## 1. Introduction

[2] 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].

[3] The best-known examples of equatorial waves are eastward propagating Kelvin waves. These were predicted theoretically by using a*β*-plane approximation to the governing equations in the shallow water regime of a one-layer reduced-gravity model (see the discussion by*Cushman-Roisin and Beckers* [2011] and *Fedorov and Brown* [2009]). 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, the*β*-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, 2*ω*cos*ϕ*, 2*ω*sin*ϕ*) is approximated by (0, 2*ω*, *βy*), while in the *f*-plane approximation one uses (0, 2*ω*, 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 the*f*-plane approximation is reasonable. The importance of the limited north–south extent is underlined by the fact that at*N*5° 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.