# ENSO complexity controlled by zonal shifts in the walker circulation

Sulian Thual, Boris Dewitte

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## To cite this version:

Sulian Thual, Boris Dewitte. ENSO complexity controlled by zonal shifts in the walker circulation. Nature Geoscience, 2023, 16 (4), pp.328-332. 10.1038/s41561-023-01154-x. hal-04747663

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# ENSO complexity controlled by zonal shifts in the Walker circulation

Received: 16 June 2022

Accepted: 21 February 2023

Published online: 30 March 2023

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Sulian Thual*<fnref n="1" />✉ & Boris Dewitte<fnref n="1" />,²,³

The El Niño/Southern Oscillation (ENSO) represents the most consequential fluctuation of the global climate system, with dramatic societal and environmental impacts. Its general dynamics are reasonably well understood in terms of ocean-atmosphere interactions that modify the Walker circulation in the equatorial Pacific. However, some of its space-time features remain stubbornly elusive, such as its event-to-event diversity and asymmetry. Here we show that zonal shifts in the Walker circulation control ENSO space-time complexity in a low-dimensional theoretical framework. We encapsulate these movements in a conventional recharge-discharge oscillator for ENSO by replacing the regionally fixed sea surface temperature index with a warm-pool edge index that displaces the Walker circulation structures. By doing so, we can model essential ingredients of ENSO diversity and nonlinear behaviour without increasing the complexity of the dynamical model. The simple framework is able to reproduce the continuum of ENSO flavours and the asymmetry in the amplitude of warm and cold events. The spatial shifting concept paves the way for a more unified understanding of ENSO and its associated climate teleconnections.

The El Niño/Southern Oscillation (ENSO), which refers collectively to the alternation between El Niño and La Niña events in the equatorial Pacific, affects extreme weather, populations and ecosystems worldwide<fnref n="1" />,². Its impacts are, however, highly variable and unpredictable due to a considerable diversity in characteristics from one event to the next³,⁴. Understanding and predicting this event-to-event diversity remains a fundamental yet unreached goal⁵.

Despite the primary importance of the ENSO and the decades of research progress, present-day general circulation models with full physics still tend to represent it with some bias in regards to its amplitude, spatial pattern or evolution, among others⁶,⁷. Similar issues arise in models of decreased complexity that, despite reasonably depicting the ENSO in terms of coupled interactions between the atmosphere and ocean, differ in terms of the processes invoked for its more advanced features⁸⁻¹⁰. A major reason for these shortcomings is, generally speaking, a lack of consensus in understanding how the ENSO dynamics control its space-time complexity. For example, it remains unclear whether the pattern diversity¹¹ between central or eastern Pacific El Niño types stems from atmosphere¹²

or ocean processes¹³ or how this relates to the general circulation of the equatorial Pacific¹⁴,¹⁵. Another lack of consensus regards the processes by which El Niño events may reach larger amplitudes yet shorter durations than La Niña events, which is referred to as ENSO asymmetry or non-normality¹⁶⁻¹⁸. Although important progress has been made, there is no generally accepted theory that provides a complete and comprehensive picture of the ENSO space-time complexity¹⁹.

Here we propose a model that captures essential ingredients of ENSO complexity discussed above, such as its diversity and asymmetry in spatial patterns and evolution, in imperfect but extremely simple fashion (Fig. 1). The present model is a minimal oscillator with only two core variables and one source of random forcing. It extends conventional recharge-discharge theory²⁰ by replacing its regionally fixed sea surface temperature (SST) index with a warm-pool edge index, which introduces ENSO diversity and nonlinearity but does not increase the complexity of the dynamical model. Hereafter we detail the model's rationale and properties and the broader implications for ENSO understanding.

<footnote n="1">¹CECI, Université de Toulouse, CERFACS, CNRS, Toulouse, France. ²CEAZA, La Serena, Chile. ³Universidad Católica del Norte, Facultad de Ciencias del Mar, Departamento de Biología Marina, Coquimbo, Chile. ✉e-mail: sthual@cerfacs.fr</footnote>
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**Fig. 1 | Observed and modelled ENSO variability.** a,b, Time–longitude Hovmöller diagrams of sea surface temperatures anomalies (SSTA) and thermocline depth anomalies (TDA) in the equatorial Pacific, as computed from observations (averaged 5° N–5° S with seasonal variations removed). Solid lines in each panel indicate the longitudinal position of the warm-pool edge (that is, 29 °C SST isotherm; Methods). c,d, Hovmöller diagrams repeated for solutions

generated by a minimal oscillator model, displayed over a hand-picked period with time relative to the beginning of numerical simulation. Major ENSO patterns are identified in both observations and model: El Niños with strong eastern Pacific warming (EP), with moderate central to eastern Pacific warming (CP) and La Niñas with moderate and lingering cooling that extends further westward (LN).

## Idealized Walker circulation

The formulation of the present oscillator model starts by proposing an idealized representation of the general circulation of the equatorial Pacific, commonly referred to as the Walker circulation (Fig. 2a)²¹⁻²³. The Walker circulation is depicted here as a quasi-balance between the atmosphere and ocean with a simple geometry based on the separation into so-called warm-pool and cold-tongue regions. An important non-linearity driving this quasi-balance is the threshold for deep convection around 29 °C at which SSTs saturate over the warm pool. Convective ascents over the warm pool reinforce the easterly trade winds over the cold tongue, which, in turn, gradually make the thermocline more shallow and cool the SSTs (Methods and Extended Data Fig. 1).

The Walker circulation is not static, and its evolution is, in fact, what constitutes ENSO variability. The present approach consists in representing such an evolution in rather simple terms, by directly shifting the entire Walker circulation back and forth in the longitudinal (that is zonal) direction (Fig. 2b,c). This generates La Niña events when the circulation shifts westward expanding the cold-tongue region, and conversely El Niño events when the circulation shifts eastward expanding the warm-pool region. Such a behaviour is observed in nature in the form of general and coherent movements that follow the warm-pool edge and that involve both the atmosphere and ocean (Extended Data Fig. 2).

**Fig. 2 | Idealized Walker circulation.** a, The equatorial Pacific is represented by a longitudinal strip divided into a warm-pool region and a cold-tongue region. The warm pool is characterized by saturating SSTs at 29 °C, a deep thermocline and increased convection (cloud symbols). The cold tongue is characterized by constant trade winds blowing to the west (wind symbols) that gradually make the thermocline more shallow and cool the SST. b,c, The present approach consists in shifting the circulation as a whole during La Niña and El Niño while conserving its general geometrical features. This results in opposite contractions and expansions of each region as described by the variable extents x and μ. The zonal mean of thermocline depth or heat content hM (blue dashed lines) is separately controlled by recharge–discharge processes.

Generally speaking, conventional ENSO models tend to focus directly on the departures from average conditions or anomalies⁸ (Fig. 1), which accounts rather indirectly for the spatial movements

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<figure><img src="image_6.png" /><figcaption>**Fig. 3 | Pattern continuum from spatial shifting.** a, Continuum of SST profiles obtained by shifting the idealized Walker circulation's warm-pool edge from 140° E (plain black) to 100° W (dashed black) in 20° increments (dashed grey and colour scale). Guidelines indicate warm-pool edge position (dotted grey). T_E indicates SST averaged over the cold tongue (dark purple, displayed at the region centre) that increases linearly with eastward shifting. b,c, Comparison to observed SST and SST anomalies (SSTA) for selected El Niño eastern Pacific (EP), central Pacific (CP), mixed EP-CP or La Niña (LN) events (December-January-</figcaption></figure>

February averages). d, Thermocline depth (TD) profiles obtained by shifting the idealized Walker circulation in 60° increments (colour scale) and controlling for heat content (that is, basin-wide average) kept at h_M = 125 m. Grey arrows indicate the span of cold-tongue trade winds that drive thermocline sloping through Sverdrup balance. h_E indicates TD averaged over the cold tongue (dark purple, displayed at the region centre) that increases nonlinearly with both eastward or westward shifting. e,f, Comparison to observed TD and TD anomalies (TDA) for selected ENSO events, with heat content adjusted to h_M = 125 m.

of the total circulation²⁴. To circumvent this difficulty, our representation focuses explicitly on the total circulation (minus seasonal effects).

## Recharge-discharge dynamics

Although fundamental, the process of Walker circulation shifting discussed above has not yet found a firm standing in ENSO theory with the exception of a few attempts at modelling the warm pool's edge position²⁵,²⁶. Here we encapsulate the process in a minimal oscillator with concise ENSO dynamics that extends the well-proven and widely used recharge-discharge theoretical framework⁹,²⁰.

The ocean adjustment to wind stress is approximately depicted in terms of a time-dependent Sverdrup balance that gradually shallows the thermocline over the cold tongue in addition to a non-equilibrium Sverdrup transport that slowly modifies the zonal mean thermocline depth or so-called heat content²⁰,²⁷ (Fig. 2). Thermocline variations warm or cool the cold-tongue SSTs as determined by a simplified heat budget, which modifies the cold-tongue span and consequently the span of the trade winds. A general form of the resulting coupled system can be expressed as:

$$ 
\begin{aligned}
d\mu/dt &= -a_1\mu - b_1h_M + c_1 - d_1\mu^2 \\
dh_M/dt &= -a_2h_M + b_2\mu + c_2 + \zeta
\end{aligned}
\qquad (1) 
$$

where t is time, μ is cold-tongue extent, h_M is basin-wide heat content²⁰,²⁷, ζ is an ad hoc stochastic forcing that mimics the effect of random fluctuations in nature (for example, wind bursts²⁸,²⁹) and the remainder a₁, a₂, b₁, b₂, c₁, c₂, d₁ consist of constant parameters (Methods and Extended Data Table 2). Working with cold-tongue extent μ or warm-pool extent χ is equivalent here as they vary oppositely and also determine warm-pool edge position (Fig. 2). The above system generates ENSO-like irregular oscillations that occasionally exhibit strong El Niño events as in nature (Extended Data Fig. 3).

The main modification of conventional recharge-discharge theory is that instead of fixing the domain of the cold tongue, we allow it to

expand and contract by solving for its variable extent μ. This allows generation of a diversity of ENSO patterns from a single continuum¹¹: La Niña (μ large), central Pacific El Niño (μ moderately small) and eastern Pacific El Niño (μ very small) (Fig. 1). μ relates to eastern Pacific SST from the original theory as an extended cold tongue is cooler on average and conversely (Fig. 2). The modification also conserves the original recharge-discharge mechanism for ENSO's reversal of conditions, that is, the more expanded the cold tongue, the more the expanded trade winds recharge heat content, which warms and eventually shrinks back the cold tongue, and conversely. Another modification is a quadratic nonlinearity (term - d₁μ²) detailed hereafter. Other processes that have been proposed for ENSO complexity are omitted here for brevity⁹,¹⁶⁻¹⁸,²⁹⁻³¹.

## Spatial shifting mechanisms

The model generates total fields by shifting the idealized Walker circulation structures over time with μ from equation (1) and adjusting heat content to h_M. Field anomalies are deduced by removing the resulting temporal mean (Methods and Extended Data Fig. 4). This generates a pattern continuum that, although imperfect, provides a compelling baseline for the observed ENSO spatial diversity. In terms of SSTs (Fig. 3a-c), the pattern continuum ranges from a weak basin-wide La Niña cooling to a weak central-eastern El Niño warming followed by a strong eastern El Niño warming, as in nature. Because the continuum is bounded here by the 29 °C saturation threshold, it fails to capture the peak warming of central Pacific El Niños.

In terms of thermocline depth (Fig. 3d-f), the pattern continuum controlled for heat content ranges from a tilted La Niña profile that gradually changes into a near-flat El Niño profile. The thermocline tilting is determined here by the ocean's Sverdrup balance response to the cold-tongue trade winds of varying span. The pattern continuum exhibits, in particular, a nonlinear deepening of the cold-tongue thermocline for both eastward and westward shifting (Fig. 3d). The deepening, in turn, amplifies cold-tongue warming during El Niño (eastward shifting) but mitigates cold-tongue cooling during La Niña (westward
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**Fig. 4 | Principal component analysis of SST.** a,b, First and second empirical orthogonal functions (EOFs) as computed from either SST anomalies generated by the model (red) or observed (Obs) SST anomalies (5° N–5° S averages in the equatorial Pacific, black), with explained variance indicated (%). c, Scatterplot between associated first and second PC timeseries from model outputs (red) or observations (black). A quadratic relationship between PC timeseries found in observations is highlighted by a second-order polynomial fit (blue line). The first PC timeseries strongly correlates with spatial shifting (x) in both the model (r = 0.99) and observations (r = 0.91). The EOFs are scaled in the figure to have comparable amplitudes, and the PC timeseries are accordingly inversely scaled.

shifting), which generates ENSO amplitude asymmetry in the model (term – *d*<sub>i</sub>*μ*<sup>2</sup> in equation (1)).

Various geometrical features of the idealized Walker circulation vary nonlinearly with spatial shifting, such as the one above, that also seem to hold reasonably well in observations and climate model experiments despite some bias (Methods and Extended Data Figs. 5 and 6, Extended Data Table 1). To which extent these nonlinearities may control the ENSO's atmosphere and ocean dynamics remains to be determined, which would first require principle calculations and detailed budget analysis beyond the present idealizations<sup>9,16,32</sup>.

**Principal component analysis**

The first principal components (PCs) of SST in the equatorial Pacific describe the leading ENSO patterns in terms of data covariance. Here we reproduce the PC's overall characteristics from the outputs of the

minimal model, not only in terms of spatial structures but also in terms of a nonlinear quadratic relationship between PCs whose interpretation still remains elusive<sup>33–37</sup> (Fig. 4).

The first PC from model outputs captures warming and cooling from the cold tongue's shifting movements over the eastern Pacific and no variability from the warm pool's saturating SSTs over the western Pacific (Fig. 4a). The second PC does not capture independent variability (as there is only one SST degree of freedom *μ*) but a weak nonlinear imprint from first PC in the central Pacific (3% explained variance) that has a quadratic dependency on first PC (Fig. 4b,c and Methods). Such an imprint can be traced back to the central Pacific being overlapped in similar ratios by warm-pool saturating or cold-tongue sloping SSTs with the resulting mixed SST characteristics being nonlinear (Extended Data Fig. 7). This rather simple feature from the model may help explain the nonlinear relationship between the PCs of observed SSTs, although the second PC (12% explained variance) also captures other phenomena<sup>38,39</sup>.

**Implications for ENSO understanding**

Here we have shown that the spatial shifting movements of the Walker circulation<sup>21</sup> control ENSO complexity in a low-dimensional theoretical framework. This allows reinterpreting some of the phenomenon's elusive features<sup>11,19</sup> such as its event-to-event diversity and asymmetry with an imperfect but simple baseline. The present findings suggest that some of these features may be simpler than previously considered to some extent, thus hopefully more predictable<sup>5</sup>. The present model should be seen not as a definitive theory but rather a proof of concept to show that spatial shifting may shape conventional ENSO dynamics<sup>20</sup>. In fact, the model's formulation remains open to improvements and should be combined and reconciled with other rationales for ENSO complexity proposed in the literature<sup>9,16</sup>, including in more realistic set-ups (Methods).

Beyond improving theoretical understanding, the spatial shifting concept may also have implications for other aspects of the ENSO phenomenon. Further work may extend to the study of teleconnections, considering that the Walker circulation's movements are tied to pan-tropical and extra-tropical variability<sup>40,41</sup>, bridging not only ENSO-induced climate disruptions across the planet but also remote influences on El Niño or La Niña onset<sup>1,2,42</sup>. Further work may also help elucidate how spatial shifting may impact conventional ENSO diagnosis, for example, when inferring core convective or dynamical feedbacks in general circulation models<sup>7,32,43</sup>. This, in turn, may help further understand how El Niño's strength and flavour are being modulated by recent decadal variability and climate change<sup>14,36,44–46</sup>.

**Online content**

Any methods, additional references, Nature Portfolio reporting summaries, source data, extended data, supplementary information, acknowledgements, peer review information; details of author contributions and competing interests; and statements of data and code availability are available at https://doi.org/10.1038/s41561-023-01154-x.

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## Methods

### Formulation of the idealized Walker circulation

We detail hereafter the model formulation starting with the representation of the Walker circulation (Fig. 2). Consider non-dimensional fields of SST $T$, thermocline depth $h$ and zonal wind stress $\tau_x$ in a zonal strip representative of the equatorial Pacific (120° E to 80° W, averages 5° N to 5° S). Let the fields describe the evolution of an idealized Walker circulation (Extended Data Fig. 1) that reads:

$$ \begin{array}{l} \text{if } x < \chi(t) \left\{ \begin{array}{l} \tau_x(x, t) = 0 \\ T(x, t) = T_W \\ h(x, t) = h_W(t) \end{array} \right. \\ \text{if } x \ge \chi(t) \left\{ \begin{array}{l} \tau_x(x, t) = \tau_E \\ T(x, t) = T_W - 2s(x - \chi) \\ h(x, t) = h_W(t) - 2a(x - \chi) \end{array} \right. \end{array} \quad (2) $$

in non-dimensional units where $x$ is zonal position (with $0 \le x \le 1$) and $\chi$ is time. The circulation is centred on the warm-pool edge at $x = \chi$ that separates the warm-pool region ($x < \chi$) from the cold-tongue region ($x \ge \chi$) with distinct properties.

The Walker circulation is depicted here as a quasi-balance between the atmosphere and ocean that is summarized in its essential terms$^{8,21-23}$. Trade winds (oriented westward) span the cold-tongue region with constant intensity $\tau_E < 0$ where they gradually shallow the thermocline at rate $2a$ consistent with a Sverdrup balance response (of general form $\partial h/\partial x$ proportional to $\tau_x$). The shallowing thermocline, in turn, gradually cools the SST at rate $2s$, representative of upwelling processes in nature. Meanwhile, SSTs saturate at $T_W = 29$ °C over the warm-pool region. In fact, SSTs nearing that threshold substantially increase deep convection in the atmosphere (for example, as indicated by decreasing outgoing long-wave radiations; Extended Data Fig. 1a), which, in turn, captures excess heating. Convective heating over the warm pool drives a large-scale Gill-type response$^{8}$ with low-level convergence that reinforces the trade winds over the cold tongue through large-scale convergence (and with a westerly component in the far western Pacific omitted here for simplicity).

The quasi-balance (or quasi-equilibrium) terminology here denotes a balance that is time dependent but evolves slowly enough to remain verified. It can also be seen as a fast-wave approximation$^{20}$ that omits fast fluctuations in both the atmosphere and ocean (for example, wind bursts$^{28,29}$ or equatorial waves generated by intra-seasonal variability).

Although the above quasi-balance is commonly invoked to explain the Walker circulation's average (or climatological) conditions$^{22,23}$, we allow here the warm-pool extent $\chi(t)$ to vary interannually, which consists of a spatial shifting movement of the entire structure in the longitudinal direction (Fig. 2). The rationale for this is that, from one state to another, the atmosphere and ocean approximately rebalance according to the same arguments discussed above and as also supported by observations (Extended Data Fig. 2 and Supplementary Information Video 1). Such a rationale can also be seen as an interpretation of the so-called Bjerknes hypothesis that, broadly speaking, refers to mutual reinforcements between the Walker circulation's atmosphere and ocean$^{21}$. The circulation may also vary in terms of $h_W(t)$, which relates to the evolution of ocean heat content as detailed hereafter.

### Regions averages

Let $\chi$ from equation (2) be the variable extent of the warm-pool region with $0 \le \chi \le 1$ in non-dimensional units. Similarly, let $\mu$ be the variable extent of the cold-tongue region with $0 \le \mu \le 1$. Longitudinal (that is, zonal) shifting results in opposite contractions and expansions of each

region as encapsulated by $\chi + \mu = 1$. Here $\chi$ denotes interchangeably warm-pool extent or warm-pool edge position ranging from 0 to 1 in non-dimensional units (warm-pool edge position ranges from 120° E to 80° W in dimensional units).

For a more concise representation, we introduce the indices:

$$ \begin{align} T_W(t) &= \chi^{-1} \int_0^\chi T(x, t) dx \nonumber \\ T_E(t) &= \mu^{-1} \int_\chi^1 T(x, t) dx \tag{3} \end{align} $$

$$ T_M(t) = \int_0^1 T(x, t) dx = \chi T_W + \mu T_E $$

that represent spatial averages of SST at a given time respectively over the warm-pool and cold-tongue regions of variable extent and over the entire basin (Extended Data Fig. 5). Similar indices are defined for zonal wind stress ($\tau_W$, $\tau_E$, $\tau_M$) and for thermocline depth ($h_W$, $h_E$, $h_M$). With this, the geometrical relationships from equation (2) can be expressed more simply as:

$$ \begin{align} \tau_M &= \mu \tau_E \nonumber \\ T_E &= T_W - s\mu \tag{4} \\ h_E &= h_W - a\mu \nonumber \end{align} $$

which encapsulates the trade winds' span over the cold-tongue region and the sloping of SST (at rate $2s$) and thermocline depth (at rate $2a$). Combining the above with the definitions of basin-wide averages (equation (3)), we obtain nonlinear quadratic relationships:

$$ \begin{align} T_W &= T_M + s\mu^2 \nonumber \\ T_E &= T_M - s\chi\mu \nonumber \\ h_W &= h_M + a\mu^2 \nonumber \\ h_E &= h_M - a\chi\mu \nonumber \end{align} \quad (5) $$

where we recall that $T_W$ is constant. We can also deduce spatial averages over any given region of the equatorial Pacific (for example, conventional regions Niño 4, 3.4 or 3) from equation (2).

The relationships in equations (2)–(5) infer the Walker circulation's response to given dynamical constraints as discussed hereafter. The characteristics of the Walker circulation at a given time are here set entirely by $\chi$ (or equivalently $\mu$) and $h_M$ from which we deduce other variables. $h_M$ is the spatial average of thermocline depth over the basin, also referred to as basin-wide heat content or warm-water volume in the literature$^{27}$. The evolution of $\chi$ and $h_M$ is determined by the oscillator model (equation (1) or equation (7) hereafter).

The above region averages can also be examined in observational data as well, which requires an estimate of the observed warm-pool edge position (equation (12) hereafter). The agreement between model outputs and observations for these indices is good overall in terms of timeseries and statistics (Extended Data Fig. 5 and Extended Data Table 1). The relationships between these indices predicted by equations (4) and (5) also hold reasonably well in observational data and an example of general circulation model experiment despite some bias (Extended Data Fig. 6).

### Model derivation

Following the recharge-discharge framework$^{9,20}$, we approximate the ocean adjustment to wind stress in terms of a time-dependent Sverdrup balance and a non-equilibrium Sverdrup transport. The Sverdrup balance here is part of the broader Walker circulation quasi-balance described above that varies with $\chi$ (equations (2)–(5) and Extended Data Fig. 1), while the non-equilibrium Sverdrup transport slowly modifies the basin-wide heat content $h_M$ through conventional recharge-discharge dynamics.

To determine the evolution of $\chi$ (or equivalently $\mu$) and $h_M$, we consider a thermodynamical budget for the cold-tongue SST and a dynamical budget for the basin-wide heat content:
$$ 
\begin{aligned}
\frac{d T_E}{dt} &= -\xi (T_E - T_W) + \eta h_E - Q - q^* \\
\frac{d h_M}{dt} &= -\sigma h_M + \gamma \tau_M + F + \zeta
\end{aligned}
\qquad (6) $$

with relaxation parameters ($\xi$, $\sigma$), thermocline feedback ($\eta$), wind stress forcing ($\gamma$) and optional stochastic forcing $\zeta$ as in conventional recharge-discharge theory$^{20}$. Modifications to this theory consist here in adding constant external sources $F$, $Q$ that set equilibrium conditions (as we are working with field totals instead of anomalies), in addition to a flux $q^* = (T_E - T_W)\mu^{-1}(d\mu/dt)$ as we allow the cold-tongue region to expand and contract (Leibniz integral rule). With the exception of $q^*$, the present starting budgets are linear as model nonlinearities will be introduced hereafter from the geometry of the problem. We also approximately omit seasonal effects for the sake of brevity. Finally, we close the above budgets with the Walker circulation's quasi-balance relationships (equations (4) and (5)).

Combining the above relationships, we obtain a minimal oscillator for the ENSO (identical to the one from equation (1)):

$$ 
\begin{aligned}
2s(d\mu/dt) &= -\xi s\mu + a\eta(\mu - \mu^2) - \eta h_M + Q \\
dh_M/dt &= -\sigma h_M + b\mu + F + \zeta
\end{aligned}
\qquad (7) $$

with $\chi + \mu = 1$ in non-dimensional units and where we note $b = -\gamma T_E$ for brevity. The total fields generated by the model are obtained by shifting the idealized structures from equation (2) over time with $\chi$, $\mu$ and adjusting heat content $h_M$ (Extended Data Figs. 2 and 3). Field anomalies are then deduced by removing the temporal mean computed over the entire simulation (Fig. 1 and Extended Data Fig. 4). The above model is nicknamed Contraction-Recharge-Expansion-Discharge Oscillator (CREDO) in reference to the original recharge-discharge oscillator$^{20}$. This is because the warm pool consecutively contracts, recharges, expands, then discharges during an ENSO cycle (Extended Data Fig. 4). The model generates a diversity of ENSO patterns as a single dynamical continuum$^{11}$: La Niña ($\mu$ large), central Pacific El Niño ($\mu$ moderately small) and eastern Pacific El Niño ($\mu$ very small) (Figs. 1–3).

## Model details

To sustain ENSO variability, we include ad hoc stochastic forcing in the otherwise dissipated system from equation (7) in the form of:

$$ d\zeta/dt = -e\zeta + r\dot{W} \qquad (8) $$

where $\zeta$ accounts for random fluctuations in nature (for example, wind bursts$^{28}$) as driven by a white noise source $\dot{W}$ with standard normal distribution$^{29}$ (Extended Data Fig. 5e).

In practice, the external sources $F$, $Q$ are deduced such as to obtain a model equilibrium $\mu_0$, $h_{M0}$ that is comparable to observed temporal averages$^{17}$. This reads:

$$ 
\begin{aligned}
F &= -b\mu_0 + \sigma h_{M0} \\
Q &= \xi s\mu_0 - a\eta(\mu_0 - \mu_0^2) + \eta h_{M0}
\end{aligned}
\qquad (9) $$

where $\mu_0$, $h_{M0}$ are imposed. These external sources here maintain the cold tongue and control its average extent. For lower values of $F$ and $Q$, it is in fact possible to obtain $\mu_0 = 0$ for which the cold tongue collapses and the equatorial Pacific is overlapped entirely by the warm pool. While these external sources may stem from a wide range of processes$^{22,23}$, we leave their discussion for future work. The model's linear solutions around the present equilibrium consist in an oscillatory mode with a period around 2.5 years and decay time around five months.

Extended Data Table 2 details model units and parameters. Regarding numerics, we integrate the model using a simple Euler scheme in time with a timestep 0.1 month. We constrain $0 \le \mu \le 1$ each timestep to deal with potential region collapses even though none occurs during the present simulation (Extended Data Fig. 3).

## Potential model modifications

The present model's formulation remains open for improvements and may be extended in future work through different avenues. Building upon the existing body of work from the literature, the present framework could be combined, for example, with seasonal effects, state-dependent noise, zonal advection of SST or other SST feedbacks that commonly extend the original recharge-discharge framework$^{9}$. First principle calculations and detailed budget analysis may also help determine a more exact form of the model's equations. For instance, other formulations of the feedback processes embedded into the SST budget equation (6) may introduce additional nonlinearities when integrated over the varying cold-tongue domain. Another improvement might consist in refining the model's Walker circulation geometry beyond the idealized structures from equation (2) considering that the displacements of these structures generate the model's spatial patterns of El Niño and La Niña (Fig. 3). These modifications might help reconcile spatial shifting with other rationales for complexity, nonlinearity and mean-state interactions from the literature$^{16-18,29,31,44}$.

Suitable model modifications may also help grasp important missing features from the observational record (Supplementary Video 1). For example, some central Pacific El Niños exhibit a localized SST warming just west of the warm-pool edge that overshoots the warm-pool saturation level of 29 °C imposed here. Some eastern Pacific El Niños exhibit a lingering of warm SSTs in their trail that is not grasped by the present idealized geometry. The model representation may also be refined to account for the gradual decaying of the trade winds towards the South American coast, the thermocline sloping in the western Pacific and so on.

## Principal components reconstruction

For the present problem, discretized SST anomalies (SSTA) in the equatorial band can be decomposed into their principal components as:

$$ 
\begin{aligned}
\text{SSTA}(x, t) &= \sum_{i=1}^{N} R_i \text{ where} \\
R_i(x, t) &= p_i(t)E_i(x)
\end{aligned}
\qquad (10) $$

with $N$ the spatial dimension and where, for each component $i$ (ordered by decaying explained variance), $p_i$ is the PC timeseries, $E_i$ is the associated empirical orthogonal functions (EOF) and $R_i$ the resulting SSTA reconstruction. SST is reconstructed in similar fashion adding the mean-state component. Here we restrict our analysis to the first and second PCs as computed from either the SSTAs generated by the model or observations (Fig. 4). In the model outputs, $R_i$ accounts for 97% explained variance and $R_2$ for 3%, while in observations, $R_1$ accounts for 75% explained variance and $R_2$ for 12%. While the first PC accounts for most of the explained variance, the second PC is tied to it by an approximately quadratic relationship$^{33–38}$ in both model outputs and observations. Thus, we may define the nonlinear imprint of first PC on second PC as:

$$ 
\begin{aligned}
R_2^*(x, t) &= p_2^*(t)E_2(x) \text{ where} \\
p_2^*(t) &= a^*p_1^*(t) + b^*p_1(t) + c^*
\end{aligned}
\qquad (11) $$

and where the coefficients $a^*$, $b^*$, $c^*$ are deduced from a polynomial fit of order 2 between $p_2$ and $p_1$ (Fig. 4c). $R_2^*$ accounts for around 3% explained variance in both model outputs and observations as deduced from mean-square error (in model outputs, $R_2^*$ is in fact almost identical to $R_2$). When combined with $R_1$, $R_2^*$ allows capturing the nonlinear SST characteristics in the central Pacific that result from the mixing of warm-pool saturating and cold-tongue sloping SST characteristics (Extended Data Fig. 7).

## Datasets

Observational data used in this paper is from the NCEP Global Ocean Data Assimilation System$^{47}$. Data is monthly over 1980–2021
and includes fields of zonal wind stress (N m⁻²), thermocline depth (m, measured from the 20 °C isotherm) and SST (°C). For Extended Data Figs. 1 and 2, we also use National Oceanic and Atmospheric Administration Interpolated Outgoing Longwave Radiation⁴⁸ (OLR, W m⁻²).

For Extended Data Fig. 6, we overcome the short observational record by also analysing SST and thermocline depth data from the 42 members of the historical simulations over 1920–2005 of the National Center for Atmospheric Research’s Community Earth System Model Large Ensemble Project (CESM-LE)⁴⁹. To overcome the CESM-LE model’s cold bias, the thermocline depth is measured from the 15 °C isotherm⁵⁰ and uniformly shallowed by 50 m, while χ is measured for a target warm-pool saturation SST of 28.5 °C instead of 29 °C (see below).

All fields in all datasets are monthly, span the equatorial Pacific 120 °E–80 °W, are meridionally averaged within 5 °N–5 °S and have their seasonal variations around temporal averages removed (which we refer to as totals in the present paper with a slight abuse in terminology).

The index χ that measures the warm-pool edge position in observations (Fig. 1 and Extended Data Figs. 1–3, 5 and 6) is defined as:

$$ \chi(t) = \int_{0}^{1} \text{bool}(T > 28.5^{\circ}\text{C})dx + \Delta_{c} \quad (12) $$

where T(x, t) is observed SST, the equatorial Pacific spans 0 ≤ x ≤ 1 (120 °E–80 °W in dimensional units), the bool function maps 1 if the boolean condition is true and 0 otherwise, and Δc is a constant. The estimation method is meant to delineate robustly the 29 °C isotherm at which warm-pool SSTs approximately saturate but requires additional workarounds to minimize thresholding sensitivity. First, we measure the index on data without seasonal variations, which simplifies the delineation. Second, a threshold value 28.5 °C is chosen that, to allow delineation, must be slightly below the actual saturation value 29 °C. To grasp the saturation value slightly more west, we add an empirical correction Δc (−10° E in dimensional units). Third, occasional cooling below 28.5 °C in the far western Pacific can bias the estimation, and to remedy this, we set bool = 1 west of the measured χ and recompute it once.

## Data availability

GODAS⁴⁷ data and OLR⁴⁸ data are provided by the National Oceanic and Atmospheric Administration Physical Sciences Laboratory, Boulder, Colorado, USA, from the website https://psl.noaa.gov. CESM-LE⁴⁹ data are provided by National Science Foundation/Computational and Information Systems Lab/Yellowstone from the website https://www.earthsystemgrid.org/dataset/ucar.cgd.ccsm4.cesmLE.html.

## Code availability

The code associated with this paper is available upon request from the corresponding author.

## References

47. Behringer, D. & Xue, Y. Evaluation of the global ocean data assimilation system at NCEP: the Pacific Ocean. In Proc. 8th Symposium on Integrated Observing and Assimilation Systems for Atmosphere, Oceans, and Land Surface, 11–15 (AMS, 2004).

48. Liebmann, B. & Smith, C. A. Description of a complete (interpolated) outgoing longwave radiation dataset. Bull. Am. Meteorol. Soc. 77, 1275–1277 (1996).

49. Kay, J. E. et al. The Community Earth System Model (CESM) large ensemble project: a community resource for studying climate change in the presence of internal climate variability. Bull. Am. Meteorol. Soc. 96, 1333–1349 (2015).

50. Deser, C. et al. ENSO and Pacific decadal variability in the Community Climate System Model version 4. J. Clim 25, 2622–2651 (2012).

## Acknowledgements

The research of S.T. is supported by Agence Nationale de la Recherche (grant ANR-18-CE01-0012). B.D. acknowledges support from Agencia Nacional de Investigación y Desarrollo (Concurso de Fortalecimiento al Desarrollo Científico de Centros Regionales 2020-R20F0008-CEAZA, Anillo Eclipse ACT210071, Núcleo Milenio UPWELL NCN19-153, Centro de Investigación Oceanográfica en el Pacífico Sur-Oriental COASTAL FB210021) and Agence Nationale de la Recherche (grant ANR-18-CE01-0012). We thank G. Bellon and O. Thual for discussions.

## Author contributions

S.T. designed and performed the research. S.T. and B.D. discussed the research and wrote the paper.

## Competing interests

The authors declare no competing interests.

## Additional information

Extended data is available for this paper at https://doi.org/10.1038/s41561-023-01154-x.

Supplementary information The online version contains supplementary material available at https://doi.org/10.1038/s41561-023-01154-x.

Correspondence and requests for materials should be addressed to Sulian Thual.

Peer review information Nature Geoscience thanks Fei-Fei Jin and the other, anonymous, reviewer(s) for their contribution to the peer review of this work. Primary Handling Editor: Tom Richardson, in collaboration with the Nature Geoscience team.

Reprints and permissions information is available at www.nature.com/reprints.
<figure><img src="image_10.png" /></figure>

**Extended Data Fig. 1 | Idealized Walker circulation structure.** Zonal profiles of zonal wind stresses (ZWS, a), sea surface temperatures (SST, b) and thermocline depth (TD, c) in a longitudinal strip spanning the equatorial Pacific (black lines and fillings). The circulation can vary in terms of its warm pool extent $\chi$, and thermocline depth can additionally vary in terms of its basin-wide average $h_M$

(blue dashed lines). We adjust these variables in the figure in order to match observed temporal averages of ZWS, SST and TD (purple dashed lines). Observed temporal average of outgoing longwave radiations (OLR) is displayed in (a) as an indicator of convective heating over the warm pool that drives the atmospheric part of the circulation.
<figure><img src="image_11.png" /></figure>

**Extended Data Fig. 2** | Time-longitude hovmollers for totals. a,b,c,d
Time-longitude hovmollers of outgoing longwave radiation (OLR), zonal wind stress (ZWS), sea surface temperature (SST) and thermocline depth (TD) in the equatorial Pacific, as computed from observations (averaged 5°N-5°S with seasonal variations removed). e,f,g, Analysis repeated for the outputs of the oscillator model (which excludes OLR), displayed over a hand-picked period.

Solid lines in each panel indicate the longitudinal position of the warm pool edge (that is, 29°C SST isotherm, see methods). The spatial shifting of the Walker circulation consists of a general and coherent longitudinal movement of the atmospheric and oceanic structures that follows the warm pool edge. The movement is less apparent on thermocline depth (panels d,g) that also varies substantially in terms of its basin-wide average.
<figure><img src="image_12.png" /></figure>

**Extended Data Fig. 3** | Phase space evolution. a,b, Timeseries of warm pool extent $\chi$ and basin-wide heat content $h_M$ obtained from a numerical integration of the oscillator model (Eq. (1)), and which represents ENSO variability. Warm pool extent $\chi$ within 0-1 corresponds to a warm pool edge position within 120°E-80°W. Shading indicates the hand-picked period used in other figures. c,d, Kernel density estimates (filled contours) and individual realizations (black lines) in phase space of warm pool extent $\chi$ and basin-wide heat content $h_M$, compared

between observations and model solutions. For clarity, density estimates are shown only above 0.04% where they hide individual realizations that are too dense to visualize. Phase space is divided into El Niño or La Niña states as well as recharged or discharged states as defined here with respect to average conditions (white dashed lines). El Niño events may reach larger amplitudes than La Niña events but are more short-lived.
<figure><img src="image_13.png" /></figure>

**Extended Data Fig. 4** | ENSO cycle in the oscillator model. a–d, Sketch of the Walker circulation at the four consecutive stages of one ENSO cycle: contracted (La Niña), recharged, expanded (El Niño) and discharged. This is shown for sea surface temperatures (SST) and thermocline depth (TD) represented as in Extended Data Fig. 1. The circulation is entirely determined by the warm pool extent $\chi$ that shifts the structures and the basin-wide heat content $h_M$ that sets

the zonal mean TD, as indicated at each stage of the cycle (blue dashed lines) and for the entire cycle (red lines, years 132 to 135 in Fig. 1)). e–h, Mean state of SST and TD (that is temporal averages computed over the entire simulation, green dashed lines) with total fields from a–d repeated (gray fillings). i–l, Sea surface temperature anomalies (SSTA) and thermocline depth anomalies (TDA) at each stage as computed with respect to the mean state.
<figure><img src="image_14.png" /></figure>

**Extended Data Fig. 5** | Indices timeseries. a-d, Timeseries from observations of warm pool edge position (WPE, a), SST spatial averages (b), thermocline depth spatial averages (c), and Niño SST indices (d). e-h, Analysis repeated for outputs of the oscillator model displayed over a hand-picked period. We also show ad-hoc stochastic forcing ζ in the oscillator model (e, green line). The indices *T*<sub>W</sub>, *T*<sub>E</sub>, *T*<sub>M</sub> average SST conditions respectively over the warm pool (120°E-WPE), cold

tongue (WPE-80°W) and entire basin (120°E-80°W). The indices *h*<sub>W</sub>, *h*<sub>E</sub>, *h*<sub>M</sub> average thermocline depth in same fashion. The conventional indices Niño 4, 3.4 and 3 SST average SST conditions respectively over the central (160°E-150°W), central-eastern (170°W-120°W) and eastern (150°W-90°W) Pacific. All averages are within the equatorial band (5°N-5°S).
<figure><img src="image_15.png" /></figure>

**Extended Data Fig. 6** | Indices scatterplots. a-o, Scatterplots between spatial averages (or combinations of them) and warm pool edge position (WPE) as measured in either model outputs (red), observations (black) or outputs from CESM-LE (see methods). The indices *T*<sub>W</sub>, *T*<sub>E</sub>, *T*<sub>M</sub> average SST conditions over the warm pool (120°E-WPE), cold tongue (WPE-80°W) and entire basin (120°E-80°W). The indices *h*<sub>W</sub>, *h*<sub>E</sub>, *h*<sub>M</sub> average thermocline depth in same fashion. The conventional indices Niño 4, 3.4 and 3 SST average SST conditions over the

central (160°E-150°W), central-eastern (170°W-120°W) and eastern (150°W-90°W)
Pacific (shaded areas). All averages are within 5°N-5°S. The relationships predicted
by the oscillator model are either constant (C), linear (L), oscillatory (O),
nonlinear quadratic (Q) or nonlinear saturating (S). The thermocline nonlinearity
from Fig. 3d is encapsulated by the relationship in panel b. The model predicts
that the Niño SST indices in panels e,j,o saturate to around 29°C when the warm
pool region overlaps their domain.
<figure><img src="image_16.png" /></figure>

**Extended Data Fig. 7** | Principal components reconstruction. a–e, Scatter plots between sea surface temperatures (SST) at different longitudes of the equatorial Pacific and first principal component timeseries of SST (PC1), as computed from observations. f–j, Analysis repeated for SSTs generated by the model. The present representation helps determine how well data is aligned with (and thus captured by) the SST reconstruction $R_1$ from the first PC (black lines, see methods).

Individual SST realizations are discriminated in terms of their belonging to the warm pool region where they saturate around 29°C (WP, red) or cold tongue region where they slope (CT, blue), with total occurrences indicated in %. Where WP and CT occurrences are more alike (that is mixed), the SST variability is no longer captured by $R_1$ but by $R_1 + R_2^*$ (green lines) where $R_2^*$ is a nonlinear imprint of the first PC on the second PC (see methods).
Extended Data Table 1 | Indices statistics

<table><thead><tr><th rowspan="2">Indice</th><th colspan="2">Mean</th><th colspan="2">Std</th><th colspan="2">Skewness</th></tr><tr><th>Model</th><th>Obs</th><th>Model</th><th>Obs</th><th>Model</th><th>Obs</th></tr></thead><tbody><tr><td>χ (1000 km)</td><td>6.61</td><td>6.56</td><td>2.43</td><td>1.96</td><td>0.92</td><td>1.44</td></tr><tr><td>τ<sub>W</sub> (10<sup>-3</sup> N.m<sup>-2</sup>)</td><td>0</td><td>-10</td><td>0</td><td>7.8</td><td>0</td><td>0.45</td></tr><tr><td>τ<sub>M</sub></td><td>-22</td><td>-26</td><td>4.7</td><td>5.8</td><td>0.92</td><td>0.09</td></tr><tr><td>τ<sub>E</sub></td><td>-35</td><td>-35</td><td>0</td><td>0.6</td><td>0</td><td>0.27</td></tr><tr><td>h<sub>W</sub> (m)</td><td>165</td><td>165</td><td>12.8</td><td>12.5</td><td>-1.1</td><td>-0.63</td></tr><tr><td>h<sub>M</sub></td><td>124</td><td>125</td><td>8.4</td><td>6.3</td><td>-0.25</td><td>-0.85</td></tr><tr><td>h<sub>E</sub></td><td>102</td><td>104</td><td>7.0</td><td>7.7</td><td>-0.33</td><td>-0.22</td></tr><tr><td>T<sub>W</sub> (°C)</td><td>29.0</td><td>29.4</td><td>0</td><td>0.25</td><td>0</td><td>-0.24</td></tr><tr><td>T<sub>M</sub></td><td>27.1</td><td>27.5</td><td>0.71</td><td>0.53</td><td>0.11</td><td>0.34</td></tr><tr><td>T<sub>E</sub></td><td>26.2</td><td>26.4</td><td>0.61</td><td>0.55</td><td>0.92</td><td>0.98</td></tr><tr><td>Niño 4 SST (°C)</td><td>28.3</td><td>28.5</td><td>0.63</td><td>0.67</td><td>-0.89</td><td>-0.45</td></tr><tr><td>Niño 3.4 SST</td><td>27.4</td><td>27.5</td><td>0.88</td><td>0.81</td><td>-0.18</td><td>0.06</td></tr><tr><td>Niño 3 SST</td><td>25.7</td><td>25.8</td><td>1.14</td><td>0.93</td><td>0.57</td><td>0.97</td></tr></tbody></table>
Extended Data Table 2 | Model units and parameters

<table><thead><tr><th>Parameter</th><th>Value</th><th>Scaling</th><th>Definition</th></tr></thead><tbody><tr><td>[x]</td><td>1</td><td>18 000 km</td><td>unit zonal position</td></tr><tr><td>[t]</td><td>1</td><td>1 month</td><td>unit time</td></tr><tr><td>[h]</td><td>1</td><td>20m</td><td>unit thermocline depth</td></tr><tr><td>[T]</td><td>1</td><td>1°C</td><td>unit SST</td></tr><tr><td>[τ<sub>x</sub>]</td><td>1</td><td>0.01 N.m<sup>-2</sup></td><td>unit zonal wind stress</td></tr><tr><td>L</td><td>1</td><td>18 000 km</td><td>equatorial Pacific length</td></tr><tr><td>T<sub>w</sub></td><td>29</td><td>29°C</td><td>saturation level of SST</td></tr><tr><td>2s</td><td>9</td><td>0.5°C.10<sup>-6</sup>m</td><td>sloping rate SST</td></tr><tr><td>2a</td><td>10</td><td>11.1 m. 10<sup>-6</sup>m</td><td>sloping rate thermocline</td></tr><tr><td>ξ</td><td>1/6</td><td>(6 mth)<sup>-1</sup></td><td>relaxation rate SST</td></tr><tr><td>σ</td><td>1/14</td><td>(14 mth)<sup>-1</sup></td><td>relaxation rate thermocline</td></tr><tr><td>b</td><td>0.7</td><td>7.8 10<sup>-7</sup> mth<sup>-1</sup></td><td>trade winds constant</td></tr><tr><td>η</td><td>0.8</td><td>0.04°C.m<sup>-1</sup>.mth<sup>-1</sup></td><td>thermocline feedback</td></tr><tr><td>Q</td><td>4.56</td><td>4.56°C.mth<sup>-1</sup></td><td>external cooling</td></tr><tr><td>F</td><td>-1.6 10<sup>-3</sup></td><td>-0.37 m.yr<sup>-1</sup></td><td>external forcing</td></tr><tr><td>μ<sub>0</sub></td><td>0.67</td><td>11520km</td><td>equilibrium value of μ</td></tr><tr><td>h<sub>M0</sub></td><td>6.25</td><td>125m</td><td>equilibrium value of h<sub>M</sub></td></tr><tr><td>e</td><td>1/3</td><td>(3 mth)<sup>-1</sup></td><td>stochastic forcing dissipation rate</td></tr><tr><td>r</td><td>0.1</td><td>2 m.mth<sup>-2</sup></td><td>stochastic forcing generation rate</td></tr><tr><td>Δ<sub>c</sub></td><td>-0.0625</td><td>-10 deg °E</td><td>correction for χ in observations</td></tr></tbody></table>