Introduction

Plant disease epidemics develop inside plant canopies, which provide the susceptible host tissue and the conditions necessary for infection by the pathogen. To cause an epidemic, however, the pathogen must be dispersed repeatedly to healthy host tissue. Without this dispersal, an epidemic would simply burn out like a forest fire starved for fuel. Fungus spores are the dispersal agents for a major class of plant diseases, and atmospheric dispersion is the predominant mechanism for transporting the spores of the pathogens to healthy tissue.

Because atmospheric dispersal drives epidemics, the release rate of spores from the plant canopy into the atmosphere Q (spores per meter squared per second) is a crucial quantity. Knowledge of Q is necessary to estimate the spread of disease or to compare management techniques for reducing disease. However, Q is a difficult quantity to measure for spores. The problem is similar to measuring trace gas fluxes from the surface. Although a suite of techniques exists for measuring gas fluxes (e.g., eddy covariance, flux gradient), these methods are generally not amenable to measuring spore emission rates because of the lack of fast-response spore sensors and because of the complexity of the spore release mechanisms (Aylor 1990).

A potential technique for estimating Q is to use a dispersion-model-based method. This method relies on a measurement of the concentration of spores (C_measured) at a point P downwind of the spore source. A dispersion model is then used to simulate the release of spores into the atmosphere (e.g., assuming an arbitrary release rate Q_model), and the spore concentration at point P is calculated (C_model). The true spore release rate is found by reconciling the modeled and measured concentrations. The method has been used successfully to estimate trace gases emissions (Wilson et al. 1983; Flesch et al. 1995). The success of the method hinges on an accurate dispersion model, applied faithfully to the actual release conditions. However, modeling the dispersion of plant pathogenic spores is a more difficult task than modeling trace gas dispersion, because Q for spores can depend directly on atmospheric turbulence and because the inertia of spores must be accounted for.

A Lagrangian stochastic (LS) dispersion model represents the best candidate for determining Q. These models numerically mimic the trajectories of “particles” (either marked fluid elements or solid particles) downwind of a source. These models have proved to be accurate and easy to use for calculating the dispersion of atmospheric tracers (Wilson and Sawford 1996). The advantages of an LS model as compared with other kinds of models are 1) the correct prediction of dispersion close to a source, 2) easy incorporation of complex wind flow, 3) grid-free models, and 4) easy incorporation of time-dependent processes. The major difficulty in using an LS model for spore dispersion is theoretical. Plant pathogens are usually transported via “heavy” particles, whose size and mass make modeling atmospheric transport more complex than for “passive” tracer particles. Current theory does not point to a correct way to calculate heavy-particle trajectories. Modeling the dispersion of spores is also made difficult by the need to account for additional physical processes: deposition onto foliage, deposition onto the ground, and, in some cases, washout of spores.

In this paper, our objective is to use an LS-model-based approach to estimate Q for plant pathogenic spores. Following the work of others, we have adapted an LS model for passive tracers to obtain a model for dispersion of heavy particles. We then added algorithms for calculating deposition onto vegetation and the ground and for calculating the washout of spores. This model was then applied to two field experiments to determine Q. The LS-model-based estimates of Q were then compared with independent estimates. We will comment on the accuracy of LS technique in our experiments and on the general applicability of the technique for estimating spore emission rates.

Experiments

The LS simulation model (described in section 3) is compared with particle concentrations from two sets of experimental data. One set of data is taken from an earlier study in which Lycopodium spores were released from line sources inside a wheat canopy (Aylor and Ferrandino 1989). The second set is new data, reported here, in which V. inaequalis ascospores were released naturally by the fungus from a ground-level source in a canopy of mixed grasses. These two case studies present notable contrasts with respect to the size of spores, the height in the canopy where spores were released, and the weather conditions during spore release.

LS model for spore dispersion

The LS models are most commonly used to simulate the trajectories of passive tracers. In a passive model, the trajectory of a background fluid element is calculated, and this trajectory will reflect the continually changing fluid velocity as the fluid element travels through the turbulent atmosphere. Complications arise when applying LS models to spore transport; we must account for the effects of inertia and gravity on spore trajectories. According to Wilson (2000), two modifications are necessary to convert a passive LS model to a spore-trajectory LS model. First, the settling velocity υ_S of the spore must be added to the vertical velocity calculated by the passive LS model. This is to represent the vertical velocity of the heavy particle as it travels through turbulence. The second modification is to reduce the turbulence timescale of the passive LS model to reflect the turbulent conditions “seen” by the spore. These simple changes create an accurate LS model for particles smaller than 300 μm in diameter (Wilson 2000). In this section, we detail our LS trajectory model.

Even with an accurate spore trajectory model, we must include other important processes that affect spore dispersal. In the following section, we outline the trajectory model and our algorithms for foliage and ground deposition of spores and for washout of spores by rain.

Spore trajectory model

A 2D Langevin model is used to increment the alongwind and vertical air velocities (u, w) surrounding a spore and to calculate spore displacement in the alongwind (x) and vertical (z) directions over a time step dt:

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where the Langevin coefficients a_u, b_u, a_w, and b_w are functions of velocity and position; dξ_u and dξ_w are random numbers selected from independent Gaussian distributions, each having average 0 and variance dt; and υ_S is the settling velocity of the spore in still air. This model is identical to a passive model, except for the addition of υ_S in Eq. (2).

In a paper on passive LS models, Thomson (1987) gives a solution for the Langevin coefficients that met two important theoretical constraints: Kolmogorov's similarity theory and the well-mixed condition. In the case of Gaussian turbulence (Eulerian velocities described by Gaussian probability functions) and in stationary and horizontally homogeneous flow, these coefficients are

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where U is the average horizontal velocity (we have assumed the average vertical velocity W = 0); σ²_u and σ²_w are the horizontal and vertical Eulerian velocity variances, respectively; uw is the Eulerian velocity covariance [more precisely ]; C₀ is a universal constant (whose value is debated); and ε is the turbulent kinetic energy dissipation rate. We use the traditional definition

C₀σ²_wτ,(6)

where τ is the turbulent decorrelation timescale for velocity. In passive models, τ is loosely referred to as the Lagrangian timescale T_L. The model equations were solved numerically using a simulation time step Δt = 0.025τ. Particles were released with an initial velocity in accordance with the Eulerian probability distribution function pdf for the source location.

For heavy particles (e.g., unit density spheres with diameters >300 μm), it would be necessary for us to solve simultaneously the equations of motion for the fluid and for the particle. Wilson (2000) and others before him found that for small particles (diameter < 300 μm) it is not necessary to solve the particle equation of motion. This is fortunate because of the extreme computational demands that would be added by the particle equation of motion due to the very small time step implied by the drop equation of motion. The equivalent aerodynamic diameter of V. inaequalis ascospores is about 8.2 μm (particle relaxation time τ_R is about 0.2 ms), and the aerodynamic diameter for Lycopodium spores is about 25 μm (τ_R about 2 ms). The diameters of both particles are well below the limit determined by Wilson (2000), and their motions in the atmosphere can be adequately described by the model given here.

Timescale correction

Sawford and Guest (1991) reasoned the fluid velocity timescale τ following a heavy particle should be less than the timescale of a passive fluid element (T_L). Following the approach of Sawford and Guest (and Csanady 1963), we assume that τ is given by

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where β is an empirical constant, supposedly related to the ratio of Lagrangian to Eulerian timescales of the flow. This equation gives the intuitive result that large spores become uncorrelated from the turbulence more quickly than small spores. We used β = 1.5, consistent with Sawford and Guest (1991).

Because of their relatively small settling speeds, the effect of f on the calculated motion of spores is small. For the V. inaequalis ascospore experiments (friction velocity u∗ values were ≥ 0.06 m s⁻¹), we estimate that f was greater than 0.99 above midcanopy height and greater than 0.95 at heights within a few centimeters above the ground. For the Lycopodium experiments (u∗ ≥ 0.30 m s⁻¹), we estimate that f is greater than 0.96 above midcanopy height and greater than 0.83 within a few centimeters above the ground.

Spore deposition on vegetation

As a spore drifts through a plant canopy, it encounters stems, twigs, leaves, and branches. Some spores will be deposited on these canopy elements. The probability of spore deposition over a model time step depends on the probability of the spore encountering vegetation during that time step and the probability that the encountered vegetation “captures” the spore (deposition efficiency). Following the approach of previous work (Legg and Powell 1979; Aylor 1982; Legg 1983; Aylor and Ferrandino 1989; McCartney and Aylor 1987), we express the probability of vegetative deposition over a time step dt as

G^Vzυ_Sf^xa_pE^xdtuf^za_pE^zdt,(8)

where f^x and f^z are respective fractions of the plant area projected onto horizontal and vertical planes; u is the horizontal velocity of the spore; E^x and E^z are the efficiencies for horizontal and vertical deposition, respectively; a_p (m⁻¹) is the plant area density. The first term on the right-hand side of Eq. (8) reflects the probability of deposition during gravitational settling, and the second term represents the probability of deposition during horizontal impaction.

After the methods of Legg and Powell (1979), Aylor (1982), and others, we assume the absorption efficiency on horizontal vegetation E^x is 1. For E^z, we use Aylor's (1982) fit to May and Clifford's (1967) data for impaction on a cylinder:

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where τ_R is the particle relaxation time, and L_V is the characteristic size of the vegetation elements. Equation (9) was evaluated by letting τ_R = υ_S/g (Davies 1966), where g is the acceleration of gravity.

In calculating deposition, we divide the plant canopy vertically into i layers and define the plant area a_i, vegetation size L_Vi, and the area projections f^z_i and f^x_i within each layer. We assume vertically uniform vegetation within a layer and a horizontally homogeneous canopy. To handle the wide range of vegetation sizes, shapes, and angular orientations, the canopy is divided into j vegetation classes based on characteristic size. If a spore is traveling within canopy layer i, the probability of being deposited during time step dt is

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The efficiency E^x_j is assumed one, and E^z_j is calculated for each vegetation class according to Eq. (9). To determine if a spore is absorbed over the time step, we chose a random number η from a uniform distribution between 0 and 1 (Aylor and Ferrandino 1989; Legg 1983). If η is less than G^V, then the spore is deposited, and the next spore is “released” from the source. Otherwise, the spore is allowed to move on to the next time step, and G^V is reevaluated.

Spore deposition on ground

When a spore trajectory crosses the ground plane (actually a reflecting height z_r just above the ground), it is either deposited on the ground or “reflected” back into the atmosphere. According to Aylor and Ferrandino (1989), the probability of deposition is

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When a spore “impacts” the ground, we chose a random number η from a uniform distribution between 0 and 1. If η is less than G^g, the spore is deposited. If the spore is not deposited, it is reflected back into the atmosphere. At reflection, the spore position and the surrounding air velocity are modified. We have used the following rules when the spore passes below our reflecting height z_r:

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where the superscript “old” refers to the position/velocity at the previous time step before reflection, and the superscript “new” refers to the position and velocity after reflection.

Calculating C and Q

The purpose of our LS model is to calculate the spore concentration C_LS at a “sensor” location(s). We calculate C_LS (spores per meter cubed) at x_sens, z_sens by defining a collector “bin” of height Δz_sens centered on z_sens. When simulating a continuous crosswind line source (e.g., the Lycopodium experiment),

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where Q_LS is the modeled source strength (spores per meter per second), N is the number of model spores released from the source, and u_n is the horizontal spore velocity for spore number n as it passes x_sens between z_sens ± Δz_sens/2 (only a fraction of the particles will pass through this bin). A formal description of Eq. (14) can be found in Flesch et al. (1995).

When applying the LS model to simulate the concentration in the center of a circular area source of radius R (e.g., V. inaequalis ascospore experiment), we treat the source as an infinite crosswind area source. We release particles at M discrete locations upwind of the sensor location:

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If N particles are released at each x_m [to represent release from a crosswind strip Δx (=R/M) meters wide in the alongwind direction], then the concentration at the center of the source is given by

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where Q_LS is the modeled source strength (spores per meter squared per second), and u_n is the horizontal spore velocity for spore number n as it passes x_sens between z_sens ± Δz_sens/2. To simulate the V. inaequalis ascospore sources, we used M = 12.

We then estimated the true spore release rate Q. To do this, we picked an arbitrary model release rate (we used a unit release rate Q_LS = 1) and then reconciled C_LS with the actual spore concentration measured at x_sens, z_sens(C_measured):

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Our LS model was used to simulate each of the spore dispersion experiments, creating vertical profiles of concentration C_LS. The overall objective of our study is to scale these profiles [using Eq. (17)] to give a Q_model that results in good agreement with the observed concentration profiles. This is then taken as our estimate of the true spore emission rate. In each experiment, we have C_measured at several heights. In determining the best Q_model estimate, we took into account where the LS model and the observations of C_measured were the most accurate.

Results

Discussion and conclusions

Modeling the spread of plant disease by aerially dispersed pathogens requires knowledge of the number of pathogenic spores deposited on susceptible plant tissue at various distances from a source. Given the aerial concentration of spores surrounding susceptible plants (or in the vertical column of air above the plants if wet deposition is involved), it is possible to quantify the number of infections that are likely to occur (Aylor and Kiyomoto 1993). Unfortunately, the ability to predict C is extremely limited because of the difficulty of determining the rate of spore release from a source, owing mainly to the microscopic nature of spores and to the lack of a fast-response sensor for discriminating fluxes of spores of specific pathogens. Our LS model gave reasonable estimates of Q both for Lycopodium spores released inside a wheat canopy from elevated line sources (Table 1) and for V. inaequalis ascospores released naturally by the fungus during rain from ground-level area sources (Fig. 4).

On both theoretical grounds and an examination of our results, we have lower confidence in our model predictions of Lycopodium concentrations within the wheat canopy than above (Fig. 1). This lack of confidence is due to the difficulty in accurately measuring winds in a plant canopy and to the unresolved complexity of the wind flow within the canopy. Also, our model predictions in the canopy were characterized by large stochastic uncertainty. We attribute this to the small number of model spores that remain airborne deep in the canopy at the sampling location and to the high turbulent intensity in the canopy. These factors combine to give a relatively large variability in C_LS predictions in the canopy. Another factor creating lower confidence is the greater uncertainty of C_measured in the canopy. Suction samplers, which were used of necessity owing to the close proximity of the canopy's various elements, are less accurate than rotorods because of anisokinetic sampling errors (Hinds 1982).

An outlying point for V. inaequalis ascospores occurred on yearday 126 (Fig. 4) during a period of rapid biological development (maturation) of the ascospores (Fig. 2). It is highly likely that our spore tower determination underestimated the peak spore release in this case. In view of the difficulty of estimating Q for ascospores from direct measurements, we strongly favor the LS model determinations of Q.

An important issue when using a model to simulate spore dispersal is the difference between actively and passively released spores. The model we have outlined here is appropriate for actively released spores, that is, spores that are released quasi-continuously, independent of the ambient wind speed (e.g., Lycopodium and V. inaequalis experiments). However, a key model assumption is invalid for passively released spores whose release rate varies with the ambient wind velocity. This is the assumption that the wind velocity is described by Gaussian statistics (pdfs). It is reasonable to suppose that actively released spores experience the entire velocity pdf. Based on the dispersion results of Flesch and Wilson (1992) and Wilson (2000), it is reasonable to use Gaussian statistics. However, passively released spores are different. The liberation and the initial stages of dispersal of this type of spore occur during wind gusts that are above a particular threshold wind speed (Aylor and Parlange 1975; Aylor 1990). Although our model can theoretically accommodate situations in which spore release occurs in such truncated (censored) wind statistics (Legg 1983), it is likely that the inaccuracy of using Gaussian pdfs will become pronounced, and a more sophisticated model will be required to quantify the dispersal of spores that are passively liberated from lesions into the air during gusts.

Gusts and lulls in the wind have another potential effect on dispersal. In the absence of vertical fluctuations of the wind, spores released near the ground would settle onto the ground in a matter of seconds. If lulls in the wind last for too long, then the dispersal of a substantial number of spores might be extremely limited. A characteristic timescale for spores to be deposited on the ground by sedimentation is τ_S = d_S/υ_s, where d_S is the vertical distance that a spore is energetically projected into the air by the fungus, and υ_S is the settling speed of the spore in still air. For V. inaequalis ascospores, the value of d_S is between 3 and 7 mm (Aylor and Anagnostakis 1991), and υ_S is 0.002 m s⁻¹ (Gregory 1973). Together, these lead to values of τ_S in the range 1.5–3.5 s. Lulls in the wind near the ground in a grass canopy rarely lasted for more than 3 s (Aylor et al. 1993). Their average length was closer to 1 s, which is considerably less than τ_S. Thus, it does not appear that lulls generally last long enough or are frequent enough to bias significantly our calculations of deposition of V. inaequalis ascospores by sedimentation. Based on this, we likely are justified in our use of Gaussian pdfs to calculate the motions of ascospores near the ground. This may not be the case for larger spores (with substantially greater settling speeds) that are released this close to the ground. In this case, many spores would be deposited right at the source, leaving only those that travel in gusts (see previous paragraph) to be dispersed.

Venturia inaequalis ascospores become airborne during periods of rain. Earlier studies (Aylor and Ducharme 1995) showed that the fluctuations in vertical and horizontal components of wind velocity within a few meters above the ground are in good agreement with those found during conditions of fair weather and near-neutral atmospheric stability. Thus, for purposes of modeling the initial aerial dispersal of V. inaequalis ascospores, it probably is reasonable for us to have used the description of velocity variances determined in studies during fair weather (appendix A). As far as we are aware, there are no studies during rain to guide us in a choice of T_L different than the one we took for our studies.

Quantifying the rate of release of spores and their escape from a canopy is a crucial first step in evaluating risk of disease spread, whether from naturally occurring pathogens or from pathogens purposely introduced as biocontrol agents (De Jong et al. 1999). The release rate Q is determined by the biology of the pathogen interacting with the environment and is a highly dynamic process. The dispersal of spores released inside a plant canopy is determined by the physical characteristics of the spores and the canopy, the height of spore release in the canopy, and the vertical profiles of wind speed and turbulence in and just above the canopy. Additional challenges of modeling spore dispersal (as compared with trace gases) arise from the spore's small but finite inertia and sedimentation, microscopic size, and proximity of the source to surfaces. Our LS model realistically describes these physical effects.

We successfully used a Lagrangian stochastic model–based approach to estimate the release of spores into the atmosphere. This problem is nearly intractable by other methods. Our estimates of Q appear to be excellent, considering the complexity of the problem. This indicates accuracy of not only the LS model but also of the deposition algorithms and shows the quality of the meteorological inputs. Thus, the application of an LS model provides a new tool that significantly advances the ability to model the aerial dispersal of microscopic biological aerosols and furthers the application of meteorology to management of plant diseases.