In most places, episodes of extremely high winds should be sufficiently rare and short that the basic photosynthetic function of leaves could be compromised with minimal long-term cost. While storms may last for hours or days, because of the irregularity of near-ground wind, severe gusts have durations comparable to the lulls previously considered. The mechanically destructive effects of high winds are said to increase with an exponent of velocity well above 1. But figures cited for that relationship should be viewed sceptically as they assume that the rigid structures predominant in human technology. For leaves, which are highly flexible, the particular relationships among wind speed, drag and likely damage should depend on their particular mechanical responses to wind.
1. Quantifying and measuring the drag of leaves
The force and torque about the base of a tree or other seed plant depends on the drag of its leaves, the drag of trunk and branches, and the virtual masses of the latter. For a fully leafed tree the drag of its leaves should be the largest force. Because the leaves center further from the ground than trunk and branches, they will impose a still greater fraction of the torque about the base. By contrast, trunks and branches displace enough volume to have significant virtual mass (the mass of the air they displace, which must be accelerated in the opposite direction; Daniel, 1984), generating forces additional to drag. Furthermore, they might develop oscillations in synchrony with gusts. Damping, though, should ordinarily be great enough to avoid any large component of relative wind from windward sway (de Langre, 2008). Leaves, by contrast, neither displace much volume nor respond with sufficiently long time constants to be especially sensitive to the unsteadiness of gusts. Thus the force exerted by wind on leaves should be unaffected by unsteady effects other than those attributable to the flag-like flexibility of the leaves themselves.
High wind might caused several distinct forms of damage. Leaves may be torn or shredded, they may be pulled off their branches, branches may be broken, or entire trees may suffer breakage or be uprooted. We have as yet too little information to compare the relative risks of these modes of failure with any confidence. Wilson (1980) found in sycamores (Acer pseudoplatanus), both in the field and in a wind tunnel, that leaves suffered greatest damage in the first few weeks after bud break even though wind speeds differed little from those later in the season. My own observations point to tearing and shredding as especially common in newly expanded leaves. Detachment of leaves proves surprisingly rare considering the dramatic reduction in drag it could afford; moreover, defoliation appears less likely than windthrow to be fatal.
A graph of drag versus speed misleadingly emphasizes the nearly inevitable rise of the drag with increase in the speed. Because, for rigid, nonstreamlined objects at moderate and high speeds, drag increases with the square of speed, a more revealing plot looks at drag divided by the square of speed, D/v2, versus speed, v. In this alternative graph, such an ordinary object gives a horizontal line; deviations in slope indicate special behavior. Thus an object that gradually assumes a low-drag configuration will yield a descending line. For instance, if drag varies linearly with speed instead of the square of speed, the slope on a logarithmic graph will be –1.0; if drag is independent of speed, the slope will be –2.0. That slope, of course, is the exponent, b, in the relationship
- (Eqn 5)
Wind tunnels large enough to accommodate trees of at least modest size have been available since the 1930s, although the earliest data for the drag of entire trees commonly cited are those of Fraser (1962) as reanalyzed by Mayhead (1973). These data, for Scots pine (Pinus sylvestris), produce an exponent of –0.72, incidentally a greater decrease with speed than that of streamlined shapes, which typically have values between –0.2 and –0.5. For a meter-high seedling of loblolly pine (Pinus taeda), the exponent is –1.13 (Vogel, 1984b); photographs suggest that this lower value results from the longer, more flexible needles of this latter species, needles that will cluster more tightly in high winds than those of P. sylvestris. (For lower speeds, exponents are higher, sometimes even positive, but such speeds present no great mechanical challenge; see Vogel, 1984b; Speck, 2003.)
Many investigations (Cullen (2005) cites copious references; see also Vollsinger et al., 2005) point to an exponent of c. –1.0, that is, to a near-linear relationship, and none to either 0.0 or –2.0. While no empirical work supports an exponent of 0.0, Cullen (2005) still recommends it for arboricultural predictions; that figure has also been used, at least tacitly, by de Langre (2008). The matter can be confusing in as much as referring drag to wind-speed-specific frontal area (‘dynamic drag coefficient’) gives an exponent nearer 0.0 (see Eqn 6), a result of the severe decrease in frontal area as speed increases. For common hardwoods that decrease reaches 70% by a speed of 20 m s−1 (Vollsinger et al., 2005).
Individual leaves, both simple and compound, as well as small clusters, ordinarily give exponents only slightly above those from work on whole trees and multi-leafed branches. Thus Vogel (1989) reported an average exponent of −0.72 ± 0.28 (SD) for 12 kinds of broad leaves and clusters (omitting single white oak leaves, as noted beneath Fig. 7) at speeds between 10 and 20 m s−1. That is consistent with the presumptions that the leaves of a fully foliated tree incur most of its drag and that in substantial winds large-scale sheltering of one leaf by another has only a second-order effect. But universality should not be presumed. The drag of a group of leaves on a branch of holly (Ilex opaca) changed with an exponent of −0.10, while a 1-m-high holly tree did much better, producing an exponent of −1.30 (Vogel, 1984b).
Figure 7. Drag coefficients for a flag, a rigid flat plate, and a variety of leaves in highly turbulent flows. One leaf has been omitted, that of a white oak (Quercus alba) that tore rather than reconfigured in strong wind. (Modified from Vogel, 1989.)
Download figure to PowerPoint
The literature in fluid mechanics commonly replaces D/v2 with the drag coefficient, Cd, defined as
- (Eqn 6)
where S is surface area and ρ is air density. It provides a fully dimensionless variable that corrects drag for surface area as well as speed, in effect giving a speed- and area-specific drag. Drag coefficients must specify reference areas to be meaningful. The common choices of reference areas reflect the usual focus on rigid objects quite unlike leaves. For present purposes, either of two possible areas might be chosen, (1) the area of one side of a leaf pressed flat or (2) the projected area as a leaf is exposed to sun or sky while subjected to wind. (1) is simplest and least ambiguous to determine as well as representing effective photosynthetic area under ordinary, near-calm conditions. (2) approximates instantaneous effective photosynthetic and aerodynamic areas, the latter being useful for comparisons with better known rigid objects. For present purposes, (1) offers the best combination of ease of measurement and functional relevance. In particular, it generates exponents directly comparable to exponents describing how drag changes with speed (D/v2) that make no reference to specific areas.
As referents for the drag of leaves, two kinds of object are especially relevant: rigid flat plates oriented parallel to the mainstream flow (weathervanes) and flexible flags. Flat plates parallel to flow produce less drag relative to exposed surface than even streamlined forms. Counterintuitively, the fluttering of flags gives them drag coefficients far higher than those of rigid plates, although the factor of increase depends on the geometry and material of the flag and the turbulence of the flow (Hoerner, 1965). A square, leaf-sized flag of flexible polyethylene in a wind tunnel gives drag coefficients about 10 times greater than those of a rigid flat plate parallel to flow (Fig. 7; Vogel, 1989). Were leaves to experience the drag of analogous flags, it is likely that both the leaves themselves and the structures that bear them would be damaged. While the mechanisms responsible for the high drag of flags have been controversial, recent work (see, for instance, Alben & Shelley, 2004; Argentina & Mahadevan, 2005) points to destructive amplification of the vortices they themselves initiate. These analyses should help to elucidate the basis of the sharply contrasting behavior of leaves.
Drag coefficients at 20 m s−1 (again, on original projected area and thus only loosely comparable to data in the engineering literature) range widely but with hints of general patterns. Individual leaves or leaflets suffer more drag (relative to area) than do clusters of leaves or compound leaves, with coefficients typically c. 0.10 compared with 0.07, respectively. The lowest values come from extensively compound leaves. Black walnut (Juglans nigra) and black locust (Robinia pseudoacacia) average 0.032 (Vogel, 1989); a graph in Niklas (1999) gives a similarly low value for Chamaedorea. That average is only about four times that of a rigid flat plate (0.06 m across) parallel to a flow of 20 m s−1 (Fig. 7; Vogel, 1994). Outliers, omitted from the figure, are white oak (Quercus alba) leaves, for which b = +0.97 and Cd = 0.35. In addition, such oak leaves suffer physical damage, tearing at an average speed below 17 m s−1. Other leaves remain intact up to at least 20 m s−1.
We have few other drag coefficients. Those of Cescatti & Marcolla (2004) have been computed rather than measured; those of Fischenich & Dudley (1999) derive from measurements in moving water rather than in air. While rigid objects can be tested in different media as long as the Reynolds number is maintained, the reconfiguration of flexible forms such as leaves will change with fluid density (Vogel, 1994).
2. Devices for reducing drag
Leaves or clusters of broad leaves in high winds are not streamlined in the accepted use of that term. Instead, they reduce drag in ways available only to highly aeroelastic structures. As noted earlier for flags, flexibility per se, which is more likely to increase than to decrease drag, confers no automatic benefit. So drag reduction through flexibility must involve specific adaptive designs. While all relevant schemes have probably not yet been identified, flow-induced reconfiguration comes in at least four fairly distinct guises, as shown in Fig. 8 (Vogel, 1984b, 1989).
Figure 8. Leaves exposed to turbulent wind at 20 m s−1. (a) Leaf of tulip poplar (Liriodendron tulipifera); (b) cluster of white poplar (Populus alba); (c) pinnate leaf of black locust (Robinia pseudoacacia); (d) branch with leaves of American holly (Ilex opaca). (Many additional photographs appear in Vogel, 1993.)
Download figure to PowerPoint
(1) Simple leaves may curl up into cones with their apices formed by the basal portions of their blades; these cones become ever tighter (more acute) as the wind increases (Fig. 8a).
(2) Similar cones may be formed by clusters of leaves in which each leaf presses against more distal ones, curling less than it would as an individual but enough to fit against the cone's surface (Fig. 8b). Leaves that can do (1) as individuals can apparently also do (2) in groups, but not all leaves that can do (2) also do (1).
(3) Pinnately compound leaves with large numbers of leaflets may form into elongate cylinders, with each leaflet again pressed against more distal ones (Fig. 8c). Leaflet curling, though, is at least partly lengthwise, and the leaflets lie against each other as do scales on a fish.
(4) Leaves pinnately arranged on a branch may bend back along the branch so that they lie with their surfaces pressed together, abaxial to adaxial, to form a stack that gets tighter as the wind increases (Fig. 8d).
All the simple leaves that as individuals curl into cones share two structural features: petioles longer than c. 2 cm and blades extending further proximally than the petiolar attachment point; that is, some degree of basal lobing. These lobes often have a slight upward curl; as upwind extremes they initiate curling upward so an upper leaf surface forms the core of a cone. Occasionally, mainly in softer, early-season leaves, one or both lateral halves of a leaf curl downward. The role of long petioles is less clear; they are usually longer than required to keep basal lobes beyond the parent branch. Also of unclear significance is the relative rarity of serrate edges among leaves that reconfigure as individuals as well as clusters.
Variation in leaf shape within individual trees suggests a better ability to reconfigure among leaves that might experience greater wind. Thus de Soyza & Kincaid (1991) noted wider leaves with more lobing among the leaves of Sassafras albidum growing in open (presumably more wind-exposed) areas, and Kincaid et al. (1998) found more pronounced basal lobes among leaves higher up in a large tree of Pourouma tormentosa in French Guiana.
Features that enable individual coning occur in some but not all members of at least 15 families, an apparent convergence (Vogel, 1998). Convergent features are likely to represent fairly direct results of selection and consequently functional significance (Endler, 1986).
Curling into cluster cones does not require more than the most minimal petiole. Thus clusters of white oak leaves (by contrast with individual ones), with very short petioles, reconfigure well, if beginning at somewhat higher speeds than, for instance, those of red maples. Cluster coning but not individual coning requires that petioles, which are flexurally stiff in order to hold their blades outward, be torsionally flexible. The shorter the petioles, of course, the greater will be the requisite twist per unit length. Niklas (1996) found that sugar maple (Acer saccharum) leaves in wind-exposed areas were not only smaller, but had more flexible petioles than those in protected areas.
Small leaves such as those of birches (Betula) and poplars (Populus) cone only as clusters. White poplars (Populus alba) do so especially readily, probably due to the same bilaterally flattened petioles involved in fluttering in light winds. Comparison of red maples, groups of which form cluster cones in low winds, with white oaks, which require higher speeds, suggests that features promoting cluster coning may incidentally cause fluttering in modest winds. A functional advantage for the conspicuous low-speed shimmering of cottonwoods, aspens, and other Populus species has been enigmatic (see, for instance, Roden & Pearcy, 1993). Quite possibly, shimmering merely accompanies effective clustering in higher winds. Populus leaves resist acute wind damage; even isolated individual leaves of P. alba were not torn in a highly turbulent wind of 31 m s−1 (Vogel, 1989).
Pinnately compound leaves, reconfiguring into elongate cylinders, achieve especially low drag coefficients. Givnish (1978), following other observers, noted their relative prevalence in the canopy flora of lowland tropical rain forest and savannah. While he pointed to factors such as seasonal drought, perhaps in addition their prevalence results from selection for low drag in intermittently high winds. Reconfiguration into stacks has so far been seen only in a holly, Ilex opaca (Vogel, 1984b). While remarkably stable (in part as a result of interlocking of marginal spines) and effective, it involves extreme strains in the short petioles and seems incongruent with a role in what is most often a low, understory tree. Perhaps the behavior becomes significant during the winter, when this evergreen loses the shelter otherwise provided by nearby deciduous trees.
Some additional points about leaves in high winds should be considered. (1) Whatever its proximate cause, the ‘flagging’ of trees exposed to strong, directionally consistent winds might facilitate drag-resisting reconfiguration, at least to the extent that branches participate in the process. (2) Hardwoods of similar habitats have lower drag coefficients than do needled conifers (Vollsinger et al., 2005), suggesting that reduced drag in high winds is not an adaptive advantage for needled forms. (3) A wind speed of 20 m s−1 seems to have become a paradigmatic choice for maximum speed. Despite higher speeds recorded by standard meteorological equipment, it represents a reasonable extreme for what leaves on most trees might experience, based on the measurements of Oliver & Mayhead (1973) during a 1-in-10-yr gale. Still, Johnson et al. (1982) cite higher speeds, albeit only for a few seconds, in a cyclone. (4) Above 20 m s−1 and approaching the point of physical destruction, drag coefficients may (Johnson et al., 1982) or may not (Mayhead, 1973) drop less rapidly with increasing speed; in short, we have too little data to judge the point at which reconfiguration has reached its maximum. (5) Leaf reconfiguration may be easily observed without a wind tunnel by attaching a leaf or group of leaves with string or wire to an appropriate support and extending the assembly out of the window of a moving automobile. Once recognized, reconfiguration becomes evident during storms by following the thrashing of a branch with hand-held binoculars.