On the modelling of equatorial waves
 The present theory of geophysical waves that either raise or lower the equatorial thermocline, based on the reduced-gravity shallow-water equations on theβ-plane, ignores vertical variations of the flow. In particular, the vertical structure of the Equatorial Undercurrent is absent. As a remedy we propose a simple approach by modeling this geophysical process as a wave-current interaction in thef-plane approximation, the underlying current being of positive constant vorticity. The explicit dispersion relation allows us to conclude that, despite its simplicity, the proposed model captures to a reasonable extent essential features of equatorial waves.
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 Equatorial waves exhibit particular dynamics due to the vanishing of the Coriolis parameter along the Equator. Also [cf. Fedorov and Brown, 2009], the vertical stratification of the ocean is greater in this region than anywhere else. Both factors facilitate the propagation of geophysical waves that either raise or lower the equatorial thermocline (the sharp boundary between warm and deeper cold waters). These equatorial waves are one of the key factors in explaining the El Niño phenomenon — an event associated with the appearance around the Christmas season of an ocean anomaly in the form of a warm equatorial water flow approaching the western coast of South America, called by early fishermen “El Niño” (Spanish for “The Christ Child”). The inherent ocean adjustment depends crucially on the existence of geophysical waves that can alter the depth of the thermocline [cf. Cushman-Roisin and Beckers, 2011].
 The best-known examples of equatorial waves are eastward propagating Kelvin waves. These were predicted theoretically by using aβ-plane approximation to the governing equations in the shallow water regime of a one-layer reduced-gravity model (see the discussion byCushman-Roisin and Beckers  and Fedorov and Brown ). While this approximation is in many respects very successful and is therefore used broadly, it fails to capture vertical variations of the flow. This aspect is of relevance since an important feature in this region is the strong eastward flowing Equatorial Undercurrent. Due to the prevalence of winds that blow westward, the surface flow is generally directed westward. However, as first discovered in 1951 [cf. Philander, 1980], the flow reverses at a depth of several tens of metres, signalling the presence of the Equatorial Undercurrent (EUC). The thinness, symmetry and length of the EUC are truly remarkable: the Pacific EUC is confined to a shallow surface layer centred on the Equator in the thermocline (less than 200 m deep [cf. Philander, 1980]), it is typically about 300 km in width and symmetric about the Equator, with a maximum speed of about 1 m/s [cf. Hughes, 1979], and extends nearly across the whole length (about 13000 km) of the ocean basin [cf. Izumo, 2005]. We propose a model for the interaction of geophysical waves with the EUC. The simplest way to capture the vertical variation of the EUC is to regard it as an eastward flowing current of constant vorticity. Due to the smallness of the width of the EUC, we neglect the variations of the Coriolis parameter and use the f-plane approximation. Ignoring the presence of the EUC, theβ-plane approximation represents a suitable setting [cf.Cushman-Roisin and Beckers, 2011], and one obtains this way a variety of equatorially trapped geophysical waves (e.g., eastward propagating Kelvin waves [cf. Fedorov and Brown, 2009]). While this approach is roughly applicable to the entire strip extending from 5°S latitude to 5°N latitude, it does not capture the peculiarity of the EUC flow. Due to the relatively small width of the EUC we advocate the use of the f-plane approximation. To justify this, note that for flows near the Equator (up to latitudes of 30° [cf.Gallagher and Saint-Raymond, 2007]) it is customary to use the equatorial β-plane approximation: in the rotating frame the Coriolis force 2Ω = (0, 2ωcosϕ, 2ωsinϕ) is approximated by (0, 2ω, βy), while in the f-plane approximation one uses (0, 2ω, 0). Here ϕ is the latitude, y measures the meridional distance from the Equator and β ≈ 2.28 · 10−11 m−1 s−1 . The magnitude of the EUC's relative vorticity (about 25 · 10−3 s−1) is much larger than that of the equatorial planetary vorticity |2Ω| ≈ 1.46 · 10−4 s−1. In the main region of wave-current interaction, within 1° latitude (111 km) from the Equator, we have
Therefore in this setting the β-plane effect on the planetary vorticity of the flow amounts to less than 1.75% and thef-plane approximation is reasonable. The importance of the limited north–south extent is underlined by the fact that atN5° the β-plane effect would be of order 9%. Note also that we can not take advantage of the geostrophic balance that proved so useful at mid-latitudes [cf.Gallagher and Saint-Raymond, 2007; Majda and Wang, 2006]. As it is usual for equatorial wave dynamics [cf. Fedorov and Brown, 2009; Sirven, 1996], we assume that a shallow layer of relatively warm (and less dense) water overlies a much deeper motionless layer of cold water (of slightly higher density). Both layers are supposed to be homogeneous and their sharp boundary is the thermocline. In the upper layer (which contains the EUC) we distinguish two sublayers: a near-surface layer to which the effect of wind waves is confined, and a centre layer. The upper boundary of this centre layer is considered to be flat, with a horizontal fluid velocity of 1 m/s moving eastward, while its lower boundary is the thermocline near which the EUC decays. We focus on the oscillation of the thermocline, modelling it as a geophysical wave interacting with a current of positive vorticity. We will derive the dispersion relation and infer some useful properties of the flow from its analysis.
2. Main Result
 In a rotating framework with the origin at a point on the Earth's surface, with the x-axis chosen horizontally due east, they-axis horizontally due north and thez-axis upward, letz = −d be the upper boundary of the centre layer and z = −η(x, y, t) be the thermocline. In the region −η(x, y, t) ≤ z ≤ −d, the governing equations in the f-plane approximation near the Equator are [cf.Gallagher and Saint-Raymond, 2007] the Euler equation
and the equation of mass conservation
Here t represents time, (u, v, w) is the fluid velocity, ω = 73 · 10−6 rad/s is the (constant) rotational speed of the Earth (taken to be a perfect sphere of radius 6371 km) round the polar axis towards the east, g = 9.8 m/s2 is the (constant) gravitational acceleration at the Earth's surface, and P is the pressure. Beneath the thermocline z = −η(x, y, t) the motionless colder water has a slightly higher density ρ + Δρ (the typical value of Δρ/ρ for the equatorial Pacific being 0.006 [cf. Fedorov and Brown, 2009]). In this region (2) is not altered and (1) holds with ρ + Δρ replacing ρ, so that here u = v = w = 0 and
for some constant P0. A peculiar feature of (1)–(2) can be inferred from the vorticity equation [see Pedlosky, 1979]: for two-dimensional flows, independent upon they-coordinate and withv ≡ 0, the vorticity γ = (0, uz − wx, 0) is preserved. By abuse of notation we identify γ with the scalar uz − wx and assume that it is a positive constant throughout the layer −η(x, t) ≤ z ≤ −d. We seek traveling waves, with the velocity field, the pressure and the thermocline exhibiting in this layer an (x, t)-dependence of the form (x − ct), where c > 0 is the propagation speed of the oscillations of the thermocline. The boundary conditions are therefore the kinematic boundary conditions
expressing the fact that a particle on this surface remains confined to it (see the discussion by Constantin ), together with the dynamic boundary condition
which in view of (3) ensures the continuity of the pressure across the thermocline. The system governing the motion becomes
in the frame of reference (x, z) moving at the constant wave speed c > 0: we performed the change of variables x ↦ x − ct. Using (2) we now introduce the stream function ψ(x, z) determined up to an additive constant by
This permits us to re-formulate(6) as
where m is the relative mass flux, given by
Note that the first two equations in (8) merely state the fact that the expression
is constant throughout the layer −η(x) < z < −d, in analogy with Bernoulli's law for gravity water waves [see Constantin, 2011]. Therefore (8) takes the simpler form
for some constant Q. Here
is the reduced gravity [cf. Fedorov and Brown, 2009]. Under the assumption of no stagnation points of the flow, that is, provided
throughout the fluid, the hodograph transform
converts the free-boundary problem(10) into the nonlinear boundary problem
for the unknown function
(representing the level beneath the upper flat boundary of the centre layer) in the fixed strip 0 < p < |m|. Note that (9) and (12) yield m < 0 so that |m| = −m. The laminar flow solutions to (14), representing parallel shear flows for which every particle moves horizontally with a speed that depends only on the depth, are given by
the parameter λ > 0 being related to the speed on the flat thermocline p = 0 by
We linearize the problem (14) by setting
where ε > 0 is a small parameter. Seeking solutions that are even and periodic in the q-variable with principal periodL > 0, we obtain at order εthe weighted Sturm-Liouville problem
where is the wavenumber and . The substitution transforms the differential equation in (16) into F″ = k2F. Taking into account the boundary condition at p = |m| in (16), we obtain
The boundary condition at p = 0 holds if and only if
The solution λ∗ of (17) yields the bifurcating laminar flow
where z = −D is the thermocline. The corresponding relative mass flux
can be computed by means of (9). Viewing this as an equation for (D − d), we get
since the alternative expression is eliminated by the requirement (12). The above permits us to write (17) as
Solving (18) for λ∗ > 0, we obtain
At the thermocline u = 0 so there. Consequently the previous relation leads us to a polynomial equation of degree two in the unknown c, the only positive root being
This is the exact dispersion relation, presenting a slight resemblance to the dispersion relation for gravity water waves with constant vorticity discussed by Constantin and Strauss  and Constantin and Varvaruca .
 The wavelengths L occurring in the context of the motion of the thermocline are in excess of 800 km [cf. Philander, 1980] so that k(D − d) is of order 10−3 since the depths d and D are both of the order of tens of m. Consequently and the dispersion relation take the simpler form
We see that c is a strictly increasing function of (D − d), so that we can estimate the slow variation in the depth of the thermocline from measurements of the wave speed and vice-versa. As a numerical example, typical values for the equatorial Pacific are d = 80 m, D = 120 m [cf. Fedorov and Brown, 2009]. A current of constant positive vorticity γ with a horizontal velocity decaying from 1 m/s to zero in 40 m depth (as it is appropriate for the EUC [cf. Wacogne, 1988]) yields γ = 25 · 10−3 s−1 and (19) gives 2.1 m/s, which agrees well with measurements. Taking u0 = 1 m/s as a typical value of the velocity, the associated Rossby number is given by . This confirms the geophysical character of this wave motion.
 The author is grateful to both referees for useful suggestions.
 The Editor thanks two anonymous reviewers for their assistance in evaluating this paper.