The standard measure of the intensity of a tornado is the
Enhanced Fujita scale, which is based qualitatively on the damage caused by
a tornado. An alternative measure of tornado intensity is the tornado path
length,

The touchdown of a tornado is a point event in space and time in analogy to
the initial point of rupture of an earthquake. The path length of tornado
touchdowns is a measure of the strength of the tornado, in analogy to the
Richter magnitude of an earthquake. In this paper, we consider the spatial
and temporal statistics of tornado touchdowns for three USA tornado outbreak
events from 1999 and 2011. We restrict our attention to severe tornadoes,
those tornadoes with path lengths

This paper takes a methodology for spatial–temporal
clustering analysis developed by Zaliapin et al. (2008) for seismicity and applies it to tornadoes. Their
methodology considers the times of occurrence and locations of point events.
All pairs of events are considered and the spatial lag

Here we consider the time and place of the touchdown of a tornado as a point
event. Our studies will be concentrated on tornado outbreaks. An outbreak is
a sequence of several to hundreds (Fuhrmann et al., 2014) of spatially correlated
tornadoes that occur in a relatively short period of time, typically a day,
with generally fewer tornadoes at night as severe convection is inhibited.
In contrast, earthquake aftershock sequences are unrestricted in time by
convective activity, and a severe earthquake of

Illustration of our clustering analysis
methodology.

In this paper we will give several examples of tornado outbreaks, including maps
of the tornado touchdown points as well as a clustering analysis of the
dependence of spatial lag

To illustrate this clustering analysis methodology applied to tornadoes, we
consider a sequence of four point events that occur at successive times

Studies of the statistics of tornadoes are limited by the problems
associated with the quantification of tornado intensity. Ideally, tornado
intensities would be based on wind speed measurements. However, as noted by
Doswell et al. (2009), high-resolution Doppler measurements of wind velocities in
tornadoes are not possible at this time. Currently, the Enhanced Fujita
scale is the standard measure of tornado intensities (Edwards et al., 2013).
Tornadoes are classified on a scale of EF0 to EF5 based on a qualitative
measure of damage. An alternative measure of tornado intensity is the
tornado path length,

The objective of this paper is to study the clustering statistics of tornado outbreaks. However, it must be recognized that the definition of a tornado outbreak is somewhat arbitrary (Mercer et al., 2009). Ideally, the definition of a tornado outbreak would be the occurrence of multiple tornadoes within a particular synoptic-scale weather system, but the spatial and temporal limits on the weather system are subject to arbitrary distinction (Glickman, 2000). Galway (1977) classified tornado outbreaks into three types: (i) a local outbreak with a radius less than 1000 miles (1609 km), (ii) a progressive outbreak moving from west to east in time and (iii) a line outbreak associated with a single moving supercell thunderstorm. Unfortunately, the NOAA (2015) NWS–SPC database does not associate individual tornadoes with a specific tornado outbreak using any of these three (or other) classifications.

There is a strong diurnal variability in tornado occurrence associated with
solar heating. For these reasons, Doswell et al. (2006) defined a tornado outbreak
to include all tornadoes in the continental USA in a convective day, i.e.
the 24 h period from 12:00 UTC (Coordinated Universal Time) of a given day
to 12:00 UTC of the following day, with 12:00 UTC (04:00 to 08:00 local time
in the continental USA depending on month and location) corresponding to the
approximate time of the daily minimum in tornado occurrence. The Severe
Weather Database that we use in our analyses list most tornadoes in Central
Standard Time (CST), so we will consider tornadoes in a convective day as
06:00–06:00 CST. However, consistent with the studies of severe tornado
outbreaks given by Malamud and Turcotte (2012), we will consider a severe
tornado outbreak to include only those tornadoes with path lengths

Tornado outbreak on 26–28 April 2011.
Times of touchdown and path lengths of severe tornadoes
(

To illustrate our clustering analysis methodology for tornadoes, we will
first consider the intense tornado outbreaks in the central United States on
26 and 27 April 2011. The tornado outbreaks in the spring of 2011 have been
discussed in detail by Doswell et al. (2012). They concluded that ideal conditions
for severe tornado outbreaks occurred during the last 2 weeks of April
2011, and that the supercell thunderstorms responsible for the tornadoes
were generated by a sequence of extratropical cyclones. In this paper, we
focus our attention on the outbreaks that occurred on 26 and 27 April 2011.
Although these outbreaks were certainly related to the same synoptic-scale
weather pattern, we will treat the two outbreaks separately for our
statistical studies. We will consider severe (

In Fig. 2 we give touchdown times

Tornado outbreak on 26–27 April 2011.
Touchdown locations of

We next consider the spatial distributions of the tornado touchdown points
for both the 26 and 27 April 2011 outbreak events. In Fig. 3a we give a map
of the tornado paths of the 45 severe (

We now turn to our clustering analyses of the two tornado outbreaks on
26 and 27 April 2011. From the times of occurrence given in Fig. 2 and the
spatial locations of tornado touchdowns given in Fig. 3a and b, we obtain
the temporal and spatial lags using the method illustrated in Fig. 1. In
Fig. 4a we give the spatial–temporal lag correlations of all pairs of the 45
severe (

Tornado outbreak on 26–27 April 2011.
Spatial–temporal lag correlations between the touchdowns for

Consider the spatial–temporal lag correlation associated with a series of
tornadoes generated by a point source at a constant velocity

We next turn our attention to one of the near-linear trends in Fig. 4a that does not pass through the origin, indicated by the rectangular region AB. We return to Fig. 3a, where in Region A we outlined a spatial cluster of the touchdowns for three severe tornadoes that occurred on 26 April 2011 and, in Region B, a spatial cluster of the touchdowns for 14 severe tornadoes. In the rectangular region AB, given in Fig. 4a there are 51 data points of which 42 (82 %) represent all of the pairs of tornado touchdowns between the two regions A and B in Fig. 3a, with none of the data points in box AB associated with pairs of tornadoes within Region A or pairs of tornadoes in Region B. We find that this explanation of correlations between tornadoes generated by two separate single cell thunderstorms (the spatial regions A and B in Fig. 3a) provides a similar explanation for the near-linear trends of spatial and temporal lags observed in Fig. 4a.

In Fig. 4b, we give the spatial lag vs. temporal lag for each of the pairs
of the 64 severe (

Tornado outbreak on 4 April 2011.

To further illustrate the relationship between tornadoes and storms, we apply our
clustering analysis to severe tornadoes that developed during a tornado
outbreak that occurred in the south-east of the USA on 4 April 2011. During
this outbreak, an extensive squall line developed along and ahead of a cold
front extending from Ohio in a south-westerly direction to Mississippi and Louisiana (Corfidi
et al., 2015). The environment proved suitable for the development of
thunderstorms within the largely linear convective band (Aon Benfield,
2011). The 4 April 2011 tornado outbreak is recognized as a derecho event (Aon
Benfield, 2011; NOAA, 2011; Corfidi et al., 2015), that is, a near-linear squall
line dominated by straight-line high winds rather than cyclonic winds
dominant in supercell thunderstorms. We consider six severe (

One hypothesis for the 4 April 2011 tornado outbreak is that the tornadoes touched down randomly along the squall line. However, this hypothesis is not consistent with the data in Fig. 5b. Random spatial and random temporal touchdowns produce the random distribution of data points seen in Fig. 4b. The data in Fig. 5 require that the tornadoes are produced at a near-stationary point on the squall line as the squall line migrates at a near-uniform velocity.

Tornado outbreak on 3 May 1999 (convective day, 06:00–06:00 CST).

As a final application of our cluster analysis, we will consider the Great
Plains tornado outbreak of 3 May 1999. On this day, multiple supercell
thunderstorms produced many large and damaging tornadoes in central
Oklahoma. With additional tornadoes in south-central Kansas and northern
Texas, over 70 tornadoes were observed during this event. This outbreak has
been discussed in detail by Thompson and Edwards (2000) and is of particular
interest to us because a detailed association of each tornado with specific
supercells has been given (NOAA, 1999). We consider only the 18 severe
(

We will now focus our attention on the four severe tornadoes associated with
supercell B (shown in red in Fig. 6a) and the five tornadoes associated with
supercell D (shown in blue in Fig. 6a). In Fig. 6b we designate
spatial–temporal lags associated with supercell B (shown in red) and
supercell D (shown in blue). Clear linear patterns for the spatial–temporal
lags associated with each of the two supercells are obtained. Also included
are the best-fit lines for the spatial–temporal lags; for supercell B
and D the velocities (slopes) are 43 and 38 km h

Five model tornado touchdown points located
randomly in time during a 6 h time window along a linear track.

In order to better understand the implications of our spatial–temporal lag
correlations, we consider two idealized models. The first is a model for a
sequence of tornadoes generated by a single supercell thunderstorm moving at
a constant velocity

Confirmation of this behaviour has been obtained in our treatment of the 3 May 1999 outbreak. Severe tornadoes previously associated with single supercell thunderstorms generate spatial–temporal lag correlations that are well approximated by straight lines as illustrated in Fig. 6b. A similar linear correlation was shown for the six severe tornadoes we studied from the 4 April 2011 outbreak illustrated in Fig. 5b. We also suggest that the strong linear trends seen in the spatial–temporal correlation data for the 26 April 2011 outbreak (Fig. 4a) may be associated with tornadoes generated by one or more single cell storms.

Two model scenarios for 40 tornadoes in an 800 km

Spatial–temporal lag diagrams for the two model scenarios given in Fig. 8.

In order to further address the large difference in spatial–temporal
correlations in the data illustrated in Fig. 4, we consider a second model,
more complex than the one just given. In our second idealized model, we
consider a quasi-linear vertical (north–south,

We now return to a discussion of the well-defined linear trends in the
spatial–temporal correlation given in Fig. 4a. The first linear trend,
extending from the origin with a slope corresponding to

We next introduce a measure of the combined spatial–temporal separation of
pairs of tornado touchdowns, for which the spatial–temporal separation

In Fig. 10 we plot the normalized cumulative probability

Tornado outbreak on 26–27 April 2011.
Normalized cumulative probability

We see that the sets of normalized cumulative probability values for the two
outbreaks given in Fig. 10 have a very different pattern, one linear and the
other a power law. For the 26 April 2011, our data set consisted of 45
severe tornadoes resulting in

We now give an explanation for the linear and power-law correlations that we
have found. If the tornado touchdowns occur randomly along a path for
relatively small values of spatial–temporal separations

Unlike many other natural hazards, it is difficult to quantify strong
tornadoes precisely. For hurricanes, there are extensive data on wind speeds
and barometric pressures along the path of the storm. For floods, flood
gauges provide a quantitative measure of the flow rate in a river. For
earthquakes, seismographs give measures of shaking intensity. Quantifying
volcanic eruptions and landslides is more difficult but volumes of material
involved can be estimated. It is not possible to reliably measure the wind
speeds or pressure changes in tornadoes. The standard measure of tornado
intensity used today is the Enhanced Fujita scale. This scale is based
qualitatively on the damage caused by a tornado. An alternative measure of
tornado intensity is the tornado path length

Malamud and Turcotte (2012) showed that records
of tornado path lengths from the 1990s to the present appear to be relatively complete for severe
tornadoes (defined to be

Another important aspect of tornado outbreaks is the distribution of
touchdown points in space and time. In terms of expectations for these data,
there are two limiting cases.

Tornadoes occur randomly in space and time during a specified spatial region and time interval. In this case the touchdown points will be randomly distributed in space by interacting supercell thunderstorms.

Tornadoes are generated by a single cell thunderstorm moving on a near-linear path at a constant velocity. In this case the touchdown points will approximately be on a linear path.

The statistics of the touchdown points of a tornado outbreak can certainly
be studied using a spatial map of the touchdown points. However, this does
not directly incorporate the time of the touchdowns. In this paper, we have
considered an alternative statistical measure for tornado touchdowns by
applying a spatial–temporal clustering analysis originally developed by
Zaliapin et al. (2008) for earthquakes. The sequence of severe tornado touchdowns
occurring during a convective day is considered to be a sequence of point
events in space and time. All pairs of these point events are considered and
a plot produced of the spatial lag

A principal focus of this paper is the application of a clustering analysis
to several observed tornado outbreaks. It is expected that a small outbreak
of severe tornadoes in a given convective day could be associated with
tornadoes generated randomly along a linear squall line progressing at a
near-constant velocity. In this case the tornado touchdowns occur randomly
both for space and time, and the cluster plot of

To further illustrate the applicability of our clustering analysis to severe
tornado touchdowns, we considered the Great Plains tornado outbreak of 3 May
1999. Careful studies have associated individual tornadoes in the outbreak
with specific supercell thunderstorms as shown in Fig. 6b. When all 18
severe tornadoes are considered the data are quite randomly distributed.
However, when two sets of tornadoes are considered that are associated with
two supercell thunderstorms, clear linear patterns are obtained with slopes
of 43 and 38 km h

We also applied our clustering analysis to the intense tornado outbreaks in
the central United States on 26 and 27 April 2011, with 45 and 64 severe
tornadoes occurring respectively (convective days) and more than 300
fatalities. For each pair of tornadoes on the two separate days, the severe
tornado touchdown spatial lags are given as a function of their temporal
lags in Fig. 4. The observed patterns are very different. The results for 26
April 2011 (convective day) in Fig. 4a are dominated by a complex sequence
of linear tracks that we have previously discussed. Knupp et al. (2013) suggest
that this 26 April outbreak of tornadoes was associated with a quasi-linear
convective squall line. The pattern seen in Fig. 4a has similarities to that
seen in Fig. 5b but is clearly more complex. We suggest that on 26 April
2016 groups of these tornadoes were associated with one large thunderstorm
or several closely spaced thunderstorms but there was a small number of
groups that generated the complexity. This pattern is consistent with the
movement of a discrete set of thunderstorms moving from the south-west to
the north-east at velocities near 70 km h

In order to better understand the roles of supercell thunderstorms in generating random and linear patterns in our spatial–temporal lag diagrams, we studied two model scenarios for tornado generation. In the first model scenario (Fig. 8a), four supercell thunderstorms originating in a squall line each generated 10 tornadoes randomly. In the second model scenario (Fig. 8b), 10 supercell thunderstorms originating in a squall line each generated four tornadoes randomly. The corresponding spatial–temporal lag diagrams for these two model scenarios are given in Fig. 9. The first scenario generated linear-type features; the second appeared random.

Although there are no physical processes directly in these two model scenarios, the statistical processes represent a variety of scales of processes that are important in tornado outbreaks. In general, the synoptic scale provides the background that leads to convection over a broad area (e.g. Knupp et al., 2014). The spacing between storms and the timing of initiation depends upon relationships between the synoptic-scale and smaller-scale features. Lilly (1979), Bluestein and Weisman (2000) and Lee et al. (2006) modelled the complexity of behaviour of storms that were initiated along lines; interaction included both constructive and destructive ones that can lead to the characteristic spacing associated with a particular event. Within a single supercell itself, the distance in time and space for repeated tornado genesis is a function of the storm motion (related to the large-scale environment in which the storm forms) and within-storm processes that lead to the distribution of precipitation and temperature leading to the birth and death of rotation features in the storm (Burgess et al., 1982; Alderman et al., 1999). For a particular tornado outbreak, the exact details depend upon the full range of atmospheric processes. Confidence is greatest in the understanding that certain large-scale environments are more likely to lead to outbreaks occurring, with details of individual storm occurrence and within-storm features becoming increasingly less certain.

We suggest that it may be possible to generate a large number of different model scenarios of this type, with corresponding spatial–temporal diagrams and compare them to spatial–temporal diagrams of observations in a semi-automated way. The objective would be to take the type of observed data illustrated in Fig. 6a and determine association of tornadoes with postulated supercell thunderstorm tracks semi-automatically. This application has the potential to provide constraints on simulated tornado outbreaks that are made by insurance and reinsurance modellers to create scenarios estimating risk of property loss. More realistic portrayals of tornado outbreaks could be important for the setting of optimal rates.

NOAA (National Oceanic and Atmospheric Administration) Storm Prediction
Centre (SPC), Tornado, Hail, and Wind Database, available at:

The authors thank J. Elsner and one anonymous referee for their helpful and
constructive suggestions.
Edited by: R. Trigo
Reviewed by: J. Elsner, H. Brooks

H. Brooks reviewed the discussion paper and was added as a co-author during the revision.

, and one anonymous referee