1. Introduction
Official estimation of peak tornadic wind speeds is performed by the National Weather Service and relies on ground and areal surveys of damage paths. The F scale, proposed and developed by T. Fujita (e.g., Fujita 1971), is the scheme most widely used to classify tornadic wind speed based on the severity of structural damage. In this scheme, observed damage patterns are compared with a series of written descriptions and photographic examples of damage to houses, vehicles, trees, and other common objects. There are six categories of damage (F numbers), each corresponding to a range of wind speeds within the overall interval from 15 to over 130 m s−1.
The F-scale classification appears to be generally accepted as a practical means of assessing wind speeds and is routinely used by survey teams. However, the method relies primarily on damage to structures rather than to natural vegetation (Marshall 2003; Schaefer and Livingston 2003). Thus, it is difficult to classify tornadoes that occur in forested areas.
Peterson (2003) has studied tornado damaged forests to determine whether there is a way to incorporate tree damage based on species and size into an improved F-scale system. His results were encouraging when looking at forests that were similar in their composition.
Another possible way to classify tornadoes that do not pass over structures is to analyze the damage done to the trees and to try to determine the maximum wind speed of the tornado. Then the tornado could be classified using the F scale by simply putting it in the correct wind speed category.
As a first step in this endeavor, this study describes a quantitative physical model that can be readily used to assess tornadic wind speeds in forested areas. It is based on a simple Rankine vortex and a modified tree model designed by Peltola et al. (1999). The analytical model is initialized with information on tree species and stand characteristics plus estimates of tangential, radial, and forward speeds of the vortex. The model then generates a pattern of downed trees that can be compared with field observations. From this comparison, one can estimate the wind speed and, therefore, the F-number classification of a tornado.
a. An early model for the wind field
Around 1923, J. Letzmann [reviewed by Peterson (1992)] suggested that when a tornado moves through a forest it produces a damage pattern that relates to the sum of vortex rotation and forward speed. By creating a simple, analytical model of a steady-state vortex, Letzmann was able to simulate tree damage patterns that compared favorably to those observed.
In his model, Letzmann used a combined Rankine vortex to estimate the tangential flow of a tornado. Once a forward speed and radial wind component were specified, Letzmann computed the resultant wind vector for every point in and around the vortex. By experimenting with different combinations of radial, tangential, and forward speeds, Letzmann developed hand-drawn schematic illustrations of several fundamental wind field patterns. To apply the wind model to the forest, he assumed that a tree would fall at some critical velocity and that fallen stems would align with the instantaneous direction of flow.
b. Recent advances in tree response
There are, of course, limitations to Letzmann’s model. It relies on hand calculations and does not include information about the physics of tree response. Much work in the latter area has been done in the last decade. Wood (1995) used wind-tunnel experiments on model trees and pulling experiments in the field to calculate the maximum stresses that different species of trees can withstand. Mattheck and Breloer (1994) developed a simple windthrow model that calculates the additive wind and gravity forces acting upon the tree. If the sum of these forces exceeds the tree’s resistance to uprooting, the tree will fall.
Field studies have been performed to calculate the bending moments of loblolly pine trees (Pinus taeda L.) (Fredericksen et al. 1993; Hedden et al. 1995). In these studies, loblolly pines were toppled using a winch-and-cable system. The force needed to break the stem, that is, the critical turning moment, was measured for trees of different sizes and ages. Building upon earlier work, Peltola et al. (1997, 1999), Peltola and Kellomäki (1993), and Peltola (1996) developed a mechanistic tree model that predicts the critical turning moment and threshold wind speed necessary to break the stems of several tree species. The model is particularly applicable to the problem at hand because it provides a physical link between wind speed and tree breakage. As a first step, the model used in this study considers only stem breakage and does not include uprooting. The latter is a much more complex physical process controlled partly by wind stress and partly by root and soil characteristics—factors that are difficult to observe and therefore are beyond the scope of this paper.
2. Method
This work essentially combines an analytical vortex with the physical tree-response model of Peltola et al. (1999), hereinafter called the Peltola model, to produce simple visual fields of tree-fall patterns. As a starting point, we have extended the Peltola model to include information on loblolly pine, a tree species common to the southeastern United States. The resultant model produces a pattern of tree-fall somewhat analogous to the hand-drawn diagrams of Letzmann.
a. Vortex model description
b. Tree model
To calculate the projected area of the crown against the wind, the Peltola model assumes a crown shape simulated by triangles. For Scots and loblolly pine, two triangles share a common base, with the vertex of one triangle extending upward and the other downward. The bases of the triangles are equal to 2 times the length of the longest branch in the crown (Hakkila 1971; Peltola and Kellomäki 1993). For loblolly pine, the upper triangle is taller than the lower to differentiate between the living and dead crowns.
The mass of each stem segment is calculated from the green density of the stem and the volume of the segment (Clark 1991; Laasasenaho 1982; Peltola and Kellomäki 1993). The vertical distribution of crown mass is assumed to be similar to that of the crown area projected against the wind and is proportional to the stem mass (Baldwin 1987; Peltola and Kellomäki 1993).
The model moves the tornado vortex over the forest, calculating the turning moment for each grid point and comparing it with the resistance to stem breakage. If the turning moment exceeds the resistance, the stem is considered to be broken.
3. Results
a. Vortex wind field
As in earlier work by Letzmann (Peterson 1992), a moving Rankine vortex with variable inflow of the type described above produces an isotach field that generally consists of a family of concentric circles that enclose a calm point and an adjacent family of crescents that enclose the area of maximum wind speeds. Somewhat as defined by Letzmann, key parameters α and Gmax control the asymmetry of the wind pattern, where Gmax is the ratio of the maximum tangential wind speed to the forward speed and α is the angle between the tangential and summed component wind vectors at the point of maximum winds (Fig. 5). For the simple vortex, it can be seen in (2)–(5) that α does not vary with radial distance from the vortex center.
For a few examples, Fig. 6 combines isotach and vector fields to show the differences in the flow pattern when α and Gmax are varied. In Figs. 6a and 6b, α is held constant at 45° such that tangential wind speed is equal to radial wind speed. For this and all subsequent illustrations, Rmax is held constant at 75 m given that it has no effect on overall patterns. Figure 6a represents a slow-moving tornado, that is, one with a rotational speed that is 4.2 times the forward speed (Gmax = 4.2). In this case, the forward speed is 10 m s−1 and the maximum wind speed relative to the ground is 69 m s−1. The isotach pattern is only slightly asymmetric, but the radial inflow causes the area of maximum winds to shift 45° rearward from immediately right of center. In Fig. 6b, where Gmax = 2.1, the vortex forward speed is doubled, causing an increase in the maximum wind speed and a further leftward shift of the area of calm winds. Note that, as the forward speed increases, the location of the center of maximum winds is not displaced outward from the center of the relative vortex (located at the origin of the diagram).
In Figs. 6c and 6d the forward speed is held constant at 10 m s−1 and α is varied. When α = 36°, the tangential wind speed is greater than radial wind speed and vice versa for α = 54°. Note the rotation of the axis of extremes, AB.
b. Critical wind speed
The data used for loblolly pine in the model were taken from Wahlenberg (1960). This table includes the height, DBH, and spacing for trees aged 20–80 yr in site indices of 60–120. The site index refers to the height of 50-yr-old dominant trees of the same species in a given index area. The site index is determined for a tree species based on the soil and climatic factors characteristic of that area (Avery and Burkhart 2002, chapter 15). For every site index, the model assumes a random distribution of tree ages from 20 to 80 yr. The model calculates the critical wind speed for any height, DBH, and spacing. For example, Fig. 7 shows how the critical wind speed varies with height and form quotient, defined as the height in meters divided by the DBH in meters. For loblolly pines of given tree height, low-taper trees can withstand greater wind speeds than can high-taper trees—a result that is not surprising because a high taper value indicates a tall, thin tree and hence one that is more susceptible to wind breakage.
Spacing and crown size can also have a large effect on critical wind speed. The model assumes that spacing is equal to crown width. Therefore, tree crowns will meet but not overlap and will leave little space between trees. As the crown area increases, there is more surface area on which the wind can act. Thus, as the crown size grows with tree spacing, the wind force increases and the critical wind speed decreases. Figure 8 is a plot of critical wind speed versus tree spacing and illustrates this effect.
c. Damage patterns
The model produces a graphical plot of downed trees. The arrows are proportional to the tree height and represent trees downed in the instantaneous direction of the wind, because the model assumes no airborne stems or interaction among the trees. The gray dots represent standing trees (Fig. 9).
Suppose, for simplicity, that the radial wind component is held at zero and the maximum absolute wind speed is held constant at 60 m s−1, equivalent to an F2 tornado. For loblolly pine with a site index of 90 and an upwind gap of 10 tree heights at the forest edge, the model produces a series of damage patterns when the forward speed of the vortex is varied.
At the slowest forward speed, a broad area of intermittently downed trees lies nearly centered along the storm track and forms a semicircular pattern, as seen in Fig. 9a (cf. with Atkins and Taber 2004, their Fig. 4). A few downed trees along the left side of the path lie nearly opposite the direction of the storm motion. The general pattern is initially roughly symmetric about the axis of the track. However, as the forward speed increases, as shown in Figs. 9b–d, the damage area moves to the right side of the track and individual trees align increasingly in the direction of the storm motion (e.g., Hall and Brewer 1959, their Fig. 3; Atkins et al. 2005).
As the absolute wind speed increases to F3 strength in Fig. 10, the same general shift from symmetry to asymmetry is observed. As expected, many more trees are downed, but of interest is that progressively fewer trees are affected even though the maximum wind speed is the same for all four examples. The reason for this curious outcome is that if the maximum wind is held constant and the forward speed is increased, the tangential component must decrease. As this vortex component decreases, the initial broad ring of supercritical wind speeds shrinks and eventually shifts to the right of the track and consequently impacts fewer trees. The outcome suggests that the areal extent of damage may decrease considerably with forward speed, even for a given F number. Note, however, that the overall form of the tree damage pattern resembles that of Fig. 9.
Adding the effect of radial winds for a slow-moving storm, one again finds along the left side of the swath a tendency for tree alignment opposite that of the storm direction (Fig. 11). This tendency increases with inflow angle. As the forward motion of the vortex is increased from 3 to 18 m s−1, a new pattern develops in Fig. 12. Here the orientation of tree fall converges along a line on the right side of the track. The convergence line becomes more pronounced and moves closer to the centerline of the track as the radial component increases (cf. Figs. 12a–d). This pattern can be described as a “herringbone” (e.g., Hall and Brewer 1959, their Fig. 5).
To explain the pattern variations, one needs to consider two key controls: 1) the wind field resulting from the vortex and its forward motion and 2) the critical wind speed for the tree species in question. The first control is similar to that found by Letzmann, who illustrated the separate effects of forward speed and vortex inflow (see also Hall and Brewer 1959). However, the second effect—that determined by the tree species—can dramatically change the tree-fall pattern.
Absolute winds in a slowly moving vortex are nearly axially symmetric. If the maximum wind is slightly greater than the tree’s critical speed, the greatest damage will be located at Rmax at the upwind edge of the forest where exposure is greatest. These trees will align with the storm motion vector (see the southern boundary in Fig. 9). As the absolute wind speed increases for this slow-moving vortex, trees will fall as they encounter the approaching ring of maximum winds. The great majority of these trees will fall on their first encounter with the ring’s critical isotachs ahead of and along the sides of the vortex center. Thus, we find the semicircular pattern seen in Figs. 9a, 10a and 11.
As a radial component is added to the vortex, the situation becomes more complicated. Referring back to Fig. 6, we recall that, for a moving vortex, radial winds cause a clockwise shift in the location of the absolute wind maximum. This shift, controlled by α, causes the convergence pattern in the tree alignment observed in Fig. 12. As α increases, trees everywhere to the left of the storm track encounter maximum winds at the left-rear quadrant of the vortex (i.e., after the vortex center has passed). If wind speeds in this rear cusp of the isotach pattern exceed critical values, trees will be downed. At this location, the winds are directed from left to right: hence, the cross-track alignment of downed trees in this region. In a similar way, adjacent trees that stand between the vortex center and the track of the absolute wind maximum will experience maximum wind exposure after the vortex center has passed. These downed trees will also trend from left to right, but they will also tend to align in the forward direction. Because all trees aligned left-to-right fall after the vortex center has passed, their stems should lie atop the others. As forward speed increases, the ultimate extreme of this pattern is a pair of adjacent semicircles: the one on the right is created before the vortex center has passed, and the one on the left is created after the vortex center has passed (see Fig. 12d).
So far, the discussion has been restricted to a single species and site index and has compared damage patterns for fixed maximum absolute wind speeds. Holding the maximum winds constant requires reducing the vortex circulation as storm speed increases. Although this procedure is reasonable if one wishes to compare damage for a constant F number, it unavoidably results in a reduction in damage as the forward speed increases. However, to discern rudimentary controls of damage patterns, it might seem best to keep the vortex circulation constant and to vary the storm speed.
Figure 13 presents the results for an example of such a constant vortex. In this case, the tangential and radial wind maxima are fixed at 70 and 20 m s−1, respectively, yielding α = 16° and a peak vortex wind speed of 73 m s−1—a threshold F3 strength. The vortex is moved at varying speeds through the same stand of loblolly pine with a site index of 90 and an upwind gap of 10 tree heights. Results are generally similar to those for the trials with fixed F number. A crisscross damage pattern emerges to the right of the centerline and becomes more pronounced as the forward speed increases. This time, however, the extent of tree damage increases with forward speed because the absolute wind maximum is allowed to increase. As before, the rightward-oriented tree vectors and damage convergence increase as the radial wind component increases, as shown in Fig. 12.
The effect of critical wind speed is illustrated well by repeating the conditions shown in Fig. 13c but changing the tree species to Scots pine, which has a lower damage threshold. For this run, trees of various ages were randomly distributed through the stand. Results in Fig. 14 show not only more overall tree damage (cf. 36% for loblolly with 81% for Scots pine), but also a clear change in the character of the damage pattern. Few Scots pine survive the initial wind, and, hence, the pattern resembles that of a stronger or slower-moving vortex in a loblolly forest.
To illustrate how the tree-fall patterns vary with site index, an F4 tornado was used. The storm characteristics are identical to those of Fig. 13c. The patterns illustrated in Fig. 15 are similar: a converging pattern of damage forms to the right of the vortex track and trees nearest the upwind edge of the stand are most affected. The only striking difference is that, as the site index increases from 60 to 120, the percentage of downed trees decreases by about 10%. This reduction is simply because a site index of 120 includes stronger, bigger trees.
4. Conclusions
This study describes a physical model that produces patterns of tree fall associated with idealized tornadoes. The tornadic wind fields are specified as steady-state analytical functions that produce lateral forces on idealized trees. Empirical data from loblolly pine and Scots pine are used to simulate stem failure and tree fall for various tree spacings and site indices for a moving modified Rankin vortex with specified radial and tangential components.
It is found that characteristic tree-fall patterns for loblolly pine result from different combinations of tangential and radial components of the vortex and its forward speed. At slow forward speeds and small radial components, vortex flow that exceeds the critical value causes a semicircular damage pattern of tree fall centered along the vortex track in which most trees fall left to right. As the forward motion of the vortex increases, the damage shifts to the right and downed trees begin to align with the track. For a fixed F number, the extent of tree damage may decrease as forward speed increases.
As the radial inflow of the vortex increases, a third damage pattern emerges—one of convergence of tree alignment along a line parallel and to the right of the track. Here a unique crisscross pattern emerges with right-to-left-aligned stems superimposed upon the others. Although site index and absolute wind speed influence the amount of tree damage, they appear to have little effect on the above patterns. However, when the tree species is changed from loblolly to Scots pine, the cross-track damage pattern is found even at rapid forward speed and large radial inflow. This result illustrates the importance of the critical wind speed in determining the damage pattern. It further suggests that in a mixed forest, a combination of patterns may be found within the same damage swath.
This study is an early step in attempting to relate tornadoes to tree damage and as such has many limitations. The type of modified Rankin vortex used in this study likely represents a vast idealization of the actual near-surface flow even for a simple tornado vortex. For example, Doppler measurements by Bluestein et al. (2003) suggest that the maximum inflow in the radial wind component may occur at a radius of about 2 times that of the maximum tangential wind. Measurements by Wurman and Gill (2000) suggest that in the outer region of the vortex the tangential velocity may decrease approximately with r−0.6 rather than r−1 as assumed here. Only a limited number of actual storms have been sampled and thus it may be premature to make generalizations.
A key related question is whether a given tornado vortex flow can be realistically represented by a single set of analytical functions. Most studies and numerous cases of video footage indicate that tornadic winds are anything but steady state. Variations in tangential, radial, and forward speeds are probably large and the vortex flow may make transitions through different flow regimes (Bluestein et al. 2003; Wurman and Gill 2000; Davies-Jones 1986).
Another aspect of actual tornadoes that has been omitted here is that of airborne debris. Some trees and tree sections are picked up and deposited far from their points of origin and do not serve as markers of the instantaneous wind direction. Within the damage swath, such behavior might superimpose randomly oriented debris lying atop the trees that fall in place. This characteristic implies that sole use of areal photography would be open to misinterpretation unless accompanied by ground surveys.
The Peltola model, although based heavily on observed tree response, still represents a first-order parameterized approach. For example, it is likely that tree-stem failure most often results from a harmonic matching of wind gustiness to the frequency of tree sway. This characteristic is parameterized in the model using a gust factor. Uprooting is not included here; nor are second-order effects included. For example, partially uprooted trees often lean on their neighbors, and gaps created by uprooted or broken trees increase the wind exposure of downwind trees. It would be interesting to include some of these effects in future work.
The model uses the logarithmic profile to compute the wind speed at the tree crown from that at some distance above the treetops. Amid the sea of uncertainty, this particular assumption may seem unimportant, but it links the wind force to the tree and should be carefully considered. Studies similar to that of Oliver and Mayhead (1974), who measured wind profiles in and above forests during high winds, may prove to be useful in obtaining profile relationships.
Even in light of uncertainties, the model presented here seems to be a reasonable starting point for simulating tree damage for a simple single-vortex tornado. The model may prove useful to compare the basic simulated damage patterns with those found in the field. For example, several variables, including the tree species, the stand characteristics, and estimates of the forward speed of the vortex, may be obtainable in a given case. Once these variables are specified, observed damage patterns can be compared with those modeled for different radial and tangential components. If successful, the results might yield reliable objective estimates of the storm strength as well as vortex characteristics in forested areas.
Acknowledgments
The authors thank Dr. Jerry Davis for his comments and advice and Dr. Heli Peltola of the University of Joensuu, Finland, for her many helpful suggestions and advice in applying her tree model to this research.
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Wind vectors and isotachs (shaded) representing the total wind field for a vortex with a maximum tangential velocity of 60 m s−1, a maximum radial velocity of 30 m s−1, and a forward speed of 15 m s−1. The radius of maximum vortex-relative components is 75 m. The center of the relative vortex is at the origin.
Citation: Journal of Applied Meteorology and Climatology 45, 12; 10.1175/JAM2413.1
Wind force (Fwind) and gravity (Fmass) producing turning moment (T) and lateral displacement (x) on an idealized tree (after Peltola et al. 1997).
Citation: Journal of Applied Meteorology and Climatology 45, 12; 10.1175/JAM2413.1
Variation of critical turning moment with distance from stand edge for Scot pine.
Citation: Journal of Applied Meteorology and Climatology 45, 12; 10.1175/JAM2413.1
Critical wind speed for stem breakage for different upwind gap sizes for Scots pine.
Citation: Journal of Applied Meteorology and Climatology 45, 12; 10.1175/JAM2413.1
Rudimentary relationships controlling the isotach pattern for a moving Rankine vortex. Isotachs are curved lines (m s−1); the vortex is centered at the origin but is moving at velocity VS. Line AB intersects the calm point at radius ro, the origin, and the location X of maximum absolute wind Vx. Vectors Vt (Vtan in text) and Vr denote the tangential and radial vortex components at X, respectively.
Citation: Journal of Applied Meteorology and Climatology 45, 12; 10.1175/JAM2413.1
As in Fig. 1, but for selected combinations of radial component Vr, tangential component Vt, and forward velocity VS. Note the AB axis of extremes.
Citation: Journal of Applied Meteorology and Climatology 45, 12; 10.1175/JAM2413.1
Variation of critical wind speed with height and taper for loblolly pine.
Citation: Journal of Applied Meteorology and Climatology 45, 12; 10.1175/JAM2413.1
Variation of critical wind speeds with tree spacing for loblolly pines with two different taper values.
Citation: Journal of Applied Meteorology and Climatology 45, 12; 10.1175/JAM2413.1
Windfall patterns resulting from varying the tangential component and forward speed for an F2 tornado of fixed intensity. Values of the radial component Vr, tangential component Vt, forward motion of the vortex VS, and maximum wind Vx (m s−1) are given for each panel. The vortex center moves northward at x = 0. For all cases, the radius of maximum vortex winds is 75 m. Standing loblolly pine trees are represented by dots, and fallen trees are represented by arrows whose lengths represent the lengths of stems that were broken (see scale) and whose orientations represent stem orientations. The site index is 90, and the upwind gap (at the south edge of the stand) is 10 tree heights. The percentage for each panel refers to the percent of trees felled out of the total of 7056.
Citation: Journal of Applied Meteorology and Climatology 45, 12; 10.1175/JAM2413.1
Same as Fig. 9, but for an F3 tornado.
Citation: Journal of Applied Meteorology and Climatology 45, 12; 10.1175/JAM2413.1
Same as Fig. 9, but for a slow-moving F3 tornado.
Citation: Journal of Applied Meteorology and Climatology 45, 12; 10.1175/JAM2413.1
Same as Fig. 9, but for a faster-moving F3 tornado.
Citation: Journal of Applied Meteorology and Climatology 45, 12; 10.1175/JAM2413.1
Same as Fig. 9, but for a fixed vortex moving at increasing forward speed, resulting in increasing intensity.
Citation: Journal of Applied Meteorology and Climatology 45, 12; 10.1175/JAM2413.1
Damage patterns for wind conditions identical to those of Fig. 13c, but for Scots pine with a random distribution of different aged trees. This combination resulted in 81% breakage of 12 100 trees.
Citation: Journal of Applied Meteorology and Climatology 45, 12; 10.1175/JAM2413.1
Damage patterns for wind conditions identical to those of Fig. 13c, but for loblolly pine with site indices of (a) 60 and (b) 120. These site indices represent totals of 13 689 and 5184 trees, respectively, with losses of 38% and 28%.
Citation: Journal of Applied Meteorology and Climatology 45, 12; 10.1175/JAM2413.1
