Simple dynamical models capturing the key features ofthe Central Pacific El NiñoNan Chena,b,1 and Andrew J. Majdaa,b,c,1

aDepartment of Mathematics, New York University, New York, NY 10012; bCenter for Atmosphere Ocean Science, Courant Institute of MathematicalSciences, New York University, New York, NY 10012; and cCenter for Prototype Climate Modeling, New York University Abu Dhabi, Saadiyat Island, AbuDhabi 129188, United Arab Emirates

Contributed by Andrew J. Majda, August 31, 2016 (sent for review August 1, 2016; reviewed by George N. Kiladis and Xiaoming Wang)

The Central Pacific El Niño (CP El Niño) has been frequently ob-served in recent decades. The phenomenon is characterized byan anomalous warm sea surface temperature (SST) confined tothe central Pacific and has different teleconnections from thetraditional El Niño. Here, simple models are developed andshown to capture the key mechanisms of the CP El Niño. Thestarting model involves coupled atmosphere–ocean processesthat are deterministic, linear, and stable. Then, systematic strat-egies are developed for incorporating several major mechanismsof the CP El Niño into the coupled system. First, simple nonlinearzonal advection with no ad hoc parameterization of the back-ground SST gradient is introduced that creates coupled nonlin-ear advective modes of the SST. Secondly, due to the recentmultidecadal strengthening of the easterly trade wind, a sto-chastic parameterization of the wind bursts including a mean east-erly trade wind anomaly is coupled to the simple atmosphere–oceanprocesses. Effective stochastic noise in the wind burst model facili-tates the intermittent occurrence of the CP El Niño with realisticamplitude and duration. In addition to the anomalous warm SSTin the central Pacific, other major features of the CP El Niño suchas the rising branch of the anomalous Walker circulation beingshifted to the central Pacific and the eastern Pacific cooling with ashallow thermocline are all captured by this simple coupled model.Importantly, the coupled model succeeds in simulating a series of CPEl Niño that lasts for 5 y, which resembles the two CP El Niño epi-sodes during 1990–1995 and 2002–2006.

nonlinear zonal advection | strengthening of the easterly trade wind |effective stochastic noise | Walker circulation

The El Niño–Southern Oscillation (ENSO) is the mostprominent interannual climate variability on earth, affecting

much of the tropics and subtropics. This variability consists of acycle of anomalously warm El Niño conditions and cold La Niñaconditions with considerable irregularity in amplitude, duration,temporal evolution, and spatial structure. The well-known tra-ditional El Niño involves unusual warming of sea surface tem-perature (SST) anomalies in the equatorial eastern Pacific Ocean.The atmospheric response to such anomalous ocean warming isthat the Walker circulation shifts eastward and results in in-creased precipitation near the west coast of America (1).In recent decades, a different type of El Niño has been fre-

quently observed (2, 3), which is called the central Pacific ElNiño (CP El Niño) [also known as El Niño Modoki (4), warmpool El Niño (5), date line El Niño (6), or S-Mode (7)]. The CPEl Niño is characterized by positive SST anomalies confined tothe central Pacific, flanked by colder waters to both east and west,where the corresponding thermocline becomes shallow. Suchzonal SST gradients result in anomalous two-cell Walker circu-lation over the tropical Pacific, with a strong convection region inthe central Pacific (see the illustrations in Fig. 1). Associated withthese distinct warming and cooling patterns, the teleconnectionsof the CP El Niño are quite different from those of the traditionalEl Niño with major societal impact (4, 8).

Although the traditional El Niño is mainly associated withthermocline variation, the CP El Niño appears more related tozonal advection and atmospheric forcing (3, 9–11). Compositeanalysis of reanalysis data shows that zonal advection has asignificant contribution to the SST tendency (5) and is par-ticularly important during the initiation phase of CP El Niño(12). On the other hand, accompanying with the increasingoccurrence of the CP El Niño since 1990s, a recent multi-decadal acceleration of easterly trade winds is observed (13–15), which is linked with the strengthening of the Walker cir-culation (16–18). Due to the atmosphere–ocean coupling, theenhanced atmospheric circulation then induces the intensificationof the westward ocean current with an increased upwelling of coldwater in the eastern Pacific ocean, which tends to prevent the fulleastward extension of the anomalous warm SST during El Niñophases and leads to the occurrence of El Niño in the centralPacific ocean.Despite the significant impact of the CP El Niño, many

general circulation models fail to distinguish this new type ofEl Niño from the traditional one (19), and very few statisticalor dynamical ENSO models are able to simulate the CP ElNiño with realistic features. In the present article, a simplemodeling framework is introduced and developed that cap-tures the key mechanisms of the CP El Niño. The startingmodel involves coupled ocean–atmosphere processes that aredeterministic, linear, and stable. Then, systematic strategiesare developed for incorporating several major causes of the CPEl Niño into the coupled system. First, simple nonlinear zonaladvection with no ad hoc parameterization of the background

Significance

The Central Pacific El Niño (CP El Niño) has been frequentlyobserved in recent decades. The phenomenon is characterizedby an anomalous warm sea surface temperature (SST) confinedto the central Pacific and has different teleconnections fromthe traditional El Niño with major societal impact. Here, asimple modeling framework is developed and shown to cap-ture the key mechanisms of the CP El Niño. In addition to theSST, other major characteristics of the CP El Niño such as therising branch of the anomalousWalker circulation being shiftedto the central Pacific and the eastern Pacific cooling with ashallow thermocline are all captured by this simple coupledmodel. Key features of the model are nonlinear advection ofSST and effective stochastic wind bursts.

Author contributions: A.J.M. designed research; N.C. and A.J.M. performed research; andN.C. and A.J.M. wrote the paper.

Reviewers: G.N.K., Earth System Research Laboratory, National Oceanic and AtmosphericAdministration; and X.W., Florida State University.

The authors declare no conflict of interest.

Freely available online through the PNAS open access option.1To whom correspondence may be addressed. Email: chennan@cims.nyu.edu or jonjon@cims.nyu.edu.

This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1614533113/-/DCSupplemental.

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SST gradient is introduced that creates a coupled nonlinearadvective mode of SST. Second, due to the recent multidecadalstrengthening of the easterly trade wind, a mean easterly tradewind anomaly is included in the parameterization of the windthat couples to the atmosphere–ocean processes. The combinedeffect of the nonlinear zonal advection and the enhanced east-erly trade wind enables the coupled model to generate regularpatterns that are associated with the CP El Niño. Then, a hier-archy of effective stochastic noise models (20) is incorporatedinto the parameterization of the wind bursts that facilitates theintermittent occurrence of the CP El Niño with realistic ampli-tude and duration.The remainder of this article is organized as follows. After

introducing the coupled model, both the regular patterns asso-ciated with the CP El Niño due to the deterministic nonlinearadvection and the role of the effective stochastic wind bursts arestudied. Details of model derivations, mathematical backgroundof the effective stochastic wind bursts and more supporting in-formation of the results are included in SI Appendix.

Basic Coupled ModelENSO Model. The ENSO model considered in this article consistsof a nondissipative atmosphere coupled to a simple shallow-waterocean and SST budget (20). This reads:

Interannual atmosphere model:

− yv−∂xθ= 0yu−∂yθ= 0

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∂τU − c1YV + c1∂xH = c1τxYU + ∂YH = 0

∂τH + c1ð∂xU + ∂YV Þ= 0,[2]

Interannual SST model:

∂τT + μ∂xðUTÞ =−c1ζEq + c1ηH, [3]

with

Eq = αqTτx = γ

�u+ up

�. [4]

In the above model, x is zonal direction and τ is interannualtime, whereas y and Y are meridional direction in the atmosphere

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Fig. 1. Illustrations of normal condition (A), traditional El Niño (B), and CP ElNiño (C). The black arrows with the dashed box show the anomalous Walkercirculation and the white arrows show the direction of ocean currentanomalies. Positive and negative SST anomalies are displayed in blue andred, respectively.

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Fig. 2. Profiles of the thermocline coefficient ηðxÞ (A), the zonal structure ofthe wind burst spðxÞ with its peak at x* (B), and surface wind response inequatorial Pacific band to a mean easterly trade wind anomaly ap (C). Here,the range in meridional direction is from 15o S to 15o N.

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and ocean, respectively. The u, v are zonal and meridional winds;θ is potential temperature; U, V, are zonal and meridional cur-rents; H is thermocline depth; T is SST; Eq is latent heating; andτx is zonal wind stress. All variables are anomalies from anequilibrium state and are nondimensional. The coefficient c1 is anondimensional ratio of time scales, which is of order Oð1Þ. Theterm up in Eq. 4 is a stochastic wind burst perturbation describedin Stochastic Wind Burst Model. The atmosphere extends over theentire equatorial belt 0≤ x≤LA with periodic boundary condi-tions uð0, y, τÞ= uðLA, y, τÞ, whereas the Pacific ocean extendsover 0≤ x≤LO with reflection boundary conditions for the oceanmodel and zero normal derivative at the boundaries for theSST model.The above model retains a few essential processes that model

the ENSO dynamics in a simple fashion. Latent heating Eq that isproportional to SST T is depleted from the ocean and forces anatmospheric circulation. The resulting zonal wind stress τx inreturn forces an ocean circulation that can have feedback on theSST through thermocline depth anomalies H. This thermoclinefeedback is maximal in the eastern Pacific, as shown by theprofile of η in Fig. 2.The model introduces unique theoretical elements such as a

nondissipative atmosphere consistent with the skeleton modelfor the Madden–Julian oscillation (MJO) in the tropics (21, 22),valid here on the interannual timescale, and suitable to describethe dynamics of the Walker circulation (23–25). In addition, themeridional axis y and Y are different in the atmosphere andocean because they each scale to a suitable Rossby radius. Suchdifference in the axis allows for a systematic meridional de-composition and truncation of the flow into the well known para-bolic cylinder functions, which keeps the system low-dimensional(26). For instance, here model solutions Eq. 1 are projected andtruncated to the first parabolic cylinder function of the atmosphere(21), whereas Eqs. 2 and 3 are projected and truncated to the firstparabolic cylinder function of the ocean (1).The coupled system Eqs. 1–4 without the nonlinear zonal

advection in Eq. 3 was systematically studied in ref. 20. Thesystem succeeds in recovering the traditional El Niño and cap-turing the ENSO statistics in the eastern Pacific as in nature.Note that if the stochastic wind burst up is further removed, theresulting coupled system is linear, deterministic, and stable. The

SI appendix in ref. 20 provides detailed derivations of the modelfrom an asymptotic expansion as well as the low-order meridi-onal truncation (27).The observational significance of the zonal advection has been

shown for the CP El Niño (5, 12). However, unlike the previousworks (28–30), where the advection is mostly linear and requiresad hoc parameterization of the background SST gradient, asimple nonlinear advection is adopted in Eq. 3 that contributessignificantly to the SST tendency. Such nonlinear advectionprovides the mechanism of transporting anomalous warm waterto the central Pacific region by the westward ocean zonal current.Importantly, when stochasticity is included in the wind burst up,this nonlinear zonal advection involves the contribution fromboth mean and fluctuation, the latter of which is usually ignoredin the previous works.

Stochastic Wind Burst Model. Stochastic parameterization of thewind bursts with speed up are added to the model that representboth the recent multidecadal strengthening of the easterly tradeand several important ENSO triggers such as westerly windbursts, easterly wind bursts, as well as the convective envelope ofthe MJO. The wind burst reads:

up = apðτÞspðxÞϕ0ðyÞ, [5]

with amplitude apðτÞ and fixed zonal spatial structure spðxÞshown in Fig. 2. Here, ϕ0ðyÞ has a Gaussian profile centeredas the equator and is equal to the first parabolic cylinder func-tion of the atmosphere (SI Appendix). Both the wind burstsperturbations (31) and the strengthening of the trade wind(13, 14) are localized over the western equatorial Pacific accord-ing to the observations and for simplicity they share the samezonal extent.The evolution of wind burst amplitude ap reads:

dapdτ

=−dp�ap − ap ðTW Þ

�+ σpðTW Þ _W ðτÞ, [6]

where dp is noise dissipation and _W ðτÞ is a white noise source.The amplitude of the wind burst noise source σp can either be aconstant or depends on TW , which is the average of SST anom-alies in the western half of the equatorial Pacific (0≤ x≤LO=2).The term ap < 0 represents the mean strengthening of the easterly

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Fig. 3. (A) Phase diagram of ap and μ, where ap ≡ ap is a negative constant,representing a mean easterly trade wind. (B–D) The three typical spatial–temporal SST patterns at the equator associated with each regime, corre-sponding to the three red dots in A. The corresponding atmospheric wind,ocean zonal current, and thermocline feedback are included in SI Appendix.Note that the range of the y axis in D is different from that in B and C.

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trade wind. Corresponding to a nonzero constant easterly tradewind ap < 0, the direct response of the surface wind associated withthe Walker circulation at the equatorial Pacific band is shown inFig. 2C, which is similar to the observed intensification of theWalker circulation in recent decades (13, 14).

CP El Niño Model, the Deterministic Nonlinear AdvectiveModesConsider an intensification of the easterly trade wind with aconstant amplitude ap ≡ ap < 0, where the stochastic effect σp isset to be zero in Eq. 6. Without the randomness in the wind burstmodel, the nonlinear advection becomes deterministic. Note thatthe nondimensional value ap =−0.25 adopted below is roughly−0.94 m/s at its maximum of the equator x*, which is comparableto the observational record (13, 14).The solution of the coupled model illustrates three different

spatial–temporal structures depending on the strength of boththe easterly trade wind ap and the nonlinear zonal advection μ.See the phase diagram in Fig. 3A.In regime I, the steady-state solution has constant values at each

longitude. Particularly, with a suitably strong ap, even without thenonlinear advection, the anomalous warm SST is shifted to thecentral–eastern Pacific region (Fig. 3B). The corresponding oceanzonal current is westward and the rising branch of the anomalousWalker circulation is shifted to the central–eastern Pacific region(SI Appendix).With a nonzero zonal advection μ and a suitably strong easterly

trade wind ap, all of the atmosphere, ocean, and SST anomalyfields become time-periodic, and the period is much longer than2 y (regime II and Fig. 3C). In each period, the positive SSTanomaly develops from central–eastern Pacific and evolves slowlytoward the western Pacific, where it arrives at the maximum

value. The corresponding ocean zonal current is westward at theanomalous warm SST phase and the rising branch of the anomalousWalker circulation is shifted toward the west accompanyingwith the warm water (SI Appendix). It is worth noting that, ex-cept at the end of each period, the eastern Pacific remains coolfrom the ocean upwelling, which is one of the features of theCP El Niño.When both μ and ap are sufficiently large, the steady-state

solution shows regular oscillation patterns with period around1.6 y (regime III and Fig. 3D). Within each period, warm water istransported westward and the maximum of anomalous warmSST is at the central Pacific.

The Effective Stochastic Wind Burst and the Occurrence ofCP El NiñoThe deterministic nonlinear advection with an intensified east-erly trade wind is able to generate regular patterns that are as-sociated with the CP El Niño. However, the irregularity of natureand the intermittent occurrence of CP El Niño are not captured.Effective stochastic wind bursts help generate a more realisticCP El Niño.

Additive Noise. First, additive noise is adopted in the stochasticwind burst model Eq. 6, where the mean easterly trade windintensification ap =−0.25 is fixed and the wind burst noise σp is aconstant that has no dependence on TW .The SST field shown in Fig. 4 becomes more irregular with a

gradual increase of the stochastic noise amplitude σp. When σp isaround σp = 1.0, most of the anomalous warm SST is located inthe central Pacific region and each single event resembles theSST pattern associated with the CP El Niño. It is shown in SIAppendix that both the nonlinear advection and the easterly

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Fig. 5. Solutions of the coupled system with an additive noise in the wind burst model Eq. 6, where ap =−0.25, σp = 1.0 and dp = 3.4. Different columns showthe wind burst upðtÞ at the peak x* of its zonal profile with its 120-d moving average (red), Hovmoller diagrams of the atmospheric wind u+ up, the oceanzonal current U, the thermocline depth H, the SST field T, the SST tendency dT=dt, the flux divergence −μ∂xðUTÞ, and the combined effect of the latent heat−c1ζEq and thermocline feedback c1ηH in Eq. 3. Here, up is defined as up = apspðxÞϕ0ðyÞ. All variables shown are at the equator.

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mean trade wind are the necessary ingredients in generating CPEl Niño. It is also shown there that the CP El Niño disappears ifthe amplitude σp of the stochastic noise is too large.In Fig. 5, the Hovmoller diagrams for different fields are

shown with ap =−0.25 and σp = 1.0. Most events with anomalouspositive SST in the central Pacific represent CP El Niño. Atthese phases, the surface westerly wind and easterly wind con-verge in the central Pacific region, where the rising branch of theanomalous Walker circulation is formed (see Fig. 7 in A Two-State Markov Jump Model). The westward zonal ocean current inthe central Pacific region serves to transport the warm water tothe central Pacific via the nonlinear zonal advection. The asso-ciated thermocline becomes deeper in the central Pacific but isshallower in the eastern Pacific, resulting in an upwelling of coldwater. All of these features are consistent with the observationalrecord during the CP El Niño years.To understand the role of the zonal nonlinear advection, the

budget of SST tendency dT=dτ in Eq. 3 is studied. The positiveflux divergence −μ∂xðUTÞ in the central–western Pacific regionindicates its dominant role in transporting anomalous warm waterto the central and western Pacific. On the other hand, the com-bined effect of the latent heat −c1ζEq and thermocline feedbackc1ηH leads to the increase of the anomalous warm SST only in theeastern Pacific region. In fact, even the positive component ofthermocline feedback itself does not extend as much to the cen-tral–western Pacific region as the flux divergence. These resultsare consistent with the observational findings that the CP El Niñoappears more related to zonal advection than thermocline feed-back (3, 9–11).

A Two-State Markov Jump Model. To obtain the occurrence of theCP El Niño with realistic duration and amplitude, a two-stateMarkov jump process (20, 32) is adopted to model the stochastic

wind burst Eq. 6. Here, both σp and ap switch between onequiescent phase (State 0) and one active phase (State 1),

State  0 : σp = 0.2, and ap = 0,State  1 : σp = 1.0, and ap =−0.25.

The transition rates between the two states are functions of TW ,where a larger (smaller) TW corresponds to a higher probabilityof transition from State 0 (1) to State 1 (0). This setup is becausewind burst activity is usually favored by warmer SST in the west-ern Pacific and conversely (31, 33, 34). The mathematical for-mulae of the transition probability and the profiles of switchingrates are shown in SI Appendix.

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In Fig. 6, Hovmoller diagrams of different fields for a 30-yperiod are shown. The stochastic switching process in the windburst model leads to the intermittent occurrence of the CP ElNiño with realistic amplitude and duration. Both a 1-y CP ElNiño event (t= 342) and a series of CP El Niño events that lastsfor 5 y (from t= 329 to t= 334) are simulated. The latter is par-ticularly important because it resembles the two CP El Niñoepisodes as observed during 1990–1995 and 2002–2006 (35),which have much longer durations than the traditional El Niño.Similar to Fig. 5, detailed analysis shows that the flux di-

vergence is the main contributor to the occurrence of the centralPacific El Niño, where the westward zonal ocean current trans-ports the anomalous warm water to the central Pacific regionand leads to the eastern Pacific cooling with a shallow thermo-cline depth.The anomalous Walker circulation at the CP El Niño phase t0

is shown in Fig. 7. It is important to note that the rising branch islocated at the central Pacific, where the atmosphere surface windu+ up is westerly in the western Pacific and easterly in theeastern Pacific. These features in the Walker circulation areconsistent with observations during CP El Niño years (5) and aredistinct from those associated with the traditional El Niño, wherethe rising branch is located at the eastern Pacific.

Conclusion and DiscussionSimple dynamical models are developed here that capture thekey mechanism of the CP El Niño. Systematic strategies aredeveloped for incorporating several major mechanisms of the CPEl Niño into simple coupled atmosphere–ocean processes thatare otherwise deterministic, linear, and stable. First, a simplenonlinear zonal advection with no ad hoc parameterization ofthe background SST gradient is introduced that contributes tothe SST tendency through a coupled nonlinear advective mode.Secondly, due to the recent multidecadal strengthening of the

easterly trade wind, a stochastic parameterization of the windbursts including a mean easterly trade wind anomaly is coupledto the simple atmosphere–ocean processes.The deterministic nonlinear advection model involving an

easterly trade wind anomaly shows regular patterns that are as-sociated with the CP El Niño. The irregularity of nature is re-covered by involving stochastic noise in the wind burst model. Tocapture the intermittent occurrence of the CP El Niño with re-alistic amplitude and duration, effective stochastic noise thataccounts for its dependence on the strength of the western Pa-cific warm pool through a two-state Markov jump process (20,32) is incorporated into the wind burst model. In addition to theanomalous warm SST in the central Pacific, other major featuresof the CP El Niño such as the rising branch of the anomalousWalker circulation being shifted to the central Pacific and theeastern Pacific cooling with a shallow thermocline are all cap-tured by this simple coupled model. Importantly, the coupledmodel succeeds in simulating a series of CP El Niño that lasts for5 y, which resembles the two CP El Niño episodes during 1990–1995 and 2002–2006.It is worthwhile mentioning the possibility of developing a

simple stochastic model for ENSO that involves both the CP ElNiño and the traditional El Niño and La Niña. This probably canbe achieved by combining the simple dynamical model for theCP El Niño developed here with the one studied in ref. 20 thatgenerates the traditional El Niño and super El Niño events withwesterly wind bursts.

ACKNOWLEDGMENTS. The authors thank Sulian Thual for useful discussion.The research of A.J.M. is partially supported by Office of Naval ResearchMultidisciplinary University Research Initiative (ONR MURI) Grant N00014-16-1-2161 and the New York University Abu Dhabi Research Institute. N.C. issupported as a postdoctoral fellow through A.J.M.’s ONR MURI grant.

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