EQUATORIAL WAVES A. V. Fedorov and J. N. Brown, Yale University, suggesting an analogy between coastally trapped and New Haven, CT, USA equatorial waves. & 2009 Elsevier Ltd. All rights reserved. The most well-known examples of equatorial waves are eastward propagating Kelvin waves and westward propagating Rossby waves. These waves are usually observed as disturbances that either raise or lower the equatorial thermocline. These thermo- Introduction cline disturbances are mirrored by small anomalies in sea-level elevation, which offer a practical method It has been long recognized that the tropical thermo- for tracking these waves from space. cline (the sharp boundary between warm and deeper For some time the theory of equatorial waves, cold waters) provides a wave guide for several types based on the shallow-water equations, remained a of large-scale ocean waves. The existence of this theoretical curiosity and an interesting application wave guide is due to two key factors. First, the mean for Hermite functions. The first direct measurements ocean vertical stratification in the tropics is perhaps of equatorial Kelvin waves in the 1960s and 1970s greater than anywhere else in the ocean (Figures 1 served as a rough confirmation of the theory. By the and 2), which facilitates wave propagation. Second, 1980s, scientists came to realize that the equatorial o since the Coriolis parameter vanishes exactly at 0 of waves, crucial in the response of the tropical ocean latitude, the equator works as a natural boundary, to varying wind forcing, are one of the key factors in (a) 30° E60° E90° E 120° E 150° E 180° 150° W 120° W90° W60° W30° W0° E30° E 5 2 − 0 N m 2 − −5 10 Zonal wind stress −10 (b) Indian Pacific Atlantic 0 28 28 26 26 24 24 26 22 16 100 22 26 24 18 24 16 22 16 20 14 14 14 200 18 14 16 Depth (m) 14 12 12 12 300 400 30° E60° E90° E 120° E 150° E 180° 150° W 120° W90° W60° W30° W0° E30° E Figure 1 The thermal structure of the upper ocean along the equator: (a) the zonal wind stress along the equator; shading indicates the standard deviation of the annual cycle. (b) Ocean temperature along the equator as a function of depth and longitude. The east– west slope of the thermocline in the Pacific and the Atlantic is maintained by the easterly winds. (b) From Kessler (2005). 3680 EQUATORIAL WAVES Pacific 0 24.0 26.0 100 22.0 20.0 18.0 16.0 14.0 200 Depth (m) 12.0 300 10.0 400 10° S0° 10° N Figure 2 Ocean temperature as a function of depth and latitude in the middle of the Pacific basin (at 1401 W). The thermocline is particularly sharp in the vicinity of the equator. Note that the scaling of the horizontal axis is different from that in Figure 1. Temperature data are from Levitus S and Boyer T (1994) World Ocean Atlas 1994, Vol. 4: Temperature NOAA Atlas NESDIS4. Washington, DC: US Government Printing Office. 28 C) ° 27 26 25 Temperature ( 24 1900 1920 1940 1960 1980 2000 Figure 3 Interannual variations in sea surface temperatures (SSTs) in the eastern equatorial Pacific shown on the background of decadal changes (in 1C). The annual cycle and high-frequency variations are removed from the data. El Nin˜o conditions correspond to warmer temperatures. Note El Nin˜o events of 1982 and 1997, the strongest in the instrumental record. From Fedorov AV and Philander SG (2000) Is El Nin˜o changing? Science 288: 1997–2002. explaining ENSO – the El Nin˜ o-Southern Oscillation The ocean adjustment associated with these phenomenon. changes crucially depends on the existence of equa- El Nin˜ o, and its complement La Nin˜ a, have torial waves, especially Kelvin and Rossby waves, physical manifestations in the sea surface tempera- as they can alter the depth of the tropical thermo- ture (SST) of the eastern equatorial Pacific (Figure 3). cline. This article gives a brief summary of the These climate phenomena cause a gradual horizontal theory behind equatorial waves, the available ob- redistribution of warm surface water along the servations of those waves, and their role in ENSO equator: strong zonal winds during La Nin˜ a years dynamics. For a detailed description of El Nin˜ o pile up the warm water in the west, causing the phenomenology the reader is referred to other thermocline to slope downward to the west and relevant papers in this encyclopedia (see El Nin˜ o exposing cold water to the surface in the east Southern Oscillation (ENSO), El Nin˜ o Southern (Figure 4(b)). During an El Nin˜ o, weakened Oscillation (ENSO) Models, and Pacific Ocean zonal winds permit the warm water to flow back Equatorial Currents) eastward so that the thermocline becomes more It is significant that ENSO is characterized by horizontal, inducing strong warm anomalies in the a spectral peak at the period of 3–5 years. The SST (Figure 4(a)). timescales associated with the low-order (and most EQUATORIAL WAVES 3681 (a) 140° E 160° E180° 160° W 140° W 120° W 100° W 0 32 28 28 100 22 24 20 18 20 16 14 200 16 12 Depth (m) 12 300 8 4 400 0 (b) 140° E 160° E180° 160° W 140° W 120° W 100° W 0 32 28 100 24 28 20 200 14 16 Depth (m) 12 12 300 8 4 10 400 0 Figure 4 Temperatures (1C) as a function of depth along the equator at the peaks of (a) El Nin˜o (Jan. 1998) and (b) La Nin˜a (Dec. 1999). From the TAO data; see McPhaden MJ (1999) Genesis and evolution of the 1997–98 El Nin˜o. Science 283: 950–954. important dynamically) equatorial waves are much The Shallow-Water Equations shorter than this period. For instance, it takes only 2–3 months for a Kelvin wave to cross the Pacific Equatorial wave dynamics are easily understood 1 basin, and less than 8 months for a first-mode Rossby from simple models based on the 1 2-layer shallow- wave. Because of such scale separation, the prop- water equations. This approximation assumes that a erties of the ocean response to wind perturbations shallow layer of relatively warm (and less dense) strongly depend on the character of the imposed water overlies a much deeper layer of cold water. The forcing. It is, therefore, necessary to distinguish the two layers are separated by a sharp thermocline following. (Figure 5), and it is assumed that there is no motion in the deep layer. The idea is to approximate the 1. Free equatorial waves which arise as solutions of thermal (and density) structure of the ocean dis- unforced equations of motion (e.g., free Kelvin played in Figures 1 and 2 in the simplest form and Rossby waves), possible. 2. Equatorial waves forced by brief wind pertur- The momentum and continuity equations, usually bations (of the order of a few weeks). In effect, referred to as the reduced-gravity shallow-water these waves become free waves as soon as the equations on the b-plane, are wind perturbation has ended, 0 x 3. Equatorial wave-like anomalies forced by slowly ut þ g hx À byv ¼ t =rD À asu ½1 varying periodic or quasi-periodic winds reflecting 0 y ocean adjustment on interannual timescales. Even vt þ g hy þ byu ¼ t =rD À asv ½2 though these anomalies can be represented mathematically as a superposition of continuously ht þ Hðux þ vyÞ¼ash ½3 forced Kelvin and Rossby waves of different modes, the properties of the superposition (such These equations have been linearized, and pertur- as the propagation speed) can be rather different bations with respect to the mean state are con- from the properties of free waves. sidered. Variations in the mean east–west slope of the 3682 EQUATORIAL WAVES describes a flow that with different velocities in two H1=H h(x,y,t) layers, barotropic flow does not depend on the ver- tical coordinate). This approximation filters out higher-order baroclinic modes with more elaborate vertical structure. For instance, the equatorial +Δ H2 undercurrent (EUC) is absent in this model. Obser- vations and numerical calculations show that to represent the full vertical structure of the currents z y and ocean response to winds correctly, both the first- x and second-baroclinic modes are necessary. Never- 1 theless, the shallow-water equations within the 1 2- layer approximation remain very successful and are Figure 5 A sketch of the 1 1-layer shallow-water system with 2 used broadly in the famous Cane-Zebiak model of the rigid-lid approximation. H1/H2{1. The mean depth of the thermocline is H. The x-axis is directed to the east along the ENSO and its numerous modifications. equator. The y-axis is directed toward the North Pole. The mean For many applications, the shallow-water equa- east–west thermocline slope along the equator is neglected. tions are further simplified to filter out short waves: in the long-wave approximation the second mo- thermocline and mean zonal currents are neglected. mentum equation (eqn [2]) is replaced with a simple The notations are conventional (some are shown in geostrophic balance: Figure 5), with u, v denoting the ocean zonal and meridional currents, H the mean depth of the 0 g hy þ byu ¼ 0 ½4 thermocline, h thermocline depth anomalies, tx and y t the zonal and meridional components of the wind The boundary condition at the western boundary is stress, r mean water density, Dr the difference be- ÀÁthenR replaced with the no-net flow requirement tween the density of the upper (warm) layer and the udy 0 .