Simple dynamical models capturing the key features of the Central Pacific El Niño

Significance

The Central Pacific El Niño (CP El Niño) has been frequently observed in recent decades. The phenomenon is characterized by an anomalous warm sea surface temperature (SST) confined to the central Pacific and has different teleconnections from the traditional El Niño with major societal impact. Here, a simple modeling framework is developed and shown to capture the key mechanisms of the CP El Niño. In addition to the SST, other major characteristics of the CP El Niño such as the rising branch of the anomalous Walker circulation being shifted to the central Pacific and the eastern Pacific cooling with a shallow thermocline are all captured by this simple coupled model. Key features of the model are nonlinear advection of SST and effective stochastic wind bursts.

Abstract

The Central Pacific El Niño (CP El Niño) has been frequently observed in recent decades. The phenomenon is characterized by an anomalous warm sea surface temperature (SST) confined to the central Pacific and has different teleconnections from the traditional El Niño. Here, simple models are developed and shown to capture the key mechanisms of the CP El Niño. The starting model involves coupled atmosphere–ocean processes that are deterministic, linear, and stable. Then, systematic strategies are developed for incorporating several major mechanisms of the CP El Niño into the coupled system. First, simple nonlinear zonal advection with no ad hoc parameterization of the background SST gradient is introduced that creates coupled nonlinear advective modes of the SST. Secondly, due to the recent multidecadal strengthening of the easterly trade wind, a stochastic parameterization of the wind bursts including a mean easterly trade wind anomaly is coupled to the simple atmosphere–ocean processes. Effective stochastic noise in the wind burst model facilitates the intermittent occurrence of the CP El Niño with realistic amplitude and duration. In addition to the anomalous warm SST in the central Pacific, other major features of the CP El Niño such as the rising branch of the anomalous Walker circulation being shifted to the central Pacific and the eastern Pacific cooling with a shallow thermocline are all captured by this simple coupled model. Importantly, the coupled model succeeds in simulating a series of CP El Niño that lasts for 5 y, which resembles the two CP El Niño episodes during 1990–1995 and 2002–2006.

The El Niño–Southern Oscillation (ENSO) is the most prominent interannual climate variability on earth, affecting much of the tropics and subtropics. This variability consists of a cycle of anomalously warm El Niño conditions and cold La Niña conditions with considerable irregularity in amplitude, duration, temporal evolution, and spatial structure. The well-known traditional El Niño involves unusual warming of sea surface temperature (SST) anomalies in the equatorial eastern Pacific Ocean. The atmospheric response to such anomalous ocean warming is that the Walker circulation shifts eastward and results in increased precipitation near the west coast of America (1).

In recent decades, a different type of El Niño has been frequently observed (2, 3), which is called the central Pacific El Niño (CP El Niño) [also known as El Niño Modoki (4), warm pool El Niño (5), date line El Niño (6), or S-Mode (7)]. The CP El Niño is characterized by positive SST anomalies confined to the central Pacific, flanked by colder waters to both east and west, where the corresponding thermocline becomes shallow. Such zonal SST gradients result in anomalous two-cell Walker circulation over the tropical Pacific, with a strong convection region in the central Pacific (see the illustrations in Fig. 1). Associated with these distinct warming and cooling patterns, the teleconnections of the CP El Niño are quite different from those of the traditional El Niño with major societal impact (4, 8).

Fig. 1.

Illustrations of normal condition (A), traditional El Niño (B), and CP El Niño (C). The black arrows with the dashed box show the anomalous Walker circulation and the white arrows show the direction of ocean current anomalies. Positive and negative SST anomalies are displayed in blue and red, respectively.

Although the traditional El Niño is mainly associated with thermocline variation, the CP El Niño appears more related to zonal advection and atmospheric forcing (3, 9–11). Composite analysis of reanalysis data shows that zonal advection has a significant contribution to the SST tendency (5) and is particularly important during the initiation phase of CP El Niño (12). On the other hand, accompanying with the increasing occurrence of the CP El Niño since 1990s, a recent multidecadal acceleration of easterly trade winds is observed (13–15), which is linked with the strengthening of the Walker circulation (16–18). Due to the atmosphere–ocean coupling, the enhanced atmospheric circulation then induces the intensification of the westward ocean current with an increased upwelling of cold water in the eastern Pacific ocean, which tends to prevent the full eastward extension of the anomalous warm SST during El Niño phases and leads to the occurrence of El Niño in the central Pacific ocean.

Despite the significant impact of the CP El Niño, many general circulation models fail to distinguish this new type of El Niño from the traditional one (19), and very few statistical or dynamical ENSO models are able to simulate the CP El Niño with realistic features. In the present article, a simple modeling framework is introduced and developed that captures the key mechanisms of the CP El Niño. The starting model involves coupled ocean–atmosphere processes that are deterministic, linear, and stable. Then, systematic strategies are developed for incorporating several major causes of the CP El Niño into the coupled system. First, simple nonlinear zonal advection with no ad hoc parameterization of the background SST gradient is introduced that creates a coupled nonlinear advective mode of SST. Second, due to the recent multidecadal strengthening of the easterly trade wind, a mean easterly trade wind anomaly is included in the parameterization of the wind that couples to the atmosphere–ocean processes. The combined effect of the nonlinear zonal advection and the enhanced easterly trade wind enables the coupled model to generate regular patterns that are associated with the CP El Niño. Then, a hierarchy of effective stochastic noise models (20) is incorporated into the parameterization of the wind bursts that facilitates the intermittent occurrence of the CP El Niño with realistic amplitude and duration.

The remainder of this article is organized as follows. After introducing the coupled model, both the regular patterns associated with the CP El Niño due to the deterministic nonlinear advection and the role of the effective stochastic wind bursts are studied. Details of model derivations, mathematical background of the effective stochastic wind bursts and more supporting information of the results are included in SI Appendix.

Basic Coupled Model

ENSO Model.

The ENSO model considered in this article consists of a nondissipative atmosphere coupled to a simple shallow-water ocean and SST budget (20). This reads:

• Interannual atmosphere model:

$$\begin{array}{l}-yv-{\partial }_x\theta =0\hfill \\ yu-{\partial }_y\theta =0\hfill \\ -\left({\partial }_{x}u+{\partial }_{y}v\right)=E_q/\left(1-\overline{Q}\right),\hfill \end{array}$$ [1]

• Interannual ocean model:

$$\begin{array}{l}{\partial }_{\tau }U-c_{1}YV+c_1{\partial }_{x}H=c_1{\tau }_x\hfill \\ YU+{\partial }_{Y}H=0\hfill \\ {\partial }_{\tau }H+c_1\left({\partial }_{x}U+{\partial }_{Y}V\right)=0,\hfill \end{array}$$ [2]

• Interannual SST model:

$${\partial }_{\tau }T+\underset{¯}{\mu {\partial }_x\left(UT\right)}=-c_1\zeta E_q+c_1\eta H,$$ [3]

with

$$\begin{array}{l}{E}_q={\alpha }_{q}T\hfill \\ {\tau }_x=\gamma \left(u+u_p\right).\hfill \end{array}$$ [4]

In the above model, x is zonal direction and τ is interannual time, whereas y and Y are meridional direction in the atmosphere and ocean, respectively. The $u,v$ are zonal and meridional winds; θ is potential temperature; U, V, are zonal and meridional currents; H is thermocline depth; T is SST; $E_q$ is latent heating; and ${\tau }_x$ is zonal wind stress. All variables are anomalies from an equilibrium state and are nondimensional. The coefficient $c_1$ is a nondimensional ratio of time scales, which is of order $O\left(1\right)$. The term $u_p$ in Eq. 4 is a stochastic wind burst perturbation described in Stochastic Wind Burst Model. The atmosphere extends over the entire equatorial belt $0\le x\le L_A$ with periodic boundary conditions $u\left(0,y,\tau \right)=u\left(L_A,y,\tau \right)$, whereas the Pacific ocean extends over $0\le x\le L_O$ with reflection boundary conditions for the ocean model and zero normal derivative at the boundaries for the SST model.

The above model retains a few essential processes that model the ENSO dynamics in a simple fashion. Latent heating $E_q$ that is proportional to SST T is depleted from the ocean and forces an atmospheric circulation. The resulting zonal wind stress ${\tau }_x$ in return forces an ocean circulation that can have feedback on the SST through thermocline depth anomalies H. This thermocline feedback is maximal in the eastern Pacific, as shown by the profile of η in Fig. 2.

Fig. 2.

Profiles of the thermocline coefficient $\eta \left(x\right)$ (A), the zonal structure of the wind burst $s_p\left(x\right)$ with its peak at $x^{*}$ (B), and surface wind response in equatorial Pacific band to a mean easterly trade wind anomaly ${\widehat{a}}_p$ (C). Here, the range in meridional direction is from ${15}^o$ S to ${15}^o$ N.

The model introduces unique theoretical elements such as a nondissipative atmosphere consistent with the skeleton model for the Madden–Julian oscillation (MJO) in the tropics (21, 22), valid here on the interannual timescale, and suitable to describe the dynamics of the Walker circulation (23–25). In addition, the meridional axis y and Y are different in the atmosphere and ocean because they each scale to a suitable Rossby radius. Such difference in the axis allows for a systematic meridional decomposition and truncation of the flow into the well known parabolic cylinder functions, which keeps the system low-dimensional (26). For instance, here model solutions Eq. 1 are projected and truncated to the first parabolic cylinder function of the atmosphere (21), whereas Eqs. 2 and 3 are projected and truncated to the first parabolic 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. The system succeeds in recovering the traditional El Niño and capturing the ENSO statistics in the eastern Pacific as in nature. Note that if the stochastic wind burst $u_p$ is further removed, the resulting coupled system is linear, deterministic, and stable. The SI appendix in ref. 20 provides detailed derivations of the model from an asymptotic expansion as well as the low-order meridional truncation (27).

The observational significance of the zonal advection has been shown for the CP El Niño (5, 12). However, unlike the previous works (28–30), where the advection is mostly linear and requires ad hoc parameterization of the background SST gradient, a simple nonlinear advection is adopted in Eq. 3 that contributes significantly to the SST tendency. Such nonlinear advection provides the mechanism of transporting anomalous warm water to the central Pacific region by the westward ocean zonal current. Importantly, when stochasticity is included in the wind burst $u_p$, this nonlinear zonal advection involves the contribution from both mean and fluctuation, the latter of which is usually ignored in the previous works.

Stochastic Wind Burst Model.

Stochastic parameterization of the wind bursts with speed $u_p$ are added to the model that represent both the recent multidecadal strengthening of the easterly trade and several important ENSO triggers such as westerly wind bursts, easterly wind bursts, as well as the convective envelope of the MJO. The wind burst reads:

$$u_p=a_p\left(\tau \right)s_p\left(x\right){\varphi }_0\left(y\right),$$ [5]

with amplitude $a_p\left(\tau \right)$ and fixed zonal spatial structure $s_p\left(x\right)$ shown in Fig. 2. Here, ${\varphi }_0\left(y\right)$ has a Gaussian profile centered as the equator and is equal to the first parabolic cylinder function of the atmosphere (SI Appendix). Both the wind bursts perturbations (31) and the strengthening of the trade wind (13, 14) are localized over the western equatorial Pacific according to the observations and for simplicity they share the same zonal extent.

The evolution of wind burst amplitude $a_p$ reads:

$$\frac{d{a}_p}{d\tau }=-d_p\left(a_p\underset{¯}{-{\widehat{a}}_p\left(T_W\right)}\right)+{\sigma }_p\left(T_W\right)\dot{W}\left(\tau \right),$$ [6]

where $d_p$ is noise dissipation and $\dot{W}\left(\tau \right)$ is a white noise source. The amplitude of the wind burst noise source ${\sigma }_p$ can either be a constant or depends on $T_W$, which is the average of SST anomalies in the western half of the equatorial Pacific ($0\le x\le L_O/2$). The term ${\widehat{a}}_p<0$ represents the mean strengthening of the easterly trade wind. Corresponding to a nonzero constant easterly trade wind ${\widehat{a}}_p<0$, the direct response of the surface wind associated with the Walker circulation at the equatorial Pacific band is shown in Fig. 2C, which is similar to the observed intensification of the Walker circulation in recent decades (13, 14).

CP El Niño Model, the Deterministic Nonlinear Advective Modes

Consider an intensification of the easterly trade wind with a constant amplitude $a_p\equiv {\widehat{a}}_p<0$, where the stochastic effect ${\sigma }_p$ is set to be zero in Eq. 6. Without the randomness in the wind burst model, the nonlinear advection becomes deterministic. Note that the nondimensional value $a_p=-0.25$ adopted below is roughly $-0.94$ m/s at its maximum of the equator $x^{*}$, which is comparable to the observational record (13, 14).

The solution of the coupled model illustrates three different spatial–temporal structures depending on the strength of both the easterly trade wind $a_p$ and the nonlinear zonal advection μ. See the phase diagram in Fig. 3A.

Fig. 3.

(A) Phase diagram of $a_p$ and μ, where $a_p\equiv {\widehat{a}}_p$ 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, corresponding 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.

In regime I, the steady-state solution has constant values at each longitude. Particularly, with a suitably strong $a_p$, even without the nonlinear advection, the anomalous warm SST is shifted to the central–eastern Pacific region (Fig. 3B). The corresponding ocean zonal current is westward and the rising branch of the anomalous Walker circulation is shifted to the central–eastern Pacific region (SI Appendix).

With a nonzero zonal advection μ and a suitably strong easterly trade wind $a_p$, all of the atmosphere, ocean, and SST anomaly fields become time-periodic, and the period is much longer than 2 y (regime II and Fig. 3C). In each period, the positive SST anomaly develops from central–eastern Pacific and evolves slowly toward the western Pacific, where it arrives at the maximum value. The corresponding ocean zonal current is westward at the anomalous warm SST phase and the rising branch of the anomalous Walker circulation is shifted toward the west accompanying with the warm water (SI Appendix). It is worth noting that, except at the end of each period, the eastern Pacific remains cool from the ocean upwelling, which is one of the features of the CP El Niño.

When both μ and $a_p$ are sufficiently large, the steady-state solution shows regular oscillation patterns with period around 1.6 y (regime III and Fig. 3D). Within each period, warm water is transported westward and the maximum of anomalous warm SST is at the central Pacific.

The Effective Stochastic Wind Burst and the Occurrence of CP El Niño

The deterministic nonlinear advection with an intensified easterly trade wind is able to generate regular patterns that are associated with the CP El Niño. However, the irregularity of nature and the intermittent occurrence of CP El Niño are not captured. Effective stochastic wind bursts help generate a more realistic CP El Niño.

Additive Noise.

First, additive noise is adopted in the stochastic wind burst model Eq. 6, where the mean easterly trade wind intensification ${\widehat{a}}_p=-0.25$ is fixed and the wind burst noise ${\sigma }_p$ is a constant that has no dependence on $T_W$.

The SST field shown in Fig. 4 becomes more irregular with a gradual increase of the stochastic noise amplitude ${\sigma }_p$. When ${\sigma }_p$ is around ${\sigma }_p=1.0$, most of the anomalous warm SST is located in the central Pacific region and each single event resembles the SST pattern associated with the CP El Niño. It is shown in SI Appendix that both the nonlinear advection and the easterly mean trade wind are the necessary ingredients in generating CP El Niño. It is also shown there that the CP El Niño disappears if the amplitude ${\sigma }_p$ of the stochastic noise is too large.

Fig. 4.

Hovmoller diagrams of SST field at the equator with different levels of additive noise. Here, $a_p=-0.25$ and $d_p=3.4$ are fixed. The seeds of random number generators in different columns are kept to be the same.

In Fig. 5, the Hovmoller diagrams for different fields are shown with ${\widehat{a}}_p=-0.25$ and ${\sigma }_p=1.0$. Most events with anomalous positive SST in the central Pacific represent CP El Niño. At these phases, the surface westerly wind and easterly wind converge in the central Pacific region, where the rising branch of the anomalous Walker circulation is formed (see Fig. 7 in A Two-State Markov Jump Model). The westward zonal ocean current in the central Pacific region serves to transport the warm water to the central Pacific via the nonlinear zonal advection. The associated thermocline becomes deeper in the central Pacific but is shallower in the eastern Pacific, resulting in an upwelling of cold water. All of these features are consistent with the observational record during the CP El Niño years.

Fig. 5.

Solutions of the coupled system with an additive noise in the wind burst model Eq. 6, where ${\widehat{a}}_p=-0.25,{\sigma }_p=1.0$ and $d_p=3.4$. Different columns show the wind burst $u_p\left(t\right)$ at the peak $x^{*}$ of its zonal profile with its 120-d moving average (red), Hovmoller diagrams of the atmospheric wind $u+{\widehat{u}}_p$, the ocean zonal current U, the thermocline depth H, the SST field T, the SST tendency $dT/dt$, the flux divergence $-\mu {\partial }_x\left(UT\right)$, and the combined effect of the latent heat $-c_1\zeta E_q$ and thermocline feedback $c_1\eta H$ in Eq. 3. Here, ${\widehat{u}}_p$ is defined as ${\widehat{u}}_p={\widehat{a}}_p{s}_p\left(x\right){\varphi }_0\left(y\right)$. All variables shown are at the equator.

Fig. 7.

The anomalous Walker circulation as a function of x over the equatorial Pacific and height z at time $t_0$ shown in Fig. 6. The Walker circulation is computed based on first baroclinic mode structure in the vertical direction (SI Appendix).

To understand the role of the zonal nonlinear advection, the budget of SST tendency $dT/d\tau$ in Eq. 3 is studied. The positive flux divergence $-\mu {\partial }_x\left(UT\right)$ in the central–western Pacific region indicates its dominant role in transporting anomalous warm water to the central and western Pacific. On the other hand, the combined effect of the latent heat $-c_1\zeta E_q$ and thermocline feedback $c_1\eta H$ leads to the increase of the anomalous warm SST only in the eastern Pacific region. In fact, even the positive component of thermocline feedback itself does not extend as much to the central–western Pacific region as the flux divergence. These results are consistent with the observational findings that the CP El Niño appears more related to zonal advection than thermocline feedback (3, 9–11).

A Two-State Markov Jump Model.

To obtain the occurrence of the CP El Niño with realistic duration and amplitude, a two-state Markov jump process (20, 32) is adopted to model the stochastic wind burst Eq. 6. Here, both ${\sigma }_p$ and ${\widehat{a}}_p$ switch between one quiescent phase (State 0) and one active phase (State 1),

$$\begin{array}{l}\begin{array}{cccc}\text{State}\hspace{0.17em}0:& {\sigma }_p=0.2,& \text{and}& {\widehat{a}}_p=0,\end{array}\\ \begin{array}{cccc}\text{State}\hspace{0.17em}1:& {\sigma }_p=1.0,& \text{and}& {\widehat{a}}_p=-\mathrm{0.25.}\end{array}\end{array}$$

The transition rates between the two states are functions of $T_W$, where a larger (smaller) $T_W$ corresponds to a higher probability of transition from State 0 (1) to State 1 (0). This setup is because wind burst activity is usually favored by warmer SST in the western Pacific and conversely (31, 33, 34). The mathematical formulae of the transition probability and the profiles of switching rates are shown in SI Appendix.

In Fig. 6, Hovmoller diagrams of different fields for a 30-y period are shown. The stochastic switching process in the wind burst model leads to the intermittent occurrence of the CP El Niño with realistic amplitude and duration. Both a 1-y CP El Niño event ($t=342$) and a series of CP El Niño events that lasts for 5 y (from $t=329$ to $t=334$) are simulated. The latter is particularly important because it resembles the two CP El Niño episodes as observed during 1990–1995 and 2002–2006 (35), which have much longer durations than the traditional El Niño.

Fig. 6.

Solutions of the coupled system with a two-state Markov jump process in the wind burst model Eq. 6. Different columns show the indicators of switching between the two states, the wind burst $u_p\left(t\right)$ at the peak $x^{*}$ of its zonal profile with its 120-d moving average (red), Hovmoller diagrams of the atmospheric wind $u+{\widehat{u}}_p$, the ocean zonal current U, the thermocline depth H, the SST field T, the SST tendency $dT/dt$, the flux divergence $-\mu {\partial }_x\left(UT\right)$, and the combined contribution due to the latent heat $-c_1\zeta E_q$ and thermocline feedback $c_1\eta H$ in Eq. 3. Here, ${\widehat{u}}_p$ is defined as ${\widehat{u}}_p={\widehat{a}}_p\left(\tau \right)s_p\left(x\right){\varphi }_0\left(y\right)$. All variables shown are at the equator.

Similar to Fig. 5, detailed analysis shows that the flux divergence is the main contributor to the occurrence of the central Pacific El Niño, where the westward zonal ocean current transports the anomalous warm water to the central Pacific region and leads to the eastern Pacific cooling with a shallow thermocline depth.

The anomalous Walker circulation at the CP El Niño phase $t_0$ is shown in Fig. 7. It is important to note that the rising branch is located at the central Pacific, where the atmosphere surface wind $u+{\widehat{u}}_p$ is westerly in the western Pacific and easterly in the eastern Pacific. These features in the Walker circulation are consistent with observations during CP El Niño years (5) and are distinct from those associated with the traditional El Niño, where the rising branch is located at the eastern Pacific.

Conclusion and Discussion

Simple dynamical models are developed here that capture the key mechanism of the CP El Niño. Systematic strategies are developed for incorporating several major mechanisms of the CP El Niño into simple coupled atmosphere–ocean processes that are otherwise deterministic, linear, and stable. First, a simple nonlinear zonal advection with no ad hoc parameterization of the background SST gradient is introduced that contributes to the 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 wind bursts including a mean easterly trade wind anomaly is coupled to the simple atmosphere–ocean processes.

The deterministic nonlinear advection model involving an easterly trade wind anomaly shows regular patterns that are associated with the CP El Niño. The irregularity of nature is recovered by involving stochastic noise in the wind burst model. To capture the intermittent occurrence of the CP El Niño with realistic amplitude and duration, effective stochastic noise that accounts for its dependence on the strength of the western Pacific warm pool through a two-state Markov jump process (20, 32) is incorporated into the wind burst model. In addition to the anomalous warm SST in the central Pacific, other major features of the CP El Niño such as the rising branch of the anomalous Walker circulation being shifted to the central Pacific and the eastern Pacific cooling with a shallow thermocline are all captured by this simple coupled model. Importantly, the coupled model succeeds in simulating a series of CP El Niño that lasts for 5 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 El Niño and the traditional El Niño and La Niña. This probably can be achieved by combining the simple dynamical model for the CP El Niño developed here with the one studied in ref. 20 that generates the traditional El Niño and super El Niño events with westerly wind bursts.