Aerial Dispersal of Pollen and Spores

12

C H A P T ER

Escape of Spores from a Canopy C H A P T E R

O U T L I N E

12.1. Escape Fraction Defined 12.2. Escape Fraction Under Near-Neutral Atmospheric Conditions 12.2.1. Time Scales for Deposition and Escape 12.2.2. Deposition Dominated by Sedimentation 12.2.3. Deposition by a Combination of Sedimentation and Impaction 12.2.4. Vertical Distribution of Spores in a Canopy 12.2.5. Use of More Complex Models

For diseases spread primarily by airborne spores, the amount of diseased tissue typically decreases rapidly with an increase in distance from a focus of infection (Gregory, 1973). This is a direct reflection of a pattern of decreasing deposition of spores with distance from a source that is determined in large part by the rate at which spores escape from a diseased crop into the fastermoving air above the canopy. Thus, the escape of spores from a canopy plays a major role in determining disease gradients near sources and in setting the stage for their long-distance dispersal.

The rate at which spores escape from a canopy depends on the relative magnitudes of two competing processes: (a) deposition on foliage and on the ground and (b) vertical transport out of the canopy by turbulence. These two processes act in parallel, and the interaction between them largely determines the pattern of spore deposition near a source. Wind gusts play a large role in liberating spores from the host plant tissue and subsequently ejecting them from the canopy. On a gusty day, as many as half of the spores originating from

12.3. Escape Fraction Under Highly Convective Conditions 12.4. Source Emission Rates Determined from Vertical Concentration Gradients 12.5. Summary

sources located above midcanopy height can escape the canopy within a downwind distance of only a few canopy heights from a source.

12.1. Escape Fraction Defined The word escape is defined here as the net upward flux density of particles passing through the horizontal plane that demarcates the top of the canopy (Fig. 12.1).

Close to a source, this flux is positive upward (white arrow in Fig. 12.1), while at some distance downwind, a certain small fraction of this flux will be downward and returned to the canopy (hatched arrows in Fig. 12.1).

For simplicity in this discussion, we will assume that spores are released from an infinite crosswind line source. Escape from the canopy is defined as the integrated vertical flux of spores through a plane at the top of the canopy (i.e., at z = hc) between the upwind edge of the source (at x = 0) and a given downwind 215

216 PART THREE Modeling Wind Transport of Pollen and Spores

distance (at x = x1) (Fig. 12.2). By conservation of material, this vertical flux must be equal to the horizontal (downwind) flux above the canopy.

Specifically, e(x1) =

1 Qx1

∫

x1

0

Fz ( x , hc ) d x =

1 Qx1

∞

∫ F ( x , z) dz , (12.1) hc

x

1

where e(x1) is the escape fraction (dimensionless), Q (spores m−2 s−1) is the spore release rate per unit area of the source, hc is the height of the canopy, and Fz and Fx are the vertical and horizontal (downwind) flux densities of spores, respectively. Referring back to Chapter 10, we see that Eq. 12.1 is just another statement of conservation of particles, in which the net cumulative

FIG. 12.1. A fraction of the total spores released by a source are transported upward (Fz > 0) through the top of the canopy, and a fraction of these spores will return to the canopy as a downward flux (Fz < 0) at various distances downwind from the source. Escape is the net of these two fluxes and a function of distance from the source. (Courtesy D. E. Aylor—© APS)

FIG. 12.2. Three regimes of spore plume spread in and above a plant canopy. In the first regime (labeled “exponential”), the

plume is confined to the canopy airspace, where the vertically integrated spore concentration decreases exponentially with distance from the source. In the second regime (“transition”), approximately equal numbers of spores are transported inside and above the canopy. In the third regime (“power law”), most of the spores released at the source that are still airborne at this point are transported in the airflow above the canopy, where the action of turbulent diffusion results in an inverse power law dispersal regime. (Based on Aylor [1987]. Courtesy D. E. Aylor—© APS)

CHAPTER 12 Escape of Spores from a Canopy 217

number of spores that exit the canopy through a plane at the top of the canopy between 0 and x1 are carried by the horizontal component of the wind through the portion of the vertical plane (located at x1) that is above the canopy.1 Close to a source, e is positive (i.e., out of the canopy), but at some point downwind from a source, e can become negative as particles are returned to the canopy from the spore-laden air above it.

Eq. 12.1 defines escape in terms of a particular downwind distance, x1. The normalized flux, Fz(x, hc )/Q, and escape, e(x), are both functions of downwind distance, x. Examples of calculations done using the two-dimensional Lagrangian stochastic (LS) model described in Chapter 11 are shown in Figure 12.3.

The remarkable feature of this figure is that almost all

the net upward flux occurs within the first few meters from the source and appears as a relatively sharp spike on either side of x = 0. The contributions in the upwind direction are a result of horizontal velocity fluctuations, σu, in the wind. The general shape of e(x) shows a rapid increase to a peak followed by a gradual decline with increasing distance from the source. This decline is due to the return (i.e., downward flux) of particles to the canopy from above. Both the initial upward flux near the source and the downward flux downwind are larger for higher values of u* ; this is simply a reflection of larger vertical transport due to the larger vertical velocity variance, σw , associated with higher u* (Chapter 9). After the initial burst, the e(x) changes rather slowly with distance.

FIG. 12.3. (A) Escape and (B) normalized vertical flux of particles through the horizontal plane, z = hc , for two cases: case 1, vs = 0.02 m s–1 and u* = 0.4 m s–1 (solid lines), and case 2, vs = 0.02 m s–1 and u* = 0.1 m s–1 (dashed lines). Particles were released from a line source positioned at the midcanopy height of a 0.7-m-tall wheat canopy at the “boot” stage of plant growth. The line representing the flux for the second case falls partly on top of the first case and is not plotted for clarity of presentation. (Note: The inset in part B is a portion of the curve shown in the larger panel plotted on an expanded scale to better show the return flux, Fz < 0, of spores to the canopy.) (Courtesy D. E. Aylor—© APS)

218 PART THREE Modeling Wind Transport of Pollen and Spores

In our examples, designated case 1 and case 2 (see Fig. 12.3 caption), the peak values of e(x) were 0.61 and 0.26, respectively, and at 66 m downwind, the abovecanopy flux was reduced only by about 15% and 8% for cases 1 and 2, respectively. Therefore, a significant fraction of spore release can be transported well beyond the boundaries of a typical experimental test plot and a typical farmer’s field. At some impractically far downwind distance (i.e., in the limit as x → ∞), e approaches 0, because all the spores will eventually be deposited.

12.2. Escape Fraction Under Near-Neutral Atmospheric Conditions The LS model, described in detail in Chapter 11, has been used to estimate the escapes of differentsized particles from a line source inside a wheat canopy (Aylor, 1999). Two main features of particle escape from the canopy were identified: 1. Escape is a hyperbolic function of vs /u*, illustrating that escape is larger both for smaller particles (smaller vs) and for higher wind speeds (larger u*).

2. Escape decreases with the relative depth of the source in the canopy (i.e., for smaller values of zs /hc).

These findings depend directly on the time scales for deposition inside and escape from a canopy.

12.2.1. Time Scales for Deposition and Escape An approximate formula for estimating the escape of particles from a crop canopy can be obtained from a heuristic argument based on an electrical circuit analogy (Aylor, 1999). Imagine a control volume (referred hereafter as a box) defined by the dashed lines and dotted area shown in Figure 12.4, part A, and extending 1 unit of depth perpendicular to the plane of the page. (Recall that we are still considering an infinite line source.) The box extends from the ground to the top of the canopy and is open to the atmosphere at the top. Q particles are continuously released into the box, and turbulence keeps the airborne particles well mixed within the box. Removal of particles from the air inside the box occurs via two parallel pathways: (a) particles are either deposited on crop elements and the ground inside the box, or (b) they escape from the box through its open top. After enough time has passed (t → ∞), an equilibrium will have been reached between particle release and particle losses due to particles either having been deposited or having escaped out the top of the box, so that the concentration reaches and maintains a constant value, C 0.

The argument proceeds by analogy with an electrical circuit (Fig. 12.4, part B), whereby particles are imagined to flow out of (be removed from) the airborne fraction of particles inside the box via either an escape “current” or a deposition “current,” designated JE and JD, respectively. The two processes for removing particles from the box—escape and deposition—act in parallel, and the relative magnitude of each “current” depends inversely on the time scales for the two processes, TE and

FIG. 12.4. Schematic of an electrical analogy used to illustrate a heuristic argument for estimating the escape of particles from a plant canopy in terms of their fall speed and the turbulent mixing of the air (see description in text). (Courtesy D. E. Aylor—© APS)

CHAPTER 12 Escape of Spores from a Canopy 219

TD.2 Continuing with the electrical circuit analogy, the escape fraction, e, is expressed as the ratio of the escape current, JE , to the total current, JE + JD, so that e=

JE 1 = .

JE + JD 1 + TE (12.2) TD

Note that e approaches logical values for two limiting cases: namely, (a) e approaches 0 when particles are deposited instantly (i.e., when TD → 0), and (b) e approaches 1 when particles are deposited extremely slowly, if at all (i.e., when TD → ∞).

To proceed, we need expressions for the time scales, TE and TD. A time scale for escape, TE , can be expressed in terms of a turbulent diffusion time scale, defined by a length squared divided by an average diffusivity: TE ∝ hc 2 / 〈 σw 2 TLw 〉 ∝ c1 ×

hc . u*

(12.3)

Here, σw2 TLw is a diffusivity (analogous to the eddy diffusivity, Kz, which we expect to hold for the far field result); σw is the standard deviation of the vertical velocity fluctuations; TLw is the Lagrangian integral time scale; and the angle brackets, < >, represent a spatial average over the height of the box. The value of the coefficient c1 is not a constant but depends on the source height, zs , which affects how the weighted average of the <σw2TLw > in the canopy is obtained. For example, spores released near the ground will necessarily be affected by smaller values of σw2 TLw during a greater proportion of the total time that they remain in the canopy compared with spores released near the canopy top.3 If we restrict our attention to the release of particles at or above midcanopy height, then a value of c1 = 7 can be justified theoretically (Aylor, 1999); using this value in Eq. 12.3 leads to values of escape that are in good agreement with LS calculations for particles released from midcanopy height and above. For spore release lower in the canopy, a rough approximation to the LS model results for escape can be obtained by replacing c1 with the values 20 and 50 for zs /hc = 0.20 and 0.05, respectively; these two values of c1 are empirical and cannot be justified theoretically.

We will consider two cases for computing the deposition time scale, TD: (a) deposition is dominated by sedimentation and (b) deposition occurs by a combination of sedimentation and inertial impaction. In both cases, we assume that the plant area is distributed uniformly with height in the canopy.

12.2.2. Deposition Dominated by Sedimentation In the simplest case, we ignore inertial impaction and assume that spore deposition occurs solely by sedimentation onto plant parts. We can write the time scale for deposition as TD ∝

hc , (12.4) vs fx LAI

where LAI is the leaf area index. Combining Eqs. 12.2– 12.4, we obtain 1

e≈

1 + c1 fx LAI

vs

.

(12.5)

u*

Escape has an inverse (hyperbolic) dependence on the parameter group, vs /u*.

Eq. 12.5 can be used to estimate escape from a grasslike canopy for any particular spore of interest as long as the settling speed, vs, and the friction velocity, u*, are specified. Two contrasting examples are the escape of Venturia inaequalis ascospores and Pyrenophora triticirepentis ascospores from a grass canopy. Both kinds of spores are released close to the ground; however, for V. inaequalis ascospores, vs is about 0.002 m s−1 and for P. tritici-repentis ascospores, it is about 0.014 m s−1 .

Nearly half the V. inaequalis ascospores released during a light wind, characterized by u* = 0.2 m s−1 (for vs /u* = 0.01), are expected to escape from our hypothetical grass canopy. By contrast, it would take a much stronger wind speed (u* = 1.4 m s−1) for half of the P. tritici-repentis ascospores (vs /u* = 0.01) to escape from the same canopy.

12.2.3. Deposition by a Combination of Sedimentation and Impaction Next, we consider the case in which deposition occurs by both impaction and sedimentation. We continue with the analogy of an electrical circuit and write expressions for time scales TDi and TDs for impaction and sedimentation, respectively. Particle “current” flows are assumed to act along parallel pathways (inside the dashed circle in Fig. 12.4, part B). In this case, the time scale for impaction is approximately TDi ∝

hc , (12.6) c 2 u * fz LAI

where fz is the fraction of the crop area that is oriented vertically and c2 is a parameter of the model. We

220 PART THREE Modeling Wind Transport of Pollen and Spores

introduced c2 because we want to express the wind speed in the canopy in terms of u* ; the product c2 u* is the characteristic wind speed in the canopy at which impaction is assumed to take place. In the examples given in Figure 12.5, the coefficient c2 takes on two values, 2.75 and 0.61, which represent, respectively, the average wind speed near the top of the canopy and the average wind speed near midcanopy height. When Eqs. 12.4 and 12.6 are combined (assuming that the two deposition “currents” are in parallel, electronically speaking), we obtain TDmod =

hc t1t 2 = t1 + t 2 vs fx LAI + c2u * fz LAI

hc / LAI . = vs fx + c2u * fz

(12.7)

Finally, when Eqs. 12.3 and 12.7 are substituted into Eq. 12.2, we get e≈

1 1 . = (12.8) TE vs 1+ 1 + c1 fx LAI + c1 c 2 fz LAI TD u *

Eqs. 12.5 and 12.8 are plotted in Figure 12.5 for three particular examples.

As can be concluded from what we learned in Chapter 5, there is essentially no difference between the calculations with and without impaction for the smallest value of vs calculated (i.e., 0.0028 m s−1). Although the curves shown in Figure 12.5 will certainly be different for different choices of parameter values, the general shapes will be similar.

12.2.4. Vertical Distribution of Spores in a Canopy

FIG. 12.5. Calculations for the fractions of spores escaping from a canopy obtained using Eqs. 12.5 and 12.8 for scenarios 1 and 2, respectively. In scenario 1, deposition is due to sedimentation alone (filled symbols, solid line), while in scenario 2, both sedimentation and inertial impaction are important (open symbols, asterisks, plus symbols, and dashed lines). Calculations used the following parameter values—LAI = 4, fx = 0.5, fz = 0.5—and a characteristic leaf size of 0.05 m. The calculations that included impaction are shown for vs (m s–1) = 0.28 (open circles for c2 = 2.75 and plus symbols for c2 = 0.61); 0.028 (open squares for c2 = 2.75 and asterisks for c2 = 0.61). Similar calculations done for vs = 0.0028 m s–1 (for both values of c2) overlay the solid triangles almost exactly and are not plotted. Impaction efficiency was calculated using Eq. 5.10 in Chapter 5.

The dashed line has a slope of –1 and is shown for comparison with the large vs /u* limit of Eq. 12.5.

(Courtesy D. E. Aylor—© APS)

As can be inferred from Eq. 12.8 (recalling that c1 and c2 are functions of height in the canopy), spore escape will differ considerably depending on the vertical distribution of spores in a canopy. This distribution can change during the course of an epidemic and have a direct effect on the dynamics of disease spread. For example, stem rust of wheat often begins in small, intense foci, for which disease severity can be as high as 10–100% when the ground is visibly covered with Puccinia graminis urediniospores (Roelfs, 1985). The disease often remains limited until it reaches the top of the canopy, after which it spreads rapidly across the field. Likewise, potato late blight can be restricted to small foci early in an epidemic and become widespread only after blight destroys much of the foliage around the foci, thereby exposing Phytophthora infestans sporangia to drier air and higher wind speeds (Van der Plank, 1963).

Depending on the nature of the crop and the disease, as well as the canopy microclimate (e.g., the vertical distribution of moisture in the canopy), the proportions of produced spores will differ at various heights inside the crop canopy (Fig. 12.6). Bottom-heavy spore distributions (such as that shown in Fig. 12.6, part A) are similar to the distribution of Peronospora tabacina sporangia in a tobacco canopy (Rotem and Aylor, 1984) and are often found for other diseases, because foliage is likely to remain wet longer in the lower canopy than in the upper canopy. Top-heavy distributions (Fig. 12.6, part C) are illustrative of wheat rust after it has moved up in the canopy.

CHAPTER 12 Escape of Spores from a Canopy 221

Spore escape will differ considerably among these three spore distribution scenarios (Table 12.1). As expected, a greater number of smaller, lighter spores escape for any comparable situation.

12.2.5. Use of More Complex Models The discussion of escape in this chapter is based on a kinematic model of airflow in a canopy, which accounts

for canopy architecture and plant area density by means of various empirical parameters (see Chapter 9). Largeeddy simulation (LES) can be used to examine the interactions of canopy structure and flow dynamics and leads to a more fundamental description of airflow in a canopy. Bailey et al. (2014) used an LES model to study the escape of weightless particles from plant canopies and found that the canopy residence time for particles increased (i.e., escape decreased) with an increase in

TABLE 12.1. Spore escape fraction, fe , from a canopy calculated using a Lagrangian stochastic (LS) model for three values of u* for each of three vertical distributions of spore release: (a) bottom heavy, (b) middle heavy, and (c) top heavy. For these calculations, it has been assumed that the local spore release rate is directly proportional to the number of spores present in each canopy layer.a

Spore Distribution Bottom heavy

Middle heavy

Top heavy

Panelb

u* (m s–1)

fe (0.01)

fe (0.02)

A

0.15

0.36

0.16

A

0.25

0.48

0.26

A

0.35

0.58

0.34

B

0.15

0.54

0.33

B

0.25

0.65

0.44

B

0.35

0.70

0.49

C

0.15

0.66

0.46

C

0.25

0.72

0.54

C

0.35

0.77

0.57

a The

crop canopy used in the calculations was a 0.7-m-tall wheat canopy in the boot stage of development.

Neutral atmospheric stability (L = –1,000 m) was assumed, and the particle settling speeds were vs = 0.01 and 0.02 m s–1 for columns fe (0.01) and fe (0.02), respectively. b See Figure 12.6.

FIG. 12.6. Three possible vertical distributions of spore release scenarios in a canopy of height hc. Spore release is weighted toward (A) the bottom, (B) the middle, or (C) the top of the canopy. (Courtesy D. E. Aylor—© APS)

222 PART THREE Modeling Wind Transport of Pollen and Spores

canopy density. Since the particles were weightless, the increased residence time in this study was due to the details of the airflow and how it was affected by canopy density. However, an increase in canopy heterogeneity (i.e., areas of both high foliage density and open areas without foliage) increased turbulence and reduced particle residence time. These results are in agreement with the ideas about escape presented in this chapter and with the discussion of heterogeneous canopies in Chapter 9 (section 9.5.1).

A detailed study by Gleicher et al. (2014) found values of the escape fraction in qualitative agreement with those given by the simple “back-of-the-envelope” estimates presented here. In contrast, Gleicher et al. found that the escape fraction decreased more slowly with increasing values of the parameter ratio vs /u* than is given by Eq. 12.5, where deposition is assumed to be due to sedimentation alone. When deposition by impaction is included in our simple model (as it is in Eq. 12.8), the estimates of escape also show a flattening of the curves (Fig. 12.5) for lower values of vs /u*. Importantly, the Gleicher et al. (2014) study also quantified the effect of source height on escape, which has been done here only qualitatively through the parameters c1 and c2.

In summary, the escape fraction is dominated by deposition and vertical components of the airflow in the canopy. Particle deposition depends on the size and density of the particle and on the size, surface characteristics, orientation, and aerodynamic characteristics of the canopy elements (see Chapter 5).

Given these complexities, it is clear that more research is needed to quantify the escape of pollen and spores from plant canopies.

for maize pollen), σw2 is the variance of the vertical fluctuations in the turbulent wind, and τ is the Lagrangian time scale. The variable a1 is an empirical constant that has a value of about 3.75 (Boehm et al., 2008), and vs ~ 0.26 m s−1 for maize pollen (Aylor, 2002b).

Referring back to Chapter 9, we see that the magnitude of τ tends to decrease while σw2 tends to increase as the atmosphere trends from unstable to neutral, where atmosphere stability can be characterized in terms of the Monin–Obukhov length, L. For unstable conditions, L is a negative number, and smaller negative numbers reflect greater instability.

The reciprocal dependence of σw2 and τ on L leads to the interesting dependence of e on L, shown in Figure 12.7. Atmospheric conditions are unstable and free convection tends to dominate particle transport for small negative values of L, while stability is more nearly neutral and forced convection is the dominant mechanism for large values of L. The convective boundary layer (CBL) model escape fractions were calculated as the fraction of particles that remained airborne for a downwind distance of at least

12.3. Escape Fraction Under Highly Convective Conditions Using an electrical circuit analogy similar to that used in the previous section, Boehm et al. (2008) arrived at the following approximate expression for the escape of heavy particles from a maize canopy: e=

1 , a1 vs hc (12.9) 1+ σ w2 τ

where vs is the settling speed of the particle, hc is the source height (i.e., the tassels at the top of the canopy

FIG. 12.7. Escape fraction versus the absolute value of the Monin–Obukhov length, |L|, for the convective boundary layer (CBL) Lagrangian stochastic (LS) model runs using the merged model parameterization (open circles, lightweight line) and the corresponding escape fractions from Eq. 12.9 (heavyweight line). (Reproduced, by permission, from Boehm, M. T., Aylor, D. E., and Shields, E. J. 2008. Maize pollen dispersal under convective conditions. J. Appl. Meteorol. Climatol. 47:291-307. © American Meteorological Society. )

CHAPTER 12 Escape of Spores from a Canopy 223

50 m. The parameters used in the model and in Eq. 12.9 were a1 = 3.75; hc = 2.5 m; model deposition height, zdep = 1.25 m; and vs = 0.26 m s−1; these parameters represent the case of maize pollen released at the top of a hybrid maize canopy (Boehm at al., 2008). The V-shaped curves in Figure 12.7 represent the interplay between σw and τ near the surface, which depend on L in opposite ways; specifically, σw increases with increasing |L|, while τ decreases with increasing |L|.

12.4. Source Emission Rates Determined from Vertical Concentration Gradients In addition to the obvious need to determine bioaerosol emission rates for plant epidemiology and pollen-mediated gene flow problems, emission rates are also needed for a range of environmental problems—for example, to help evaluate the potential indirect effects of bioaerosols on the earth’s climate through their potential effects on cloud formation, cloud duration, and cloud optical depth (Möhler, 2007; Morris et al., 2011; Després et al., 2012). A generally applicable method is needed for determining aerosol emission rates in terms of primary particle production in the vegetation layer.

The methods described in this book have potential usefulness in this regard.

Historically, bioaerosol emission rates have been estimated using three basic methods: 1. methods that involve modeling the production of spores or pollen (often based on crop or host phenology) and using particle production as a surrogate for an emission rate (Hoose et al., 2010) 2. gradient methods, which use vertical gradients of concentration and horizontal wind speed to estimate emission rates (Lindemann et al., 1982; Aylor and Taylor, 1983; Lindemann and Upper, 1985; Lighthart and Shaffer, 1994; Lighthart, 1997; Sesartic and Dallafior, 2011) 3. conservation methods using measurements of integrated horizontal flux (Aylor and Qiu, 1996; Aylor, 1998) (see also Fig. 10.5 in this book) It would be advantageous to have a fourth method, in which bioaerosol production rates, ground cover characteristics, and micrometeorology are all seamlessly integrated into a single model, as was described in Chapter 1 (see Fig. 1.4).

The most prevalent method for estimating emission rates employed in the literature is the gradient method

or some variation thereof. Included in this class of models are two subclasses or submethods: (a) similarity methods based on the Bowen ratio energy balance method (Lighthart and Shaffer, 1994) and (b) methods that combine a single concentration measurement at a fixed reference height with an estimate for the length of time required for vertical mixing to occur in the layer below the reference height. The second method can be expressed as (Sesartic and Dallafior, 2011) Fz = C (zref )

∆ zref ∆ zref = C (zref ) ∆t (∆ zref 2 / 2Kz )

(12.10)

= 2 Kz C (zref ) / ∆ zref , where the mixing time is based on an estimate of turbulent diffusivity (Kz) in the atmospheric surface layer (ASL). Unfortunately, wind conditions are rarely included with concentration measurements reported in the literature, so the value of Kz to be used in each instance must be guessed. Sesartic and Dallafior (2011) used Eq. 12.10 to summarize data from the literature.4 They assumed that all the measurements reported were made at a reference height of 10 m. (Specifically, they used Δzref = 10 m and Kz = 1 m2 s−1, which is approximately the value expected at 10 m above the surface in a neutral atmosphere with u* = 0.25 m s−1.) Given these values, Eq. 12.10 can be reduced to Fz = 0.2 C(zref ).

One-dimensional gradient methods, such as Eq. 12.10, should be restricted to measurements made either over a large uniform source or close to the ground plane in smaller source areas. For the ideal case of a uniform source with a large horizontal extent, we can approximate the particle emission rates by solving a one-dimensional (vertical) diffusion problem. We examine the case in which spore concentrations are measured at various heights near the middle of a large uniform source. In particular, we assume that the source is large enough in lateral extent and that concentrations are measured close enough to the surface so that the variation of the measured concentrations in the horizontal direction is negligible (i.e., we have an ideal one-dimensional situation). In this highly idealized case, the net upward flux of particles, Fz (spores m−2 s−1), can be expressed as5 Fz = −Kz (z)

∂C − vsC ∂z

∂C = −k u * (z − d ) − vsC . ∂z

(12.11)

224 PART THREE Modeling Wind Transport of Pollen and Spores

We illustrate the use of Eq. 12.11 by examining some vertical profiles of P. tabacina sporangia measured above a broadleaf tobacco crop (Aylor and Taylor, 1983).

Values of Fz can be obtained from the concentration profiles in Figure 12.8 by following these three steps: 1. Use regression analysis to fit a power law function of the form C = a(z−d )−b to the data, where a and b are regression coefficients and d is the zeroplane displacement height.

2. Evaluate the derivative of the C(z) function at z = 1.5 m.

3. Multiply this gradient by the turbulent diffusivity at z = 1.5 m for neutral atmospheric conditions.

Recall from Chapter 9 that Kz(z) = ku*(z−d ) for a neutral atmosphere; therefore, at z = 1.5 m for a 1-m-tall crop, Kz (at z = 1.5) = 0.4u* (1.5−0.7) = 0.32u*. Following through on this procedure gives the sporangia emission rates listed in Table 12.2.

12.5. Summary

FIG. 12.8. Concentrations of Peronospora tabacina

sporangia at various heights above an approximately 1-m-tall broadleaf tobacco crop for wind conditions in which u* was equal to (a) 0.19, (b) 0.33, and (c) 0.12 m s–1.

(Data from Aylor and Taylor (1983). Figure courtesy D. E. Aylor—© APS)

The turbulent transport of mass and of momentum are highly intermittent inside and just above a plant canopy (Finnigan, 1979; Shaw et al., 1983). Nearly half the spores that become airborne during periods of brisk wind can escape from the canopy within a distance of one to two canopy heights from a source. The LS random flight spore trajectory simulation model, described in Chapter 11, gives reasonable predictions of the escape of particles from a plant canopy. The model works best for particles released above midcanopy.

The LS model uses a constant value of TL in the upper half of the canopy, given by TL = 0.5hc /σw (hc ) (Reid, 1979; Wilson et al., 1981c; Leclerc et al., 1988). This corresponds to a mixing length at the top of the crop that is about five times larger than would be expected using K-theory and underscores the importance of large-scale coherent motions in the flow field for vertical mixing. Reasonable estimates for the escape fraction can be determined using a “back-of-the-envelope” calculation based on the analogy of an electrical circuit,

TABLE 12.2. Emission rates of Peronospora tabacina sporangia (sporangia m–2 s–1) from

a diseased tobacco crop.a The values in the table were calculated using Eq. 12.11 and evaluated at z = 1.5 m. Column 3 contains the upward diffusive flux, and column 4 contains Fz, which is the net of the upward diffusive flux and downward settling flux under gravity.b

a Data

Profilec

u* (m s–1)

Diffusive Term (m–2 s–1)

Fz(z = 1.5 m) (m–2 s–1)

a

0.19

0.88

0.77

b

0.33

7.75

7.04

c

0.12

3.06

1.91

from Aylor and Taylor (1983). settling speed, vs, was set equal to 0.01 m s–1 in these calculations. c Profiles a, b, and c refer to the concentration profiles so labeled in Figure 12.8. b The

CHAPTER 12 Escape of Spores from a Canopy 225

which partitions the flux of particles into a deposition stream and an escape stream. In many practical cases, these simple estimates might offer a sufficient degree of accuracy for the problem at hand.

E N D N O T E S 1. For the purposes of this discussion, we have neglected crosswind fluctuations represented by, say, σv. Including σv would tend to lengthen the particle paths (and therefore the time spent) in the canopy, which would tend to slightly increase deposition and reduce escape; thus, by ignoring σv, we will have slightly overestimated escape.

2. The time scales are somewhat akin to flow resistances.

The analogy can be expanded to include a separate time scale for deposition on the ground (Ferrandino, 2008). We will not pursue this, because the time scale for escape would need to be modified in a more complicated way, which is best handled by a full solution of the LS model.

3. The airflow in the lower part of the canopy is highly intermittent, which will tend to enhance deposition there.

Particle residence time in the lower canopy will be relatively long during lulls in wind speed, which can allow more deposition by sedimentation.

4. Sesartic and Dallafior (2011) included in their analysis measurements of total pollen and spores collected at a height of 2 m “in the open air in dry weather” several times during the day, as had been reported by Gregory (1952). The putative sources of pollen and spores sampled by Gregory were not in the immediate vicinity of the spore sampler, and the use of a one-dimensional model is questionable.

5. In the literature, Fz has sometimes been estimated from measurements of the differences in C and in U made at two heights above a canopy and assuming that the wind speed above the source can be represented by logarithmic wind profile (e.g., Lindemann and Upper, 1985).