1496 VOLUME 13J O U R N A L O F C L I M A T E

q 2000 American Meteorological Society

The Energetics of El Nino and La Nina

LISA GODDARD* AND S. GEORGE PHILANDER

Program in Atmospheric and Oceanic Sciences, Princeton University, Princeton, New Jersey

(Manuscript received 13 January 1998, in final form 4 August 1999)

ABSTRACT

Data from a realistic model of the ocean, forced with observed atmospheric conditions for the period 1953–92, are analyzed to determine the energetics of interannual variability in the tropical Pacific. The work done bythe winds on the ocean, rather than generating kinetic energy, does work against pressure gradients and generatesbuoyancy power, which in turn is responsible for the rate of change of available potential energy (APE). Thismeans interannual fluctuations in work done by the wind have a phase that leads variations in APE. Variationsin the sea surface temperature (SST) of the eastern equatorial Pacific and in APE are highly correlated and inphase so that changes in the work done by the wind are precursors of El Nino. The wind does positive workon the ocean during the half cycle that starts with the peak of El Nino and continues into La Nina; it doesnegative work during the remaining half cycle.

The results corroborate the delayed oscillator mechanism that qualitatively describes the deterministic behaviorof ENSO. In that paradigm, a thermocline perturbation appearing in the western equatorial Pacific affects thetransition from one phase of ENSO to the next when that perturbation arrives in the eastern equatorial Pacificwhere it influences SST. The analysis of energetics indicates that the transition starts earlier, during La Nina,when the perturbation is still in the far western equatorial Pacific. Although the perturbation at that stage affectsthe thermal structure mainly in the thermocline, at depth, the associated currents are manifest at the surface andimmediately affect work done by the wind. For the simulation presented here, the change in energy resultingfrom adjustment processes far outweighs that due to stochastic processes, such as intraseasonal wind bursts, atleast during periods of successive El Nino and La Nina events.

1. Introduction

Although the dynamics of El Nino (and La Nina) havebeen studied extensively [see, e.g., Neelin et al. (1998)and references therein], little attention has been paid tothe energetics of interannual variability in the tropicalPacific Ocean. The few brief discussions of El Ninoenergetics (Yamagata 1985; Hirst 1986) have only ad-dressed the event growth. We present here the energeticsof the full quasiperiodic fluctuation between El Ninoand La Nina states, and the identification of air–seainteraction important to the evolution of El Nino–LaNina that has not been fully appreciated by previousstudies.

Earlier studies treated El Nino as an episodic phe-nomenon. Precursors were sought: Cane and Zebiak

* Current affiliation: Experimental Climate Forecast Group, Inter-national Research Institute for Climate Prediction, Lamont–DohertyEarth Observatory of Columbia University, Palisades, New York.

Corresponding author address: Dr. Lisa Goddard, InternationalResearch Institute for Climate Prediction, Lamont–Doherty Earth Ob-servatory of Columbia University, 61 Route 9W, Palisades, NY10964.E-mail: goddard@iri.ldgo.columbia.edu

(1985) pointed to anomalously high upper-ocean heatcontent, and Wyrtki (1975) watched for persistentlystrong trade winds that would inexplicably relax. Littleattention was paid to La Nina, which was described asan ‘‘overshoot’’ of El Nino conditions as the PacificOcean adjusted back toward normal (Cane and Zebiak1985). However, the indices used to identify El Ninoepisodes exhibit a distinct spectral peak near 4 yr (e.g.,Rasmusson and Carpenter 1982; Jiang et al. 1995), sug-gesting that the phenomenon is part of a continuousoscillation, albeit an irregular one.

Since the 1980s the commonly accepted view of thiscoupled air–sea variability is that of a quasiperiodic os-cillation between cold and warm extremes, for whichthe delayed oscillator (Schopf and Suarez 1988) hasbecome the dynamical paradigm. In the delayed oscil-lator, as originally described by Suarez and Schopf(1988) and Schopf and Suarez (1988), SST anomaliesin the eastern Pacific associated with either La Nina orEl Nino initiate wind stress anomalies in the centralPacific that feed back positively to the SST anomaliesmainly through wind-forced changes in the local, east-ern equatorial, thermocline depth. The wind stressanomalies also induce thermocline anomalies in thewestern Pacific of the opposite sign to those in the eastthat are uncoupled from the surface and that are ex-

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pected to adjust in the tropical Pacific basin as Rossbyand Kelvin waves. Thus, when the adjusting Rossbywaves reflect from the western boundary as Kelvinwaves and arrive in the east, they terminate the currentextreme phase and initiate growth of the opposite phase.In short, SST anomalies in the east are influenced pos-itively by local processes acting now and negatively bynonlocal processes that will not be realized until aftersome delay.

In spite of its general acceptance as a conceptual mod-el, considerable debate has arisen over the degree towhich the delayed oscillator operates in nature. Criticsuse nature’s poor agreement with the details of this sim-ple model to question the deterministic nature of the ElNino–La Nina cycle. Li and Clarke (1994) showed thatalthough equatorial wind stress anomalies are highlycorrelated with ocean signals arriving at the westernboundary a few months later, those western Pacificocean signals are only weakly correlated with the anom-alous wind stress 12–18 months later. However, Mantuaand Battisti (1994) point out that such a low correlationwould result if the delayed signals were efficient at ter-minating the current event but less so at initiating thenext one.

We adopt a loose interpretation of the delayed oscil-lator, allowing for reconciliation with other studies thatsuggest the delayed oscillator is not a good model forthe El Nino –La Nina cycle. For example, Kessler et al.(1995) and McPhaden (1999) point to the importanceof atmospheric intraseasonal [Madden–Julian oscillation(MJO)] variability in forcing equatorial Kelvin wavesthat played an important role in El Nino events of the1990s. Also, Picaut and Delcroix (1995) and Picaut etal. (1997), although they do not wholly discount thedelayed oscillator, call for substantial modification of it,by giving primary importance to zonal SST advection,particularly of the 288C isotherm in the western–centralPacific, in the initiation of El Nino events. These resultsare not necessarily at odds with the delayed oscillatormechanism; they merely imply that SST can change bymore varied processes than upwelling on the anomaloustemperature gradient of remotely forced thermoclineanomalies. Surely, varying combinations of these pro-cesses lead to the uniqueness of each event’s evolution.Our results are not inconsistent with these studies, butwe do maintain that the ocean’s memory of previousair–sea interaction exerts a strong influence on futurevariability. We show that during periods of active in-terannual variability when cold events follow warmevents, and so on, the delayed oscillator is responsiblefor the deterministic nature of the variability. However,the delayed oscillator does run out of energy eventually,at which time stochastic forcing, such as the MJO, maybe the only possible mechanism.

This study examines the energy balances as well asthe temporal and spatial structures of the energeticsterms important to interannual variability in the tropicalPacific Ocean. We focus on the ocean component of the

coupled air–sea system because of its large inertia andthus potential memory of previous air–sea interaction.Hirst (1986) recognized that the ultimate energy sourcefor a growing event comes from latent heating of theatmosphere, and that the growing energy of the atmo-sphere feeds energy into the ocean. The subsequent im-portance of energy gained by the ocean on evolution ofthe current event and genesis of the next event lies atthe heart of the delayed oscillator debate.

Following our energetics analysis, one can comparethe change in oceanic energy resulting from the redis-tribution of previously acquired energy with the changein oceanic energy due to the growth of new perturba-tions. What is evident in our results is that the adjustingperturbations influence the basinwide energy throughair–sea interaction long before their associated temper-ature anomaly is realized at the surface. This again isnot inconsistent with the delayed oscillator mechanism,which describes the ocean current anomaly of the ad-justing thermocline perturbations (eastward for a warm,downwelling Kelvin signal). However, the original vi-sion of the adjusting ocean signals was that they werecompletely uncoupled from the surface until arriving inthe east, where the depth anomaly of the thermoclinewould be translated into SST anomalies via equatorialupwelling.

The results presented in this paper are based on datafrom an OGCM forced with observed atmospheric data.In section 2 we describe the model configuration anddetails of the simulation, which covers the period 1953–92. The kinetic and available potential energies of thesimulated interannual fluctuations, and the terms in theenergetics equations responsible for the fluctuations, aredescribed first in section 3. Section 3 then focuses onthe spatial and temporal details of the energetics for aspecific case study: the period 1970–75, during whicha full cycle, from La Nina to El Nino and back to LaNina, was completed. Section 4 summarizes the results.

2. The model

The model used for this analysis is the modular oceanmodel (MOM1) (Pacanowski et al. 1991) produced atGeophysical Fluid Dynamics Laboratory and based onthe primitive-equation ocean model, developed by Bry-an and Cox (1967), and described by Cox (1984). Thedomain is the Pacific Ocean between 458S and 558N,bounded by walls on the poleward edges of the mid-latitude gyres and by realistic coastlines to the east andwest, and realistic bottom topography is given by a mod-erately smoothed version of the Scripps topography. Themodel resolution is 18 in longitude and variable in lat-itude, with ⅓8 resolution within 108 of the equator, inorder to resolve equatorial waves; the latitudinal reso-lution increases to 18 between 108 and 308N and S, thenremains 18 poleward of 308. There are 27 levels in thevertical with 10 in the top 100 m to better resolve thethermocline, and the maximum depth is 3830 m.

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The tracer fields have zero flux through the lateraland bottom boundaries. Instead, sponge layers placedwithin the northern and southernmost 108 of the domaindamp the model temperature and salinity fields towardLevitus climatology using a latitude-dependent time-scale of (2–40 days)21. The model calculates a full sur-face heat flux based on observed atmospheric conditionsand model-diagnosed oceanic conditions, as describedby Rosati and Miyakoda (1988). The source term forthe salinity field is a linear damping of the sea surfacesalinity to Levitus (1982) climatology at the rate of (120days)21.

For vertical mixing of both tracers and momentum,the Pacanowski and Philander (1981) scheme was cho-sen. This Richardson number-dependent method resultsin less mixing for more stable local conditions. Thesurface momentum flux is given by the Pacanowskiwind stress (Pacanowski 1987) parameterization, whichuses the bulk aerodynamic formula, taking the surfacewind vectors relative to the velocity of the ocean sur-face. This formulation is appropriate to the Tropics andespecially near the equator where the surface currentscan exceed speeds of 1 m s21 (Gill 1983; Richardsonand McKee 1984). A complete description of the modelsetup and parameters chosen for the simulation pre-sented here are given in Goddard (1995).

Initially, the model was assigned zero currents andgiven temperature and salinity fields from Levitusmonthly climatology. The atmospheric data—monthlymean surface winds, air temperature, relative humidity,and cloudiness—applied to the OGCM are from theComprehensive Ocean–Atmosphere Data Set (COADS)Release 1 (Slutz et al. 1985) and Release 1a (updatedpost-1979 data). Surface conditions from the COADSclimatology, based on the period 1951–79, were appliedfor 5 yr to spin up the ocean. Beginning with year 1952,the full (annual cycle plus interannual perturbation)COADS atmosphere was applied. The model has beenintegrated for 40 yr, 1953–92, plus spinup. The monthlyclimatology of the model is based on the model years1953–79, the approximate climatology period definedin the COADS data.

In order to establish the realism of the OGCM sim-ulation, we compare the simulated and observed oceanfields. COADS contains sea surface temperature (SST)data, but this is not used in forcing the model. BecauseSST anomalies lead air temperature anomalies inter-annually (e.g., Battisti 1988), the observed air temper-ature used in the model’s calculation of surface heatfluxes is not sufficient to produce the simulated SSTanomalies. Therefore, comparing the OGCM’s SST withCOADS SST tests the consistency of the model withits forcing data. As shown in Fig. 1a, the correlation isr 5 0.83 between the observed and simulated SSTanomaly in Nino-3 (58S–58N; 908–1508W) after smooth-ing with a 3-month running mean filter (r 5 0.77 formonthly means, with no additional smoothing), wellabove the 99% confidence level for significance. The

model fails to capture the full magnitude of warmevents, a problem that is likely due to a thermoclinethat is somewhat more diffuse than observed. The modelalso exhibits occasional brief cold anomalies not presentin the observations, such as after the 1965 El Nino andbefore the 1982/83 El Nino. The phasing of the SSTvariability is correct, however, and in general, the modelis able to reproduce the magnitude of warm and coldevents. The map of anomaly correlations (Fig. 1b) basedon 39 yr of monthly mean SST anomalies shows sta-tistically significant agreement, exceeding 99% confi-dence where shaded, between the observations and sim-ulations for most of the tropical Pacific region, exceptin the western Pacific where the variance of SST anom-alies is low (Evans et al. 1998), perhaps below the levelof observational error. Also, cloud forcing, which playsan important role in the surface heat fluxes over thewestern Pacific, was crudely prescribed in our simula-tion and may be partly to blame.

The model produces a discernible thermocline (Fig.1c). However, like many ocean models that do not in-corporate subsurface observations, the thermocline dif-fuses slowly as the integration proceeds. COADS doesnot contain any subsurface data with which to comparethe simulation. Thus a monthly averaged ‘‘snapshot’’ isprovided from the brief period of overlap between Trop-ical Ocean and Global Atmosphere (TOGA)–TropicalAtmosphere–Ocean (TAO) array buoy data and our sim-ulation. Even at nearly 40 model years into the inte-gration, the 208C isotherm representing the core of thethermocline agrees well with the observed data alongthe equator (Fig. 1c), although the vertical gradient oftemperature around 208C is weaker than observed. Ear-lier, such as during the 1970–75 period to be examinedlater as a case study, the thermocline is tighter (notshown) although still not to the degree seen in recentTOGA–TAO data.

The OGCM also reproduces reasonable climatologi-cal mean oceanic fields. [See Goddard (1995) for figuresof the time mean simulated thermal and dynamic fields.]The model has an equatorial undercurrent, that reachesa maximum time mean speed of 80 cm s21 and variesseasonally in agreement with the literature (Philanderet al. 1987). The climatological vertical velocity fieldsindicate strong upwelling along most of the equator, anddownwelling off the equator with the strongest down-welling areas in the central–western Pacific.

3. Results

a. Available potential energy

As will be shown throughout this section, the mostrelevant energy quantity to El Nino–La Nina is the grav-itational available potential energy (APE). The gravi-tational APE measures the energy potentially availableto the system from a horizontal redistribution of mass(Lorenz 1955). Thus the portion of the oceanic mass

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contributing to the APE is defined relative to a referencestate by separating out the time mean component of thepotential density that is hydrostatically balanced andvarying only with depth over the tropical Pacific region.That is,

r(x, y, z; t) 5 1 .r(z) r(x, y, z; t) (1)

Following this notation, represents the referencer(z)state of the mass field, and represents ther(x, y, z; t)perturbations to that mass field including the structureof the mean state as well as interannual variability.Quantitative values for APE are then calculated follow-ing the formulation of Oort et al. (1989),

21 rA 5 2 . (2)

22 N

This equation describes energy contained in the verticalperturbations to the potential density field, by using thedepth-dependent stability factor, N 2 5 /g, to weight2rz

the density variance. However, for purposes of discus-sion, an alternate but more conceptual form is adopted:

2h 1 22A 5 5 (h 1 2hh9 1 h9 ). (3)

2 22S 2S

This formulation is equivalent to Eq. (2), applied to anisopycnal or shallow-water model. Here the depth dis-placement h (positive for downward displacements) ofa constant density surface from its mean depth H sub-stitutes for the vertical perturbations to the density field;and S 2 [5 H/(gredr0)] accounts for the gravitational sta-bility, where gred is the reduced gravity, and r0 is thebackground density. The mass field in Eq. (3) has beenfurther decomposed into a mean component h and aperturbation component h9. Thus the constituent termsin Eq. (3) are interpreted as mean state energy (Amm),mean perturbation energy (Amp), and perturbation energy(App).

Figure 2 illustrates how changes in the ocean’s massfield, manifested as changes in thermocline slope alongthe equator, contribute to these terms during El Ninoand La Nina events. The mean state energy, Amm, isnecessarily positive, resulting from the mean east–westslope of the thermocline maintained by the mean zonalwinds. The perturbation energy, App is also always pos-itive since it is proportional to the interannual varianceof the mass field. The sign of Amp, on the other hand,depends on the placement of the interannual perturba-tions relative to the mean state perturbations. When SSTis anomalously warm in the eastern equatorial Pacific,as during El Nino, the cause and consequence is a deeperthan average thermocline in the east (i.e., h9 . 0, h ,0), which contributes to a flatter mass field and con-sequently less oceanic APE. The opposite is true for LaNina.

The importance of the mean state to the character ofEl Nino and La Nina is well recognized (e.g., Battistiand Hirst 1989; Neelin et al. 1998). The structure of the

mean state, including the air–sea interaction that estab-lishes it (Dijkstra and Neelin 1995), is why deep (shal-low) thermocline perturbations in the eastern equatorialPacific result in warm (cold) SST anomalies. Thus itcomes as no surprise that the mean state is also centralto the interannual energetics. If there were no structurein the mean state (i.e., if the mean mass field werehorizontally leveled), the location of thermocline per-turbations would not matter, and Amp would be zero. Thediscussion will therefore highlight both the generationof perturbations (perturbation energy) and their adjust-ment against the mean state (mean perturbation energy),and how these processes relate to what is already knownabout El Nino and La Nina.

b. The energetics of interannual variability

Interannually, APE far outweighs kinetic energy (KE)for both perturbation energy (Fig. 3a) and mean per-turbation energy (Fig. 3b). On large space and time-scales it is not surprising that the APE should dominate.Averaged over the tropical Pacific (from 158S to 158N),the APE of the mean state is about 15 times greaterthan the KE, and it is still 7 times greater than the KEfrom 58S to 58N, where the zonal flow and thus the KEdensity is greatest (Goddard 1995). Furthermore, An-derson and Moore (1989) showed that although the en-ergy of free Kelvin waves is equipartitioned betweenKE and APE, that of Rossby waves is stored almostentirely as APE. Therefore, one should expect APE toplay a more significant role in the energetics of El Nino–La Nina. However, it was not expected that the KEshould constitute only a few percent of the anomalousenergy in the tropical Pacific.

As evidenced by Fig. 4, APE correlates highly withthe SST variability that defines the phase and amplitudeof El Nino and La Nina. Thus, the SST anomalies as-sociated with El Nino and La Nina are just the surfacemanifestation of a change in oceanic energy occurringthroughout the entire upper ocean in the tropical Pacific.Averaged over a narrow equatorial zone, Amp varies inphase with the Nino-3 SST anomaly [i.e., the anomalousSST averaged over the Nino-3 region (58S–58N; 1508–908W), hereafter referred to as SSTa], shown in Figs.4a,c. Thermocline changes in the eastern–central Pacificare associated with local SST changes [e.g., McPhadenet al. (1998) and references therein], and these ther-mocline changes constitute the majority of Amp near theequator since the mean thermocline depth in the westernequatorial Pacific is close to the (158S–158N) basin av-erage. Because these thermocline changes in the eastoccur as part of a basinwide variation, which is largelyin balance with the trade winds, it has proved difficultto separate out the portion of the subsurface variabilitythat is not in balance with the winds, since the adjust-ment of mass along the equator happens at relativelyshort timescales (Neelin 1991). Considering the largertropical region (158S–158N), Amp still correlates highly

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FIG. 1. (a) Time series of SST anomaly averaged over Nino-3 (58S–58N; 1508–908W) for GCM simulation (solid line) and COADSobservations (dashed line), correlation between time series: r 5 0.83. Data have been smoothed with 3-month running mean filter (r 5 0.77for unfiltered data). (b) Correlation map of GCM vs COADS anomalous monthly SST. Shading indicates statistically significant correlation

with SSTa (Fig. 4b) but now lags it by approximately3 months (Fig. 4c), as seen in observations of sea level(McPhaden et al. 1998). Thus, once the warm (cold)event is underway, more energy is lost (gained) by theocean off the equator, in the western Pacific (as will beshown later), in response to the anomalous winds as-sociated with the event.

The in-phase relationship between SSTa and Amp im-plies that the rate of change of the basinwide Amp will

lead SSTa. Balances of the energy equations are usedto investigate whether there is a dominant term con-tributing to this change in APE, one that might be usefulto monitor to anticipate future evolution of El Nino andLa Nina events. The full form of the energy equationsderived from the OGCM primitive equations is pre-sented in appendix A. Here, only the distilled energyequations are discussed, those obtained from integratingthe monthly mean energies over the tropical Pacific from

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FIG. 1. (Continued ) at the 99% confidence level. (c) Snapshot of longitude–depth thermal structure along equator for the month of Jun1991 comparing total temperature field from GCM (top) to total temperatures from the TOGA–TAO observational buoy array (bottom).

158S to 158N, and to a depth of about 300 m. Thedistilled equations will not always represent completebalance, but certainly the majority of it on this scale.Other terms in the full equations may yield substantialcontributions locally that cancel when averaged over thelarger domain.

The primary balance seen in the APE equation is

]A dV 5 g rw dV, (4)EEE EEE]t

where g is gravity, and w is the vertical component ofthe oceanic velocity. The left-hand side is the rate of

change of the volume integrated APE, and the right-hand side is the vertical motion of the mass field, herecalled ‘‘buoyancy power’’ [52hw, in the isopycnal no-tation of Eq. (3)]. This simple equation represents themain balance of energy for both the perturbation energy(Fig. 5a) and the mean perturbation energy (Fig. 6a).However, there are no external source or sink terms inEq. (4), which merely describes a redistribution of mass.Heat fluxes through the surface, and flux convergencethrough the boundaries of the domain, are too small toaffect the APE significantly and therefore do not appearin Eq. (4). Only on a few occasions, toward the end of

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FIG. 2. Schematic of APE concept and how decomposition of APEis viewed. The top picture illustrates the mean state. Here h refersto the vertical deviation of the uppermost density surface from itshorizontally averaged depth and is defined to be positive downward(note: h is therefore negative in the eastern Pacific, at the locationindicated); h represents the mean climatological state and is the samefor each sketch. Here h9 is the interannual perturbation to h—0 forthe mean state, negative (positive) for La Nina (El Nino), in theeastern Pacific.

some large events, does a sizable amount of energy flowthrough the eastern boundary of the integration domain.

The simplicity of Eq. (4) is its appeal. Obviously, onecould examine, instead, the rate of change of SSTa, butin that case many processes are involved, such as up-welling in the presence of changing thermocline depth(e.g., Philander et al. 1984), anomalous zonal advection(e.g., Picaut and Delcroix 1995), and changes in air–sea heat fluxes. The weakness of Eq. (4) is that up-welling is difficult to calculate from observations. How-ever, data from high quality OGCMs forced with ob-served winds may be used as a surrogate (OCGMs thatassimilate subsurface data lack consistency betweentheir dynamical and thermodynamical fields).

The source term for the dynamical energy is foundin the KE equation, where most terms in the full equa-tion (A3) are again negligible on this basin-averagedscale. Thus, the primary balance of KE in the tropicalPacific Ocean on interannual timescales distills to

0 ø vt ds 2 ( p 1 p )u · n dsEE o R s

z50

2 g rw dV, (5)EEEwhere v represents the horizontal surface currents, t o

is the surface wind stress vector, p is the portion of theoceanic pressure field not in hydrostatic balance withthe reference density field [see Eq. (1)], ps is the at-mospheric surface pressure, and u is the three-dimen-sional oceanic velocity field. The first term on the right-hand side represents the work delivered by the atmo-sphere to the ocean, the second term is the work doneagainst internal and surface pressure gradients by theageostrophic flow, and the third term quantifies the workgiven to vertical motion of the mass field. As was seenin Eq. (4), the buoyancy power appears, but now withopposite sign, representing a conversion between KEand APE.

The balance shown in Eq. (5) is the energetics coun-terpart of the well-known momentum balance in whichthe winds primarily maintain pressure gradients. Usingthe KE equation instead of the momentum equations,however, it becomes possible to separate out the workdone against pressure gradients, which is a relativelylarge quantity near the equator where the trade windsmaintain the strong east–west slope in the thermocline.This leaves the buoyancy power, which quantifies thecreation of thermocline perturbations through the hor-izontal convergence and divergence of the mass field.Using the kinetic energy equation also yields the explicitimpact of the dynamical air–sea interaction by couplingthe wind stress and ocean currents.

Whereas the momentum equations tell us that, at lowfrequencies, the winds maintain pressure gradients, theenergy equations indicate that the wind, by doing work

on the ocean, creates APE. Eqs. (4) and (5), written interms of the perturbation energy, describe the creationand destruction of thermocline perturbations (App). Asshown in Fig. 5 (note that the buoyancy power is plottedin Figs. 5a,b with the sign as it appears in the APEequation), positive perturbation wind power,

Wpp } u9t9, (6)

contributes positively to the perturbation buoyancy,

Bpp } 2h9w9, (7)

generating thermocline perturbations and increasing App.Thus oceanic energy grows while the Wpp is positive, con-sistent with the simple energetics analysis of Hirst (1986).Figure 5b further illustrates why the ‘‘tropical Pacific’’region was chosen to extend to 158S–158N; over this do-main, the rate of ageostrophic pressure work is alwaysseen as a sink of energy. Thus wind power is the onlysource of energy for the volume. The energy gained fromWpp eventually will be radiated out of the volume throughthe pressure power or dissipated by wave diffusion.

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FIG. 3. (a) Perturbation available potential energy (heavy line) av-eraged over tropical Pacific Ocean (158S–158N; 1508E–1008W; 30–280 m) and perturbation kinetic energy (light line) averaged overtropical Pacific Ocean (158S–158N; 1508E–1008W; 0–280 m). (b)Same as in (a), except for the mean perturbation energy densities.

Once the perturbations are created, they can adjust with-in the basin, via equatorial wave dynamics. The signatureof their adjustment can be seen in the mean perturbationenergetics. The anomalous fields generated and/or main-tained by Wpp do interact with the mean fields, and it isthe temporal and spatial characteristics of this interactionthat is meaningful for interannual variability in the tropicalPacific Ocean. As seen for the perturbation energetics, thetime series for the mean perturbation energetics (Fig. 6)show a direct relationship between the mean perturbationwind power (Wmp) and the mean perturbation buoyancypower (Bmp). Both of these terms therefore relate directlyto changes in Amp, and because of the close agreementbetween Amp and SSTa, these terms may prove to be usefulprecursors to the termination and initiation of El Nino andLa Nina events.

Comparing the mean perturbation energetics terms,Bmp and Wmp to their dynamics counterparts, upper-oceanheat content, and zonal wind stress (respectively), onefinds that the energetics are indeed better indicators offuture variability. Figure 7a shows the relationship be-tween SSTa and Bmp. The two time series exhibit peaksof similar shape and magnitude, with Bmp leading byapproximately 3 months (Fig. 7c). The anomalous up-

per-ocean heat content (e.g., Cane and Zebiak 1985),equivalent to the mass anomaly, also shows agreementwith the SSTa (Fig. 7b), and the correlation betweenthem does have a peak at 3-month lead. However, thecorrelation of SSTa with Bmp is significantly larger atthis lead time. The mass anomaly actually correlates toSSTa slightly better at lag times of about 12 months(although the difference is not significant), suggestingthat anomalous heat content is more a reaction to theEl Nino–La Nina event than a cause.

As was just shown for the buoyancy power, Wmp rep-resents a more reliable precursor of El Nino–La Ninaevolution than does the zonal wind stress anomaly (t x9)originally proposed by Wyrtki (1975). The correlationbetween the dynamical variables peaks at r 5 0.4 whent x9 is in phase with SSTa (Figs. 8b,c). The negativecorrelation Wyrtki referred to, where stronger easterliesprecede warm SSTa by 1–2 yr, is weak and insignificant.To his credit, he does acknowledge that not all El Ninosfollow such sustained periods of intensified trade windsas occurred before the 1957/58 and 1972/73 El Ninos(Wyrtki 1975).

The Wmp time series correlates with SSTa at betterthan r 5 0.4 for lead times up to 9 months, on average(Figs. 8a,c). Eventually the correlation peaks at r 5 0.7with Wmp leading SSTa by about 2 months. It should benoted that Wyrtki’s region of interest was restricted to48S–48N and 1808–1408W, and in that case he obtaineda much stronger relationship between t x9 and SSTa, butover that region the correlation for Wmp increases sig-nificantly too (r 5 0.9) and still exceeds that of the windstress anomaly. The peaks in the time-lagged correla-tions of both Figs. 7 and 8 at lead/lag times greater than18 months merely reflect the quasiperiodic nature of ElNino–La Nina and are therefore not useful as precursors.

c. Case study: 1970–75

With the energetics of interannual variability presentedand its relevance to the El Nino–La Nina cycle demon-strated, we now look in detail at a complete cycle ofvariability in the tropical Pacific. The years 1970 to 1975saw a large-amplitude El Nino event in 1972, precededand followed by La Nina events. The time series for SSTa,Amp, and App (Fig. 9) exemplifies much of the previousdiscussion for a clearly identified sequence of El Nino andLa Nina events: first, Amp has a strong negative correlationwith SSTa; and second, App grows during the mature phaseof an El Nino or La Nina event. Note, that the magnitudeof Amp is comparable to, but generally larger than, that ofApp, such that the total anomalous oceanic APE is positiveduring La Nina and negative during El Nino, as was il-lustrated schematically in Fig. 2.

By examining the timing, location, and processes bywhich the ocean gains energy and redistributes it, the-ories or paradigms such as the delayed oscillator can betested. Treating first the perturbation energy, a series ofmaps are presented at selected points through the 1970–

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FIG. 4. (a) Mean perturbation APE averaged over equatorial Pacific Ocean (58S–58N; 1508E–1008W; 30–280 m), andSST anomaly averaged over Nino-3 region (58S–58N; 1508–908W). (b) Same as (a), except energy is averaged over(158S–158N; 1508E–1008W; 30–280 m). (c) Time-lag correlations between SSTa and Amp; SSTa lags Amp for positivelag values. (Solid line: Amp averaged 58S–58N; dashed line: Amp averaged 158S–158N.)

1 MAY 2000 1505G O D D A R D A N D P H I L A N D E R

FIG. 5. (a) Primary balance of perturbation APE equation, averaged over subsurface tropical Pacific Ocean (158S–158N; 1508E–1008W;30–280 m), showing rate of change of perturbation APE (light line) and perturbation buoyancy power, Bpp (heavy line). (b) Primary balanceof perturbation KE equation, averaged over tropical Pacific Ocean (158S–158N; 1508E–1008W; 0–280 m), showing rate of perturbation windpower, Wpp (light line); perturbation buoyancy power, Bpp (heavy solid line); and perturbation pressure power, Ppp (heavy dashed line).

75 period indicated by the 3s on Fig. 9a. The previousdiscussion has shown that energy is gained during themature phase of the event and provided by the windpower [as theorized by Yamagata (1985) and Hirst(1986)]. The structure of Bpp generated by Wpp agreeswith the simple models that show thermocline reactionto zonal wind anomalies in the central basin (e.g.,McCreary 1978). Thus, the large-scale easterly windanomaly in December 1970 accelerated westward sur-face currents and fed energy into the ocean (Fig. 10a,left). Positive Bpp occurred off the flanks of Wpp (Fig.10a, right), with anomalous upwelling and a shoalingthermocline on the equator to the east, and downwellingand a deepening thermocline off the equator to the west.All these features are common to the discussions of thedelayed oscillator (e.g., Schopf and Suarez 1988; Bat-tisti 1988). Although sizable cold SST anomalies con-tinued though 1971, the wind power did not (Fig. 10b,left). Thus, inconsistent with many theoretical and ideal-ized discussions of the delayed oscillator model, the windstress, and thus the wind power, are not smoothly varyingat the low frequency displayed in the SSTa time series.

As the wind power weakens or disappears, the portionof the mass field that is no longer balanced by the large-scale wind stress will adjust via equatorial wave dy-namics—westward off the equator (Rossby waves) andeastward on the equator (Kelvin waves). The Bpp mapsuggests movement of thermocline perturbations (Fig.10b, right), as perturbations gained between 1608E and1608W (Fig. 10a, right) propagate toward the westernboundary. Mere translation of perturbations will not af-fect the basinwide energy; as one region loses App, an-other gains it. However, when perturbations of oppositesign meet, such as when adjusting perturbations origi-nally from the western Pacific arrive in the eastern equa-torial Pacific, they can cancel each other, and net de-struction of App occurs. As adjusting thermocline per-turbations begin to arrive in the east during late 1971/early 1972, App decreases rapidly. During the early stageof development of the 1972/73 El Nino, some areas ofWpp appear (Fig. 10c, left), particularly in the westernPacific, but they do not contribute to the growth of App,which continues to decrease at this time. It appears thatsome App remains after the adjustment of perturbations

1506 VOLUME 13J O U R N A L O F C L I M A T E

FIG. 6. (a) Primary balance of mean perturbation APE equation, averaged over subsurface tropical Pacific Ocean (158S–158N; 1508E–1008W; 30–280 m), showing rate of change of mean perturbation APE (light line) and mean perturbation buoyancy power, Bmp (heavy line).(b) Primary balance of mean perturbation KE equation, averaged over tropical Pacific Ocean (158S–158N; 1508E–1008W; 0–280 m), showingrate of mean perturbation wind power, Wmp (light line) and mean perturbation buoyancy power, Bmp (heavy line).

generated during the 1970/71 La Nina (Fig. 9b), in theform of a deepened thermocline in the east (not shown),leading to warm SSTa and the start of the 1972/73 ElNino. At the peak of the 1972/73 El Nino App is againdelivered to the ocean (Figs. 9b, 10d,e), which is thendepleted during the transition to La Nina conditions inmid-1973 (Fig. 9b). And again, residual energy existsin the ocean as the 1973/74 La Nina begins to grow.Notice, at the end of the 1973/74 La Nina, App is nearzero, and another El Nino did not occur until 1976.

The important issue is to what degree the perturbationenergy gained during one phase (El Nino or La Nina)impacts future variability. It is now well accepted thatsubsurface ‘‘memory’’ (i.e., the perturbation energy) doesinfluence the subsequent evolution of the tropical Pacific[see Neelin et al. (1998) and references therein]. However,recent works (Kessler et al. 1995; McPhaden 1999) havesuggested a more important role for nondeterministic forc-ing, such as intraseasonal westerly wind bursts and theMadden–Julian oscillation. The relative contributions ofall these processes is certain to vary from one event to

the next. An examination of the mean perturbation ener-getics potentially can ascertain which mechanism domi-nates, in general or for a specific period.

The mean perturbation buoyancy power, Bmp, con-tributes directly to changes in Amp [Eq. (4); Fig. 6],which mirror changes in SSTa (Figs. 4 and 9). Whendecomposed, it is evident that Bmp is affected by twoprocesses:

Bmp } 2h9w 2 hw9, (8)

where, as described in Fig. 2, h is defined positive down-ward. The adjustment of existing thermocline perturba-tions dominates the first term on the right, which will becalled ‘‘adjustment buoyancy.’’ The spontaneous actionof the wind initiating areas of anomalous upwelling dom-inates the second term, which will be called ‘‘wind-drivenbuoyancy.’’ Figure 11 illustrates schematically how boththese terms contribute to changes in Amp for the case ofa shallow, upwelling perturbation freely propagating east-ward along the equatorial thermocline. Assuming that w9, w and h9 ø h, h9w will monopolize Bmp in the ad-

1 MAY 2000 1507G O D D A R D A N D P H I L A N D E R

FIG. 7. Mean perturbation buoyancy power Bmp averaged over equatorial Pacific Ocean (58S–58N; 1508E–1008W; 30–280 m), and SST anomaly averaged over Nino-3 region (58S–58N; 1508–908W). (b) Anomalous mass (r 5 2aT 1 bS)integrated over equatorial Pacific Ocean (58S–58N; 1508E–1008W; 30–280 m), and SST anomaly averaged over Nino-3region (58S–58N; 1508–908W). (c) Time-lag correlations between SSTa and Amp, and SSTa and near-equatorial massanomaly. SSTa lags the other quantities for positive lag values. Also shown are the time-lag correlations between SSTaand the two terms that compose Bmp: gh9w (light solid line) and ghw9 (light dashed line).

1508 VOLUME 13J O U R N A L O F C L I M A T E

FIG. 8. (a) Mean perturbation wind power Wmp averaged over equatorial Pacific Ocean (58S–58N; 1508E–1008W;0–280 m), and SST anomaly averaged over Nino-3 region (58S–58N; 1508–908W). (b) Anomalous zonal wind stressaveraged over equatorial Pacific Ocean (58S–58N; 1508E–1008W), and SST anomaly averaged over Nino-3 region (58S–58N; 1508–908W). (c) Time-lag correlation between SSTa and Wmp and SSTa and t x9. SSTa lags the other quantities forpositive lag values. Also shown are the time-lag correlations between SSTa and the two terms that compose Wmp: u9t x

(light solid line) and ut x9 (light dashed line).

1 MAY 2000 1509G O D D A R D A N D P H I L A N D E R

FIG. 9. (a) Amp } hh9 averaged over tropical Pacific subsurfaceocean (158S–158N; 1508E–1008W; 30–280 m), and SST anomaly av-eraged over Nino-3 region. (b) Perturbation APE density averaged oversubsurface tropical Pacific (158S–158N; 1508E–1008W; 30–280 m).

justment process depicted in Fig. 11. However, when theanomalous wind forcing is strong, hw9 will contributesubstantially; because of the typical spatial distributionof Bpp seen in Fig. 10, anomalous upwelling will oftenbe generated in areas where mean upwelling is weak (e.g.,off equator) or where h is large (e.g., eastern equatorialPacific or off-equatorial western Pacific).

The time series of both the adjustment buoyancy andthe wind-driven buoyancy are presented in Fig. 12b for1970–75. As suggested above, the adjustment buoyancydisplays the low-frequency character of slowly adjustingperturbations. Maps of this term through the 1970–75period (not shown) exhibit strong values usually confinednear the equator, where the mean upwelling is strong,and often extending from the date line to the easternPacific. Also, the lead time of strongest correlation be-tween 2h9w (Fig. 7c, thin solid line) is 3 months, con-sistent with equatorial wave theory that predicts a similartimescale for equatorial Kelvin waves to propagate fromthe western to eastern boundary. Thus, the temporal be-havior and spatial structure of the adjustment buoyancypower fit cleanly within the delayed oscillator paradigm,where mass perturbations adjusting from west to eastalong the equatorial thermocline lead initially to the de-cay of the current event and eventually to the genesis ofthe subsequent event. The wind-driven buoyancy acts ona much higher temporal frequency (Fig. 12b) and spatialfrequency (not shown), as was the case of the wind-drivenperturbations in Fig. 10. During much of the time series,these two terms contribute constructively to Bmp. How-

ever, as was shown in Fig. 7c (thin dashed line), the wind-driven buoyancy generally participates only during theonset and peak of an event, the demise of the event beingcontrolled by the adjustment buoyancy.

The mean perturbation wind power Wmp similarly iscomprised of two terms with differing physical inter-pretations:

Wmp } u9t 1 ut9. (9)

Again, the first term on the right-hand side is associatedwith oceanic adjustment, and the second term is relatedto anomalous wind forcing. However, the distinctionbetween these terms is much greater than was the casefor Bmp. The partitioning of energy between these termscan be used, for example, to assess the importance ofadjusting thermocline perturbations relative to impor-tance of intraseasonal (‘‘wind burst’’) forcing in the on-set, growth, or decay of an event. Note, however, thatthese results are from an ocean simulation forced withmonthly mean winds and may not properly account forthe energy input due to the higher-frequency couplingbetween atmosphere and ocean from wind burst forcing.

During the 1970–75 period, u9t dominates the windpower (Fig. 12a). This is particularly true during thetransition and early growth phases of the El Nino andLa Nina events, when this term is actually in compe-tition with ut9. The anomalous currents associated withthe adjusting perturbations (eastward, for deep, down-welling thermocline perturbations) interact with themean wind stress to change the energetic state of theocean long before changes in SSTa nullify or reversethe wind stress anomaly. Even though wind bursts ofthe appropriate sign appear in the time series (Fig. 12),they are sporadic and generate only a small fraction ofthe change in energy delivered by u9t . As with theadjustment buoyancy, the adjustment wind power (Fig.13, left) acts on a much larger scale than does ut9 (Fig.13, right), and during the decay phase is focused nearthe equator. The appearance of anomalous zonal currentsdue to adjusting thermocline perturbations is also in-herent in the delayed oscillator theory; they are the cur-rent anomalies of the equatorial waves. Picaut and Del-croix (1995) have also heralded the importance of thesecurrents for shifting the edge of the western Pacificwarm pool. However, those discussions neglected theimpact of the dynamical air–sea interaction between thecurrent anomalies and mean wind stress that is seen toeffect the oceanic energy so strongly.

4. Summary

Changes in gravitational available potential energy(APE) dominate the anomalous energy of the tropicalPacific Ocean interannually. Moreover, these changes inAPE are highly anticorrelated and in phase with thechanges in sea surface temperature used to index ElNino and La Nina [anomalous SST averaged over theNino-3 region (58S–58N; 1508–908W)]. This implies that

1510 VOLUME 13J O U R N A L O F C L I M A T E

FIG. 10. Perturbation wind power Wpp calculated at surface. Positive values greater than 2 mW m22 are shaded. Contour intervalis 8 mW m22, starting at 62. (right) Perturbation buoyancy power Bpp averaged 30–280 m. Positive values greater than 1 mWm22 are shaded. Contour interval is 4 mW m22, starting at 61.

1 MAY 2000 1511G O D D A R D A N D P H I L A N D E R

FIG. 11. Schematic of adjustment buoyancy showing contributionsto buoyancy power due to movement of thermocline perturbation onmean state of equatorial Pacific. Mean state is represented as uniformupwelling w and a thermocline sloping upward from west to east h ,measured relative to its average depth H (basin assumed to extendfarther west than shown). Perturbation is moving eastward.

FIG. 12. (a) Decomposition of mean perturbation wind power av-eraged over tropical Pacific (158S–158N; 1508E–1008W). Wmp ø t u91 t9u (light solid line); t u9: adjustment wind power (heavy solidline); and t9u : mature event wind power (heavy dotted line). (b)Decomposition of mean perturbation buoyancy power averaged overtropical Pacific subsurface ocean (158S–158N; 1508E–1008W; 30–280m). Bmp ø 2ghw9 2 gh9w (light solid line); 2 gh9w : adjustmentbuoyancy power (heavy solid line); and 2ghw9: wind-generatedbuoyancy power (heavy dotted line).

the ocean gains energy as the air–sea system evolvesfrom El Nino to La Nina and loses energy in going fromLa Nina to El Nino.

The energy is separated into two parts for this anal-ysis, the perturbation energy and the mean perturbationenergy. The perturbation energy equations describe thelife cycle—effectively the generation and destruction ofperturbations to the mass field (i.e., thermocline per-turbations). The mean perturbation energy equations de-scribe how the newly generated perturbations evolveagainst the mean state. In both cases, the primary bal-ance in the energetics equations yields a simple one-to-one relationship between the evolution of APE and thebuoyancy power (2hw), which quantifies vertical mo-tion of the mass field. The ultimate energy source comesfrom the work done by the wind (ut) in the kineticenergy equation—which has little effect on the basin-wide kinetic energy but, instead, is delivered to buoy-ancy power, feeding the APE—and the work againstpressure gradients by the ageostrophic flow.

The energetics analysis presented for the 1970–75period, in which the tropical Pacific air–sea system cy-cled between cold and warm extremes and back again,suggests a strong role for the delayed oscillator mech-anism. The results from the energetics support the visionof Suarez and Schopf (1988) and Battisti (1988) (al-though with slight differences in the details), as follows.

1) Perturbation energy is fed to the ocean during thegrowth phase of El Nino or La Nina by the pertur-bation wind power, u9t9 (Hirst 1986).

2) The perturbation wind power is strongest in the centraland western tropical Pacific, and generates pertur-bation buoyancy power on the equator to the east andoff the poleward flanks of the wind power to the west.

3) The gained energy subsequently adjusts, against awind forcing that is neither spatially or temporallycoherent, via equatorial wave dynamics. Energy fedinto the off-equatorial western Pacific travels to thewestern boundary where it is channeled into theequatorial wave guide and then propagates along theequator to the eastern Pacific, eroding the current ElNino–La Nina event and often initiating the next one.

The analysis of the mean perturbation energetics ofEl Nino–La Nina highlights the role of the air–sea in-teraction occurring while the perturbations are adjusting,before they have begun to influence SST anomalies. Thevelocity field associated with the adjusting perturbationsaffects surface currents. The concept of a zonal velocitycomponent to the ‘‘delayed’’ Kelvin wave signal cer-tainly exists in the delayed oscillator theory and hasalso been suggested to contribute substantially to thechange of SST anomalies through zonal advection (Bat-

1512 VOLUME 13J O U R N A L O F C L I M A T E

FIG. 13. Maps through La Nina–El Nino cycle for mean anomalous wind power terms. (left) Adjustment wind power 5 t u9. (right)Mature event wind power 5 t9u . Contour interval is 8 mW m22, starting at 64. Positive values are shaded.

1 MAY 2000 1513G O D D A R D A N D P H I L A N D E R

tisti 1988; Picaut et al. 1997). What the energetics anal-ysis makes more clear is that these delay signals beginto influence conditions in the tropical Pacific, duringtheir adjustment from west to east, generally beginningnear the peak of an El Nino or La Nina event.

The signature of this adjustment in the mean pertur-bation energetics is found in the adjustment wind power(u9t ) as well as the adjustment buoyancy power (h9w)and is generally coherent from peak La Nina to peakEl Nino and from peak El Nino to peak La Nina (Fig.12). The observational measurements from TOGA–TAO already exist for real-time monitoring of the ad-justment wind power, which at the very least, may serveas a useful predictor of event transition. It may alsoprove useful in the debate over which process dominatesevent initiation: deterministic or stochastic forcing. Notethat to evaluate properly the relative importance of sto-chastic processes, daily or at least pentad data shouldbe used in order to resolve the higher-frequency cou-pling between atmosphere and ocean.

During the sequence of El Nino and La Nina events of1970–75, the tropical Pacific appears to be largely deter-ministic in our simulation, although higher-frequency (i.e.,stochastic) behavior is also present. During other periods,stochastic forcing may play a more important role. Twoother periods, 1963–67 and 1980–84 (not shown), havebeen analyzed using energetics, and the results presentedherein apply to them as well. This approach has not yetbeen applied to the unusual period of the early 1990s(Goddard and Graham 1997) nor to the recent ‘‘El Ninoof the century’’ for which westerly wind bursts have beenimplicated as playing a vital role (McPhaden 1999).

Air–sea interaction associated with El Nino and LaNina is intimately related to the air–sea interaction re-sponsible for the mean state of the tropical Pacific. Themean state plays a critical, but implicit, role in concep-tual models such as the delayed oscillator. In this studythe contribution of the mean state was made explicit.Indeed, examination of the energetics involving inter-action between the mean atmospheric winds and anom-alous ocean currents uncovers a dynamic air–sea cou-pling that significantly influences the variability, partic-ularly during the transition between events. Apprecia-tion of this mean perturbation coupling may be used tomonitor future variability or to mediate the debate overthe relative influences of deterministic and stochasticforcing in initiating El Nino and La Nina events. It mayeven be used to understand how lower-frequency chang-es in the mean state might effect the character of ElNino and La Nina events such as those in the 1990s.

Acknowledgments. The authors thank L. Rothstein andanother anonymous reviewer for their thoughtful and in-sightful comments on an earlier draft of this manuscript.We are also grateful for thought-provoking conversationswith K. Bryan, N. Mantua, K. Miyakoda, J. D. Neelin,and A. Oort. This work was funded by a NASA GlobalChange Research Fellowship Grant NGT-30197, and alsoNA86GP0338(343-6083) and NA56GP0226(343-6100).

APPENDIX

Energy Equations

The energy equations used for this study are derivedstarting from the primitive form of the momentum andthermodynamic equations for a Boussinesq, hydrostatic,and incompressible fluid. These equations are discussedby Bryan (1969) for the ocean GCM used here.

a. Kinetic energy

The KE equation is obtained by taking the inner prod-uct of the three-dimensional momentum equations andvelocity vector, (Du/Dt) · u. The resulting equation,written in flux form, where K represents KE, is

Kt 5 2= · (uK ) 2 = · (up) 2 rgw 1 kMHKxx

1 (kMHKy)y 2 rokMH(vx · vx 1 vy · vy) 1 (kMV Kz)z

2 rokMV (vz · vz), (A1)

where, because of the hydrostatic approximation, onlythe horizontal velocities contribute to KE [i.e., K 5(ro/2)(u2 1 y 2)]. All variables have their usual meaning:u 5 (u, y , w) is the full oceanic velocity field; v 5 (u, y)is the two-dimensional horizontal velocity; p and r arethe three-dimensional pressure and density fields, re-spectively; ro is the mean background density; and g isgravity. Subscripts indicate differentiation with respectto space [(x, y, z) 5 zonal, meridional, vertical] or time(t). All prescribed and parameterized values are givenin Goddard (1995).

The horizontally averaged, time mean portion of thedensity and pressure fields is in hydrostatic balance. Thus,

2= · (up) 2rgw 5 2= · [u( 1 ps)] 2 gw, (A2)p r

where the (˜) signifies deviation from the horizontallyaveraged field, and ps is the surface air pressure.

Substituting (A2) into (A1) and integrating them overa fixed volume in the ocean, the final form of the fullkinetic energy equation appears as

]K dV 5 2 Ku · n ds 2 ( p 1 p )u · n ds 2 g rw dVEEE R R s EEE]t

1 v · t ds r 2 k (u · u ) dV r 2 k [(u · u ) 1 (u · u )] dV. (A3)EE o o EEE MV z z o EEE MH x x y y

z50

1514 VOLUME 13J O U R N A L O F C L I M A T E

Unit surface area is denoted by ds, and n points out ofthe volume. The diffusion of energy through the bottomof the volume is neglected because of the weak verticalshear of the horizontal currents at depths much belowthe thermocline. The terms on the right-hand side ofthis equation are identified as

1) ) K u · n ds [ the advection of kinetic energythrough the ‘‘walls’’ of the volume;

2) ) (p 1 ps)u · n ds [ the change of kinetic energydue to work done against pressure gradients by theageostrophic flow;

3) g ### w dV [ the kinetic energy lost to verticalrmovement of the mass field (buoyancy power);

4) ##z50 v · to ds [ the source of kinetic energy due tothe wind stress (to) acting on the surface currents; and,

5) and 6) ro ### kMH[(ux · ux) 1 (uy · uy)] 1 kMV(uz · uz)dV [ energy losses within the volume from workdone by stresses due to horizontal and vertical shearsin the flow, respectively.

b. Available potential energy

The derivation of available potential energy (APE),specifically the available gravitational potential energy,proceeds from the equation for density conservation:

r 1 u · =r 1 wrt z

5 k (r 1 r ) 1 [k (r 1 r )] 1 Q .TH xx yy TV z z z r (A4)

The variable is the vertical gradient of potential den-rz

sity as opposed to in situ density (Oort et al. 1989;Neumann and Pierson 1966), where is the time meanrpotential density horizontally averaged over the analysisregion; and contains the spatially and temporally vary-ring components of the density field. The source term Qencompasses both the thermal and salinity fluxes im-pacting the density field.

The OGCM used here does not calculate density asa prognostic tracer; rather it calculates the temperatureand salinity fields using the tracer conservation equa-tion, calculating the density only when needed (e.g., forthe Richardson number-dependent mixing coefficients).This model approximates density from a third-orderpolynomial involving temperature, salinity, and depth;thus density is not necessarily conserved in the GCM,even though the temperature and salinity are.

The density field here is approximated as a linearcombination of T and S, and so will be conserved:

r 5 ro 2 aT 1 bS. (A5)

The values of the expansion coefficients are given by

24 23 21a 5 2.60 3 10 g cm (8C) ; (A6)24 23 21b 5 7.65 3 10 g cm ppt , (A7)

based on values supplied by Gill (1980). This linearrepresentation for density has shown to be a good first-order approximation for the range of temperatures andsalinities of the world ocean (Bryan and Cox 1972), andparticularly so for the even narrower range of temper-atures and salinities of the tropical upper ocean.

The APE equation is derived to leading order by ig-noring nonlinear changes in temperature and salinity.According to Reid et al. (1981), ‘‘the collective effectsof these [neglected] terms in an [global] oceanwidesense is that they contribute ,10% to the total APE,’’and again, the narrow range of T and S of the tropicalPacific upper ocean yields an even smaller estimatederror for both the mean and interannual conditions.

Equation (A4) is then multiplied through by/ , where the ‘‘stability-like’’ weighting has been2gr rz

chosen to illuminate the conversion between KE andAPE. The equation for the APE, after integrating overa large oceanic volume, is then

] rzzA dV 5 2 (uA) · n ds 1 g rw dV 1 Aw dV 1 k =A · (i, j ) dsEEE R EEE EEE R TH21 2]t rz

k r rTH 2 22 [(r ) 1 (r ) ] dV 1 (k r ) dV 1 Q dV, (A8)EEE x y EEE TV z z EEE r2 2 21 2[ ]N N N

where the APE is defined (Oort et al. 1989) as2g 1 r

2A 5 2 r 5 , (A9)22r 2 Nz

and N 2 5 /g is the depth-dependent stability factor.2rz

The terms on the right-hand side of the APE equationcan be identified as1) ) (uA) · n ds [ the advection of APE through the

walls of the volume;

2) g ### w dV [ vertical motion of the mass fieldr(buoyancy power);

3) ### Aw( / ) dV [ the apparent source or sink due2r rzz z

to shear in the stability profile;4) ) kTH=A · (ı, j ) ds [ the horizontal diffusion of APE

through the walls of the volume; and5) ### (kTH/N 2)[( )2 1 ( )2] dV [ the dissipation ofr rx y

APE within the volume due to horizontal densitygradients.

1 MAY 2000 1515G O D D A R D A N D P H I L A N D E R

6) ### [ /N 2)(kTVrz)z] dV [ the vertical diffusion and(rdissipation of APE. They are not separated since thevertical dependence of the diffusivity coefficient andthe stability lead to a long and messy form. As withthe KE, the diffusion/dissipation terms are smallerand more noisy than other terms in the balance, sothey are lumped into a single term.

7) ### [( /N 2)Qr] dV [ the change in APE due to sur-rface fluxes of density connected with thermal andfresh water sources (i.e., Qr 5 2aQheat 1 bQsalt).

As described by Lorenz (1955), the concept of APErefers to energy that may be stored or extracted by aredistribution of the mass field, which implies that ananalysis of APE should be applied to a fixed mass offluid. In the preceding derivation of APE, three ap-proximations have been made concerning the ocean’smass field. First, density is a linear combination of tem-perature and salinity. Second, mass is conserved in our‘‘open-boundary’’ volume. And third, APE can be de-scribed by its leading-order Taylor-expanded term (Reidet al. 1981). Each of the three assumptions is valid forour data to within a few percent of the total energyseasonally and interannually.

Using this formulation for APE, the goal is not togive an exact value in a localized sense, but rather tosee how the APE changes in a more regional, basinwidesense. The simplifications allow the APE equation tobe matched against the KE equation in a straightforwardway in order to see the dynamical forcing of the APEdirectly. If one desires more precise numbers for theenergy densities at particular times and places, these canbe retrieved by more exact formulations later.

EXCLUSION OF THE MIXED LAYER

Since the concept of gravitational APE rests on thepossible adiabatic redistribution of mass, the mixed lay-er, where the fluid is well mixed and changes in energyare primarily diabatic, should be excluded. The stabilityis much weaker near the surface, and horizontal gra-dients in density are effectively nonexistent (Rudnickand Ferrari 1999); thus large-scale gravitational pro-cesses are unimportant to interannual variability in themixed layer. However, due to the vanishing vertical den-sity gradient and the present formulation for APE [Eq.(A9)], surface processes may be weighted more heavilyin the volume-integrated APE equation than are sub-surface ones. If the mixed layer were to be included inthe volume integral, the surface would dominate, andthe APE anomaly would measure merely the strengthof the SST anomaly. In the real world, a conversionbetween potential and internal forms of energy wouldoccur such that the energy in the mixed layer is primarilyinternal energy (IE; measuring temperature changes).The conversion between APE and IE is not possible,however, in the hydrostatic and Boussinesq OGCM usedin this study.

To isolate the dynamically forced and physicallymeaningful portion of the upper-ocean APE, the mixedlayer is discarded for volume integrations of APE. Al-though the actual mixed layer, generally defined as Tsub

5 Tsrf 2 0.58C, has significant spatial variation in theTropics (Levitus 1982), a constant value, based on themodel’s stability profile, is adequate for this analysis.The mixed layer depth here is taken as 30 m, based onthe tenfold increase in the potential density gradient withrespect to the surface (not shown).

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