Large Deviation Theory Lends Clarity to Atmospheric Persistent Events

By Lina Sorg

The Earth’s climate is just one example of a complicated, chaotic dynamical system with multiple factors and geophysical inputs. During a minisymposium presentation at the 2021 SIAM Conference on Applications of Dynamical Systems, which is taking place virtually this week, Vera Melinda Galfi of Uppsala University used large deviation theory (LDT) to analyze temperature-related atmospheric persistent events, such as heat waves or cold spells.

Galfi framed her talk in the context of two high-impact persistent temperature events: the 2010 Russian heat wave and the 2010 Mongolian dzud (a severe cold spell). The heat wave affected a very large geographic area and led to both drought and wildfires. The extreme temperatures—which lasted for nearly two months and were responsible for roughly 55,000 deaths—have also been linked to devastating floods in Pakistan and the Indian peninsula. Although the Mongolian dzud is perhaps less familiar, it was no less severe. The resulting harsh winter conditions prevented livestock from grazing, causing many animals to die of starvation and negatively affecting the country’s economy.

Researchers frequently utilize extreme indices to study persistent climate events that fall above a fixed threshold (i.e., the 95th percent quartile). Although these indices are flexible and useful, subjective decision-making contributes to threshold identification. Instead, one could employ extreme value theory—which uses limit laws to discern probabilities of extreme events—to minimize subjectivity. However, the correlation of persistent extreme events complicates the application of this theory. LDT provides a solution. “Large deviation is based on limit laws for probabilities of sample averages,” Galfi said. “Based on LDT, we can identify ‘typical’ heatwaves or cold spells over land regions and estimate their probability of occurrence.” In this context, persistent events are the extremes of the averages. Galfi defines “typical” based on spatial patterns; similar patterns in time between events; and shared, well-defined physical characteristics.

To apply LDT, researchers average random variables over blocks of equal length. Galfi addressed both the large deviation principle and the rate function, which provides the probabilities’ speed of decay. “LDT is more than the law of large numbers or the central limit theory because it also gives us large deviations from the mean,” she said. “We call a large deviation the most likely of all the unlikely events.” She then noted that geophysical time series have dependence and correlations; under weak dependence, the sample averages are almost uncorrelated if the average block size is large enough.

Figure 1. The nine different regions in the Northern Hemisphere for the MPI-ESM-LR Earth system model.

Next, Galfi and her team ran simulations with the MPI-ESM-LR Earth system model. To do so, they selected nine different regions in the Northern Hemisphere—including parts of Europe, the Americas, the Mediterranean, Asia, and two oceanic areas (see Figure 1)—and computed a refraction for these different regions for two simulations: a pre-industrial control run and a scenario with four times the concentration of CO₂. But first they eliminated the seasonal cycle by subtracting the long-term daily mean, then defined an extended summer and extended winter to separate the seasons. 

Using the simulations, Galfi examined the effect of increased CO₂ levels on the probability of heat waves. She obtained the rate functions for the nine different regions by averaging over increasing averaging block sizes (see Figure 2). “After an averaging block size of one or two months, these rate functions seem to converge over land regions,” Galfi said. “However, we don’t see any convergence over oceanic regions. We think that in the case of the oceanic regions, because of the long-range dependence of temperature over the oceans, one cannot get a convergence at seasonal timescales.” She discerned that increasing CO₂ levels have a threefold effect on heat waves in North America and Europe: (i) larger deviations will be more probable, (ii) these deviations will have a longer duration, and (iii) they will occur around a warmer mean.

Figure 2. Rate functions for the nine regions in the Northern Hemisphere.

The same simulations in the context of cold spells yielded the opposite result; cold spells seem to become less frequent with an increasing CO₂ concentration. This conclusion is in agreement with global warming.

Next, Galfi applied the concept of large deviation to the real anomaly patterns that stem from the model. “The most important thing is that this typical pattern corresponds to large deviation and resembles the observed anomaly pattern during the 2010 Russian heat wave,” she said. “If we look at more severe or longer events, we again obtain very similar spatial patterns. The patterns related to large deviation seem to be stable in the case of larger or longer events.”

A comparable comparison for the 2010 Mongolian dzud produced a similar pattern and the same conclusion as the heat wave. Galfi noted the pattern’s large spatial expansion in the context of the dzud, which affects almost two continents. “Heat waves seem to become more frequent and longer lasting in Europe and North America as an effect of CO₂ increase,” Galfi said. “The 2010 Russian heave wave an the Mongolian dzud seem to be typical persistent events, compared to other persistent events with similar magnitude.”

Lina Sorg is the managing editor of SIAM News.