Climate science has relied for decades on average values: global mean temperature, regional trends, anomalies. This is convenient, but averages hide details. In the first decade of the 21st century, warming slowed, and average trends showed almost no change, even though shifts were occurring in the system: extremes were changing, Pacific currents were reorganizing. Alongside this work in the archive lie studies that also seek signals beyond the average: some separate weather noise from climate signal, others look at how two factors drive weather on a lone planet, and still others catch early signs of climate shifts. All move in one direction: from "what is the average" to "what is happening to the distribution." This work takes the next step and offers a tool that directly measures how much the entire shape of the distribution has changed, not just its center.
The authors took surface air temperature data from the ERA5 reanalysis from 1998 to 2012, regridded it to a 2.5 by 2.5 degree grid, and split the period into two halves: 1998–2005 and 2005–2012. For each point on the planet, they built probability distributions of temperature for these two periods. They then applied the Wasserstein distance (also known as the "earth mover's distance"): a metric that shows how much "work" is needed to transform one distribution into another. To determine whether the difference is significant, they used a Monte Carlo test: they added white noise to the original data 500 times and checked whether the real distance falls within the confidence interval of the noise distances. If it does not, the distribution changed not by chance. This approach they named Wasserstein Stability Analysis (WSA).
WSA found two large significant signals. The first is in the central-eastern Pacific Ocean: there the Wasserstein distance between the distributions of 1998–2005 and 2005–2012 ranged from 0.08 to 0.24, even though the linear temperature trend in that region was absent. Decomposing the distributions showed that hot extremes (above two standard deviations, about 27 °C) became less frequent by more than 10 days per year, while cold extremes (below minus two standard deviations, about 19 °C) became more frequent by the same amount. This resembles a shift toward La Niña conditions and agrees with earlier findings that during the warming slowdown the Pacific was in the negative phase of the Pacific Decadal Oscillation. The second signal is in the Barents and Kara Seas. There WSA showed a significant change in the distribution at point B (66° E, 79° N), where the mean temperature rose from −10.2 °C to −6.5 °C, and the standard deviation decreased. The cause is melting sea ice: ice previously imposed a hard limit on the maximum air temperature near the freezing point, so the distribution had a sharp peak. When ice diminished, the cap weakened and the peak smoothed out. At point C (131° E, 79° N), where the ice cover shrank less, the distribution change was weaker and did not pass the significance threshold.
This work differs from neighboring studies in that it offers not another way of averaging or filtering, but a direct quantitative test for changes in the entire shape of the distribution. Neighboring works often search for signals in mean values or in individual extremes; here the measure is the distance between whole distributions, which allows detecting shifts that are invisible in either the mean or the variance alone. For climate science, this means a tool emerges for detecting changes in weather regimes, not just in the average background. Furthermore, the method revealed a physical constraint that is usually not discussed in the context of trends: sea ice acts as a "ceiling" for air temperature, and its disappearance changes not only the average but the very structure of variability.
Currently WSA can only say that the distribution has changed, but it does not show in which direction: whether there is more heat or more cold. The authors call this limitation the main task for the future. In addition, the method has so far been applied only to surface air temperature, but it could be used for precipitation, wind, pressure, and other variables. If directionality can be added, WSA could become a standard tool for real-time monitoring of climate shifts.
The method could be useful for evaluating climate models: if a model does not reproduce the distribution change that WSA detects, that signals a systematic error. It could also help in the analysis of extreme events and in sea ice research, where temperature constraints play a key role.
Immediate next steps include developing a way to determine the direction of the distribution change and applying WSA to other climate variables. The authors also plan to examine in more detail the scattered signals in the Indian Ocean and South America that remained outside the scope of this paper.
An open question in climatology is how to separate internal variability of the climate system from external forcing. The warming slowdown showed that average trends can be misleading, and methods sensitive to distribution reshuffling are needed. WSA directly addresses this challenge: it finds changes that do not reduce to a shift in the mean, thereby helping to see which processes actually alter the climate and which merely add noise.
The Wasserstein distance is sometimes called the "earth mover's distance" because it quantifies the minimal work needed to move a pile of earth from one shape to another. In climate, the "pile of earth" is the temperature distribution, and the "work" is the number of days by which extremes have shifted.