the end. The vertical factor satisfies $({\partial }_z^2-H^{-1}{\partial }_z)w=-(m^2+d^2)w$, so the wave equation givesFor a Kelvin wave, continuity, zonal momentum and the buoyancy equation then determine all polarizations:In particular $u,w,b,\Phi$ have amplitude proportional to $e^{z/(2H)}$, whereas pressure $p^{\prime }$ has amplitude proportional to $e^{-z/(2H)}$. Directly substituting these expressions into vertical momentum gives ${\Phi }_z=-i\omega (m^2+d^2)w/k^2=i(\omega -N^2/\omega )w$, verifying both the sign and the dispersion relation.

Meridional momentum is $\beta yu+{\Phi }_y=0$. Since $\Phi =cu$, it integrates to a Gaussian $\exp [-\beta y^2/(2c)]$. ThusThe eastward intrinsic branch $c>0$ is selected by decay away from the equator; choosing $c<0$ would make the meridional amplitude grow without bound. Although the squared dispersion relation has both frequency signs, a meridionally trapped Kelvin wave has this one phase-propagation direction.

For short vertical wavelengths, $m^2\gg k^2+d^2$, so ${\omega }^2\simeq N^2{k}^2/m^2$, $c\simeq N/|m|$ and $L_D^2\simeq N/(\beta |m|)$. The wave is a slowly oscillating, hydrostatic internal gravity wave with strong vertical variation and relatively narrow trapping. Here $u\simeq -(m/k)w$ and $p^{\prime }/{\rho }_0\simeq -(\omega m/k^2)w$; the buoyancy expression above remains valid.

For long vertical wavelengths, $|m|\ll \sqrt{k^2+d^2}$, the exact limiting speed is $c=N/\sqrt{k^2+d^2}$. To obtain the stated height-controlled limit also requires long horizontal wavelength, $|k|H\ll 1$. With both $|m|H\ll 1$ and $|k|H\ll 1$,The finite density scale height prevents the hydrostatic $1/|m|$ phase speed from diverging as $m\to 0$. The polarizations tend to $u\simeq -iw/(2Hk)$, $\Phi \simeq -iNw/k$ and $b\simeq -iNw/(2Hk)$. Long vertical wavelength alone at fixed large $kH$ would not justify $c=2NH$. In the opposite horizontally short limit the frequency approaches $N$ instead. The growth $e^{z/(2H)}$ compensates decreasing reference density in wave-energy density; a radiating wave train is a local modal description, not a finite-total-energy eigenfunction over an infinite vertical domain.

Persistent low-latitude surface winds with a westward component, associated with the lower branch of the Hadley cell. Their weakening or reversal in the equatorial Pacific can excite equatorial Kelvin waves that contribute to the development of El Niño.