Quantifying the role of variability in future intensification of heat extremes

Global warming has advanced rapidly since the industrial era, with impacts varying widely across regions due to multiple physical processes and regional vulnerabilities. Critical regions such as the Arctic already exceed conservative warming limits and exhibit dramatic changes in extremes. Climate models project further acceleration of changes, producing heterogeneous patterns where some regions will see large increases in the number of heat extremes and others faster increases in their severity than mean-climate changes would suggest. A central question is how changes in extremes relate to background regional and seasonal warming and to the statistical properties of temperature distributions. Amplification mechanisms involve large-scale circulation, meridional gradients, and cryosphere–land–atmosphere feedbacks, which can alter leading moments (mean, variance, skewness) of temperature distributions. Evidence suggests higher-order moment changes can amplify or damp extreme responses, with expected decreases in midlatitude variability and local-to-regional increases (notably over tropical lands) affecting hot events. Observational constraints on historical higher-moment changes remain limited. Intrinsic non-Gaussian properties, such as skewness, can also accelerate or inhibit increases in hot-event probabilities depending on tail behavior. This study provides a global quantitative analysis of how leading distribution moments, especially variability, shape past and future trajectories of heat extremes, explaining global patterns in their frequency and severity and offering elements to better diagnose model performance and projections.

Prior work documents robust but heterogeneous increases in temperature extremes under anthropogenic forcing and emphasizes linear scaling of many extreme metrics with global mean temperature. Studies have highlighted the role of mean warming in explaining increased hot extremes, while others identify contributions from changes in variability and higher-order moments. Decreases in midlatitude temperature variability are projected and observed, linked to Arctic amplification and weakened meridional gradients, contributing to weaker cold extremes. In contrast, variability increases over tropical lands can enhance hot extremes. Land–atmosphere interactions, particularly soil moisture–temperature feedbacks, are shown to amplify hot extremes and can affect variance and skewness. Non-Gaussianity of temperature distributions, including asymmetric tails, influences extreme probabilities, with longer warm tails accelerating hot-event frequency increases. However, observational evidence for historical changes in higher moments is sparse and model representations vary, complicating attribution and projection.

Data: Daily near-surface (2 m) maximum (TX) and minimum (TN) temperatures from a multimodel CMIP6 ensemble (historical 1850–2014; scenarios SSP1-2.6, SSP2-4.5, SSP5-8.5; one realization per model). Annual mean temperatures from monthly data yielded GSAT trajectories. For the high-end scenario SSP5-8.5, model-specific 21-year windows centered on years when GSAT crosses +2 K and +3 K (relative to 1851–1900) were used to compute expected values at fixed global warming levels (GWLs). Anomalies: Daily TX and TN anomalies were computed relative to early-industrial (1851–1900) day-of-year normals estimated using moving windows and Fourier filtering to remove noise; anomalies thus have zero mean in the early-industrial period. Moments: For each grid point, year, and season, leading moments were computed from daily anomalies: mean (µ1), variance (µ2), standard deviation (σ = sqrt(µ2)), and skewness (γ1 = µ3/µ2^(3/2)). Extremes: Magnitudes examined were the hottest daytime anomaly (TXx) and coldest nighttime anomaly (TNn) per year and per season. Probabilities of fixed-magnitude extremes used exceedance thresholds (global 1st, 99th, 99.9th percentiles) based on early-industrial all-days anomaly distributions. Decomposition of extreme changes: Changes in extreme magnitudes were related to changes in the mean and standard deviation of the underlying seasonal anomaly distributions via a Taylor-like expansion, e.g., ΔTXx/ΔT ≈ ΔTXm/ΔT + (TXx−TXm)·ΔTXsd/ΔT + higher-order terms, with an analogous expression for TNn. Here ΔT is local annual mean warming, and TXm/TNm are seasonal mean anomalies. Exceedance probability scaling: A Gaussian-shift approximation with fixed early-industrial variability (TXsd) and a shifting mean (TXm) was used to derive theoretical rates of change of exceedance probability P(x≥x*) with warming: dP/dT = (1/(√(2π) TXsd)) (dTXm/dT) exp(−(x−TXm)^2/(2 TXsd^2)). This was integrated over +2 K to predict spatial patterns of fractional probability change. Evaluation: Agreement between simulated patterns and theoretical expectations was quantified by area-weighted coefficient of determination R² = 1 − SSres/SStot, with SS terms computed over remapped N32 grids. Spatial remapping and averaging: Native grid results were remapped to a common Gaussian N32 grid for multimodel averaging and zonal means. Aggregations: Global and regional means were area-weighted; selected hotspot regions included Euro-Mediterranean (30–55 N, 10 W–40 E, land), Central South America (25–30 S, 40–73 W, land), Pan-Arctic (67–90 N), and northern mid-to-high latitudes (45–67 N). Uncertainty: Intermodel spread (across 20 models) was used as the primary uncertainty estimate; where available, intramodel spread comparisons confirmed intermodel spread dominates. Observational check: A preliminary comparison of early-industrial land TXsd with Berkeley Earth (1881–1910) indicated notable regional biases in simulated variability. Code and data availability are stated.

- Extreme magnitudes scale approximately linearly with GSAT across scenarios and models, but with strong regional and seasonal heterogeneity. Over land, TXx and TNn increase faster than GSAT by about 30% and 70%, respectively. - Variability is a dominant control on extreme warming rates. Decreases in daily temperature variability (standard deviation) at mid-to-high latitudes in cool seasons bring extremes closer to the mean, amplifying TNn warming and suppressing TXx warming. Increases in variability over lower latitudes (notably tropical lands) enhance TXx warming. - The inclusion of variability changes explains most spatial variability in extreme warming rates: coefficients of determination R²_full (first two moments) reach 0.82–0.87 for TXx and 0.87–0.95 for TNn across seasons, versus ≤0.58 (often ≤0) when considering mean changes alone. - Zonal analyses show winter TNn warming at northern mid-to-high latitudes and over the Southern Ocean exceeding the seasonal background by up to ~30–50% due to variability decreases; boreal summer shows minimal higher-moment contributions at high latitudes. - Physical drivers: Wintertime weakening of meridional and zonal temperature gradients, linked to Arctic amplification and faster continental warming, aligns with widespread TN variability decreases. Summer increases in variability over some midlatitude and tropical regions are consistent with reinforced gradients and land–atmosphere (soil moisture) feedbacks. - Regional hotspots: Euro-Mediterranean (JJA) and Central South America (SON) exhibit TXx warming ~30% faster than global land; in EMD this closely follows mean warming (TXsd change small), while in CSA a significant TXsd increase adds ~10% amplification. Pan-Arctic winter TNn increases about three times the global change; in mid-to-high northern latitudes, TNsd decreases account for about one-third of the TNn increase regionally; over the Southern Ocean, roughly half of winter TNn increase stems from TNsd decrease. - Exceedance probabilities: Native (historical) variability overwhelmingly controls spatial patterns of future hot-day frequency. A Gaussian fixed-variance, shifting-mean model using early-industrial TXsd and projected TXm changes explains ~90% of spatial variation in hot-day probability increases at +2 K; removing spatial variation of TXsd collapses explanatory power. - Tropical oceans, with the lowest TXsd (≤1 K), experience the largest fractional increases in hot-day frequency: ~50-fold at +2 K despite relatively slow mean warming. Tropical lands see ~12-fold increase in hot days and ~34-fold in very hot days at +2 K; increases are much smaller at higher latitudes due to larger variability. - At +3 K global warming, hot extremes over large continental areas warm ~50% more than at +2 K, and Arctic cold extremes warm by an additional ~4 K. - Model biases in native variability (e.g., overestimated subtropical/high-latitude variability; underestimated near the equator) imply projections may under/overestimate hot-day frequency regionally, underscoring the need to constrain models with observations.

The study demonstrates that the leading statistical moments of temperature distributions, especially variability, strongly modulate how heat extremes respond to greenhouse warming. While mean warming sets a baseline, reductions in variability at mid-to-high latitudes accelerate wintertime warming of cold extremes and suppress hot extremes, whereas increases in variability at lower latitudes enhance hot extremes. These statistical effects reconcile the observed and projected heterogeneity of extreme-temperature scaling with global warming. The Gaussian-shift framework reveals that native (historical) variability governs future changes in exceedance probabilities, with small variability producing large fractional increases in hot-event frequency, particularly over tropical oceans and lands. Physical consistency is provided by links between Arctic amplification, weakened thermal gradients, and reduced variability in winter, and by land–atmosphere feedbacks and strengthened gradients contributing to increased summer variability regionally. These insights imply that accurate simulation of historical variability is critical: biases in native variability translate nonlinearly into probability projection errors, especially in the tropics. The findings advocate for targeted evaluation, weighting, or bias correction of models based on their representation of historical distribution moments to improve the reliability of extreme-event projections.

Daily temperature variability is a key determinant of the future intensification of heat extremes. Changes in variability, in concert with mean warming, explain most of the spatial heterogeneity in extreme warming rates, causing cold extremes to warm much faster than hot extremes in many regions and amplifying hot extremes where variability increases. Native (historical) variability largely controls future changes in the frequency of unusual hot events; a simple Gaussian, fixed-variance shift model explains about 90% of spatial variation in hot-day probability increases. These results provide a robust, quantitative link between distribution moments and extreme trajectories, offering a pathway to diagnose physical drivers and to constrain climate model projections. Future work should: - Improve observational constraints on historical higher moments globally and quantify their uncertainties. - Investigate physical drivers of higher-moment changes (including skewness and kurtosis) via targeted model experiments. - Develop and apply model weighting and bias-correction approaches focused on distribution moments to reduce projection uncertainty, particularly for regions with high societal risk such as the tropics. - Explore compound effects of circulation changes and land–atmosphere feedbacks on variability and extremes under higher warming levels.

Observed data coverage for early-industrial daily variability is limited and uncertain, hindering comprehensive validation of simulated historical distribution moments. The Gaussian fixed-variance approximation, while informative, neglects non-Gaussianity and changes in higher moments that can matter regionally. There is substantial intermodel spread, and a single realization per model was used; internal variability and scenario uncertainty persist. Model biases in native variability are evident and may cause regional under- or overestimation of future exceedance probabilities. The study primarily relies on CMIP6 models and coarse-grained diagnostics; process-level attribution (e.g., of skewness changes) remains incomplete.