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Pollen dispersal models in quaternary plant ecology: assumptions, parameters, and prescriptions.

II. Introduction

Fossil pollen assemblages are the primary source of information on past vegetation composition and pattern at timescales of 100 to 1 million years. Vegetational inferences from pollen assemblages are not straightforward, however; distortions are introduced by widespread pollen dispersal and by differential pollen productivity and dispersal among plant taxa. Paleoecologists have long applied informal, "subjective" weightings in interpreting pollen data to correct for these distortions. The development of formal models of pollen dispersal (Tauber, 1965; Kabailiene, 1969; Prentice, 1985, 1988) and representation (Davis, 1963; Prentice, 1986; Sugita, 1994) has led to increased precision of paleo-vegetational inferences and enhanced design of paleoecological studies aimed at specific questions and spatial scales. The pollen-representation models are empirical models in which parameters (e.g., slope, y-intercept) are estimated using calibration sets consisting of modern pollen assemblages and quantitative vegetation data. The most critical challenge in parameter estimation is expression of the vegetation data in a form that resembles the vegetation as sampled by the pollen assemblages. In particular, the widespread dispersal of airborne pollen requires that some form of weighting be applied to the vegetation data. Such weighting can take a variety of forms, ranging from expanding the vegetation-sampling area (to incorporate more-distant pollen sources) to upweighting nearby vegetation (inverse-distance; inverse squared-distance) to application of formal pollen-dispersal models.

Formal models of pollen dispersal applied in paleoecological contexts to date (Tauber, 1965; Kabailiene, 1969; Prentice, 1985, 1988; Sugita, 1993) are all derived from Sutton's (1947a, 1947b, 1953) equations for diffusion in turbulent air within the planetary boundary layer. Sutton's equations are particular forms of the general class of Gaussian-plume models, which have been widely applied in atmospheric-diffusion studies, particularly smokestack emissions and atomic-energy hazards (Gifford, 1968; Hanna et al., 1982; Pasquill & Smith, 1983). Application to pollen dispersal assumes that individual plants are analogous to ground-level or elevated smokestacks, in which pollen is emitted continuously as a plume. Sutton's equations predict the dispersion of a cloud of gas or fine particulate matter from a continuously emitting point-source (ground-level or elevated) as a function of source-strength, wind speed, turbulence, a vertical diffusion coefficient, and height of the emitting source. Both the diffusion coefficient and the turbulence factor depend on atmospheric conditions, particularly stability as related to vertical temperature gradients (Sutton, 1947b). Application of Sutton's models to dispersal of pollen and other small particles with non-trivial mass requires an additional parameter, the deposition velocity of the particles (Chamberlain, 1953). This last parameter varies among plant taxa depending upon pollen-grain size, shape, and density.

Prentice's (1985, 1988) model, based on Sutton's ground-level model as modified by Chamberlain (1953) for particle deposition, incorporates an additional parameter, basin radius (i.e., distance of the depositional site from the nearest emitting sources), and predicts the pollen source area of a basin as a function of basin size and deposition velocity of pollen. A modified version of this model integrates pollen deposition over an entire lake surface based on lake surface-area (Sugita, 1993). These models have been widely applied; their predictions match existing data sets better than alternative models (Jackson, 1990, 1991, 1994; Sugita, 1994; Calcote, 1995). All applications to date have treated the three atmospheric parameters (wind speed, turbulence, vertical diffusion) as constants, using prescribed values for neutral atmospheric conditions (Sutton, 1947a, 1947b; Chamberlain, 1975; Prentice, 1985). Because pollen dispersal may occur under other atmospheric conditions, implications of alternative atmospheric parameters for pollen dispersal and representation need to be explored. Furthermore, most applications have used a relatively small set of the available measured and calculated data on terminal velocity of pollen grains; most values have been taken from Table III of Gregory (1973), which derive in turn mainly from the measurements of Knoll (1932) and Dyakowska (1936). Alternative data sets are available, and comparison among data sets indicates a wide range of estimates within taxa. Available measurements of terminal velocity need to be compared with each other and with values predicted from physical theory to assess which datasets are most reliable, and the implications of measurement uncertainties on predictions of Prentice's model for individual taxa need to be explored.

In this paper, we review Prentice's (1985) model and evaluate its parameters. We argue that the atmospheric parameters specified for neutral conditions are not typical for most anemophilous pollen liberation and dispersal, and present results of simulations using more-realistic parameter-values. We then discuss the physical factors influencing deposition velocity, and review the available datasets. We compare published measurements of terminal velocity with predictions from physical theory, examine the apparent assets and liabilities of the available data sets, and discuss implications of variability in deposition-velocity measurements for models of pollen dispersal. We also assess the contribution of impaction to pollen deposition, argue that deposition velocity may vary inversely with wind speed, and discuss effects of variation in wind speed on pollen dispersal and representation. Finally, we summarize areas of theory and measurement that are in our view high priorities for study. Our review is aimed at bringing important issues, data, and literature to the attention of botanists and paleoecologists, and to stimulate additional research in pollen dispersal, which is at the heart of paleoecological inference.

III. Sutton's Equation and Prentice's Model

In Sutton's equation for dispersion and deposition of particulate material from a continuously emitting point source at ground level,

[Mathematical Expression Omitted] (1)

where Q(0) is the particle source-strength (total number of particles emitted from the point source), Q(x) is the number of particles that remain airborne at distance x from the source, [v.sub.g] is the velocity of deposition of particles (cm/sec), u is the wind speed (cm/sec), n is a dimensionless turbulence parameter, and [C.sub.z] is a vertical diffusion coefficient ([m.sup.1/8]). The last two terms depend on atmospheric stability. Sutton specified these parameters (n [similar to] 0.25, [C.sub.z] [similar to] 0.12) based on neutral atmospheric conditions, under which turbulence was minimized and consistent field measurements were most easily obtained (Sutton, 1947a). Neutral conditions, in which the adiabatic lapse rate ([approximately]1 [degrees] C/100 m) obtains and a rising or falling mass of air undergoes no heat exchange with its surroundings, are most common during cloudy days and nights with strong, steady winds. Turbulence is relatively low owing to the lack of buoyant forces.

Prentice's (1985) model is based on Equation 1, incorporating distance of the deposition site from the nearest emitting sources:

F(x) = 1 - ex(b([r.sup.[Gamma]] - [x.sup.[Gamma]])) (2)

where F(x) = the proportion of pollen grains deposited at the site originating within distance x of the basin center, r = radius of the depositional basin, b = 4[v.sub.g]/nu[-square root of] [Phi][C.sub.z] ([approximately] 75[v.sub.g]/u under neutral conditions), [Gamma] = n/2 (0.125 under neutral conditions), and x [greater than] r. In application to depositional basins, Prentice's model assumes that all pollen deposited at the center of the basin falls directly from the atmosphere (i.e., there is no transport and redeposition of pollen from shallow, near-shore sediments to the basin center). Sugita (1993) has modified Prentice's model to integrate pollen deposited over the entire surface of a lake. In Sugita's model, plants growing near a lake margin have greater representation than in Prentice's model because the pollen is redeposited from near-shore sites to the center. Sugita's (1993) model assumes complete mixing and redeposition of pollen deposited over an entire lake surface; Sugita's and Prentice's models may therefore be regarded as end members spanning a continuum from no redeposition to complete mixing and redeposition. In this paper, we will work with Prentice's model because of its relative simplicity, although our results and conclusions apply to Sugita's model as well.

IV. Atmospheric Parameters and Their Consequences

A. ATMOSPHERIC STABILITY

Atmospheric stability depends on vertical temperature structure, and affects the degree of turbulence and diffusive spread (Singer & Smith, 1953; Gifford, 1968, 1976). Under neutral conditions, the observed relationship between temperature and height above the ground follows the adiabatic lapse rate, buoyant forces are minimized, and turbulence is modest. These conditions are most common under cloudy skies with steady, strong winds. In stable atmospheric conditions, temperature decreases with height more slowly than the adiabatic lapse rate. In this case, a rising mass of air will be increasingly cooler (and denser) than its surroundings, and hence will sink. Conversely, a descending mass of air will be increasingly warmer than surrounding air and will reascend to an altitude appropriate to its temperature. Thus, turbulence is suppressed in stable conditions, which prevail most nights and the first hours after sunrise. Extremely stable conditions are associated with temperature inversions.

Unstable conditions occur when air temperature decreases with height at a faster rate than the adiabatic lapse rate. Rising air masses will be warmer than their surroundings, and the resulting buoyancy will lead to further rise of the air mass. The buoyant effects result in overturning and high turbulence, including gusting with high-frequency variation in wind speed and direction. Unstable conditions are characteristic of warm days under clear skies. Atmospheric stability in clear weather typically alternates diurnally between stable nighttime and unstable daytime conditions, with greatest instability occurring between midmorning and late afternoon (Singer 8,: Smith, 1953; Gifford, 1968).

B. ATMOSPHERIC CONDITIONS DURING POLLEN RELEASE

Empirical studies of pollen dispersal for a wide variety of herbs (Ambrosia, Poaceae), deciduous trees (Betula, Quercus), and coniferous trees (Abies, Picea, Pinus, Pseudotsuga, Thuja, Tsuga) indicate that most pollen release occurs between midday and late afternoon (1000-1800 hr) on clear, warm days with low relative humidity (Buell, 1947; Sarvas, 1952, 1962, 1968; Bianchi et al., 1959; Wang et al., 1960; Sharp & Chisman, 1961; Silen, 1962; Ebell & Schmidt, 1964; Pande et al., 1972; Curtis & Lersten, 1995). The lower atmosphere is likely to be moderately to highly unstable under such conditions owing to development of thermally generated vertical eddies (Gifford, 1968).

Evaporation is required for dehiscence of anthers and microsporangia prior to pollen release, and onset of cool, cloudy, and especially humid conditions leads to temporary suppression of pollen release (Sarvas, 1952, 1962; Sharp & Chisman, 1961; Silen, 1962; Ebell & Schmidt, 1964). Although few pollen-release studies have included details on atmospheric stability, several specifically describe conditions typical of instability. Sarvas (1952: 13) noted that, although release of Betula verrucosa and B. pubescens pollen was suppressed during cold, cloudy, and rainy periods, "short [rain] showers in particular are frequent at the time of flowering of birch." Brief showers on sunny days are characteristic of convective activity resulting from thermally induced atmospheric instability. Silen (1962: 791) described maximum Pseudotsuga pollen release as occurring when weather was "predominantly clear with variable winds." Ebell and Schmidt (1964) observed that pollen release of Abies grandis, Pinus contorta, Pseudotsuga menziesii, Thuja plicata, and Tsuga heterophylla in British Columbia was concentrated during extended periods of high pressure, and suppressed when low-pressure maritime air characterized by "a normal temperature lapse rate" passed into the study area. They further noted that convective cells, which arise during unstable conditions, were apparent during much of the pollen-release period.

C. GUSTINESS AND THE PARADOX OF POLLEN LIBERATION

Consideration of the physical force required to entrain pollen grains from anthers, microsporangia, or foliage also supports the role of wind gusts and turbulent conditions in pollen dispersal. A pollen-sized particle at rest on a surface will resist movement by simple gravitational forces; this resistance increases with particle density, particle size, deviation from sphericity, and surface roughness (Grace & Collins, 1976; Niklas, 1985). The threshold wind speed (u[prime]) required to set a pollen grain in motion is defined as

u[prime] = (2[r.sup.2][Rho]g tan[Theta])9[Mu] (3)

where r = particle radius, [Rho] = particle density, g = gravitational acceleration constant (981 cm/[sec.sup.2]), [Mu] = viscosity of air (1.8 x [10.sup.-4] g/cm/sec for air at 18 [degrees] C), and [Theta] = angle between the particle radius perpendicular to the surface and the particle's furthest point of contact with the surface from that radius (Grace & Collins, 1976). The angle [Theta] increases with deviation from particle sphericity and with increasing roughness of the surface on which the particle is resting. Grace and Collins (1976) used this model to estimate the threshold wind speed for Lycopodium spores on a moderately smooth surface ([Theta] = 30 [degrees]) at 0.018 m/sec. This is a minimum estimate; surface roughness of approximately the same scale as pollen size will increase [Theta] up to 90 [degrees]. Furthermore, resistance to particle motion may also be increased by electrostatic forces and by surface adhesion. The latter may be particularly important for grains in newly opened anthers and microsporangia.

Because of boundary-layer effects, mean wind speeds corresponding to u[prime] for a particular pollen type are typically insufficient to liberate pollen grains from anthers, microsporangia, or vegetative surfaces. For example, although u[prime] for Lycopodium spores on a moderately smooth surface is 0.018 m/sec, mean wind speeds of 5-7 m/sec were required to entrain 50% of the spores in experimental situations, and 20% of the spores remained at rest in winds of 14 m/sec (Grace & Collins, 1976). Aylor (1978) calculated that average wind speeds of 25 m/sec would be required to produce spore-level wind speeds of 5 m/sec on a leaf surface.

The paradox of wind-entrainment of small particles - the gale-force-or-greater average wind speeds apparently required to overcome boundary-layer resistance - has been resolved by Aylor (Aylor & Parlange, 1975; Aylor, 1978), who proposed that brief but strong wind gusts are sufficient to overcome the inertial and adhesive forces keeping particles in position. Aylor (1978) noted a time lag between inception of a wind gust across a surface and development of a boundary layer; during this lag period the surface, and particles upon it, are exposed to the full force of the moving air. This lag-time, while brief (ca. [10.sup.-3]-[10.sup.-4] sec), is sufficient to set particles in motion provided the gust has sufficient velocity (Aylor & Parlange, 1975; Aylor, 1978). Thus, relatively low average wind speeds ([less than]5 cm/sec) can liberate pollen grains and other particles from surfaces as long as the wind is characterized by energetic gusts, which are common during unstable atmospheric conditions but minimal in neutral or stable conditions.

Surprisingly little is known about the aerodynamics of pollen liberation from anthers and microsporangia, in contrast to fungal-spore liberation (e.g., Aylor, 1978) and pollen interception by ovulate organs (e.g., Niklas, 1985). Individual anthers and microsporangia certainly have reduced boundary layers compared to leaf Surfaces. The architecture of microstrobili (Pinaceae) and catkins (Juglandaceae, Fagaceae, Betulaceae) may lead to fine-scale eddy patterns which might contribute to pollen entrainment. Neither boundary-layer effects nor functional morphology of pollen-bearing organs has been studied from a pollen-entrainment standpoint.

Although boundary-layer effects will likely be reduced for anthers and microsporangia, it seems likely that wind gusts play a strong role in pollen liberation, via agitation of shoots bearing polleniferous organs and overcoming the reduced but still-present boundary-layer effects. Faegri and van der Pijl (1979) suggest that wind agitation is required for pollen release from catkins and microstrobili. Observation of microsporangial development and pollen release in several conifers (Abies concolor, Picea engelmannii, Pinus contorta, P. ponderosa, P. strobus) in Arizona, New York, and Wyoming indicate that little pollen is shed from dehisced microsporangia without agitation, and that newly dehisced microstrobili are emptied of virtually all pollen after a single dry, sunny day with energetic wind gusts (S. T. Jackson, pers. obs.).

In some cases, pollen liberation is directly analogous to the entrainment of particles from leaf surfaces, where boundary-layer effects will be strong. In Ambrosia, pollen drops by gravity from anthers onto underlying leaves, where it is then entrained into wind currents (Bianchi et al., 1959). Reentrainment of pollen sedimented or impacted onto vegetative surfaces in forest or other canopy will also require wind gusts.

D. PARAMETER SPECIFICATION FOR SUTTON'S EQUATIONS

In Sutton's equation (Eq. 1), the vertical diffusion coefficient [C.sub.z] depends primarily on the turbulence parameter (n) and on the "vertical gustiness factor," defined as the ratio between mean vertical eddy velocity and mean wind velocity (Sutton, 1947b). Sutton (1947b) suggested that under unstable conditions, n [similar to] 1/5 (0.20). Using this value and a postulated vertical gustiness factor, Sutton (1947b) calculated that [[C.sup.2].sub.z] in unstable air is 3.03 times [[C.sup.2].sub.z] under neutral conditions; thus [C.sub.z] for unstable air should be [approximately]0.21.

Only a handful of attempts have been made to estimate atmospheric parameters for Sutton's equations from empirical observations of gaseous or particulate diffusion. Estimates from the Harwell Reactor are of limited application because of the height of emission (68 m) and the high temperature and high stack-velocity of the effluent (Islitzer & Slade, 1968). However, the estimates obtained under different atmospheric conditions confirm theoretical expectations (Sutton, 1947b) that [C.sub.z] will be higher in unstable conditions (0.25-0.32) than neutral (0.20) or stable ([less than]0.04-0.11) conditions (Stewart et al., 1958). Elevated-source (108 m) releases of oil fog-droplets at Brookhaven National Laboratory yield [C.sub.z] estimates of 0.32 for neutral conditions, and 0.46-0.58 for unstable conditions (Singer & Smith, 1966).

The only [C.sub.z] estimates for unstable conditions based on diffusion from ground-level sources are from the Green Glow experiments at Hanford, Washington, where fluorescent zinc sulfide was released (Barad & Fuquay, 1962). [C.sub.z] estimates from these experiments indicated greater vertical diffusion in unstable air [[C.sub.z] ranged from 0.28 (u = 10 m/sec) to 0.35 (1 m/sec)] than neutral conditions [0.13 (10 m/sec) to 0.17 (1 m/sec)] (Barad & Fuquay, 1962).

We used the neutral-condition parameters (n = 0.25, [C.sub.z] = 0.12) (Sutton, 1947b; Prentice, 1985) as a control in our simulations using Equation 1. In simulating unstable conditions, we set n = 0.20 (Sutton, 1947b) and used two different values of [C.sub.z]: 0.21 and 0.35. These [C.sub.z] values correspond respectively to Sutton's (1947b) theoretical calculation and to the measured estimates from Hanford (mean wind speed 1 m/sec) (Barad & Fuquay, 1962).

In simulations using Prentice's model (Eq. 2), we used Prentice's (1985) neutral-condition parameter values (b = 75[v.sub.g]/u; [Gamma] = 0.125) as a control. We set [Gamma] = 0.1 for unstable-air simulations, and used values of b corresponding to the two unstable-air scenarios simulated using the Sutton equation (b [approximately equal to]54[v.sub.g]/u where [C.sub.z] = 0.21; b [approximately equal to]32[v.sub.g]/u where [C.sub.z] = 0.35). We held u constant at 3 m/sec, and used values of [v.sub.g] (1.5, 3, 6, 9 cm/sec) characteristic of most wind-dispersed pollen grains (see [ILLUSTRATION FOR FIGURE 3 OMITTED] and associated discussion).

E. POLLEN DISPERSAL IN UNSTABLE ATMOSPHERIC CONDITIONS

Results of our simulations using Sutton's equation (Eq. 1) show a dramatic difference in pollen-dispersal patterns between neutral and unstable atmospheric conditions [ILLUSTRATION FOR FIGURE 1 OMITTED]. Pollen is predicted to be dispersed farther under unstable conditions than neutral conditions, regardless of sedimentation velocity [ILLUSTRATION FOR FIGURE 1 OMITTED]. The magnitude of the difference is substantial. For example, the dispersal-distance curve for a moderately large pollen type ([v.sub.g] = 6 cm/sec) under unstable conditions is similar to that predicted for a small pollen type ([v.sub.g] = 3 cm/sec) under neutral conditions [ILLUSTRATION FOR FIGURE 1 OMITTED]. The greater vertical diffusion (i.e., high [C.sub.z]) in an unstable atmosphere leads to increased lofting of pollen grains near the source, and hence to greater dispersal.

The more-widespread dispersal of pollen in unstable conditions has a strong effect on predicted pollen source-areas for basins of different sizes [ILLUSTRATION FOR FIGURE 2 OMITTED]. The general predictions of Prentice's (1985) model (increasing pollen source-area with increasing basin size; increasing pollen source-area with decreasing [v.sub.g]) still obtain, but the predicted pollen source-areas for a given pollen type and basin size are significantly larger [ILLUSTRATION FOR FIGURE 2 OMITTED]. For example, for a small hollow (r = 2 m), the radius within which 20% of the pollen of a moderately large pollen type ([v.sub.g] = 6 cm/sec) originates is predicted to be 6 m under neutral conditions and [approximately]30 m under unstable conditions ([C.sub.z] = 0.35) [ILLUSTRATION FOR FIGURE 2 OMITTED]. For the same basin and pollen type, 50% of the pollen is predicted to come from within 30 m under neutral conditions, and 1150 m under unstable conditions [ILLUSTRATION FOR FIGURE 2 OMITTED]. The effect is still substantial using Sutton's value for [C.sub.z] (0.21); the predicted 20% and 50% radii are [greater than]10 m and 200 m, respectively.

Similar effects of instability are seen regardless of pollen size [e.g., given a small pollen type ([v.sub.g] = 1.5 cm/sec), the 20% radius for a small hollow is [approximately]60 m under neutral conditions, and [greater than]10 km under unstable conditions]. Larger basins also show sensitivity [ILLUSTRATION FOR FIGURE 2 OMITTED]. For a small pond (r = 100 m) and small pollen grain ([v.sub.g] = 3 cm/sec), for instance, the 20% radius is predicted to be [approximately]250 m and [approximately]4 km under neutral and unstable conditions, respectively [ILLUSTRATION FOR FIGURE 2 OMITTED]. In all cases, the greater pollen-source areas predicted under unstable conditions are consequences of the displacement of the dispersal-distance functions [ILLUSTRATION FOR FIGURE 1 OMITTED] owing to greater vertical diffusion of pollen grains near the source.

V. Depositional Parameters and Their Consequences

A. DEPOSITION VELOCITY, SEDIMENTATION VELOCITY, AND PARTICLE SIZE

The ratio [v.sub.g]/u in Equations 1 and 2 (implicit in the definition of b) is defined as a capture efficiency representing the proportion of particles passing near a receptive surface (i.e., canopy or ground) that are captured by (i.e., deposited upon) the surface (Chamberlain, 1975; Chamberlain & Little, 1981). Capture efficiency is essentially a ratio between two forces acting upon particles: a capture force ([v.sub.g]) and an escape force (u); the greater this ratio, the greater the rate of deposition of particles from the air mass.

Velocity of deposition ([v.sub.g]) is formally defined as the deposition rate (particles/[cm.sup.2]/sec) at the surface divided by the airborne particle concentration (particles/[cm.sup.3]) immediately above the surface (Sehmel, 1980). Velocity of deposition must be at least as great as the velocity of sedimentation (or terminal velocity), [v.sub.s], which is the rate at which particles descend in still air owing to gravitational effects. When sedimentation is the only force responsible for deposition of particles, [v.sub.g] = [v.sub.s], and the capture efficiency [v.sub.g]/u is simply the ratio between the vertical force (gravitational settling) and the horizontal force (horizontal wind speed).

Velocity of sedimentation ([v.sub.s]) is related to particle size and density by Stokes's Law, which for a spherical particle is expressed as

[v.sub.s] = 2[r.sup.2]g([p.sub.o] - p)/9[Mu] (4)

where r = particle radius (cm), g = gravitational acceleration constant (981 cm/[sec.sup.2]), [p.sub.o] = particle density (g/[cm.sup.3]), p = density of fluid (1.27 x [10.sup.-3] g/[cm.sup.3] for air), and [Mu] = viscosity of fluid (1.8 x [10.sup.-4] g/cm/sec for air at 18 [degrees] C). An alternative formulation is convenient for non-spherical particles:

[v.sub.s] = mg/[c.sub.d][Mu][v.sup.1/3] (5)

where m = particle mass (g), V = particle volume ([cm.sup.3]), and [C.sub.d] = a dimensionless viscous drag coefficient dependent on the particle shape and fall orientation (Vogel, 1981). [C.sub.d] can range from [approximately]11 (spheroids) to [greater than]36 (disks or rods with the long axis perpendicular to the direction of fall) (Vogel, 1981).

Equations 4 and 5 show that [v.sub.s] decreases with decreasing particle size, decreasing particle mass or density, and increasing irregularity of shape (i.e., deviation from sphericity). These properties vary substantially between wet and dry pollen grains. Harrington and Metzger (1963) showed that density of Ambrosia pollen grains ranges from 0.84 to 1.28 g/[cm.sup.3], depending on relative humidity; the grain imbibes water and fills cell contents and air spaces as humidity increases. Ambrosia is unusual in that its pollen grains show little volumetric change ([approximately]8%) with hydration and dehydration owing to the unique exine architecture (Payne, 1972, 1981). In a survey of 150 angiosperm genera, Payne (1981) demonstrated extensive volume reduction with dehydration in most genera, averaging 46%. SEM photographs in Payne, 1981, and in Blackmore & Barnes, 1986, show the extent to which pollen-grain shape and dimension can be altered by dehydration. Payne (1981) studied relatively few anemophilous genera; representatives included Poaceae (Bambusa at 14% volume loss), Liquidambar (28%), Carya (34%), and Myrica (34%). Heslop-Harrison (1979) noted that Poaceae (Secale cereale) pollen loses 20-35% of its weight upon release into dry air. Blackmore and Barnes (1986) described changes in shape and volume for several anemophilous angiosperms with drying. Cursory observations of other anemophilous pollen types (Quercus, Acer, Pinus, Picea, Juniperus, Larix) indicate volumetric and dimensional changes with dehydration (S. T. Jackson, pers. obs.). In some cases, pollen-grain density may remain constant, while mass and volume decrease with dehydration.

B. EXPERIMENTAL ESTIMATES OF POLLEN SEDIMENTATION VELOCITY

Numerous attempts have been made to measure sedimentation velocity of pollen grains and spores directly, using a variety of apparatus. Of these, measurements based on sedimentation in aqueous media (e.g., Brush & Brush, 1972, 1994; Heathcote, 1978) are not directly applicable to atmospheric dispersal, for reasons discussed above. Dry-air sedimentation velocity cannot be recovered from these measurements without simultaneous correction for differences in pollen-grain density, size, and shape.

Still-air measurements of pollen sedimentation-velocity are mainly based on fall-towers, in which pollen is released from the top of a cylinder and the time required to reach the bottom is measured. Fall-towers used in pollen studies vary widely in material (glass, iron, steel, tin, plastic) and dimension (length 98-250 cm; diameter 2.5-30 cm) (Bodmer, 1922; Knoll, 1932; Dyakowska, 1936; Durham, 1946; Eisenhut, 1961; Niklas, 1982; Niklas & Paw U, 1983; Ferrandino & Aylor, 1984; Di-Giovanni et al., 1995). These differences may affect measured sedimentation velocities owing to convection and electrostatic effects (Durham, 1946). Niklas (1984) used a rectangular box (1 m x 30 cm x 30 cm).

We compiled published still-air measurements of sedimentation velocity (Appendix 1), and plotted the sedimentation-velocity data to assess variability in estimates within and among studies for specific taxa [ILLUSTRATION FOR FIGURE 3 OMITTED]. Measured sedimentation velocities for pollen grains vary widely, from [less than]0.5 to 39 cm/sec ([ILLUSTRATION FOR FIGURE 3A OMITTED]; Appendix 1). Much of this variation is attributable to variation among taxa in grain size and shape ([ILLUSTRATION FOR FIGURE 3A OMITTED]; see also Jackson, 1994: fig. 14.7). For example, the lowest sedimentation velocities are for small, spheroidal pollen grains (Urtica, Ambrosia, Juniperus, Artemisia), and the highest are for large grains (Pseudotsuga, Larix, Abies, Tsuga) [ILLUSTRATION FOR FIGURES 3 AND 4 OMITTED]. However, there is also substantial variation among and within data sets (e.g., [ILLUSTRATION FOR FIGURES 3A AND 3B OMITTED], respectively). At least some of the within-study variation may be attributable to grain-size variation. For example, Eisenhut (1961) studied many species of Pinus, Picea, and Abies [ILLUSTRATION FOR FIGURE 4 OMITTED], each of which varies in grain-size within and among species. Eisenhut (1961) obtained lower sedimentation-velocity estimates for Pinus strobus and P. banksiana than for P. rigida [ILLUSTRATION FOR FIGURE 4 OMITTED]; Whitehead (1964) observed that P. rigida grains were typically 2-8 [[micro]meter] larger than P. strobus and P. banksiana grains.

Some of the variation within studies is undoubtedly attributable to measurement error. Variation among studies may result in part from differences in grain size (authors typically studied different species of a genus than other authors studied), but the wide range of variability within genera and systematic differences among workers [ILLUSTRATION FOR FIGURE 3A OMITTED] suggests that much of the variation stems from biases related to the particular experimental apparatus and procedures. For example, the estimates of Dyakowska (1936) and Dyakowska and Zurzycki (1959) are consistently higher than those of others, Durham's (1946) estimates are consistently lower, and Eisenhut's (1961) are in the mid-range [ILLUSTRATION FOR FIGURE 3A OMITTED]. Estimates of Niklas (1982) and Niklas and Paw U (1983) for Picea, Abies, and Pinus are substantially lower than those of other workers, and the Picea and Larix estimates of Niklas (1984), based on a different apparatus than his earlier studies, are even lower [ILLUSTRATION FOR FIGURE 3A OMITTED].

C. COMPARISON WITH THEORETICAL ESTIMATES OF SEDIMENTATION VELOCITY

We have no objective basis for identifying any of the experimental studies of sedimentation velocity as authoritative; none of the sampling apparatus were calibrated using standards (e.g., microspheres of known dimension and density). However, we can evaluate the estimates by comparing them with predictions from Stokes's Law. We compiled available data on dry-pollen dimension, volume, mass, and density (Appendix 2), used them to calculate theoretical sedimentation velocities using Equations 4 and 5, and compared the predicted sedimentation velocities with the measured data to assess bias and determine which data sets most nearly match "ideal" theoretical values [ILLUSTRATION FOR FIGURE 5 OMITTED]. The theoretical standards are not perfect, however, owing to deviations from sphericity, measurement errors in parameter estimates (size, mass, volume, density), and necessary approximations in the calculations. In applying Equation 5, we assumed that non-spheroidal particles tend to fall with the long axis perpendicular to the direction of fall. This assumption is consistent with theoretical expectations (McNown & Malaika, 1950; McNown et al., 1951) and with the majority of observations of Eisenhut (1961).

Eisenhut's (1961) sedimentation-velocity measurements were related to theoretical estimates [ILLUSTRATION FOR FIGURE 5 OMITTED]. Linear regression analysis indicated a significant relationship ([r.sup.2] = 0.83, p [less than] 0.001). Measured values were consistently higher than predicted estimates (y-intercept = 2.56), and the slope of the relationship was gentle (0.77) [ILLUSTRATION FOR FIGURE 5 OMITTED]. These results suggest a modest bias in Eisenhut's sedimentation-velocity measurements toward overestimation, with the bias greatest for small pollen grains. Alternatively, Eisenhut's mass and volume measurements may be biased toward underestimation.

Figure 5 also includes measured sedimentation velocities from other studies (Knoll, 1932; Durham, 1946; Niklas, 1982) that used the same species as Eisenhut (1961). We plotted these measured values against those predicted by Eisenhut's mass and volume estimates using Equation 5. These data were not used in the statistical analyses. All five points fall within the point-cloud of Eisenhut's data, although all measurements are lower than Eisenhut's (1961) corresponding estimates.

We used Durham's (1943) weight and volume measurements to estimate grain density for seven taxa (Appendix 2). All of these pollen types are spheroidal, so we used the density estimates together with Durham's (1943) grain-radius measurements to predict sedimentation velocities based on Equation 4. We compared these predicted values with Durham's (1946) measured sedimentation velocities [ILLUSTRATION FOR FIGURE 5 OMITTED]. All fall near the line predicted by a perfect relationship, although most of the measured values are modestly higher ([less than]0.5 cm/sec) than predicted values. We also used Durham's (1943) weight and volume measurements to calculate predicted sedimentation velocities based on Equation 5 (Appendix 2; [ILLUSTRATION FOR FIGURE 5 OMITTED]). All the estimates from Equation 5 are modestly higher than the corresponding estimates from Equation 4 ([ILLUSTRATION FOR FIGURE 5 OMITTED]; Appendix 2).

The overall good fit between measured and predicted estimates of sedimentation velocity [ILLUSTRATION FOR FIGURE 5 OMITTED] provides a basis for some confidence in the measured values of Eisenhut (1961) and especially Durham (1946). However, none of these measurements have been calibrated against a standard, and both studies show substantial variation in measured estimates within genera [ILLUSTRATION FOR FIGURE 4 OMITTED]. Furthermore, the magnitude of within-genus variation for these studies is of the same order as the variation among different studies [ILLUSTRATION FOR FIGURE 4 OMITTED].

D. CONSEQUENCES OF MEASUREMENT UNCERTAINTY IN SEDIMENTATION VELOCITY

We assessed the effects of within-taxon variation in measured sedimentation-velocity estimates on dispersal-distance patterns by applying the minimum, maximum, and median values for selected taxa to Equation 1. We assumed neutral conditions (n = 0.25; [C.sub.z] = 0.12) and set wind speed at 3 m/sec in these simulations. Results varied among taxa [ILLUSTRATION FOR FIGURE 6 OMITTED]. The dispersal-distance functions were more sensitive to the relative magnitude of variation than to the absolute magnitude of variation, because [v.sub.g] is an exponent in Equation 1. For example, the dispersal-distance curves for Abies varied little despite the high magnitude of variation (4 cm/sec), while Quercus varied more with a smaller magnitude of variation (2.2 cm/sec) [ILLUSTRATION FOR FIGURE 6 OMITTED]. Estimates of sedimentation velocity for Quercus varied twofold, while those for Abies varied by a factor of 1.5.

Overall, results of this analysis demonstrate substantial uncertainty in predicted pollen dispersal patterns and, accordingly, predicted pollen source-areas from Prentice's model (Eq. 2) for several important taxa (Picea, Pinus, Quercus, Betula, Alnus) owing to measurement uncertainties in sedimentation velocity [ILLUSTRATION FOR FIGURE 6 OMITTED]. Thus, choice of [v.sub.g] will have non-trivial consequences, although the effects on pollen dispersal distances are of smaller magnitude than those imposed by choice of assumptions concerning atmospheric conditions [ILLUSTRATION FOR FIGURES 6 AND 1 OMITTED].

E. IMPACTION, WIND SPEED, AND DEPOSITION VELOCITY

Observed velocities of deposition for small particles often exceed velocities of sedimentation (Sehmel, 1980), because velocity of deposition is the sum of velocity of sedimentation ([v.sub.s]) and velocity of impaction ([v.sub.i]). In impaction, particles approaching an obstacle are carried by their inertia to the obstacle surface rather than around it with the airstream. The physical theory underlying impaction is more complex and less certain than that for sedimentation. Velocity of impaction ([v.sub.I]) is proportional to impaction efficiency ([E.sub.I]), defined as the ratio between the number of particles impacting an obstacle and the number that would have passed through the space occupied by the obstacle in its absence. Impaction efficiency is non-linearly related to a parameter P, defined as

P = u[v.sub.s]/gR (6)

where R = the radius of the obstacle (assumed to be a cylinder or sphere) (Chamberlain, 1967a; Gregory, 1973). Thus, impaction will increase with particle size and wind speed, and decrease as size of the intercepting surface increases. The relationship between impaction efficiency and P is approximately linear when 0.3 [less than] P [less than] 5.0, and impaction efficiency is [less than]25% when P [less than or equal to] 0.5 (Chamberlain, 1967a). Theoretical impaction efficiencies are typically low for pollen-sized particles. For example, a relatively large pollen grain ([v.sub.s] = 12 cm/sec) approaching a moderate-sized obstacle (r = 0.5 cm) requires a wind speed of [greater than]20 m/sec for P to reach 0.5 (and thus for impaction efficiency to reach as high as 25%).

The contribution of impaction to deposition velocity is difficult to measure directly (Sehmel, 1980; Chamberlain & Little, 1981). Validation of theoretical predictions of impaction efficiency is further complicated by non-retention of impacted particles by obstacles; particles may bounce off the surface or be blown off later (Gregory, 1973; Chamberlain, 1975). Anemophilous pollen grains typically have non-sticky surfaces (Hesse, 1981), so they are especially prone to rebounding and reentrainment. The limited field data available suggest that impaction of pollen grains on vegetative surfaces is minor except for very fine and/or sticky surfaces (e.g., stigmas)(Chamberlain, 1967b; Raynor et al., 1974, 1975; Aylor, 1975; Chamberlain & Chadwick, 1972; Chamberlain & Little, 1981). Laboratory studies indicate that impaction strongly depends on the nature of the intercepting surface; Chamberlain (1967a, 1967b) noted substantially higher impaction on wet grass leaves than on dry leaves, and highest impaction on sticky artificial surfaces with the same aerodynamic properties as the grass leaves. The differences in impaction among these different surfaces are attributable to differences in surface retention of particles (Chamberlain, 1967b). Rebounded particles, of course, are returned to the air mass for continued transport. Although Chamberlain (1967a, 1975) suggested that reentrainment of impacted particles (i.e., blow-off) is minor, his conclusion is based on wind-tunnel experiments with constant wind speed (Chamberlain, 1967b). Gusts may reentrain impacted particles, for reasons discussed earlier.

Chamberlain (1967a, 1967b) observed that the ratio between deposition velocity and friction velocity (u*) may be approximately constant for a given particle size and an intercepting surface of given aerodynamic properties. Friction velocity is proportional to wind speed, so the capture efficiency ratio [v.sub.g]/u should also be constant. As wind speed increases, so may deposition velocity owing to increased impaction (Eq. 6). Chamberlain's wind-tunnel experiments with Lycopodium spores indicated that the relationship between [v.sub.g] and u* is approximately linear (Chamberlain, 1967b: fig. 5). A positive linear relationship between capture efficiency and wind speed has been well documented for particles [approximately]5/[[micro]meter] in diameter (Carter, 1965; Little, 1977).

Prentice (1985) suggested that the ratio [v.sub.g]/u can be considered constant for a given particle type in applying Sutton's model (Eq. 1) and Prentice's model (Eq. 2) to pollen dispersal. However, Chamberlain (1967b) showed that the ratio varied substantially depending on the intercepting surface; the ratio was higher for wet grass than dry grass, and highest for sticky artificial grass. The differences stem from differential rebound and reentrainment. Furthermore, Gregory (1973: tab. XIV; see also Chamberlain, 1975: fig. 8a) showed that capture efficiency of Lycopodium spores on broad-bean and potato leaves decreased with increasing wind speed. Chamberlain (1975) made similar observations using Ambrosia pollen and barley stalks, as did Starr (1967) using Broussonetia pollen and fungal spores on paper targets. Capture efficiency of glycerine droplets (15 and 32 [[micro]meter] diameter) on Abies balsamea foliage showed a complex relationship with wind speed, increasing at first and then decreasing (Thorne et al., 1982). Chamberlain and Chadwick (1972) observed that deposition velocity of Lycopodium spores to dry soil was similar to sedimentation velocity regardless of wind speed.

From a theoretical standpoint, likelihood of rebound increases with increasing particle size and with decreasing obstacle size (Chamberlain, 1975). Particles of [less than or equal to]5 [[micro]meter] are unlikely to rebound under most circumstances, so [v.sub.g]/u is likely to be constant at different wind speeds. Larger particles within the size range of pollen grains are more subject to rebounding. The experiments of Chamberlain and Gregory were with particles in the low end of the pollen size-range (Ambrosia pollen at 19 [[micro]meter] and Lycopodium spores at 32 [[micro]meter] diameter), so their impaction-efficiency estimates are likely to be higher than for most pollen types.

F. EFFECTS OF WIND SPEED AND IMPACTION ON POLLEN DISPERSAL

The preceding discussion suggests two "end-member" hypotheses concerning the roles of impaction and wind speed in pollen deposition. First, we can assume that [v.sub.g]/u is constant for a given pollen type, use [v.sub.s] as an estimator of [v.sub.g] at moderate wind speeds, and set u at some moderate level (e.g., 3 m/sec) to estimate [v.sub.g]/u. This is essentially what Prentice (1985) proposed. Alternatively, we can assume that impaction is negligible in vegetative canopy and other natural surfaces, owing to rebounding and reentrainment. In this second hypothesis, we assume that [v.sub.g] = [v.sub.s] under most circumstances, and therefore that [v.sub.g]/u will vary inversely with wind speed. These hypotheses presumably bound reality, although we view the second hypothesis as more likely accurate based on the preceding discussion.

The effect of wind speed variations in Equations 1 and 2 is reciprocal to variations in deposition velocity. For instance, a doubling of wind speed has the same effect on pollen dispersal and pollen source-area as a halving of deposition velocity. Prentice (1985, 1988) presented a series of figures showing the effects of variation in deposition velocity [ILLUSTRATION FOR FIGURE 6 OMITTED], so we do not present simulation results in this paper. Essentially, increasing wind speed has the effect of transporting pollen grains of a given size greater distances, and increasing the pollen source-area of a basin of given size.

VI. Prescriptions for Model Application and Parameter Specification

Prentice's (1985, 1988) application of Sutton's equation was a fundamental step forward in understanding pollen dispersal and representation in sedimentary settings. Building on earlier work by Tauber (1965) and Kabailiene (1969), Prentice's model comprised the first formal and quantitative linkage among pollen dispersal properties, pollen source-area, and basin size. Empirical studies of pollen-vegetation relationships in small lakes (Jackson, 1990, 1991, 1994) and forest-floor deposits (Jackson & Wong, 1994; Calcote, 1995; Jackson & Kearsley, 1998) showed that Prentice's model predicted pollen source-areas and spatial resolution far more accurately than alternative models of Tauber (1965) and Jacobson and Bradshaw (1981).

The success of Prentice's model should not block perception of its limitations and uncertainties. Sutton's equations and Prentice's model represent imperfect solutions to a complex set of problems. Gifford (1968: 88) noted that Sutton's model "should not be accorded the unequivocal status of a law of nature." Sutton's equations have been verified for a limited set of circumstances, and proved useful in many applications, but they have many ad hoc features and do not necessarily apply to all dispersion phenomena at all spatial scales (Gifford, 1968; Hanna et al., 1982; McCartney & Fitt, 1985). Similarly, Prentice's model is consistent with a limited number of available data sets, has yielded important insights about the vegetation-sensing properties of pollen assemblages, and is clearly better than existing alternative models of pollen dispersal and source-area. The model provides an excellent vehicle for exploring the effects of basin size, deposition velocity, and wind speed on pollen source-area and representation, as demonstrated by Prentice (1988), Jackson (1991, 1994), Sugita (1994), and our analyses in this paper.

Prentice's model has two further potential applications: (1) determination of pollen-source areas for specific taxa and basin sizes in design and interpretation of paleoecological studies, and (2) calculation of distance-weightings to apply to vegetation data in calibrating pollen-vegetation relationships. These applications place greater demands on the validity of the model's specific assumptions and parameters. Our review and analysis show that there are substantial uncertainties in the specification of model-parameters but that the parameters probably fall within specifiable bounds. Atmospheric parameters should assume unstable conditions; thus n should be 0.20 and [C.sub.z] should fall within the range 0.20-0.35. The neutral-stability parameters of Sutton (1953) and Prentice (1985) (i.e., n = 0.25, [C.sub.z] = 0.12) can be used to set an absolute lower limit to model predictions, but these parameters are likely to be unrealistic for most pollen dispersal.

The depositional parameters [v.sub.g] and u should be assumed to vary independently (i.e., the ratio [v.sub.g]/u is not constant for a taxon). Wind speed (u) can be specified as the mean wind speed during the pollination season or (preferably) during unstable daytime conditions. Prentice's (1985) recommended value of 3 m/sec is typical of unstable daytime conditions near the canopy surface (Singer & Smith, 1953; Raynor, 1971). The measured sedimentation velocities of Durham (1946) and Eisenhut (1961) are consistent with predictions from Stokes's Law, and are closer to theoretical values than are other data sets. The Durham and Eisenhut data sets cover most of the common anemophilous pollen types of temperate and boreal regions of the Northern Hemisphere (Appendix 1), and so can be applied widely.

VII. Prescriptions for Further Research on Pollen Dispersal

The very limited experimental data and field observations provide only fleeting glimpses of the processes that intervene between source vegetation and derivative pollen-assemblage. Anemophily is the "poor relation" of pollination biology; more is known about insect pollination of old-field and alpine-meadow herbs than about wind pollination of the trees, shrubs, and graminoids that dominate vegetation in most of the temperate and boreal regions. The deficiencies in knowledge of wind pollination are matched by those in understanding of dispersal and deposition processes in vegetative canopy. We suggest several areas as being in particular need of further study.

A. PHYSICS OF POLLEN ENTRAINMENT

Pollen entrainment can be studied profitably from the dual perspectives of adaptive morphology of pollen-bearing organs and the micrometeorology of pollen liberation. Morphological features of ovule-bearing organs of anemophilous plants have been shown to play important roles in pollen interception and pollination (Whitehead, 1983; Niklas, 1985; Tomlinson, 1994). Similarly, morphology of anemophilous pollen grains plays a clear adaptive role in pollination biology (Crane, 1986). We lack comparable knowledge of the functional morphology of pollen-bearing organs. What forces are required to remove pollen grains from anthers and microsporangia, and what roles do microstrobili and staminate flowers play in regulating those forces? Do the microsporophylls of microsporangia and the bracts and bracteoles of catkins play a functional role in the aerodynamics of pollen entrainment? What is the adaptive significance of the flexibility of staminate catkins? What wind speeds are required to overcome the boundary-layer effects of anthers and associated organs? Are wind gusts of a particular magnitude and frequency required for pollen liberation? Is branch flexure in gusty conditions necessary as a "shaker" mechanism for pollen liberation? Studies of functional morphology coupled with micrometeorological studies would help identify the conditions necessary for pollen liberation in anemophilous plants.

B. SEDIMENTATION AND IMPACTION IN VEGETATIVE CANOPY

Pollen dispersal and deposition in vegetative canopies present unique challenges for field and laboratory investigation. Most experimental studies of particles in vegetated landscapes have involved pollen and spore releases in monospecific croplands (e.g., Aylor, 1975, 1989; Aylor et al., 1981; Aylor & Ferrandino, 1989). Raynor et al. (1974, 1975) studied dispersal and deposition of artificially released pollen and spores in a plantation of young (25-30 yrs) Pinus strobus and P. resinosa (Raynor, 1971). Koski (1970) injected 32P into the xylem of three P. sylvestris trees growing in a mature (ca. 120 yrs) monospecific stand, and used autoradiography to measure pollen caught in an array of traps within 50 m of each tree. These studies, while revealing, obviously comprise a limited set of circumstances. In particular we lack data on pollen behavior in canopies of springtime deciduous forest, conical-crowned coniferous forest, mixed forest, and shrublands.

In the study by Raynor et al. (1974, 1975), most pollen was sedimented or impacted on foliage, and relatively little was sedimented on the forest floor. Raynor (1971) also observed a rapid change in wind speed near the canopy surface. In Koski's (1970) study, pollen deposition at canopy level within 10-20 m downwind of the sources was similar in magnitude to that at the source trees. Pollen deposition beneath the canopy (2 m above ground) was negligible for all traps regardless of distance from the source (Koski, 1970), indicating that relatively little pollen reached the forest floor near the source trees via sedimentation.

These results may not be applicable to deciduous forests, which in springtime provide a smaller total area for interception and impaction within the canopy, and which also have a less-steep gradient in wind speed across the canopy surface. Is impaction efficiency greater for coniferous and other evergreen canopies than for deciduous canopies during the peak periods of pollen release? Is sedimentation within the canopy greater for coniferous canopies than deciduous canopies? What roles do crown geometry and canopy-surface roughness play in turbulent motion and particle deposition? Does impaction play a significant role in particle deposition, and does this role differ among canopy types?

An additional complication is introduced by the relationship between wind speed, canopy density, and sedimentation. In canopies with high leaf-area index, high turbulence within the canopy at high wind speeds can cause increased transfer of particles to the lower canopy and ground level, resulting in greater sedimentation at these levels and greater deposition velocities than would be predicted from sedimentation velocity alone (Legg & Price, 1980). This phenomenon would impart some stability to [v.sub.g]/u as wind speed increases, but whether this is sufficient to justify assuming constancy of [v.sub.g]/u at different wind speeds and especially for different vegetation types is not clear. Legg and Price used dense crop canopies (leaf-area index of 5.5-6.0) in their study; canopy density is likely to be much lower in deciduous and mixed forests during the primary season of pollen release. In coniferous forests, wind speed, turbulent transfer, and impaction decrease as leaf-area index increases (Lovett, 1981; Lovett & Reiners, 1986).

Aylor (1982) and McCartney and Fitt (1985) outline a theoretical basis for predicting sedimentation and impaction rates within a canopy from sedimentation velocity, wind speed, and within-canopy geometry (obstacle area/unit volume, and vertical and horizontal components of same). This theory assumes no particle rebound, so it will overestimate deposition rates. However, it can provide a framework for studying deposition processes within canopy, and for exploring effects of variations in canopy architecture. Particle rebound can be incorporated into simulations using the model.

C. VEGETATIVE CANOPIES: PARTICLE SINKS OR TRANSFER STATIONS?

A related area of study concerns the fate of particles deposited within vegetative canopy by sedimentation and impaction. Pollen grains deposited on canopy organs may be retained within the canopy until they decompose or are consumed, redeposited on the forest floor below, or reentrained in the atmosphere and transported farther. Secondary entrainment of many deposited grains by windgusts is highly likely, at least in the upper tiers of forest canopy. Reentrained grains are as likely to diffuse upward by atmospheric turbulence as those liberated directly from anthers and microsporangia. Raynor et al. (1975) demonstrated considerable vertical diffusion from sources deep within the canopy of dense pine plantations. Transport of many pollen grains may consist of alternating entrainment, atmospheric transport, canopy deposition, and reentrainment.

D. SEDIMENTATION OF POLLEN GRAINS

Examination of the available data sets confirms that measured estimates of sedimentation velocity correspond generally to theoretical expectations ([ILLUSTRATION FOR FIGURES 3-5 OMITTED]; see also Jackson, 1994: fig. 14.7). However, our analyses indicate that considerable variation exists within and among data sets, and that this variation has non-trivial effects on predicted dispersal-distance functions and pollen source-areas. The existing data sets, especially those of Durham (1946) and Eisenhut (1961) are adequate to investigate the relative differences in dispersal and source-area among pollen grains of different size and shape, as Prentice (1985, 1988) demonstrated. However, they may be inadequate for precise characterization of dispersal functions and pollen source-areas for specific taxa. Estimates of dry-pollen sedimentation-velocities using calibrated fall-towers or other devices, together with measurements of dry-pollen size and mass, not only would contribute not only to a more-precise understanding of pollen dispersal but would also help understand the functional morphology of pollen grains (e.g., Tomlinson, 1994). For example, how effective are the sacci of Pinaceae and Podocarpaceae in increasing drag and decreasing particle density? What are the trade-offs among size, shape, and density in ensuring pollen dispersal and pollination success?

E. SOURCE-HEIGHT EFFECTS

Prentice (1985) used Sutton's equation for dispersion from a ground-level source as the basis for his model, observing that pollen emission occurs at canopy level and pollen deposition is mediated by the canopy, making it analogous to a ground surface. His use of the ground-level equation had the important consequence of predicting more-locally weighted pollen source-areas for smaller basins. Previous applications of Sutton's elevated-source equation by Kabailiene (1969) predicted only minor effects of basin size, because all pollen emitted by a given source was deposited some minimum distance away owing to source-height effects (an elevated plume will travel some distance before it hits the ground).

The analogy between canopy surface and ground surface is imperfect, as is the analogy between trees in a forest and smokestacks on a landscape. Unlike the ground surface, the canopy is porous and has depth, allowing air and particle movement within and below it. For this reason, the canopy does not behave as a perfect source-level particle sink. The canopy is a source-level air-volume partly occupied by obstacles in the form of branches, shoots, and leaves. These obstacles retard air movement and provide source-level surfaces for particle deposition, but do not retain or transfer to ground-level all particles that pass through.

Actual pollen deposition patterns in a forest are likely to be intermediate between those expected from Sutton's ground-level equation and Sutton's elevated-source equation. Application of the latter equation (using height of the canopy surface as the source height) is likely to severely underestimate pollen deposition near a source; canopy obstacles do play a role in local pollen deposition on the forest floor, and pollen released in clusters (Andersen, 1970; Tonsor, 1985; DiGiovanni et al., 1995) or retained in abscised flowers or microstrobili will not travel far from the source. However, application of the ground-level equation will likely overestimate deposition near a source; the model will lack any source-height effect, and will not account for reentrainment from the canopy. Spore-release experiments indicate substantial differences between transport patterns over bare ground and crop canopy (Aylor, 1989). In spore releases within crop canopy, maximum airborne flux was well above the source, and in fact was above the canopy (Aylor, 1989: fig. 4). This pattern contrasts with releases at the same height over bare ground, where maximum airborne flux was at source-height (Aylor, 1989: fig. 3).

Empirical data are limited on whether pollen dispersal from trees is more like elevated-source or ground-source dispersion. Wright's (1952) sampling array is not dense enough to assess presence or absence of a skip-distance effect (i.e., a deposition maximum some distance from the source), which would be expected from an elevated source. Skip distances on the order of 20-50 m have been observed for isolated individuals of Pseudotsuga menziesii (Silen, 1962) and Pinus palustris (Boyer, 1966). Artificial release of Pinus coulteri pollen from an elevated source ([approximately]4 m) in an open park yielded a deposition maximum 5-7 m from the source, with little deposition within 3.5 m (Colwell, 1951). When Ambrosia pollen was artificially released from a source 1-2 m above Pinus forest canopy (total source height of 14 m), very little pollen was deposited on the forest floor below the source; maximum ground-level concentrations were observed 15 m away (Raynor et al., 1975). Koski (1970) found negligible pollen deposition beneath Pinus forest canopy within 50 m of the source tree, while canopy-surface deposition was high within 10-20 m before tapering off. The latter are the only studies done to date in forest canopy. Koski's (1970) results, although based on a limited sample, suggest that a ground-source model is appropriate for canopy-surface deposition but not for ground deposition, while an elevated-source model may be inappropriate for ground deposition owing to deposition and reentrainment within the canopy. A few carefully designed observational or experimental studies of near-source pollen dispersal in a forested landscape would contribute greatly to clarification of source-height effects.

Paleoecologists are faced with a dilemma: Both the ground-level and elevated-source models appear to be unrealistic in application to forest canopy, yet they are the only simple models available. A single physical model capable of simultaneously describing both the ground-like and elevated properties of canopy in terms of pollen dispersal and deposition may be elusive. A solution may lie in application of an elevated-source model coupled with theoretical or empirical functions to account for near-source deposition owing to gravity (clusters and unreleased grains) and to redeposition from near-source canopy deposition. The latter functions will require empirical assessment of the magnitude of the near-source depositional processes. An alternative solution may be a model similar to one developed by Aylor (1987) for crop canopy, which consists of coupled equations to describe particle transport within the canopy and above the canopy.

F. POLLEN DISPERSAL: GAUSSIAN PLUMES OR PUFFS?

Application of Sutton's equations, whether ground level or elevated, to pollen dispersal assumes that pollen grains are emitted continuously at the source, forming a plume that disperses downwind in a manner that depends on atmospheric stability. Aylor (1978) suggests that a Gaussian puff model is more appropriate for dispersal of fungal spores that require wind gusts for entrainment. Because entrainment is episodic, dependent upon individual wind gusts that exceed the inertial threshold required to entrain the particles, spores will be emitted from a source as a series of time-separated puffs rather than continuous plumes (Aylor, 1978). The same reasoning should hold if pollen entrainment depends on wind gusts.

Patterns of particle diffusion and deposition differ between plumes and puffs (Gifford, 1968; Hanna et al., 1982; Aylor, 1978). In particular, puff models predict substantially less sedimentation near the source and relatively minor effects of average wind speed on deposition patterns (Aylor, 1978). The high initial wind speeds associated with puff dispersal may increase particle impaction in the immediate vicinity of the source (Aylor et al., 1981), but rebounding will counteract this trend in dry canopy. Gaussian-puff analogues to Sutton's equations could be explored as the basis for an alternative model of pollen dispersal and source-area. However, puff models are mathematically more complicated, and application will require field or experimental determination of some key parameters (threshold wind speed for entrainment, initial puff size, time-dependent diffusion coefficients). Similarly, application of "random-walk" (Lagrangian) models, which have several advantages over Gaussian models in both gusty and steady wind conditions (Burrows, 1975; Legg, 1983; Aylor, 1990), would require specification of many parameters from observations or experiments.

VIII. Summary

Prentice's (1985) model has the twin virtues of mathematical simplicity and small number of required parameters, and provides a useful first approximation to pollen dispersal and deposition. The model provides an excellent vehicle for understanding the vegetation-sensing properties of sedimentary pollen assemblages (Prentice, 1988; Jackson, 1994; Sugita, 1994). Simulation studies based on Prentice's model can provide insights into application and interpretation of pollen data in a variety of contexts (Sugita, 1994). However, applications of the model to infer pollen source-areas for specific taxa in specific basin-sizes and to assign distance-weightings to vegetation data for pollen-vegetation calibration should be approached with caution, given the uncertainties in parameter specification.

The model is probably adequate for estimation of pollen source-areas for moderate-sized to large basins, at least to the precision typically required by paleoecologists working at subregional and larger scales. For these applications, the potential errors introduced by uncertainties in the atmospheric parameters (n and [C.sub.z]) and depositional parameters ([v.sub.g] and u) have relatively minor consequences for paleovegetational and paleoclimatic interpretation. The use of alternative parameter sets in these contexts can provide a check on the spatial scales at which the data are interpreted. Similarly, Sugita's (1993) model helps constrain interpretation by setting a lower boundary to spatial precision (by recycling pollen initially deposited near the shore to the lake center).

The greatest uncertainties in application of Prentice's model are for small basins and closed-canopy deposition. The alternative atmospheric parameter sets we have applied, based on unstable atmospheric conditions at the time of pollen release, indicate more widespread dispersal from individual sources and hence poorer representation of local pollen sources in small basins. The dispersal curves are still strongly leptokurtic, indicating that a local vegetational signal (i.e, within a 20-100 m radius) is represented in the pollen assemblages. However, the monotonic decline in pollen deposition with distance from a source is strictly dependent on use of Sutton's ground-level equation. Application of an elevated-source equation would displace the zone of maximum deposition some distance from the source, smoothing vegetational patterns as represented in small-basin pollen assemblages.

Existing data sets of modern forest composition coupled with modern forest-floor pollen assemblages (Jackson & Wong, 1994; Calcote, 1995; Jackson & Kearsley, 1998) may be inadequate for discriminating between the ground-level and elevated-source assumptions. Most of the patches studied by Jackson and Wong (1994) and Jackson and Kearsley (1998) are more than 50-60 m in diameter, so good fit between pollen and vegetation data within a 20 m radius is ambiguous. Equally good fit is often obtained using unweighted vegetation data within a 120 m radius (Jackson & Kearsley, 1998). Similar studies of smaller patches or isolated trees within extensive forests of different composition would be desirable. Comparative studies of modern pollen-vegetation relationships can only provide us with secondary information on processes, however. A series of well-designed studies aimed at identifying dominant mechanisms and validating parameters would be invaluable in guiding future development and application of theoretical models, and bring us closer to Prentice's (1985) goal of a well-validated "unified theory of pollen analysis."

IX. Acknowledgments

This research was supported by the National Science Foundation (Ecology Program), and benefited from discussions with Don Aylor, Jennifer Kearsley, Colin Prentice, Bill Reiners, Tom Webb, Chengyu Weng, and Jerry Winslow. Barbara Strauss provided German translations.

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