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Forced and Internal Variability of Ocean Temperature Diagnosed via Multi-Model Ensemble Linear Dynamical Mode (M-ELDM) Decomposition

This paper introduces the Multi-Model Ensemble Linear Dynamical Mode (M-ELDM) decomposition method to objectively compare historical ocean temperature simulations across Earth System Models, revealing shared forced and internal variability patterns while highlighting a significant absence of pronounced multidecadal signals in the models compared to observational data.

Original authors: Aleksei Seleznev, Maria Buyanova, Sergey Kravtsov, Andrei Gavrilov, Evgeny Loskutov, Alexander Feigin

Published 2026-09-21

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CC BY 4.0

Original authors: Aleksei Seleznev, Maria Buyanova, Sergey Kravtsov, Andrei Gavrilov, Evgeny Loskutov, Alexander Feigin

Original paper licensed under CC BY 4.0 (https://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

The ocean acts as the Earth's primary heat sponge, absorbing the vast majority of excess warmth generated by human activities and natural shifts in the atmosphere. This stored heat does not simply rise uniformly; instead, it moves in complex patterns, creating regions of warming and cooling that shift over time. Scientists have long tried to separate two distinct drivers of these changes: the steady, global warming caused by greenhouse gases, and the natural, churning variability that occurs within the climate system itself, such as the periodic warming of the Pacific known as El Niño. Understanding the difference between these two forces is critical. If we cannot accurately distinguish the human-made signal from the natural noise, our predictions for future climate change remain uncertain, leaving us unsure of how fast the planet will warm or how extreme weather might evolve.

A team of researchers has developed a new way to untangle these overlapping signals using data from six of the world's most advanced climate models. Rather than treating each model as a separate entity, they created a unified method to look at all six simulations simultaneously. This approach, called Multi-Model Ensemble Linear Dynamical Mode decomposition, allows the scientists to identify patterns that are common across all the models while isolating the unique quirks of each one. By applying this technique to sea-surface temperatures and the heat stored in the top 300 meters of the ocean from 1950 to 2014, the team was able to reconstruct the history of climate change with a new level of clarity, separating the steady march of global warming from the erratic jumps of natural variability.

The analysis revealed that the models are quite good at capturing the overall story of human-induced warming. The dominant pattern across all six models shows a clear, steady rise in ocean heat, driven by increasing greenhouse gases, with temporary dips caused by major volcanic eruptions that briefly cooled the planet. However, the models also showed significant differences in how fast they warmed and how strongly they reacted to volcanic events. For instance, one model predicted a much slower rate of warming than the others, while another showed an exaggerated cooling response to volcanic ash. These differences highlight that while the general direction of climate change is consistent, the specific speed and intensity of that change remain uncertain in current simulations.

When the researchers looked at the natural, internal variability of the ocean—specifically the slow, multi-decadal swings that occur without human influence—they found a troubling gap between the models and reality. In the real world, the ocean exhibits a massive, slow-moving wave of temperature change that spans the globe, connecting the North Atlantic, the North Pacific, and the Southern Hemisphere. This phenomenon, which the authors refer to as the "Global Stadium Wave," is clearly visible in observational data. It involves a distinct pattern where the North Atlantic warms while the Southern Hemisphere cools, and vice versa, creating a rhythmic pulse that lasts for decades.

In stark contrast, the climate models failed to reproduce this specific global rhythm. While the models did generate some slow, natural variations, their patterns looked nothing like the real-world observations. The simulated waves were weaker, lacked the specific connection between the Atlantic and Pacific oceans, and missed a crucial signature of the Pacific Decadal Oscillation that is evident in the actual data. The researchers tested whether this missing pattern might be an artifact of how they calculated the human-made warming signal, but even after adjusting for potential errors in the forced response, the models still could not generate the observed global wave. This suggests that the current generation of climate models is missing a fundamental piece of the puzzle, likely failing to capture the deep, complex interactions between different parts of the ocean that drive these long-term cycles.

The study also examined the faster, year-to-year variations associated with El Niño and La Niña. Here, the models performed better, successfully capturing the basic timing and location of these events. However, a closer look revealed that the models still struggle with the true nature of these phenomena. In the real ocean, El Niño events are highly asymmetric; strong warm events are much more intense than cold events are strong. The models, however, tend to treat warm and cold events as more balanced and symmetrical. They also underestimate the non-linear complexity of the system, failing to fully replicate the sharp, irregular behavior seen in nature. This indicates that while models can predict when an El Niño might happen, they may not be accurately simulating the extreme intensity or the specific flavor of the event.

Ultimately, this work provides a powerful new lens for comparing climate models, moving beyond simple averages to understand the specific dynamics that drive them. The findings suggest that while we have a solid grasp of the broad, human-caused warming trend, our ability to simulate the natural, multi-decadal rhythms of the climate system is still limited. The absence of the observed global wave in the models implies that we are missing key mechanisms that link different ocean basins together. Until these internal dynamics are better understood and represented, our confidence in long-term climate projections will remain constrained by this gap between what the models simulate and what the Earth actually does.

Technical Summary: Forced and Internal Variability of Ocean Temperature Diagnosed via Multi-Model Ensemble Linear Dynamical Mode (M-ELDM) Decomposition

Problem Statement
State-of-the-art Earth System Models (ESMs) generally reproduce observed historical climate behavior but exhibit divergent trajectories in future projections. This divergence creates significant uncertainty in climate forecasting. A primary challenge in reducing this uncertainty is the accurate delineation of the forced climate response (driven by external agents like greenhouse gases and aerosols) from the background internal variability (driven by chaotic internal dynamics). Traditional methods for separating these signals often rely on a priori assumptions regarding spatiotemporal differences, such as distinct time scales or specific spatial patterns, which may not hold universally. Furthermore, existing intercomparison efforts often struggle to identify robust, shared patterns across diverse models while simultaneously quantifying model-specific deviations.

Methodology
The authors develop and apply a new objective methodology termed Multi-Model Ensemble Linear Dynamical Mode (M-ELDM) decomposition. This approach extends the previously established Single-Model Ensemble LDM (S-ELDM) method (Gavrilov et al., 2024) to a multi-model context.

Core Framework: The M-ELDM decomposes a multi-model ensemble of climate simulations into a set of spatiotemporal modes. It postulates that:
1. Spatial Patterns: Both forced and internal variability modes share common spatial patterns across all models in the ensemble (though these patterns are a priori unknown).
2. Temporal Components: The time series (components) associated with these patterns differ across models. Specifically, for a given model, the forced signal is identical across all its ensemble realizations, while internal variability modes exhibit different time dependence across realizations.
Mathematical Formulation: The decomposition is formulated as a dimensionality reduction technique within a Bayesian probabilistic framework. The model represents the data $x^{(m,k)}_n$ (where $k$ is the model index and $m$ is the realization index) as:
$x^{(m,k)}_n = A p^{(m,k)}_n + B f^{(k)}_n + c + \sigma \xi^{(m,k)}_n$
Here, $A$ and $B$ are matrices of spatial patterns for internal and forced modes, respectively; $p^{(m,k)}_n$ and $f^{(k)}_n$ are the corresponding time series; and $\xi$ is Gaussian white noise.
Optimization: All parameters, including the number of modes, their time scales, and spatial patterns, are determined objectively via Bayesian optimization. The method minimizes a loss function comprising a negative log-likelihood term, a smoothing regularization term (enforcing linear stochastic process priors with specific time scales), and an $L_2$ regularization term. This avoids overfitting and eliminates the need for assumptions about characteristic time scales separating forced and internal signals.
Data Application: The method was applied to Sea Surface Temperature (SST) and Upper-Ocean Heat Content (OHC, 0–300m) anomalies from a multi-model ensemble of six ESMs (including CMIP6 models and INMCM6P) with 10 realizations each (1950–2014). Observational data (ERSSTv5 and IAPv4) were analyzed using a related LDM approach where the forced signal time series was prescribed based on the multi-model ensemble mean.

Key Contributions

M-ELDM Methodology: The paper introduces a unified framework for analyzing multi-model ensembles that identifies robust spatial patterns common to all models while encapsulating model differences in their respective time series.
Objective Separation: Unlike dynamical adjustment or signal-to-noise maximizing pattern methods, M-ELDM does not assume distinct time scales or spatial structures between forced and internal modes, relying instead on the consistency of the forced signal across ensemble members of a single model.
Ranking Strategy: The authors devise a strategy to rank model behaviors against observations by comparing the reconstructed forced and internal modes derived from the M-ELDM decomposition.

Results

Forced Response: The M-ELDM identified that the dominant forced response in both SST and OHC is captured by a single leading mode (explaining 95% of SST and 85% of OHC forced variance). This mode reflects the global warming trend modulated by anthropogenic aerosols (pre-1980 cooling) and volcanic eruptions. While the simulated patterns match observations in the Atlantic and Indian Oceans, they diverge significantly in the Pacific, lacking the pronounced Pacific Decadal Oscillation (PDO)-like signature evident in observational data. The method successfully highlighted inter-model differences in climate sensitivity (e.g., INMCM5 showing the lowest warming rate, CanESM5 the highest) and volcanic response.
Multidecadal Internal Variability: The analysis of low-frequency internal modes revealed that models fail to reproduce the pronounced multidecadal signals observed in the data. Specifically, the observed "Global Stadium Wave" (GSW)—characterized by an AMO-like component and an AMO-orth component linked to PDO-like variability—was not robustly captured by the models. The simulated multidecadal patterns were spatially distinct and exhibited lower signal-to-noise ratios.
Hypothesis on GSW Origin: The authors tested whether the observed GSW could be an artifact of forced-signal biases. By constructing a "F+I" dataset (simulated low-frequency modes minus the ensemble-mean forced response), they found that the resulting patterns still differed significantly from observations. This suggests that current climate models are missing or misrepresenting the dynamics underlying the observed GSW, rather than the signal being purely an artifact of forced-response estimation errors.
ENSO Variability: The method effectively separated quasi-biennial and quasi-quadrennial ENSO signals. While models captured the leading spatial patterns, they substantially underestimated the nonlinearity and asymmetry of ENSO events (specifically the skewness between El Niño and La Niña). Regarding ENSO diversity, the study found that while most models exhibited biases in reproducing Central Pacific (C) and Eastern Pacific (E) ENSO flavors, the HadGEM3 and MPI-ESM1 models were able to reproduce the spatial patterns and intensity of these observed modes relatively well.

Significance
The paper claims that the M-ELDM decomposition provides an efficient, objective tool for intercomparing climate models. By identifying robust low-dimensional subsets of forced and internal modes shared across models, it allows for a direct comparison of model behaviors and time series. The study demonstrates that while models capture the general forced warming trend, they exhibit significant deficiencies in representing specific internal variability features, particularly the multidecadal GSW and the nonlinear asymmetry of ENSO. The authors conclude that these discrepancies suggest current models underestimate the intensity and/or coupling of internal climate "oscillators," leading to a misrepresentation of both internal variability and the associated dynamical forced response. The method is presented as a versatile framework applicable to any number of models and climatic fields, useful for model intercomparison, isolating forced signals in observational data, and constructing data-driven climate emulators.

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