The core insight of the new work is that when only a partial view of a complex system is available, the observed variables become history-dependent. In the language of dynamical systems, the SST-only evolution is effectively non-Markovian: knowing the temperature field today is not sufficient to determine the field next month, because hidden variables—the atmospheric winds, the deeper ocean, unresolved eddies—carry memory that leaks into the surface record. Traditional approaches either ignore this memory or approximate it with time-delay coordinates, the classic Takens embedding, in which past snapshots are stacked into a vector. Sugiura and colleagues take a different and arguably more faithful route: instead of collapsing history into an unordered vector, they treat each year of SST data as an ordered path, a continuous trajectory through a high-dimensional space that preserves the temporal sequence of monthly anomalies within the year.

Representing dynamics as paths, however, creates a new mathematical challenge. Koopman analysis, the framework the authors adopt, offers a way out of nonlinearity by shifting attention from states to observables. First proposed by Bernard Koopman in 1931, the Koopman operator is a linear operator that acts on scalar-valued functions of the state, propagating them forward in time. The remarkable property is that even when the underlying dynamics are strongly nonlinear, the evolution of observables can be exactly linear—an infinite-dimensional linearity, but one that can be approximated in finite dimensions. Extended dynamic mode decomposition, or EDMD, and its kernelized variant kEDMD have become the workhorse tools for this approximation, projecting dynamics onto a chosen function space or a reproducing kernel Hilbert space. The question is which function space to use when the states themselves are entire annual trajectories.

This is where path signatures enter the picture. Rooted in the theory of rough paths developed by Terry Lyons, the signature of a path is a canonical collection of iterated integrals that captures the temporal order, area, and higher-order interactions along a trajectory. Signatures form a feature representation so expressive that sufficiently regular functionals of paths can be approximated by linear functionals of these features. Computing signatures explicitly, however, becomes prohibitive for high-dimensional data—the number of features grows combinatorially with truncation depth—and SST fields, with their vast spatial grids, are nothing if not high-dimensional. The authors sidestep this bottleneck by kernelizing the signature: following work by Kiraly and Oberhauser, they use a signature kernel that implicitly compares two paths through their iterated-integral features, computing inner products in an RKHS without ever constructing the features themselves.

The resulting pipeline is a single, coherent procedure. Monthly SST fields are first converted into anomalies using a strictly past-only rolling climatology, ensuring that no future information contaminates the preprocessing—a crucial discipline for honest out-of-sample testing. Twelve consecutive monthly anomalies are then grouped into an annual segment and embedded as a piecewise-linear path via cumulative summation, so that the path’s increments are precisely the monthly anomalies. The one-year shift operator—the map that carries one annual path to the next—is learned by applying kEDMD to Gram and cross-Gram matrices built from the truncated signature kernel, with hyperparameters tuned through a kernel-alignment objective that measures the normalized similarity between predicted and true paths. The output is a finite-dimensional Koopman matrix whose spectrum and iterates serve double duty: multiplying it forward yields multiyear forecasts, while its eigendecomposition yields oscillatory modes with well-defined periods and amplitudes.

Before confronting real ocean data, the team validated the approach on a controlled synthetic benchmark: the two-scale Lorenz–96 system, a standard testbed in which slow variables are coupled to fast ones, mimicking exactly the partial-observability situation of SST. Only the slow variables were used as observables and prediction targets; the fast variables acted as unresolved stochastic-like forcing, formally inducing memory and random forcing in the reduced slow-only description, in the spirit of the Mori–Zwanzig formalism. Across three coupling regimes, the signature-kernel method matched an explicit truncated-signature EDMD almost perfectly—confirming the kernel implementation—and outperformed baselines that used either block-mean states or conventional snapshot DMD, particularly at intermediate and longer lead times. The message was clear: the ordered structure within trajectory segments carries predictive information that averages and instantaneous snapshots destroy.

Applied to observed SST, the method delivered on its promise. Forecast experiments were conducted under two rigorous time-ordered protocols: leave-future-out evaluation for forecasting skill and leave-start-out splits for spectral diagnostics, both ensuring that model training never touched data from the verification period. Against a climatology baseline that simply repeats the anchor-year month-of-year climatology, the learned Koopman operator improved out-of-sample multiyear forecast skill, with performance measured both by area-weighted RMSE on anomaly fields and by a novel kernel-based path correlation, the kPC, which scores the similarity between predicted and true annual paths in the signature-kernel space. Equally important, the eigenspectrum of the learned one-year operator revealed coherent spectral modes organized in band-like structures in period–amplitude space, with representative modes at interannual and longer periods—the kind of structured oscillatory signal that climate scientists associate with phenomena such as the El Niño–Southern Oscillation and other basin-scale variability.

The contrast with the other dominant paradigm in modern geophysical forecasting—neural operators—is instructive. Architectures such as Fourier neural operators, and systems like OceanNet, which demonstrates competitive seasonal prediction for regional ocean dynamics, learn powerful nonlinear mappings between input and output fields and transfer well across grid resolutions. But their predictions emerge from compositions of linear and nonlinear layers, so no native eigenvalues, eigenfunctions, or Koopman modes fall out of the model; spectral diagnostics require additional post-processing or are simply unavailable. The signature-kernel kEDMD approach, by contrast, yields an explicit linear operator from which forecasting and spectral analysis follow from the same object. For scientists seeking not just predictions but physical interpretation—the identification of oscillatory patterns, their decay rates, their amplitudes in physical temperature units—this linearity is a decisive advantage.

The method also highlights a subtle conceptual shift in how memory is handled. Where delay embeddings represent the past as a static vector of coordinates, the path representation retains explicit order structure: the signature distinguishes a year in which temperature anomalies rose sharply in spring and plateaued in autumn from one in which they drifted gradually upward, even if the set of monthly values were identical. Higher-order signature terms are sensitive to increment magnitude as well as order, which is why the authors normalize cumulative paths by a data-dependent scale parameter before feature construction, ensuring that the kernel reflects relative temporal structure rather than absolute amplitude. In the SST experiments, the base kernel on the spatial field is a radial basis function with a bandwidth fixed by the dataset itself, tying the geometry of path space to the physical statistics of the ocean.

What emerges from this study is a template that could extend well beyond sea surface temperature. Any climate or geophysical variable observed only partially—ocean heat content, sea ice extent, atmospheric composition—suffers from the same effective non-Markovianity that has limited purely snapshot-based statistical models. By lifting the state along the time axis into path space and then lifting nonlinearity away through the Koopman operator, the signature-kernel pipeline offers a way to learn linear dynamics from data that is simultaneously memory-robust, scalable to high dimensions, and transparent to spectral interpretation. The authors’ demonstration that a single learned operator can both beat climatology at multiyear horizons and expose coherent modes of variability suggests that the trajectory-based view of climate dynamics may be more than a mathematical curiosity—it may be the form that the ocean’s memory actually takes.

The research, led by Michael Aich of the Technical University of Munich and the Potsdam Institute for Climate Impact Research, together with Sebastian Bathiany, Philipp Hess, Yu Huang and Niklas Boers, addresses one of the most persistent weaknesses in modern climate simulation: precipitation. Earth system models, or ESMs, are the workhorses of climate projection, numerically solving the equations of atmospheric motion on discretized grids. But their grids are coarse, typically on the order of one degree or more, and the processes that actually generate rain — convection, cloud microphysics, the organization of thunderstorm complexes — unfold at scales far smaller than a single grid cell. To cope, modelers rely on parameterizations: simplified, column-based schemes that approximate subgrid physics independently at each vertical column of the atmosphere. These schemes are computationally expensive, depend on subjective calibration, and ignore horizontal interactions between neighboring columns, which makes it difficult to capture phenomena such as mesoscale convective organization. The result is a suite of systematic biases, the most famous being the “double Intertropical Convergence Zone” problem, in which models paint excessive rainfall across the southern tropics.

The new framework sidesteps these column-based assumptions entirely. Instead of treating each atmospheric column as an isolated regression problem, the researchers’ models operate on global two-dimensional fields, allowing information to flow between distant locations and between different physical variables. The inputs are deliberately minimal: specific humidity at the 850 hectopascal pressure level, the eastward and northward wind components at ten meters above the surface, and mean sea-level pressure — all at a coarse one-degree resolution. Remarkably, the authors show that these four surface and near-surface fields contain enough implicit information about the three-dimensional state of the atmosphere to reconstruct detailed rainfall patterns. Mean sea-level pressure, for instance, captures the integrated mass of the entire air column, while surface winds are intrinsically coupled to the larger-scale circulation.

The pipeline works in two stages. The first is a deterministic UNet regression model, a convolutional neural network of roughly 24 million parameters with four downsampling stages built from residual blocks and a single bottleneck attention layer. Trained on the ERA5 reanalysis — a state-of-the-art dataset produced by the European Centre for Medium-Range Weather Forecasts that fuses observations with numerical weather prediction — the network learns to map the four coarse atmospheric variables to one-degree daily precipitation. A single forward pass through this network takes about nine milliseconds on a modern Nvidia H100 graphics card.

The second stage is where the generative magic happens. A conditional denoising diffusion probabilistic model, an architecture descended from the same family of techniques behind modern image generators, is trained to transform the coarse one-degree precipitation estimate into a genuine quarter-degree field — a sixteenfold increase in the number of grid cells, from a 180-by-360 global grid to 720 by 1440. Diffusion models work by learning to reverse a gradual noising process; here, the model is conditioned on a degraded, noisy version of the coarse precipitation field and learns to regenerate the fine-scale spatial texture that such coarse inputs lack. The researchers inject Gaussian noise of a carefully calibrated magnitude into the conditioning field during training, chosen by analyzing where the spatial power spectra of the coarse and fine datasets intersect, at a threshold scale of roughly 900 kilometers. Scales larger than this threshold are preserved from the conditioning input; scales below it are regenerated stochastically in a manner consistent with the ERA5 distribution. This design turns the diffusion model into both a downscaling engine and a bias-corrector: it wipes away the flawed, overly smooth small-scale features of the coarse input and replaces them with statistically realistic texture, while anchoring the result to the trustworthy large-scale structure.

Bridging the gap between reanalysis data and climate model output posed a further challenge, because the statistics of ESM fields differ from those of ERA5. The researchers solved this with quantile delta mapping, a bias-correction technique fitted on the historical overlap between the model and the reanalysis, applied to the regression output before it enters the diffusion stage. The noise-injection step then serves double duty as a domain adaptation device: by corrupting the small-scale information of the bias-corrected field to the same degree as during training, it renders the inference-time data distribution approximately identical to the training distribution, allowing the diffusion model to operate on climate model data it has never seen.

The results are striking. When applied to the GFDL-ESM4 model from the U.S. National Oceanic and Atmospheric Administration, the framework reduced the mean absolute precipitation bias from 0.561 millimeters per day in the raw climate model to 0.164 millimeters per day after the regression stage, and further to 0.111 millimeters per day after the diffusion stage. The notorious double ITCZ bias, glaring in the GFDL fields, largely disappears. Spatial power spectral analysis reveals why: the climate model’s precipitation is blurry, lacking variability at scales below about 400 kilometers, whereas the diffusion model’s output aligns closely with the observed spectrum down to the finest resolved scales. The generated fields also reproduce the distribution of rainfall intensities far more faithfully, capturing both the frequency of moderate events and the rare extremes exceeding 150 millimeters per day.

Extreme events, arguably where accurate precipitation matters most for society, received particular attention. Using the R95p index — the annual total precipitation falling on days above the 95th percentile — the team found that the AI-generated fields track ERA5 closely during the historical period, while the raw GFDL model overestimates tropical extreme rainfall. Under the SSP3-7.0 high-emissions scenario, all approaches agree that extreme precipitation will intensify in the tropics by the late twenty-first century, but the diffusion model projects a more moderate amplification than either the raw climate model or the regression stage alone, suggesting it partially corrects the wet bias baked into the driving model. Indices of consecutive wet and dry days, critical for flood and drought assessment, are likewise reproduced more faithfully than by the original ESM.

A distinctive advantage of the generative approach is uncertainty quantification. Because the diffusion model is stochastic, running it fifty times on the same coarse input yields fifty equally plausible high-resolution realizations, forming an ensemble that reflects the inherent ambiguity of inferring fine detail from coarse data. Evaluating this ensemble with the continuous ranked probability score over a validation year, the researchers found a mean score of 0.56 millimeters per day — better than both a bi-linearly interpolated ERA5 baseline at 0.73 and the interpolated regression output at 1.42 millimeters per day. The spread-skill relationship of the ensemble roughly follows the ideal one-to-one line, indicating that the model’s stated uncertainty is well calibrated rather than overconfident or underconfident.

Perhaps the most consequential test involved the future. The SSP3-7.0 scenario is, from the model’s perspective, an out-of-distribution challenge: a warming world with systematically higher specific humidity. The regression stage alone produced precipitation with unrealistically inflated inter-annual variance, yielding a projected rainfall trend far steeper than that of the driving climate model — a known artifact of applying quantile mapping to overly smooth deterministic outputs. The diffusion stage corrected this, damping the spurious variance and aligning the global average trend with the GFDL projection while refining the spatial pattern of change. Crucially, when the pre-trained framework was applied without retraining to a second, independent model — the Max Planck Institute’s MPI-ESM-HR — it preserved that model’s own climate signal rather than forcing the data toward the ERA5 historical trend, demonstrating that the framework adapts to the input climate model rather than overfitting to its training distribution. The authors caution, however, that the approach implicitly assumes the statistical relationship between large-scale circulation and small-scale precipitation, learned from present-day observations, remains valid under future warming.

The computational economics of the method are a major part of its appeal. Running a full Earth system model at quarter-degree resolution for large ensembles remains prohibitive; generating a single global high-resolution precipitation field with the new framework takes roughly two seconds on one GPU, and each of the two networks contains fewer than 25 million parameters. This opens the door to the large ensembles of high-resolution precipitation projections that impact assessments, flood management and water resource planning urgently need but have never had. It also enables a practical decoupling: climate models can continue running at coarse resolution while precipitation is generated afterwards as a post-processing step, coupling directly to land surface and impact models without inflating the cost of the underlying simulation.

The authors are careful to frame the work as a proof of concept rather than an operational replacement for fully coupled three-dimensional parameterizations. The model is trained frame by frame, so the stochastic fine-scale details it generates are not explicitly temporally consistent from one day to the next — a limitation that matters less for daily data than for sub-daily applications, but one the team hopes to address with video diffusion architectures. ERA5 itself carries known biases, including a tendency toward wet conditions and an underestimation of extreme intensities, and the framework would need retraining on other datasets for specific operational uses. Integrating the scheme directly into a running climate model, extending it to predict three-dimensional tendencies, enforcing physical conservation laws, and accelerating the sampling process through distillation all remain open challenges, as does the risk of instability and drift that has plagued earlier machine learning parameterization efforts.

Even so, the study marks a significant step in the ongoing convergence of generative artificial intelligence and climate science. It follows a wave of machine learning successes in weather forecasting, but tackles a harder problem: not predicting the atmosphere’s next state, but reconstructing the hidden fine structure of rainfall from its coarse, resolved circulation — and doing so in a way that reduces bias, quantifies uncertainty, and respects the climate change signal. If the approach can be hardened for fully coupled deployment, the blurry rain maps of tomorrow’s climate projections may acquire the crisp, physically plausible texture of the real thing.

Traditionally, ocean turbulence has been understood mainly in coastal zones or near the surface, where wind and currents create visible churn. However, this research delves into the abyssal reaches of the ocean, exploring turbulence at scales of meters or less, millions of times smaller than the ocean itself. By deploying advanced numerical simulations combined with in situ measurements, researchers were able to characterize how these minute motions affect mixing processes that regulate temperature, salinity, and chemical distributions.

The study highlights that small-scale turbulence isn’t just a local oceanographic curiosity but a dynamically significant contributor to large-scale ocean circulation patterns. These turbulent eddies, though ephemeral and spatially limited, facilitate vertical mixing that transports heat downward and nutrients upward, fostering biological activity even in the remote interior. Crucially, the work demonstrates that this mixing mediates how heat absorbed at the surface penetrates ocean depths, impacting the ocean’s capacity to store and release thermal energy over climate-relevant timescales.

Moreover, the coupling between small-scale turbulence and larger ocean currents appears to modulate the ocean-atmosphere feedback loops integral to climate variability. The enhanced mixing enabled by turbulence can alter the stratification of the water column, influencing how the ocean communicates heat back to the atmosphere. This dynamic informs not only short-term weather patterns but also long-term trends such as ocean warming and carbon cycling.

One of the key technical advances underpinning the study is the integration of high-resolution computational fluid dynamics models with observational data from autonomous underwater vehicles equipped with finely calibrated turbulence sensors. This synergy allowed the team to quantify turbulence intensity, vertical fluxes, and the resultant impact on tracer distributions with unprecedented detail.

Importantly, the findings urge a reconsideration of how climate models incorporate ocean mixing processes. Current global climate models often parameterize small-scale turbulence in simplified ways due to computational constraints, potentially underestimating the ocean’s role in modulating surface temperatures and carbon sequestration. By providing a mechanistic framework, this study offers pathways to refine these parameterizations, potentially improving climate predictions and policy-relevant projections.

The implications extend beyond theoretical climate science. Understanding the nuanced role of small-scale turbulence could inform the design of climate interventions or geoengineering efforts aimed at enhancing ocean carbon uptake. It also sharpens our picture of how climate change might feedback into ocean dynamics, possibly altering turbulence patterns themselves and creating complex feedbacks.

Ultimately, this pioneering research illuminates the hidden microcosm of ocean turbulence as a vital cog in the Earth’s climate machinery. As climate challenges intensify, unraveling such subtle yet powerful processes will be critical for forecasting and mitigating the impacts on our planet’s future.

Subject of Research: Oceanic small-scale turbulence and its influence on climate processes.

Article Title: Climatic reach of small-scale turbulence in the ocean interior.

Article References:
Cimoli, L., Mashayek, A., Naveira Garabato, A.C. et al. Nat Commun 17, 5212 (2026). https://doi.org/10.1038/s41467-026-73809-3

Image Credits: AI Generated

DOI: https://doi.org/10.1038/s41467-026-73809-3

This turbulence facilitates the complex mixing of heat, nutrients, and carbon between the ocean surface and seafloor, which plays a crucial role in regulating sea level rise, marine ecosystems, extreme weather events, and global carbon absorption. Until now, the temporal scale of these turbulent processes as embedded in climate models underestimated their speed and effect, resulting in significant gaps in climate projections.

To probe these dynamics, researchers combined comprehensive chemical and physical data sets, including the tracking of chlorofluorocarbon (CFC) concentrations—an anthropogenic tracer released before the 1980s—and innovative dye dispersal experiments. CFC measurements revealed that Antarctic deep waters transported these compounds to regions as far as the mid-Pacific and northern Indian Ocean within just four decades, reflecting a much swifter circulation than climate models had foreseen. Similarly, dye experiments near the Rockall Trough showed that deep ocean flows can ascend at rates close to 100 meters per day—a stark contrast to model predictions lagging by a factor of 10,000.

These unexpected findings highlight the urgent need to refine climate models to accurately represent deep ocean microphysics. Lead author Dr. Laura Cimoli emphasizes that the microphysical processes in the ocean, akin to those in cloud physics, are pivotal yet extraordinarily challenging to observe and simulate. The current lack of fidelity threatens the reliability of predictions related to ocean circulation changes, ecosystem dynamics, and coastal flooding risk.

The consequences extend beyond academic concern. Altered turbulence patterns can disrupt nutrient cycling, destabilizing marine food webs and imperiling fisheries vital for global food security. Furthermore, how heat moves through deep ocean currents directly impacts the melting of polar ice sheets, which in turn accelerates sea level rise and intensifies storms. Dr. Ali Mashayek notes the geopolitical and climate ramifications stemming from these rapid ocean-atmosphere interactions.

Despite these insights, the infrastructure supporting ocean observation is under threat. The partial dismantling of the United States’ Ocean Observatories Initiative jeopardizes critical data streams that undergird the advancement of physical oceanography. As Professor Colm-cille Caulfield warns, comprehensive understanding and computationally efficient modeling of turbulence require sustained investment and enhanced observational efforts.

Ultimately, this research underscores a paradigm shift: the deep ocean is not a slow-moving, isolated system but one intimately connected to atmospheric processes on timescales impacting human society. Future climate strategies hinge on integrating these turbulent processes into models to better anticipate and mitigate climate change impacts.

Subject of Research: Ocean turbulence and its climatic implications
Article Title: Climatic Reach of Small-Scale Turbulence in the Ocean Interior
News Publication Date: 9-Jul-2026
Web References: https://www.nature.com/articles/s41467-026-73809-3
References: DOI: 10.1038/s41467-026-73809-3
Keywords: Oceans, Ocean physics, Ocean circulation, Turbulence, Climate change, Climate change effects