Intermediate Coupled MJO–ENSO Model

• The paper presents an intermediate coupled climate model that integrates full intraseasonal MJO dynamics with interannual and decadal ENSO variability.
• It employs explicit two-way air–sea feedbacks, state-dependent multiplicative stochasticity, and meridional truncation to simulate realistic non-Gaussian SST statistics and ENSO diversity.
• Implications include enhanced seasonal forecasting, improved extreme event risk assessment, and a low-cost framework bridging conceptual models with data-driven approaches.

Searching arXiv for the specified topic and related papers to ground the article in current preprints. The Intermediate Coupled MJO–ENSO Model is an intermediate coupled climate model class designed to represent multiscale interactions between the Madden–Julian Oscillation (MJO), the El Niño–Southern Oscillation (ENSO), and slower background modulation associated with the Walker circulation. In the formulation developed in "A Simple Intermediate Coupled MJO-ENSO Model: Multiscale Interactions and ENSO Complexity" (Zhang et al., 18 Jul 2025), the model explicitly bridges intraseasonal, interannual, and decadal variability through two-way air–sea feedbacks, stochastic parameterizations, and a reduced but dynamically structured representation of equatorial atmosphere–ocean coupling. Its stated purpose is to address ENSO complexity, including spatial diversity between Eastern Pacific (EP) and Central Pacific (CP) events, temporal diversity between single-year and multi-year episodes, and intensity diversity from moderate to extreme events (Zhang et al., 18 Jul 2025).

1. Concept and classification

An intermediate coupled MJO–ENSO model lies between fully coupled general circulation models and simple conceptual oscillators. In the 2025 formulation, the model is described as a statistically accurate, computationally efficient intermediate coupled model that explicitly bridges three separated but interacting tropical climate time scales: intraseasonal Madden–Julian Oscillation, interannual El Niño–Southern Oscillation, and decadal modulation by the Walker circulation (Zhang et al., 18 Jul 2025). A related stochastic conceptual formulation likewise classifies the model as intermediate coupled because it combines a dynamical ocean component with a low-order atmospheric component built from the stochastic skeleton model for the MJO, while also incorporating decadal variability and state-dependent noise (Moser et al., 2024).

The defining feature of this model class is explicit multiscale coupling. In (Zhang et al., 18 Jul 2025), the coupling is operationalized by two-way air–sea feedbacks between interannual sea surface temperature and the atmosphere, injection of intraseasonal MJO wind anomalies into interannual ocean wind stress forcing, and a slowly varying decadal index that modulates the zonal advective feedback and the statistics of intraseasonal moisture production. This architecture is intended to alternate regimes favorable to EP versus CP El Niño diversity. A plausible implication is that the model is not merely a reduced ENSO oscillator with stochastic forcing; rather, it is a structured multiscale system in which the source of intraseasonal variability is dynamically represented.

Relative to earlier reduced-order models, the novelty emphasized in (Zhang et al., 18 Jul 2025) is the integration of an MJO skeleton with an intermediate ENSO ocean–atmosphere core using state-dependent multiplicative stochasticity to capture observed non-Gaussian statistics and intermittency. The paper also introduces a simple decadal Walker index that modulates zonal advection and intraseasonal moisture variability, producing realistic alternation of CP and EP regimes and extreme EP events (Zhang et al., 18 Jul 2025). The earlier stochastic conceptual model in (Moser et al., 2024) makes the same broad point in a lower-order setting, stating that explicit MJO skeleton dynamics, stochastic state-dependent coupling, and slow decadal modulation extend beyond the scope of Zebiak–Cane-type and recharge-oscillator frameworks.

2. Multiscale architecture and governing variables

The 2025 model separates the tropical climate system into intraseasonal, interannual, and decadal components with distinct state variables and equations (Zhang et al., 18 Jul 2025). The intraseasonal MJO component, on $30$–$90$ day time scales, uses full prognostic atmospheric skeleton dynamics for zonal wind $u'(x,y,t)$, meridional wind $v'(x,y,t)$, potential temperature $\theta'(x,y,t)$, lower-tropospheric moisture $q'(x,y,t)$, and convective envelope $a'(x,y,t)$. The interannual ENSO component, on $2$–$7$ year time scales, consists of a deterministic Matsuno–Gill steady response for atmospheric background fields $\bar{u}(x,y,t)$, $90$0, $90$1, and $90$2, coupled to shallow-water long-wave ocean dynamics for zonal and meridional currents $90$3, $90$4, thermocline depth $90$5, and a reduced mixed-layer SST anomaly $90$6. The decadal component is a scalar Walker circulation index $90$7 governed by an Ornstein–Uhlenbeck-type stochastic evolution (Zhang et al., 18 Jul 2025).

The atmosphere and ocean occupy distinct meridional coordinates, with $90$8 reflecting different deformation radii. The atmospheric domain is periodic in $90$9, $u'(x,y,t)$0, whereas the Pacific ocean domain spans $u'(x,y,t)$1 with reflecting boundary conditions (Zhang et al., 18 Jul 2025). Primes denote intraseasonal anomalies and bars denote interannual components. This notation enforces time-scale separation while preserving coupling among the components.

The full intraseasonal atmospheric skeleton equations are given as

$u'(x,y,t)$2

Here $u'(x,y,t)$3, $u'(x,y,t)$4, and $u'(x,y,t)$5 are linear damping rates; $u'(x,y,t)$6 is the convective heating rate factor; $u'(x,y,t)$7 is the background vertical moisture gradient; and $u'(x,y,t)$8 is the envelope growth or decay coefficient (Zhang et al., 18 Jul 2025). The presence of multiplicative noise in both moisture and convective-envelope tendencies is central to the model’s treatment of intermittency and extremes.

The interannual atmospheric background is represented by a steady Matsuno–Gill response to diabatic heating: $u'(x,y,t)$9 The ocean obeys the shallow-water long-wave approximation,

$v'(x,y,t)$0

and the SST tendency is

$v'(x,y,t)$1

The coupling closures are

$v'(x,y,t)$2

In this SST budget, $v'(x,y,t)$3 is a thermocline feedback kernel that peaks in the EP, $v'(x,y,t)$4 is a zonal advective feedback kernel that peaks in the CP, $v'(x,y,t)$5 is a seasonally modulated and weakly nonlinear latent-heating exchange or damping coefficient, and $v'(x,y,t)$6 increases with Walker strength $v'(x,y,t)$7 (Zhang et al., 18 Jul 2025).

3. Reduced wave formulation and physical reconstruction

For analysis and efficient simulation, the 2D equations are projected onto the leading parabolic cylinder modes, yielding a one-dimensional $v'(x,y,t)$8–$v'(x,y,t)$9 wave-equation form for equatorial Kelvin and Rossby modes in both atmosphere and ocean (Zhang et al., 18 Jul 2025). This meridional truncation preserves zonal propagation and Kelvin–Rossby teleconnections while substantially reducing complexity.

The truncated intraseasonal atmosphere is written in terms of Kelvin and Rossby amplitudes $\theta'(x,y,t)$0 and $\theta'(x,y,t)$1, truncated moisture $\theta'(x,y,t)$2, and convective envelope $\theta'(x,y,t)$3: $\theta'(x,y,t)$4 The interannual atmospheric background becomes a steady Kelvin–Rossby response,

$\theta'(x,y,t)$5

with periodic boundary conditions and

$\theta'(x,y,t)$6

The ocean truncation yields

$\theta'(x,y,t)$7

with reflecting boundaries

$\theta'(x,y,t)$8

The reduced SST equation becomes

$\theta'(x,y,t)$9

and the reduced coupling terms are

$q'(x,y,t)$0

Physical fields are reconstructed from the wave variables by

$q'(x,y,t)$1

where $q'(x,y,t)$2 and $q'(x,y,t)$3 are atmospheric and oceanic parabolic cylinder functions (Zhang et al., 18 Jul 2025). This preserves a direct diagnostic link between the reduced wave dynamics and the underlying physical variables.

A plausible implication is that the model’s interpretability comes largely from this wave-based reduction. The papers explicitly state that spatial structure is preserved through meridional truncation, while computational cost remains much lower than in fully coupled GCMs (Zhang et al., 18 Jul 2025).

4. Coupling mechanisms across intraseasonal, interannual, and decadal scales

The central scientific purpose of the model is to represent bidirectional MJO–ENSO interaction rather than unidirectional forcing (Zhang et al., 18 Jul 2025). Three mechanisms organize the coupling.

First, MJO to ENSO coupling occurs through intraseasonal wind anomalies entering the oceanic wind stress. In (Zhang et al., 18 Jul 2025), intraseasonal winds force equatorial ocean Kelvin and Rossby waves, thereby adjusting $q'(x,y,t)$4 and $q'(x,y,t)$5. Westerly wind events deepen the thermocline through downwelling Kelvin waves and accelerate eastward currents, initiating EP warming, while easterly wind events can stall or terminate events. The same mechanism appears in the earlier multiscale stochastic model, where MJO-related wind bursts are represented as an Ornstein–Uhlenbeck process whose amplitude is modulated by state, season, and decadal background (Chen et al., 2022). That earlier formulation does not contain explicit MJO propagation dynamics, whereas (Zhang et al., 18 Jul 2025) embeds full intraseasonal skeleton dynamics and injects those wind anomalies directly into interannual ocean forcing.

Second, ENSO to MJO coupling occurs through SST-modulated latent heating. In (Zhang et al., 18 Jul 2025), $q'(x,y,t)$6 increases diabatic heating, strengthens the interannual convective envelope $q'(x,y,t)$7, and modulates the intraseasonal stochastic coefficients. The convective-envelope noise is

$q'(x,y,t)$8

which enhances convective-envelope variability when SST and $q'(x,y,t)$9 are large. The moisture noise $a'(x,y,t)$0 is described as approximately tanh-type in $a'(x,y,t)$1, with spatial dependence through $a'(x,y,t)$2 and decadal dependence through $a'(x,y,t)$3, enhancing moisture variability and intraseasonal winds when SST is warm and Walker circulation is weak (Zhang et al., 18 Jul 2025). This mechanism is closely aligned with the state-dependent noise formulation in (Moser et al., 2024), where warmer CP SST increases latent heat $a'(x,y,t)$4, directly strengthening atmospheric convective background and moisture noise.

Third, decadal modulation enters through the Walker circulation index $a'(x,y,t)$5. In (Zhang et al., 18 Jul 2025), $a'(x,y,t)$6 follows

$a'(x,y,t)$7

with $a'(x,y,t)$8 interpreted as a nonnegative $a'(x,y,t)$9–$2$0 Walker strength index measuring lower-level trades or easterlies. The index modulates the zonal advective SST feedback through $2$1 and also modulates the variance of intraseasonal moisture noise. Large $2$2 amplifies CP zonal advection and favors CP events, whereas small $2$3 suppresses advection and boosts moisture noise, favoring EP events through thermocline feedback (Zhang et al., 18 Jul 2025). The 2024 conceptual model makes the same regime argument in a three-box setting, stating that small $2$4 corresponds to an EP ENSO regime and large $2$5 to a CP regime (Moser et al., 2024).

An additional mechanism identified explicitly in (Zhang et al., 18 Jul 2025) is warm pool edge extension. Enhanced MJO activity and westerly wind events drive downwelling Kelvin waves that push the $2$6 isotherm eastward. In the model, increased wind stress deepens $2$7 in the west and central Pacific and accelerates $2$8, so the $2$9 term warms the CP; persistent westerly events then advect warm water into the EP, engaging the $7$0 thermocline term. Since $7$1 strengthens the Matsuno–Gill response and helps maintain intraseasonal wind variability over the warm pool, the model reproduces warm pool edge extension as a precursor to El Niño (Zhang et al., 18 Jul 2025).

5. Stochastic parameterizations, non-Gaussianity, and ENSO diversity

State-dependent stochasticity is not an auxiliary feature but a defining component of the model class. The 2025 paper states that effective stochastic parameterizations are incorporated to improve the characterization of multiscale MJO–ENSO interactions and the emergence of intermittency and extremes (Zhang et al., 18 Jul 2025). The intraseasonal moisture noise is multiplicative and state dependent on $7$2, spatially dependent through $7$3, and decadal-state dependent through $7$4. Physically, it represents unresolved moist convection and variability in wind bursts or coupled waves. The convective-envelope noise is also multiplicative and is said to be mathematically justified for skeleton dynamics (Zhang et al., 18 Jul 2025).

The role of these stochastic terms is diagnosed most clearly through their statistical consequences. The paper reports that multiplicative noise is essential for heavy-tailed SST probability density functions and extreme-event intermittency. Explicit Fokker–Planck analysis is not carried out, but the importance of multiplicative noise is emphasized repeatedly (Zhang et al., 18 Jul 2025). The earlier stochastic conceptual model likewise states that removing moisture noise weakens ENSO–MJO lagged correlation, reduces westerly wind bursts, underrepresents extreme EP events, and causes ENSO PDFs to lose heavy tails; removing convective noise makes MJO variability too regular and overly deterministic (Moser et al., 2024).

ENSO diversity in the 2025 model emerges from competition between zonal advection and thermocline feedback. The diagnostic ratio is

$7$5

CP events are favored when $7$6, typically for large $7$7, whereas EP events are favored when $7$8, typically for small $7$9 (Zhang et al., 18 Jul 2025). The linear interannual core is described as a damped oscillator with two slowly decaying leading eigenmodes: an EP-like mode with period about $\bar{u}(x,y,t)$0 years for $\bar{u}(x,y,t)$1 and $\bar{u}(x,y,t)$2, and a CP-like mode with period about $\bar{u}(x,y,t)$3 years for $\bar{u}(x,y,t)$4 and $\bar{u}(x,y,t)$5 (Zhang et al., 18 Jul 2025). In the earlier ENSO-diversity model, a closely related eigenmode interpretation is given, with EP-dominant and CP-dominant regimes corresponding to distinct leading eigenvectors of the linear operator, with characteristic periods of approximately $\bar{u}(x,y,t)$6 and $\bar{u}(x,y,t)$7 years respectively (Chen et al., 2022). This suggests a persistent structural idea across the model lineage: diversity is represented as competition between thermocline-dominated and advection-dominated oscillatory backbones.

The model also distinguishes single-year from multi-year ENSO events. In (Zhang et al., 18 Jul 2025), multi-year episodes arise when the interannual oscillator’s relaxation is slowed by sustained wind forcing, constructive decadal modulation, and seasonally synchronized $\bar{u}(x,y,t)$8 enabling winter peaks and carryover. Extreme events are reported to be more frequent under small $\bar{u}(x,y,t)$9, when Walker circulation is weakened, thermocline feedback dominates, and $90$00 is large enough to permit sequences of westerly wind events that deepen $90$01 and advect warm anomalies eastward (Zhang et al., 18 Jul 2025). Weakly nonlinear $90$02 provides effective cubic damping that prevents unbounded growth while allowing fat-tailed SST PDFs.

6. Diagnostics, validation, and numerical implementation

The model is validated against a set of statistical and dynamical diagnostics intended to reproduce observed MJO and ENSO behavior. In (Zhang et al., 18 Jul 2025), the model is reported to capture non-Gaussian statistics, seasonal cycles, energy spectra, and spatial event patterns. Seasonal phase locking is imposed through

$90$03

which yields winter-peaked SST variance and observed timing of ENSO maxima. Power spectra of SST and wavenumber–frequency spectra of $90$04 and $90$05 show intraseasonal peaks at $90$06–$90$07 days for wavenumbers $90$08–$90$09, consistent with MJO dispersion, and interannual SST peaks at $90$10–$90$11 years (Zhang et al., 18 Jul 2025).

Probability density functions of SST are strongly non-Gaussian. The 2025 model reports negative skew in Niño4, slight positive fat tail in Niño3.4, and a large positive fat tail in Niño3, corresponding to extreme events (Zhang et al., 18 Jul 2025). The earlier ENSO-diversity model reports the same qualitative structure and explicitly attributes suppressed CP extremes to cubic-like damping in the CP region (Chen et al., 2022). The 2024 stochastic conceptual MJO–ENSO model also reproduces positive skewness and one-sided fat tails in Niño3, negative skewness in Niño4, and seasonal variance peaking in boreal winter (Moser et al., 2024).

Air–sea lag relationships constitute another key diagnostic. In (Zhang et al., 18 Jul 2025), modeled lag correlations of $90$12, $90$13, $90$14, and $90$15 against Niño3.4 are said to match observations: westerly wind events precede warming, thermocline signals lead SST changes in the west and central Pacific, and the MJO peak shifts east during El Niño. Wind-event statistics are also reproduced, including winter prevalence, concentration in the western and central Pacific, and eastward shift during El Niño (Zhang et al., 18 Jul 2025).

The paper further states that event counts over $90$16-year windows agree with observations in total El Niño occurrence, EP or CP partition, multi-year episodes, and extreme-event frequencies within spread. For extremes, the observed value is approximately $90$17–$90$18, while the model produces $90$19 with the full decadal range (Zhang et al., 18 Jul 2025). Sensitivity experiments indicate that constraining or fixing $90$20 reduces EP and extreme events: with $90$21, EP events drop from $90$22 to $90$23 and extremes to $90$24; with fixed $90$25, EP events fall further to $90$26 and extremes become very rare at $90$27 (Zhang et al., 18 Jul 2025).

Numerically, the implementation is based on 1D $90$28–$90$29 equations for $90$30, $90$31, $90$32, $90$33, $90$34, $90$35, $90$36, and $90$37, after meridional truncation (Zhang et al., 18 Jul 2025). The paper specifies a uniform grid in $90$38, time stepping with explicit or semi-implicit schemes such as leapfrog or Adams–Bashforth for advection, forward Euler for stochastic terms, and an occasional Robert–Asselin filter for leapfrog. Gaussian white-noise increments are generated for $90$39, $90$40, and $90$41 at each step, and multiplicative amplitudes are evaluated from the current state. The stated algorithm advances the interannual atmosphere, forms $90$42, advances ocean waves, updates SST, advances the intraseasonal atmosphere, and then advances the Walker index (Zhang et al., 18 Jul 2025).

7. Relation to prior models, implications, and limitations

The intermediate coupled MJO–ENSO model is part of a progression from reduced ENSO diversity models with stochastic wind bursts toward explicit multiscale MJO–ENSO coupling. The 2022 model in (Chen et al., 2022) already combined a deterministic linear interannual atmosphere–ocean–SST core, a stochastic intraseasonal wind-burst process, a decadal Walker SDE, and cubic-like damping to reproduce ENSO diversity and complexity. However, that model represented MJO influence statistically through an Ornstein–Uhlenbeck wind-burst module rather than through full prognostic MJO skeleton dynamics. The 2024 conceptual model (Moser et al., 2024) moved closer to explicit coupling by adopting a low-order Fourier representation of Kelvin, Rossby, moisture, and convective activity modes for the atmosphere, a three-box ENSO ocean component, and state-dependent stochasticity. The 2025 model (Zhang et al., 18 Jul 2025) advances this line further by embedding full stochastic MJO skeleton dynamics within an intermediate coupled ocean–atmosphere framework and by linking intraseasonal, interannual, and decadal processes within a unified reduced model.

Several implications are stated directly in (Zhang et al., 18 Jul 2025). For seasonal forecasting, explicitly resolving intraseasonal wind impacts and decadal modulation can improve ENSO diversity predictions and extreme-event risk assessment. For extreme ENSO prediction, multiplicative noise combined with seasonal phase locking reproduces fat tails and intermittency, which are described as crucial for probabilistic risk. For climate resilience, the model’s computational efficiency enables long simulations for statistical characterization and machine-learning training, while air–sea lag diagnostics clarify precursors such as westerly wind events and warm pool edge extension (Zhang et al., 18 Jul 2025). This suggests that the model is intended not only as a mechanistic research tool but also as a bridge between conceptual understanding and statistical or data-driven applications.

The limitations are equally explicit. The 2025 model uses reduced physics: a 1D meridional truncation, long-wave shallow-water ocean dynamics, simplified moisture-mode dynamics, and a steady Matsuno–Gill interannual atmosphere rather than a prognostic one. It omits full 3D oceanic processes, mixed-layer depth variability, detailed radiation and cloud feedbacks, explicit extratropical teleconnections, and interactive Indo-Pacific or anthropogenic forcing pathways (Zhang et al., 18 Jul 2025). The decadal variability is stylized as OU-type noise rather than a mechanistically resolved mean-state evolution. Proposed future improvements include explicit mixed-layer heat capacity and vertical entrainment $90$43, prognostic interannual atmospheric momentum and thermodynamics, refined stochastic parameterization of westerly and easterly wind bursts with colored noise, interactive warm water volume, and teleconnections (Zhang et al., 18 Jul 2025).

A common misconception is that intermediate coupled models necessarily sacrifice realistic statistics for simplicity. The model results summarized in (Zhang et al., 18 Jul 2025, Moser et al., 2024), and (Chen et al., 2022) argue against that view within their stated scope: these models reproduce seasonal phase locking, non-Gaussian SST PDFs, intraseasonal and interannual spectra, EP–CP diversity, and realistic event-frequency statistics. The counterpoint, stated in the same papers, is that this success relies on carefully chosen state-dependent stochastic closures and stylized feedbacks rather than on comprehensive physical resolution. The appropriate interpretation is therefore not that the model replaces comprehensive climate models, but that it provides an interpretable, low-cost framework for analyzing multiscale tropical variability and its extremes (Zhang et al., 18 Jul 2025).