Tornado outbreak variability follows Taylor's power law of fluctuation scaling and increases dramatically with severity

Abstract

Tornadoes cause loss of life and damage to property each year in the United States and around the world. The largest impacts come from 'outbreaks' consisting of multiple tornadoes closely spaced in time. Here we find an upward trend in the annual mean number of tornadoes per US tornado outbreak for the period 1954-2014. Moreover, the variance of this quantity is increasing more than four times as fast as the mean. The mean and variance of the number of tornadoes per outbreak vary according to Taylor's power law of fluctuation scaling (TL), with parameters that are consistent with multiplicative growth. Tornado-related atmospheric proxies show similar power-law scaling and multiplicative growth. Path-length-integrated tornado outbreak intensity also follows TL, but with parameters consistent with sampling variability. The observed TL power-law scaling of outbreak severity means that extreme outbreaks are more frequent than would be expected if mean and variance were independent or linearly related.

ARTICLE

Received 22 Jul 2015 | Accepted 8 Jan 2016 | Published 29 Feb 2016

Tornadoes cause loss of life and damage to property each year in the United States and around the world. The largest impacts come from outbreaks consisting of multiple tornadoes closely spaced in time. Here we nd an upward trend in the annual mean number of tornadoes per US tornado outbreak for the period 19542014. Moreover, the variance of this quantity is increasing more than four times as fast as the mean. The mean and variance of the number of tornadoes per outbreak vary according to Taylors power law of uctuation scaling (TL), with parameters that are consistent with multiplicative growth. Tornado-related atmospheric proxies show similar power-law scaling and multiplicative growth. Path-length-integrated tornado outbreak intensity also follows TL, but with parameters consistent with sampling variability. The observed TL power-law scaling of outbreak severity means that extreme outbreaks are more frequent than would be expected if mean and variance were independent or linearly related.

DOI: 10.1038/ncomms10668 OPEN

Tornado outbreak variability follows Taylors power law of uctuation scaling and increases dramatically with severity

Michael K. Tippett1,2 & Joel E. Cohen3,4

1 Department of Applied Physics and Applied Mathematics, Columbia University, New York, New York 10027, USA. 2 Center of Excellence for Climate Change Research, Department of Meteorology, King Abdulaziz University, Jeddah 21589, Saudi Arabia. 3 Laboratory of Populations, Rockefeller University, New York, New York 10065, USA. 4 The Earth Institute, Columbia University, New York, New York 10027, USA. Correspondence and requests for materials should be addressed to M.K.T. (email: mailto:[email protected]

Web End [email protected] ) or to J.E.C. (email: mailto:[email protected]

Web End [email protected] ).

NATURE COMMUNICATIONS | 7:10668 | DOI: 10.1038/ncomms10668 | http://www.nature.com/naturecommunications

Web End =www.nature.com/naturecommunications 1

ARTICLE NATURE COMMUNICATIONS | DOI: 10.1038/ncomms10668

Hazardous convective weather (tornadoes, hail and damaging wind) associated with severe thunderstorms affects large portions of the United States. Tornadoes

cause particularly intense damage. Over a recent 10-year period (20052014), tornadoes in the United States resulted in an average of 110 deaths per year and annual losses ranging from $500 million to $9.6 billion1. The largest societal impacts from tornadoes are from outbreaks in which multiple tornadoes occur in a single weather event. Tornado outbreaks across the eastern two-thirds of the United States were associated with 79% of all tornado fatalities over the period 19722010 (ref. 2) and are routinely responsible for billion-dollar loss events3.

Whether US tornado activity will change in the future remains uncertain. Climate change projections indicate that environments favourable to severe thunderstorms will be more frequent in a warmer climate4, and high-resolution convection-permitting numerical modelling indicates increased activity and year-to-year variability of MarchMay US tornado occurrence as measured by severe weather proxies derived from explicitly depicted storms5. However, to date, there is no upward trend in the number of reliably reported US tornadoes per year6. Interpretation of the US tornado report data requires some caution. For instance, the total number of US tornadoes reported each year has increased dramatically over the last half century, but most of that increase is due to more reports of weak tornadoes and is believed to reect changing reporting practices and other non-meteorological factors rather than increased tornado occurrence7. The variability of reported tornado occurrence has increased over the last few decades with more tornadoes being reported on days when tornadoes are observed8,9. In addition, greater year-to-year variability in the number of tornadoes reported per year has been associated with consistent changes in the monthly averaged atmospheric environments favourable to tornado occurrence10. Likewise, US large-event severe thunderstorm losses and the frequency of the most extreme environments have increased11. Regional changes in the seasonality of tornado occurrence have also been reported12,13.

Despite their importance, relatively little is known about the current or projected statistics of tornado outbreak severity beyond their climatology2. Most studies have considered only the statistics of tornadoes occurring during a single day14 and have not considered outbreaks over multiple dates. Here we show that the annual mean number of tornadoes per outbreak increased during 19542014, and the annual variance increased more than four times faster than the mean. We show that the mean and variance of the number of tornadoes per outbreak are related by Taylors power law of uctuation scaling (TL)15,16 with parameters that are consistent with multiplicative growth17. TL scaling in tornado outbreak statistics was not previously known. Although power-law scaling is present in the probability distributions of various tornado characteristics18,19, the presence of power-law scaling in such probability distributions is neither necessary nor sufcient for TL scaling of the variance in relation to the mean (see also Supplementary Discussion for examples of TL scaling without power-law probability distributions). We also nd that a tornado-related atmospheric proxy shows a similar power-law scaling and multiplicative growth. Path-length-integrated tornado outbreak intensity follows TL as well, but with parameters predicted by sampling variability20. The ndings are similar when we restrict the analysis to the more recent period 19772014 and to more intense tornadoes. The observed TL scaling of outbreak severity means that extreme outbreaks are more frequent than would otherwise be expected if mean and variance were independent or linearly related.

ResultsData 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 19772014 to test the robustness of the results (Supplementary Figs 25). We exclude the weakest tornadoes from our analysis and denote the remaining tornadoes as F1 tornadoes (Meth

ods section). We repeat some of the analysis restricted to more intense tornadoes (F2 ; Supplementary Figs 710).

Number of tornadoes per outbreak. The annual number of F1

tornadoes shows no signicant trend over the period 19542014 (Fig. 1a). Generally, trends have not been found in the number of severe tornadoes when severity is dened using the Fujita scale, but upward trends have been found when severity is dened 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% condence 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 19772014, the number of F1 tornadoes also shows no signicant 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 signicant trend in the frequency of tornado outbreaks (Fig. 2a). Since the number of

a

1,000

Number of F1+ tornadoes per year

Slope=0.811.80

1960 1970 1980 1990 2000 2010

800

600

400

200 1960 1970 1980 1990 2000 2010

b

80

Percent of F1+ tornadoes occurring in outbreaks

Slope=0.34pp0.13pp

70

60

50

40

Figure 1 | Time series of counts and clustering of F1 tornadoes

19542014 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 signicantly 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 signicantly greater than 0. In both a and b, intervals are 95% condence intervals.

2 NATURE COMMUNICATIONS | 7:10668 | DOI: 10.1038/ncomms10668 | http://www.nature.com/naturecommunications

Web End =www.nature.com/naturecommunications

NATURE COMMUNICATIONS | DOI: 10.1038/ncomms10668 ARTICLE

a

d

Number of tornadoes per outbreak

LS b=4.330.44, log a=6.741.12 LC theory b=3.98, log a=5.84 95% CI LS

60

Number of tornado outbreaks per year

Slope=0.060.10

1960 1970 1980 1990 2000 2010

40

20

0 1960 1970 1980 1990 2000 2010

103

101

b

25

Mean number of tornadoes per outbreak

Slope=0.66%0.26%

1960 1970 1980 1990 2000 2010

20

15

10

Variance

102

c

Variance of the number of tornadoes per outbreak

Slope=2.89%1.22%

102

10 15 20 25

Mean

Figure 2 | Numbers of F1 tornadoes per outbreak 19542014. (a) Number of tornado outbreaks per year. The rate of decline is not statistically

signicantly 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 signicantly 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 signicantly 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 (Taylors power law of uctuation 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 rst half of the twenty-rst century. In all the panels, intervals are 95% condence intervals.

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

signicant 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 19542014. The growth rates are greater over the recent period 19772014, with similar ratio between the growth rates of mean and variance (Supplementary Fig. 3b,c).

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 coefcient 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. Taylors law (TL) relates the mean and variance of a probability distribution by

variance a mean

b; a40; 1

where a and b are constants15,16. A value of b41 indicates that the coefcient of dispersion increases with the mean. The annual mean and annual variance of the number of tornadoes per outbreak approximately satisfy TL with b 4.30.44 and

log a 6.741.12 (Fig. 2d); consistent values are seen over

the period 19772014 (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 exceeds2. 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 reecting the 34 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

N t 1

A t

Nt; 2 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 tZ0 with nite mean M40 and nite variance V . If Ma1, N(t)

follows TL asymptotically with17

a

E N 0

logV M2 log M : 3

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 19772014 (Supplementary Fig. 3d). 95% condence 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

2

E N 0 b

and b

NATURE COMMUNICATIONS | 7:10668 | DOI: 10.1038/ncomms10668 | http://www.nature.com/naturecommunications

Web End =www.nature.com/naturecommunications 3

ARTICLE NATURE COMMUNICATIONS | DOI: 10.1038/ncomms10668

a

d

104

3

Outbreak F-km per year

Slope =4.4256.46

1960 1970 1980 1990 2000 2010

107

F-km per outbreak

LS b =2.770.30, log a=3.751.71 95% CI LS

2

1

0 1960 1970 1980 1990 2000 2010

b

106

800

Mean F-km per outbreak

Slope=0.10%0.61%

1960 1970 1980 1990 2000 2010

600

Variance

400

200

105

c

Variance of F-km per outbreak

Slope=0.43%1.83%

106

104

150 200 250 300 400 500 750

Mean

Figure 3 | F-km per outbreak 19542014. (a) Total outbreak F-km per year. The rate of decline is not statistically signicantly 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 signicantly 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 signicantly 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 (Taylors power law of uctuation scaling). The two-digit number following the plotting symbol o gives the calendar year in the second half of the twentieth century or rst half of the twenty-rst century. In all the panels, intervals are 95% condence intervals.

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 tornados 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 signicant trends over the period 19542014 (Fig. 3ac). The mean number of F-km per outbreak and the variance of F-km per outbreak show marginally signicant trends over the recent period 19772014 (Supplementary Fig. 4ac). The TL parameters relating the mean and variance of F-km per outbreak are b 2.770.30 and log a 3.751.71 (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 g1 and coefcient of variation CV, theory20 predicts

a log v

g1CV log m and b

g1CV : 4

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 ndings 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 19792013 (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 signicant 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 to0.88 (Supplementary Fig. 6b). The annual mean of the occurrence proxy, a surrogate for number of tornadoes per year, shows a marginally signicant upward trend (Fig. 4a). The annual mean and annual variance of the proxy for number of tornadoes per occurrence show upward trends of 0.630.30% and2.431.12%, 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 3.540.42 and log a 5.871.43, 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 reect changes in the physical environment.

Sensitivity to outbreak denition. Another concern is that the results are sensitive to the details of the outbreak denition. We

4 NATURE COMMUNICATIONS | 7:10668 | DOI: 10.1038/ncomms10668 | http://www.nature.com/naturecommunications

Web End =www.nature.com/naturecommunications

NATURE COMMUNICATIONS | DOI: 10.1038/ncomms10668 ARTICLE

Annual mean of occurrence proxy

Slope = 0.880.59

1980 1985 1990 1995 2000 2005 2010

Environmental proxy for number/occurrence

LS b = 3.450.42, log a = 4.411.12 LC theory b = 3.31, log a = 4.19 95% CI LS

a

d

2

1.5

1

0.5 1980 1985 1990 1995 2000 2005 2010

b

18

Mean of proxy for number/occurrence

Slope = 0.66%0.32%

1980 1985 1990 1995 2000 2005 2010

16

14

Variance

102

12

c

300

Variance of proxy for number/occurrence

Slope = 2.47%1.14%

200

100

11 12 13 14 15 16 17 18 19

Mean

Figure 4 | Environmental proxies 19792013. (a) Annual mean of occurrence proxy in per cent. The rate of increase is statistically signicantly 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 signicantly 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 signicantly 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 rst half of the twenty-rst century. In all the panels, intervals are 95% condence intervals.

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 710). We use the period

19772014 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 insignicantly negative trend during

19772014 (Supplementary Fig. 7a), and the percentage of F2

tornadoes occurring in F2 outbreaks has a signicant positive

trend (Supplementary Fig. 7b). Although the number of F2

outbreaks shows no signicant trend, the mean number of tornadoes per F2 outbreak and its variance both have

signicant upward trends (Supplementary Fig. 8ac). 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 signicant trend

(Supplementary Fig. 9a). Mean F-km per F2 outbreak does

have a signicant 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).

DiscussionThese ndings have important implications for tornado risk in the United States and perhaps elsewhere, though we have examined only the US data. First, the number of tornadoes per outbreak is increasing. However, there is less evidence that F-km per outbreak are increasing. Both the number of tornadoes per outbreak and F-km per outbreak follow TL, which relates mean and variance. We nd that TL scaling for the number of tornadoes per outbreak is compatible with multiplicative growth, and that TL scaling for F-km per outbreak could be due to

sampling variability. Finally, the key implication of TL scaling is that both number of tornadoes per outbreak and F-km per outbreak exhibit extreme over-dispersion which increases with mean. When the average tornado outbreak severity gets worse, the high extreme of severity rises even faster and the low extreme falls even faster, by either measure of severity.

Methods

Outbreak data. Tornado reports come from the NOAA Storm Prediction Center (http://www.spc.noaa.gov/wcm/#data

Web End =http://www.spc.noaa.gov/wcm/#data). There are no reports from either Alaska or Hawaii; one report from Puerto Rico is excluded. Tornadoes are rated by estimated or reported damage using the Fujita (F) scale, introduced in the mid-1970s, and since March 2007, the enhanced Fujita (EF) scale, with 0 being the weakest and 5 the strongest. Only reports of tornadoes rated F/EF1 or greater are used and are denoted as F1 . To compute outbreak statistics, tornado reports are

rst sorted in chronological order taking into account the time zone. Outbreaks are sequences of 6 or more F1 tornadoes (regardless of location in the contiguous

United States) whose successive start times are separated by no more than 6 h (ref. 2). A total 1,361 outbreaks are found over the period 19542014. Outbreaks spanning more than 1 day are possible with this denition. The median outbreak duration is about 8 h, and 95% of the outbreaks last less than 24 h. Additional climatological features can be found in ref. 2 The number of tornadoes and F-km are computed for each outbreak. Then the annual mean and variance of two outbreak severity measures, number of tornadoes per outbreak and F-km per outbreak, are calculated. The distribution of outbreak tornadoes by F/EF-scale shows considerable variation before 1977 (Supplementary Fig. 1) and may well reect lower reliability in the earlier period23 before damage surveys were a routine part of the rating procedure. Therefore, we repeat some of our analysis using the recent period 19772014 (Supplementary Figs 2,3).

The same outbreak calculation procedure is repeated but considering only reports of tornadoes rated F/EF2 or greater, denoted as F2 . These outbreak

events are referred to as F2 outbreaks (Supplementary Figs 710). Only

F2 tornadoes are used to calculate tornado numbers and F-km of F2

outbreaks.

Trends. All trends and 95% condence intervals are assessed using linear regression and ordinary least squares, assuming approximately normal distributions of residuals. The growth rates of the annual mean and variance of the outbreak severity measures in Fig. 2b,c, Fig. 3b,c and Supplementary Figs 3b,c; 4b,c; 8b,c and 9b,c are

NATURE COMMUNICATIONS | 7:10668 | DOI: 10.1038/ncomms10668 | http://www.nature.com/naturecommunications

Web End =www.nature.com/naturecommunications 5

ARTICLE NATURE COMMUNICATIONS | DOI: 10.1038/ncomms10668

computed by assuming exponential growth and tting a linear trend to the logarithms of the data. All the other trends are tted using untransformed data.

TL parameters. The TL parameters in Figs 2d, 3d and 4d and Supplementary Figs 3d, 4d, 8d and 9d and 95% condence intervals are estimated using ordinary least-squares regression with the logarithms of the mean and variance.

TL parameters implied by LC theory. The growth factor in the LC model is computed from A(t) N(t 1)/N(t), where N(t) is the average number of

tornadoes per outbreak in calendar year t. The 95% condence intervals for the mean and variance of A(t) are computed from 10,000 bootstrap samples and reported in Supplementary Table 1. The mean and variance of A(t) are used to compute a prediction of the slope b using equation (3). The TL parameter a is estimated from equation (1) evaluated at the initial year, either 1954 or 1977.

TL parameters implied by sampling variability. There are 1,361 cases of outbreak F-km, and their distribution is highly right-skewed (Supplementary Figs 5a and 10a). The F-km values for the 1974 Super Outbreak and the 2528 April 2011 tornado outbreak are more than 26 standard deviations above the mean of the data on an arithmetic scale and more than six standard deviations above the mean of the log-transformed data, when means and standard deviations are calculated after withholding the two extreme values. These outliers (values that are far from other observations) have a substantial impact on the estimates of the mean, variance and skewness of the F-km distribution. Despite their rarity, about 86% (1 (1 2/1,361)1361) of the bootstrap samples will contain one or both of these

two events. The presence of the outliers results in bimodal distributions of the TL slope and intercept estimates (Supplementary Figs 5b,c and 10b,c) computed from equation (4), depending on whether or not the outlier values are in the particular bootstrap sample. Removal of the outliers results in unimodal distributions, whose ranges are consistent with the least-squares estimates of the TL slope and intercept for F-km (Supplementary Figs 5d,e and 10d,e).

Environmental proxies. We use two environmental proxies: one for tornado occurrence and one for the number of tornadoes. Tornadoes are often associated with elevated values of convective available potential energy (CAPE; J kg 1) and a measure of vertical wind shear, called storm relative helicity (SRH; m2 s 2). Here 0180 hPa CAPE and 03,000 m SRH data are taken from the North American

Regional Reanalysis24. The data are interpolated to a 1 1 degree latitude

longitude grid over the continental United States and daily averages are computed. The environmental proxy for tornado occurrence is dened using the energy-helicity index25 (EHI), which is the product of CAPE and SRH divided by 160,000 J kg 1 m2 s 2. Values of EHI greater than one indicate the potential for supercell thunderstorms and tornadoes26, and accordingly we take our proxy for tornado occurrence to be the condition that EHI is greater than 1. Selection of an environmental proxy for the number of tornadoes is more challenging. The only example of a proxy calibrated to predict the number of tornadoes from the surrounding environment uses monthly averaged environment and monthly number of tornadoes27,28, and therefore is not suited for outbreaks, which last much less than a month. However, previous research does provide some indications for the functional form of such a proxy. For instance, on subdaily timescales the likelihood of signicant severe weather is nearly twice as sensitive to vertical wind shear as to CAPE29,30. This sensitivity would argue for a proxy based on the product of CAPE and the square of vertical wind shear or equivalently the product of the square root of CAPE and vertical wind shear31. Likewise, proxies for the monthly number of tornadoes contain SRH with an exponent ranging from 1.89 to 4.36 depending on region27,28. The supercell composite parameter32 is used in weather forecasting and is the product of CAPE, SRH and vertical shear, again indicating a scaling of severe weather with the square of vertical wind shear measures. On the basis of this evidence, we take as the proxy for number of tornadoes

CAPE SRH2

3; 600; 000 m4 s 4 ; 5

which differs from the EHI in that the square of SRH appears. The normalizing factor is chosen to match overall annual outbreak numbers. The proxy for the number of tornadoes per occurrence is therefore the mean of equation (5) conditional on the occurrence proxy of EHI being greater than one.

References

1. Data from Summary of U.S. Natural Hazard Statistics 2005-2014 compiled by the National Weather Service Ofce of Climate, Water and Weather Services and the National Climatic Data Center. http://www.nws.noaa.gov/om/hazstats.shtml

Web End =http://www.nws.noaa.gov/om/ http://www.nws.noaa.gov/om/hazstats.shtml

Web End =hazstats.shtml .

2. Fuhrmann, C. M. et al. Ranking of tornado outbreaks across the United States and their climatological characteristics. Wea. Forecasting 29, 684701 (2014).

3. Smith, A. & Matthews, J. Quantifying uncertainty and variable sensitivity within the US billion-dollar weather and climate disaster cost estimates. Nat. Hazards 77, 18291851 (2015).

4. Diffenbaugh, N. S., Scherer, M. & Trapp, R. J. Robust increases in severe thunderstorm environments in response to greenhouse forcing. Proc. Natl Acad. Sci. USA 110, 1636116366 (2013).

5. Gensini, V. & Mote, T. Downscaled estimates of late 21st century severe weather from CCSM3. Clim. Change 129, 307321 (2015).

6. Tippett, M. K., Allen, J. T., Gensini, V. A. & Brooks, H. E. Climate and hazardous convective weather. Curr. Clim. Change Rep. 1, 6073 (2015).

7. Verbout, S. M., Brooks, H. E., Leslie, L. M. & Schultz, D. M. Evolution of the U.S. tornado database: 1954-2003. Wea. Forecasting 21, 8693 (2006).

8. Brooks, H. E., Carbin, G. W. & Marsh, P. T. Increased variability of tornado occurrence in the United States. Science 346, 349352 (2014).

9. Elsner, J. B., Elsner, S. C. & Jagger, T. H. The increasing efciency of tornado days in the United States. Clim. Dyn. 45, 651659 (2015).

10. Tippett, M. K. Changing volatility of U.S. annual tornado reports. Geophys. Res. Lett. 41, 69566961 (2014).

11. Sander, J., Eichner, J. F., Faust, E. & Steuer, M. Rising variability in thunderstorm-related U.S. losses as a reection of changes in large-scale thunderstorm forcing. Wea. Climate Soc. 5, 317331 (2013).

12. Long, J. A. & Stoy, P. C. Peak tornado activity is occurring earlier in the heart of Tornado Alley. Geophys. Res. Lett. 41, 62596264 (2014).

13. Lu, M., Tippett, M. & Lall, U. Changes in the seasonality of tornado and favorable genesis conditions in the central United States. Geophys. Res. Lett. 42, 42244231 (2015).

14. Doswell, III C. A., Edwards, R., Thompson, R. L., Hart, J. A. & Crosbie, K. C. A simple and exible method for ranking severe weather events. Wea. Forecasting 21, 939951 (2006).

15. Taylor, L. R. Aggregation, variance and the mean. Nature 189, 732735 (1961).16. Eisler, Z., Bartos, I. & Kertsz, J. Fluctuation scaling in complex systems: Taylors law and beyond. Adv. Phys. 57, 89142 (2008).

17. Cohen, J. E., Xu, M. & Schuster, W. S. F. Stochastic multiplicative population growth predicts and interprets Taylors power law of uctuation scaling. Proc.R. Soc. B 280, 20122955 (2013).18. Malamud, B. D. & Turcotte, D. L. Statistics of severe tornadoes and severe tornado outbreaks. Atmos. Chem. Phys. 12, 84598473 (2012).

19. Elsner, J. B., Jagger, T. H., Widen, H. M. & Chavas, D. R. Daily tornado frequency distributions in the United States. Environ. Res. Lett. 9, 024018 (2014).20. Cohen, J. E. & Xu, M. Random sampling of skewed distributions implies Taylors power law of uctuation scaling. Proc. Natl Acad. Sci. USA 112, 77497754 (2015).

21. Vitolo, R., Stephenson, D. B., Cook, I. M. & Mitchell-Wallace, K. Serial clustering of intense European storms. Meteor. Z. 18, 411424 (2009).

22. Karremann, M. K., Pinto, J. G., Reyers, M. & Klawa, M. Return periods of losses associated with European windstorm series in a changing climate. Environ. Res. Lett. 9, 124016 (2014).

23. Agee, E. & Childs, S. Adjustments in tornado counts, F-scale intensity, and path width for assessing signicant tornado destruction. J. Appl. Meteor. Climatol. 53, 14941505 (2014).

24. Mesinger, F. et al. North American regional reanalysis. Bull. Am. Meteorol. Soc. 87, 343360 (2006).

25. Davies, J. M. in 17th Conference on Severe Local Storms, 107111 (Amer. Meteor. Soc., St. Louis, MO, 1993).

26. Rasmussen, E. N. & Blanchard, D. O. A baseline climatology of sounding-derived supercell and tornado forecast parameters. Wea. Forecasting 13, 11481164 (1998).

27. Tippett, M. K., Sobel, A. H. & Camargo, S. J. Association of U.S. tornado occurrence with monthly environmental parameters. Geophys. Res. Lett. 39, L02801 (2012).

28. Tippett, M. K., Sobel, A. S., Camargo, S. J. & Allen, J. T. An empirical relation between U.S. tornado activity and monthly environmental parameters. J. Clim. 27, 29832999 (2014).

29. Brooks, H. E., Lee, J. W. & Craven, J. P. The spatial distribution of severe thunderstorm and tornado environments from global reanalysis data. Atmos. Res. 6768, 7394 (2003).

30. Brooks, H. E. Severe thunderstorms and climate change. Atmos. Res. 123, 129138 (2013).

31. Gilleland, E., Brown, B. G. & Ammann, C. M. Spatial extreme value analysis to project extremes of large-scale indicators for severe weather. Environmetrics 24, 418432 (2013).

32. Thompson, R. L., Mead, C. M. & Edwards, R. Effective storm-relative helicity and bulk shear in supercell thunderstorm environments. Wea. Forecasting 22, 102115 (2007).

Acknowledgements

M.K.T. was partially supported by a Columbia University Research Initiatives for Science and Engineering (RISE) award, the Ofce of Naval Research award N00014-12-1-091, NOAAs Climate Program Ofces Modeling, Analysis, Predictions and Projections program award NA14OAR4310185, and the Willis Research Network. J.E.C. was partially supported by U.S. National Science Foundation grant DMS-1225529. The views

6 NATURE COMMUNICATIONS | 7:10668 | DOI: 10.1038/ncomms10668 | http://www.nature.com/naturecommunications

Web End =www.nature.com/naturecommunications

NATURE COMMUNICATIONS | DOI: 10.1038/ncomms10668 ARTICLE

expressed herein are those of the authors and do not necessarily reect the views of any of the sponsoring agencies.

Author contributions

All the authors were involved in conducting the research and the scientic discussion equally throughout the study. The data were prepared by M.K.T. The manuscript was written, edited and reviewed by both the authors.

Additional information

Supplementary Information accompanies this paper at http://www.nature.com/naturecommunications

Web End =http://www.nature.com/ http://www.nature.com/naturecommunications

Web End =naturecommunications

Competing nancial interests: The authors declare no competing nancial interests.

Reprints and permission information is available online at http://npg.nature.com/reprintsandpermissions

Web End =http://npg.nature.com/ http://npg.nature.com/reprintsandpermissions

Web End =reprintsandpermissions /

How to cite this article: Tippett, M. K. & Cohen J. E. Tornado outbreak variability follows Taylors power law of uctuation scaling and increases dramatically with severity. Nat. Commun. 7:10668 doi: 10.1038/ncomms10668 (2016).

This work is licensed under a Creative Commons Attribution 4.0 International License. The images or other third party material in this article are included in the articles Creative Commons license, unless indicated otherwise in the credit line; if the material is not included under the Creative Commons license, users will need to obtain permission from the license holder to reproduce the material. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/

Web End =http://creativecommons.org/licenses/by/4.0/

NATURE COMMUNICATIONS | 7:10668 | DOI: 10.1038/ncomms10668 | http://www.nature.com/naturecommunications

Web End =www.nature.com/naturecommunications 7