Data and sensitivity analyses

We use data from 1954 to 2014, which are generally considered reliable. Because of concerns regarding the data before 1977 (‘Methods’ section, Supplementary Fig. 1), we repeat some analysis using the more recent period 1977–2014 to test the robustness of the results (Supplementary Figs 2–5). We exclude the weakest tornadoes from our analysis and denote the remaining tornadoes as F1+ tornadoes (‘Methods’ section). We repeat some of the analysis restricted to more intense tornadoes (F2+; Supplementary Figs 7–10).

Number of tornadoes per outbreak

The annual number of F1+ tornadoes shows no significant trend over the period 1954–2014 (Fig. 1a). Generally, trends have not been found in the number of severe tornadoes when severity is defined using the Fujita scale, but upward trends have been found when severity is defined using path length18. The percentage of F1+ tornadoes that occur in outbreaks (‘Methods’ section) is increasing by 0.34 percentage points ±0.13 percentage points per year (Fig. 1b), consistent with upward trends in the proportion of tornadoes occurring on days with many tornadoes8,9. Here and in all results, ± intervals are 95% confidence intervals. The fraction of F1+ tornadoes that occur in outbreaks is less than one because not all F1+ tornadoes occur in outbreaks. During the period 1977–2014, the number of F1+ tornadoes also shows no significant trend, and the percentage of F1+ tornadoes occurring in outbreaks is also increasing, at a larger estimated rate (Supplementary Fig. 2). The US tornado reports show no statistically significant trend in the frequency of tornado outbreaks (Fig. 2a). Since the number of F1+ tornadoes and the number of outbreaks are not changing (on average, over time), the increasing percentage of F1+ tornadoes occurring in outbreaks means that the number of F1+ tornadoes per outbreak must be increasing, and indeed, the annual mean number of F1+ tornadoes per outbreak shows a significant upward trend (Fig. 2b). The annual mean number of tornadoes per outbreak is increasing by 0.66% ±0.26% per year, and the variance is increasing more than four times as fast, 2.89% ±1.22% per year (Fig. 2b,c) over the period 1954–2014. The growth rates are greater over the recent period 1977–2014, with similar ratio between the growth rates of mean and variance (Supplementary Fig. 3b,c).

Figure 1: Time series of counts and clustering of F1+ tornadoes 1954–2014 in the contiguous US. (a) Number of F1+ tornadoes per year. The slope of the least-squares regression indicates that the number of F1+ tornadoes per year declined by 0.81 per year on average from 1954 to 2014 inclusive. This rate of decline is not statistically significantly different from 0 (no change). (b) Annual percentage of F1+ tornadoes occurring in outbreaks. The slope of the least-squares regression indicates that the percentage of F1+ tornadoes per year that occurred as part of outbreaks increased by 0.34 percentage points (pp) per year on average from 1954 to 2014 inclusive. This increase is statistically significantly greater than 0. In both a and b, ± intervals are 95% confidence intervals. Full size image

Figure 2: Numbers of F1+ tornadoes per outbreak 1954–2014. (a) Number of tornado outbreaks per year. The rate of decline is not statistically significantly different from 0 (no change). (b) Annual mean number of tornadoes per outbreak. Vertical axis is on logarithmic scale, so the rate of increase in the annual mean is expressed as a percentage per year. This rate of increase is statistically significantly greater than 0. (c) Annual variance of the number of tornadoes per outbreak. Vertical axis is on logarithmic scale, so the rate of increase in the annual mean is expressed as a percentage per year. This rate of increase is statistically significantly greater than 0. (d) Scatter plot of the annual mean number of tornadoes per outbreak versus the annual variance of the number of tornadoes per outbreak. Both axes are on logarithmic scale. The solid red line is the least-squares (LS) regression line (Taylor’s power law of fluctuation scaling) and the dashed yellow line has the slope and intercept predicted by LC theory17. The two-digit number following the plotting symbol 'o' gives the calendar year in the second half of the twentieth century or first half of the twenty-first century. In all the panels, ± intervals are 95% confidence intervals. Full size image

The fact that the variance is increasing several times faster than the mean is especially noteworthy: it indicates a changing distribution in which the likelihood of extreme outbreaks is increasing faster than what the trend in mean alone would suggest. The coefficient of dispersion of a probability distribution with a positive mean is the ratio of its variance to its mean. Values greater than one (over-dispersion) indicate more clustering than a Poisson variable. For instance, European windstorms exhibit over-dispersion and serial clustering that increases with intensity21 with implications for the return intervals of rare events22. Taylor’s law (TL) relates the mean and variance of a probability distribution by

where a and b are constants15,16. A value of b>1 indicates that the coefficient of dispersion increases with the mean. The annual mean and annual variance of the number of tornadoes per outbreak approximately satisfy TL with b=4.3±0.44 and log a=−6.74±1.12 (Fig. 2d); consistent values are seen over the period 1977–2014 (Supplementary Fig. 3d). (Throughout log is the natural logarithm.) The value of b here is remarkable since in most ecological applications, the TL exponent seldom exceeds 2. The TL exponent can be greater than 2 for lognormal distributions with changing parameters (Supplementary Discussion and Supplementary Fig. 11). The TL scaling of tornado outbreak severity reveals a remarkably regular relation between annual mean and annual variance that extends over the full range of the data, even for years like 2011 which are extreme in mean and variance. The data from 1974 deviate most from TL scaling, with the excessive variance reflecting the 3–4 April ‘Super Outbreak.’

The upward trend in the number of tornadoes per outbreak provides an interpretation for the observed TL scaling since TL scaling arises in models of stochastic multiplicative growth17. In such models, the quantity N(t+1) at time t+1 is related to its previous value N(t) by

where A(t) is the random multiplicative factor by which N(t) grows or declines from one time to the next. Here N(t) is the annual average number of tornadoes per outbreak, and each integer value of t represents one calendar year. The Lewontin–Cohen (LC) model for stochastic multiplicative growth assumes that the A(t) are independently and identically distributed for all t≥0 with finite mean M>0 and finite variance V . If M≠1, N(t) follows TL asymptotically with17

Here we estimate (‘Methods’ section) M=1.03 and V=0.068, which leads to TL parameters b=3.98 and log(a)= −5.84. Both values are consistent with the least-squares (LS) estimates of the corresponding parameters of TL (Fig. 2d). The LS estimates are also consistent with the values from LC theory during 1977–2014 (Supplementary Fig. 3d). 95% confidence intervals for M and V show that the hypothesis of no growth (M=1) under which equation (3) is not valid cannot be rejected (Supplementary Table 1). The Supplementary Discussion provides additional description of how the LC model leads to TL scaling with exponent approximately 4.

Fujita-kilometers per outbreak

Another measure of outbreak severity is Fujita-kilometers (ref. 2; F-km) which is the sum (over all tornadoes in an outbreak) of each tornado’s path length in kilometers multiplied by its Fujita or Enhanced Fujita rating (‘Methods’ section). Annual totals of outbreak F-km, mean number of F-km per outbreak and the variance of F-km per outbreak do not show significant trends over the period 1954–2014 (Fig. 3a–c). The mean number of F-km per outbreak and the variance of F-km per outbreak show marginally significant trends over the recent period 1977–2014 (Supplementary Fig. 4a–c). The TL parameters relating the mean and variance of F-km per outbreak are b=2.77±0.30 and log a=−3.75±1.71 (Fig. 3d). The lack of robust trends means that LC theory is not appropriate to explain the TL scaling of F-km. However, TL scaling also arises from the sampling of stationary skewed distributions20. For a distribution with mean m, variance v, skewness and coefficient of variation CV, theory20 predicts

Figure 3: F-km per outbreak 1954–2014. (a) Total outbreak F-km per year. The rate of decline is not statistically significantly different from 0 (no change) among F1+ tornadoes. (b) Annual mean F-km per outbreak. Vertical axis is on logarithmic scale, so the rate of increase in the annual mean is expressed as a percentage per year. This rate of increase is not statistically significantly greater than 0. (c) Annual variance of F-km per outbreak. Vertical axis is on logarithmic scale, so the rate of increase in the annual mean is expressed as a percentage per year. This rate of increase is not statistically significantly greater than 0. (d) Scatter plot of the annual mean of F-km per outbreak versus the annual variance of F-km per outbreak. Both axes are on logarithmic scale. The solid red line is the least-squares (LS) regression line (Taylor’s power law of fluctuation scaling). The two-digit number following the plotting symbol 'o' gives the calendar year in the second half of the twentieth century or first half of the twenty-first century. In all the panels, ± intervals are 95% confidence intervals. Full size image

Here, excluding two outlier outbreaks from the calculation of the distribution parameters (‘Methods’ section, Supplementary Fig. 5), equation (4) gives b=2.71 and log a=−3.05, both of which are consistent with the LS estimates of the TL parameters for F-km per outbreak. (We use ‘outlier’ to indicate values far from other observations, not to suggest that the unusual values are the result of measurement error.) Therefore TL scaling of F-km per outbreak could be explained by sampling variability.

A tornado environment proxy

A reasonable concern is that the findings here represent properties of the tornado report database that are not meteorological in origin, especially since other prominent features of the tornado report database are not meteorological in origin7. Environmental proxies for tornado occurrence and number of tornadoes per occurrence provide an independent, albeit imperfect, measure of tornado activity for the period 1979–2013 (‘Methods’ section). At a minimum, the environmental proxies provide information about the frequency and severity of environments favourable to tornado occurrence. The correlation between the annual average number of tornadoes per outbreak and the proxy for number of tornadoes per occurrence is 0.56 (Supplementary Fig. 6a). This correlation falls to 0.34, still significant at the 95% level, when the data from 2011 are excluded. Applying a 5-year moving average to the data highlights their common trends and increases the correlation to 0.88 (Supplementary Fig. 6b). The annual mean of the occurrence proxy, a surrogate for number of tornadoes per year, shows a marginally significant upward trend (Fig. 4a). The annual mean and annual variance of the proxy for number of tornadoes per occurrence show upward trends of 0.63±0.30% and 2.43±1.12%, respectively (Fig. 4b,c), values strikingly similar to those for number of tornadoes per outbreak (Fig. 2b,c). Moreover, the TL parameters of the proxy for number of tornadoes per occurrence are 3.54±0.42 and log a=−5.87±1.43, which are consistent with the LC multiplicative growth theory estimates for the proxy (Fig. 4d) and quite similar to those for the number of tornadoes per outbreak. Extreme environments associated with tornado occurrence display the TL scaling and multiplicative growth similar to those of the number of tornadoes per outbreak. This similarity plausibly suggests that the changes in the number of tornadoes per outbreak reflect changes in the physical environment.

Figure 4: Environmental proxies 1979–2013. (a) Annual mean of occurrence proxy in per cent. The rate of increase is statistically significantly greater than 0. (b) Annual mean of the environmental proxy for number of tornadoes per occurrence. Vertical axis is on logarithmic scale, so the rate of increase in the annual mean is expressed as a percentage per year. This rate of increase is statistically significantly greater than 0. (c) Annual variance of the environmental proxy for number of tornadoes per occurrence. Vertical axis is on logarithmic scale, so the rate of increase in the annual mean is expressed as a percentage per year. This rate of increase is statistically significantly greater than 0. (d) Scatter plot of the annual mean of proxy for number of tornadoes per occurrence versus the annual variance of proxy for number of tornadoes per occurrence. Both axes are on logarithmic scale. The solid red line is the least-squares (LS) regression line and the dashed yellow line has slope and intercept predicted by LC theory17. The two-digit number following the plotting symbol 'o' gives the calendar year in the second half of the twentieth century or first half of the twenty-first century. In all the panels, ± intervals are 95% confidence intervals. Full size image

Sensitivity to outbreak definition

Another concern is that the results are sensitive to the details of the outbreak definition. We assess the robustness of the results to the E/F1 threshold by repeating the analysis with tornadoes rated E/F2 and higher, denoted F2+ (Supplementary Figs 7–10). We use the period 1977–2014 because the annual number of F2+ tornadoes display a substantial decrease (not shown) around the 1970s that is likely related to the introduction of the F-scale. Overall the F2+ results are remarkably similar to the F1+ ones. The annual number of F2+ tornadoes has an insignificantly negative trend during 1977–2014 (Supplementary Fig. 7a), and the percentage of F2+ tornadoes occurring in F2+ outbreaks has a significant positive trend (Supplementary Fig. 7b). Although the number of F2+ outbreaks shows no significant trend, the mean number of tornadoes per F2+ outbreak and its variance both have significant upward trends (Supplementary Fig. 8a–c). The TL exponent for number of tornadoes per F2+ outbreak is 3.65 and is consistent with LC theory (Supplementary Fig. 8d). Annual totals of F2+ outbreak F-km have no significant trend (Supplementary Fig. 9a). Mean F-km per F2+ outbreak does have a significant upward trend, but variance does not (Supplementary Fig. 9b,c). The TL scaling of F2+ outbreak F-km (Supplementary Fig. 9d) is consistent with sampling variability when 2011 is excluded (Supplementary Fig. 10).