DECAYING SCALARS EMITTED BY A FOREST CANOPY: A NUMERICAL STUDY. 1. Introduction

Transcription

1 DECAYING SCALARS EMITTED BY A FOREST CANOPY: A NUMERICAL STUDY EDWARD G. PATTON and KENNETH J. DAVIS Department of Meteorology, The Pennsylvania State University, University Park, PA, U.S.A. MARY C. BARTH and PETER P. SULLIVAN National Center for Atmospheric Research, Boulder, CO, U.S.A. (Received in final form 12 December 2000) Abstract. A large-eddy simulation is modified to include multiple scalars emitted by a plant canopy. Each of these scalars is subjected to varying rates of chemical loss. Presented is a detailed comparison between conserved species and species undergoing first- and second-order chemical loss. Profiles of mean mixing ratio, mixing-ratio variance and vertical mixing-ratio flux reveal the influence of chemical reactivity. Distribution of the scalar source through the depth of the canopy is shown to locally reduce the reaction rate for second-order species. Transport efficiencies, diffusion coefficients, and mean source heights also exhibit chemical dependencies. Budgets of mixing-ratio variance and flux elucidate the mechanisms through which chemistry modifies each. Instantaneous fields show the existence of intermittently occurring coherent structures that are thought to enhance species segregation. Keywords: Canopy, Chemistry, Forest, Isoprene, Large-eddy simulation, Turbulence. 1. Introduction The atmospheric boundary layer (ABL) is that part of the lower atmosphere in which we live and the interface which moderates biosphere-atmosphere exchange processes. As a result, ABL processes are of importance in both environmental quality and regional to global atmospheric chemistry. Vegetation emits a wide variety of reactive gases into the ABL. Some of the most reactive of these, biogenic volatile organic carbon (BVOC) compounds, have lifetimes that are similar to the time scale of convective mixing in the unstable ABL, or convective boundary layer (CBL). These BVOCs, in particular isoprene and monoterpenes, have lifetimes of minutes to hours and are typically oxidized by the hydroxyl radical (OH). Ozone (O 3 ) is also an important oxidant for some of the monoterpenes. As well as having a large impact on the oxidizing capacity of the lower atmosphere, the BVOCs play an important role in the photochemistry of ozone in the boundary layer (e.g., Chameides et al., 1992). Corresponding address: National Center for Atmospheric Research, P. O. Box 3000, Boulder, Colorado, U.S.A ned@patton.net NCAR is sponsored in part by the National Science Foundation Boundary-Layer Meteorology 100: , Kluwer Academic Publishers. Printed in the Netherlands.

2 92 E. G. PATTON ET AL. The biosphere-atmosphere exchange of non-reactive species is largely driven by biological and physical processes (Gao et al., 1993). To reach the ABL, these constituents diffuse through the stomata, diffuse through the laminar leaf boundary layer, and are then transported by turbulence through the canopy layers and into the overlying surface layer (Baldocchi et al., 1995). While biogenic chemically reactive species, such as isoprene, typically travel similar avenues in order to reach the ABL, they must also combat oxidation in the air during transport. Because the lifetime of BVOCs is often the same order of magnitude as the turnover time of the ABL, the interplay between turbulence and chemistry becomes an important question. Wyngaard (1985) suggested that non-reactive boundarylayer trace gas concentrations and fluctuations can be viably simulated with physically based turbulence diffusion models. However, numerous studies have suggested that for chemical species like BVOCs, which undergo rapid chemical reactions, the assumption that the species are conserved over the mixing length scale can be invalid (e.g., Lenschow, 1982; Fitzjarrald and Lenschow, 1983). For chemical species that have strong emission sources and are non-uniformly mixed, Stockwell (1995) showed that atmospheric chemistry models that ignore the effects of turbulence are limited in their accuracy. This is especially true for species whose source is highly localized, like that of isoprene (C 5 H 8 ), which is predominantly emitted by deciduous plants (Zimmerman et al., 1988). In the field, measuring boundary-layer profiles of BVOCs species is quite difficult. Tethered balloon techniques based on the whole air sampling principle have been regularly used to sample mean BVOC profiles (e.g., Zimmerman et al., 1988; Andronache et al., 1994; Davis et al., 1994; Guenther et al., 1996a, b). Helmig et al. (1998) used a newly developed solid adsorption technique to circumvent some of the drawbacks of the whole air sampling procedure. Within and above a plant canopy, these methods are not viable for estimating fluxes since they measure only mean mixing ratios and must rely on the product of an eddy diffusivity and the vertical mixing-ratio gradient. Within a forest, counter-gradient fluxes occur regularly and thus cannot be described by down-gradient diffusion. Above tall plant canopies these methods must be used with caution as eddy diffusivity methods will underestimate directly measured flux values by factors as great as two to three (Raupach, 1979; Cellier and Brunet, 1992; Baldocchi et al., 1995). K-theory fails to an even greater degree with decaying scalars since the eddy-diffusivities are typically calculated from measurements of non-reactive tracers and are assumed to be the same for reactive species. Baldocchi et al. (1995) and Guenther et al. (1996a) used the relaxed eddyaccumulation method (Businger and Oncley, 1990) in an attempt to address the problems inherent with using gradient methods. Guenther et al. (1996a) found that the flux-gradient method underestimated relaxed eddy-accumulation flux measurements by nearly twenty percent. Recently Guenther and Hills (1998) presented a new chemiluminescent technique that allows for direct measurements of isoprene fluctuations via the eddy-correlation technique. Bowling et al. (1998) found good

3 DECAYING SCALARS EMITTED BY A FOREST CANOPY 93 agreement between isoprene fluxes measured using the relaxed eddy-accumulation technique and those measured directly using Guenther and Hills s (1998) new instrument. Within and above a plant canopy, the Guenther and Hills (1998) method may soon prove to be a useful instrument as it has a fast enough response time to properly capture the small fluctuation scales in and around the plants. However, at this point, we are unaware of detailed profile measurements of rapidly decaying species within and above a plant canopy. Given the difficulties and expense in collecting high space- and time-resolution BVOC measurements within and above a plant canopy, researchers have begun investigating the nature of BVOC atmospheric chemistry with numerical models to gain insight into the effects of turbulent mixing. In the ABL, realistic numerical modelling of turbulent transport and interactions with rapidly reacting chemical constituents began with the work of Schumann (1989). Schumann used the large-eddy simulation (LES) technique attempting to realistically describe the turbulent transport of bottom-up and top-down diffusing species. Schumann (1989) concluded that spatial segregation of species by turbulence played an important role in moderating reaction rates for rapidly reacting species. Sykes et al. (1994) also used the LES technique to evaluate terms appearing in higher-order closure models and showed that higher-order closure models could be extended to include segregation effects. Higher-order closure models proved insightful through numerous investigations (Fitzjarrald and Lenschow, 1983; Thompson and Lenschow, 1984; Lenschow and Delany, 1986; Kramm, 1989; Gao et al., 1991; Hamba, 1993; Vilà-Guerau de Arellano and Duynkerke, 1993; Gao and Wesely, 1994; Vilà-Guerau de Arellano et al., 1995; Galmarini et al., 1997; Verver et al., 1997). For example, Gao et al. (1991) used a one-dimensional second-order closure model to show the influence of rapid chemical reactions on mean concentration gradients, eddy diffusivities, and subsequently on flux profiles. Gao and Wesely (1994) extended their earlier work by incorporating more realistic chemistry, but neglected segregation. Verver et al. (1997) implemented a scalar covariance equation in a higher-order closure model with simplified chemistry and showed the ability of these types of models to depict segregation effects. Beets et al. (1996) implemented a subgrid scalar covariance closure in an LES and showed that for reactions with time scales larger than subgrid turbulent time scales, subgrid scalar covariance could be reasonably ignored. Petersen et al. (1997) presented a mass-flux scheme for the full CBL that includes a parameterization of species segregation. However, since the species of interest are biogenic and we are herein focusing on the interplay between chemistry and turbulence in the lowest reaches of the ABL, of most importance is the work by Gao et al. (1993). Gao et al. (1993) implemented a relatively detailed chemical mechanism within a one-dimensional first-order closure model for turbulence within and above a forest canopy. Their work provided a first look at the perturbation of both the mean isoprene and isoprene flux profiles by chemistry within and just above a

4 94 E. G. PATTON ET AL. plant canopy. While higher-order closure models are inexpensive to operate and perform reasonably well, they rely upon closure approximations that affect all turbulent scales of motion. In addition, one-dimensional models of this sort are unable to include the influence of spatial separation on reaction rates, i.e., the fact that two spatially separated chemical reactants might not mix sufficiently to react. Due to the three-dimensional and time-dependent nature of turbulence within and above a plant canopy, the LES technique has become a useful tool for investigating turbulence transport within and above a forest canopy (e.g., Shaw and Schumann, 1992; Kanda and Hino, 1994; Patton, 1997; Su et al., 1998, 2000). These investigations demonstrated that the important interactions between passive scalars and their transport above and within plant canopies via both mean and turbulent fields could be simulated given adequate resolution. In the current investigation, we use the LES technique to investigate the transport of decaying scalars released from a forest canopy. Comparisons between first-order chemical loss and simple second-order reactivity provide a mechanism for investigating the influence of species segregation. The presence of a forest canopy complicates the turbulence-chemistry interaction due to distributed scalar sources and the influence of elevated canopy-induced drag. Typical CBL photochemical models calculate chemical loss based upon mean concentrations. LES allows for the calculation of all terms in the scalar equations (for the resolved scales) and thus provides a tool for investigating the potential over-prediction of chemical loss by ignoring the influence of segregation. In addition, we use an extremely fine grid resolution, which allows us to capture the within-canopy turbulence responsible for transporting these biogenic species from the emission region to regions aloft. Due to the interception of solar radiation by the plant parts, the atmospheric layers within forest canopies are typically stably stratified and thus biogenic compounds can be retained within the canopy. We investigate the intermittent passage of canopy-scale coherent structures as a potential source of the variability seen in measured profiles of these compounds. 2. The Large-Eddy Simulation For this study we use the LES originally described in Moeng (1984) and Sullivan et al. (1996), which was subsequently modified by Patton (1997). Since the majority of the code has been previously described, we present a limited discussion of how the scalars and the vegetation are specified within the current code NUMERICAL METHOD AND CANOPY DESCRIPTION Following Shaw and Schumann (1992) and Patton (1997), we include a term in the Navier-Stokes equations representing the drag imposed by the forest canopy, which is written as the product of a drag coefficient, a leaf area density, and the

5 DECAYING SCALARS EMITTED BY A FOREST CANOPY 95 Figure 1. Vertical profiles of the prescribed horizontally homogeneous leaf area density normalized by the canopy height (h) solid, and the scalar source from the canopy (S τ, S I ) normalized by χ /t dashed. square of the instantaneous velocity. Hence, the drag imposed on the flow by the vegetation is three-dimensional and time-dependent, and is written as F i = C d auu i, (1) where C d is an isotropic drag coefficient, a is leaf area density (one-sided leaf area per unit volume), and U represents the current wind speed, (u i u i ) 1 2. The canopy specified here is horizontally homogeneous, and varies with height similar to a deciduous forest, with a relatively dense overstorey atop a relatively open trunk space (Figure 1). The canopy also warms the surrounding air with a specified canopy top heat flux that exponentially decreases with depth into the canopy in a manner consistent with Brown and Covey (1966). Integrated through the canopy, the total heating by the canopy (H c )isequalto0.34mks 1 (about 41 W m 2 ). The heat flux from the ground is zero. We use nodes each of which is centered in a m 3 grid box. Thus, we simulate a total domain m 3. The canopy resides in the lowest one-third of the domain and is resolved vertically by ten grid points, therefore the canopy height (h) is 20 m. Recently, Su et al. (2000) suggested that these dimensions and resolutions are reasonable to reproduce adequate flow statistics and structure.

6 96 E. G. PATTON ET AL. A pseudo-spectral differencing technique is used to estimate horizontal derivatives (Fox and Orzag, 1973). For velocity fields, we use a second-order centered-in-space finite-difference scheme to approximate the vertical derivatives; however for all scalars vertical differences are calculated using the monotone scheme described in Koren (1993). Spalart et al. s (1991) third-order Runge Kutta scheme advances all fields in time using a fixed Courant Friedrichs Lewy (CFL) number of 0.63 for a complete three-stage time step. Therefore, the time step for this simulation is variable, but averages about 0.1 s. To drive the simulation, the wind speed averaged across the upwind y,z plane is fixed at 1 m s 1 by an adjustment of the uniform streamwise pressure gradient at every time step. The friction velocity at the canopy top (u ) for the simulationis0.28ms 1. Combining this with the canopy heating, suggests that the non-dimensional stability parameter (ζ = h/l ) is equal to 0.4, or moderately unstable stability conditions. Periodic boundary conditions are imposed in the horizontal directions. No-slip conditions are imposed at the ground surface, where the stress is calculated from a prescribed surface roughness length and the velocity at one-half grid point above the surface. We use this condition at the surface with the understanding that the important sink for momentum is in the upper layers of the canopy, and that what happens at the soil surface is much less important. The upper boundary is set as a frictionless rigid lid, with zero vertical flux of mass, momentum, subgrid-scale (SGS) energy, and scalars. This somewhat harsh boundary condition is imposed necessarily due to computational restraints and eliminates any interaction with scales of motion larger than three times the canopy height. If the mixing-layer analogy of Raupach et al. (1996) is correct, and the formation of canopy scale structures is triggered by PBL-scale motions acting to increase the vertical streamwise velocity shear at the canopy-top, this boundary condition may influence features of the flow such as the frequency of occurrence and intensity of the canopy scale structures. With this in mind, the reader should consider these results to come from a numerically simulated wind-tunnel study with a free-slip upper boundary at three times the canopy height. Ensemble averaged statistics are calculated for the LES data by averaging over all x,y for a given z location, and to ensure stable statistics, by averaging over the last thirty time realizations that are each 250 time steps apart. Here we introduce the notation to denote such an averaging process and to denote deviations from that average. We use the friction velocity at the canopy top (u ) as a determining velocity scale. We also define a characteristic turbulent time scale t as h/u.note that averaging over thirty time realizations is equivalent to averaging over roughly 95t (Figure 2). L is the Obukhov length, defined here as L = u 3 /(kgβh c), wherek is the von Kármán constant taken as 0.4, g is the gravitational acceleration and β is the volumetric expansion coefficient.

7 DECAYING SCALARS EMITTED BY A FOREST CANOPY 97 Figure 2. Time evolution of volume-averaged scalar mixing ratio divided by χ. First-order species are lines only, while second-order species are lines and symbols. TABLE I Prescribed scalar lifetimes and Damköhler numbers for the first-order decay species. Scalar (χ τ ) Lifetime (τ, s) Damköhler number SCALARS Constant Decay Rate Scalars In addition to the form drag imposed on the flow by the vegetation, the trees also emit eight scalars. For the first six scalars, a sink term is included in the scalar conservation equation, which acts as a simple decay mechanism. Each species decays with a fixed lifetime (τ) that ranges from non-decaying down to 50 seconds (see Table I, where τ = 1/k and k is the decay rate for each species). In this formulation, the conservation equation for each resolved scalar quantity (χ) decaying at a rate k is written as: χ τ t + u i χ τ x i = τ iχ x i + S τ kχ τ. (2)

8 98 E. G. PATTON ET AL. Here, we separate the variable (χ τ ) into its resolved plus subgrid parts (χ τ = χ τ + χ τ ). u i is the resolvable-scale velocity in the x i direction (where, i =1,2,or3 for x, y and z). S τ is a source of each scalar distributed through the depth of the canopy similarly to that of heat. τ iχ is the subgrid-scale (SGS) flux of the scalar (= u i χ u i χ). If the time scale for the chemistry is large compared to the time scale for the SGS turbulence (τ SGS ), we can ignore the influence of chemistry on SGS scalar transport. Estimating τ SGS as the subgrid-scale energy divided by the dissipation (e/ɛ), the maximum τ SGS is found near the top of the canopy and is 0.02 s. Since the fastest reaction we simulate has a time scale which is 2500 times slower than τ SGS, we presume that neglecting SGS chemistry is reasonable. Thus, to parameterize τ iχ, we assume local down-gradient diffusion from the resolved scales and estimate a representative eddy-diffusivity, assuming that the SGS turbulence is equally efficient at transporting scalars as it is for heat (Moeng, 1984). In addition, we have assumed molecular transport to be negligible. The Damköhler number is a dimensionless parameter that relates the time scale of the turbulence to the time scale of the chemistry (D a = t /τ) and helps to classify whether the scalars are dominated by their chemical decay or turbulent mixing. For slow chemistry, this ratio is less than one, and for fast chemistry this ratio is greater than one. The Damköhler numbers for the first-order species are presented in Table I. Note that each of the scalars presented here is subjected to identical turbulence and therefore any variation in Damköhler number is solely a result of varying chemical decay rates Simplified Biogenic Hydrocarbons (Isoprene) The seventh and eighth scalars decay via a second-order loss mechanism, where we consider the species to be a simplified version of isoprene (C 5 H 8 ) reacting solely with the hydroxyl radical (OH). This is reasonable since 95% of the isoprene that is oxidized during the day is destroyed by OH (Zimmerman et al., 1988; Baldocchi et al., 1995). Verver (1999) and Petersen (1999) have suggested that interaction between turbulence and realistic isoprene chemistry is substantially modified by the inclusion of reactions with nitrogen oxides. The results presented here represent an idealized second-order chemistry that is similar to that of isoprene, but are not exact. Inclusion of NO x chemistry appears to reduce the isoprene-oh segregation. The mechanism for this effect has not yet been isolated. Turbulent budgets such as those presented later, but with more complete chemical schemes, will resolve this issue. However, due to computational limitations, we approximate the chemistry and focus on the interplay between second-order chemical species and turbulence, both within and above a plant canopy. In the current simulation, the equation for the resolved isoprene mixing ratio (I) is written as: I t + u I i = τ ii + S I k I OH I. (3) x i x i

9 DECAYING SCALARS EMITTED BY A FOREST CANOPY 99 In accordance with Schumann (1989), we assume the SGS contribution from the reaction of OH and I to be negligible. More specifically, we approximate OH I = OH I. (4) Note that we are only neglecting the quantity (OHI OH I) on the scales smaller than the grid resolution. The LES resolves structures down to its grid size, which for this study is (2 m) 3, so we can analyze features that would be far below the grid resolution in a typical CBL photochemical model. Assuming the SGS covariance to be negligible is valid if the time scale for the chemistry τ is large compared to the time scale for the SGS turbulence τ SGS. As discussed earlier, we have chosen to confine ourselves to that regime (τ > τ SGS ) and can therefore also ignore the influence of chemistry on the SGS transport, by solving for τ ii similarly as we do for temperature (Moeng, 1984). Molecular processes have again been assumed negligible. We assume that OH equilibrates rapidly with its surroundings therefore an OH rate equation is not needed, rather an algebraic equilibrium equation is sufficient. Thus, the resolved OH mixing ratio is three-dimensional and time-dependent, but is calculated directly from the current local isoprene mixing ratio and prescribed constants. The assumed algebraic relationship is OH = P OH k CO CO + k I I, (5) where k CO and k I are rate constants for reactions of OH with CO and I respectively. This should be a reasonable first approximation for areas where BVOCs such as isoprene are the dominant OH sink, which is thought to be true of rural, heavily forested regions. Initially, we prescribed the photochemical source of OH (denoted as P OH )at kg OH kg 1 air s 1, which is based on the summer-solstice, noontime O 3 photolysis rate at 34 N, O 3 = 50 ppbv and H 2 O = 13.7 g H2 O kg 1 air. While this initial choice of P OH is appropriate for real-life PBL chemistry, we are not simulating a full PBL and therefore our turbulence time scales are very different. In order to shorten the chemical time scale so that we are able to investigate the turbulence/chemistry interaction, our two isoprene-like scalars interact with hydroxyl radicals that have production rates (P OH ) of four and eight times larger than what would be considered realistic. Therefore P OH = kg OH kg 1 air s 1 for the first second-order scalar and P OH = kg OH kg 1 air s 1 for the second second-order scalar. To calculate the OH mixing ratio, we also include the contribution from the reaction with CO. This constrains the OH mixing ratio to a constant value if isoprene were to disappear completely (I 0). The product k CO CO is fixed at s 1, assuming CO = 200 ppbv.

10 100 E. G. PATTON ET AL. For the second-order species, the Damköhler number is defined as D a = t k I OH V. Note that the Damköhler number now depends on the availability of OH in the total spatial domain (V ). The two species attain a mean Damköhler number over the averaging period of 0.02 and 0.17, respectively. These scalars will be referred to by their Damköhler number from here on Scalar Initial Conditions and Sources The conserved and constant decay rate scalars begin with zero mixing ratio throughout the domain. The initial isoprene mixing ratio is specified as kg I kg 1 air (1 ppbv). At the canopy top (h), we impose specified scalar fluxes from the canopy for each species. The flux of each scalar from the canopy to the air is assumed to decay exponentially with depth into the canopy based upon the downward integrated leaf area density. Since the canopy is horizontally homogeneous, the vertical derivative of these prescribed fluxes is the total source of each scalar in a particular grid volume (S τ = 0.68 [unit] m s 1 ). A normalized profile valid for each scalar source is given in Figure 1. Likewise, we impose a scalar flux for the isoprene scalars at the canopy top that is constant in time and decays with descent into the canopy like the constant decay rate species. Integrated through the depth of the canopy, the total source of the second-order scalars (S I )is kg I m 2 s 1 (2.3 mg I m 2 hr 1 ). The chosen canopy-source simulates the attenuation of incoming solar radiation by the plant elements and the resultant decrease in leaf temperature descending into the canopy, which is consistent with the emission of species such as isoprene, which has been shown to be largely dependent upon leaf temperature. Due to the computational requirements of the LES method, we sacrifice some physics by choosing not to include a complete radiative transfer scheme, leaf energy budget, or photosynthesis model that would provide for more realistic canopy emission schemes. It should be noted that this choice might cause the results presented here to depart somewhat from reality if the plant canopy were able to absorb the species of interest or if the emission of that species were to depend upon atmospheric concentrations. However, we feel that for our interests, this simple scheme (Brown and Covey, 1966) is sufficient to capture the vertical structure of an isoprene emitting canopy. We do not allow for deposition of any scalar to the plants because we focus on daytime conditions when C 5 H 8 deposition is negligible (Guenther et al., 1996a). There are also no emissions from or deposition to the soil surface.

11 DECAYING SCALARS EMITTED BY A FOREST CANOPY Results and Discussion 3.1. MEAN SCALAR MIXING RATIO Time-Evolving Volume Mean Owing to the combination of conservation of mass, our time-independent scalar source, and our frictionless rigid lid at the upper boundary, the volume mean scalar mixing ratio for the conserved scalar must always increase in time. However, due to the presence of the chemistry for the first-order decay scalars, the scalars each reach a chemically stationary state with different volume-integrated scalar amounts in the domain (Figure 2). Therefore, the total amount of scalar available for chemical reaction is not equal for each species. In order to compare vertical profiles of scalar statistics we must look for a scaling parameter that accounts for the unsteady nature of the problem. Integration of the scalar conservation Equation (2) over the entire domain volume (V ) reveals that χ t τ dv = χ τ dv. V V V (6) If we say that the volume integrated scalar is χ τ V and the volume integrated scalar source is S τ V, then an analytical solution can be determined for this S τ dv k τ situation which is of the form: χ τ V t = S τ V k τ + ( χ τ V t S τ V k τ ) e k τ (t t ), (7) where t means: evaluated at some initial time t = t. This equation provides a normalizing parameter that accounts for the time-evolution of the volume mean mixing ratio. For the conserved species, in the limit k τ 0, (7) shows that χ τ V is unbounded and must always grow at a constant rate proportional to S V. For this reason, researchers typically use the emission flux as the representative scaling parameter for conserved species. However, for first-order decay scalars, the chemical loss eventually comes into balance with the source of each scalar (Figure 2). Equation (7) confirms this by revealing that there are two important quantities, (1) the emission flux divided by the decay rate, and (2) the initial mixing ratio. At large time, the dependency on the initial mixing ratio becomes negligible and the determining parameter becomes solely the ratio of the emission flux to the decay rate. However, since not all of our species have reached chemical equilibrium (Figure 2), in order to compare time-averaged mean quantities we will subtract the time-evolving volume mean from the horizontal mean before time-averaging. Each scalar will then be normalized by its characteristic scale χ, which we define equal to the total integrated source strength per unit horizontal area (the vertical integral of S) divided by the velocity scale, S τ z /u, S I z /u.

12 102 E. G. PATTON ET AL Time- and Horizontally-Averaged Mean Profiles Vertical profiles of normalized scalar mixing ratio deviation from the volume mean for each decay rate are shown in Figure 3a. Within the canopy, where the scalar source is distributed, the horizontally- and time-averaged scalar is always larger than the volume mean. The scalar values peak at z/h = 0.65, a height intermediate between the peak of the scalar source and the highest leaf area density (Figure 1). With respect to the volume mean, mean mixing ratios increase with decreasing lifetime within the canopy. An equation for the normalized deviation from the volume mean is ( χ τ χ τ V )/χ t = 0 = 1 w χ. (8) χ dz Therefore, any difference in the profile of ( χ τ χ τ V )/χ must be due to a difference in the horizontal- and time-averaged profile of the scalar flux divergence. Investigation of the flux divergence (Figure 5b) reveals that, within the canopy, the flux divergence for rapidly decaying species exports less scalar from the withincanopy region than it does for the less rapidly decaying or conserved scalars. Similar arguments explain the fact that above the canopy, the more rapidly decaying scalars are smaller in magnitude than the conserved (less source at these levels via advection). Since the modification of flux-divergence profile is height-dependent, the vertical gradient of the scalars is also modified (Figure 3b). The mechanisms responsible for the modification of the flux profile will be discussed in Section Deriving an equation for the chemical contribution to the budget of the secondorder species mean mixing ratio reveals that, compared to the first-order species, two terms arise, which are of the form: I t = k I ( OH I + OH I ) }{{}}{{} A B = k I OH I (1 + I s ) (9) where I s is defined as I OH I OH. I s is a measure of the covariance of the reacting species. Positive I s represents a situation where turbulent fluctuations in reactant mixing ratios are positively correlated, thus the reaction proceeds more quickly than in the well-mixed (I s = 0) case. This would be true if both reactants were emitted at the surface, and updrafts were rich in both reactants while downdrafts were poor in both. Negative I s represents anti-correlation between reactants, thus a slower rate of reaction. This would be true of a reactant carried by a downdraft that is segregated from a reactant emitted at the earth s surface. Typically, photochemical modellers ignore Term B in Equation (9) and estimate chemical loss using solely the contribution from Term A. To investigate the repercussions from ignoring this term, profiles of the intensity of segregation for the two

13 DECAYING SCALARS EMITTED BY A FOREST CANOPY 103 Figure 3. Vertical profiles of horizontal- and time-averaged (a) scalar mixing ratio for each decay rate and the simplified isoprene normalized by χ, and (b) the normalized vertical derivative of the scalar mixing ratio. First-order species are lines only, while second-order species are lines and symbols. Figure 4. (a) Vertical profiles of horizontal- and time-averaged resolved-scale intensity of segregation, I s, for the two second-order species, and (b) the local reaction rate of the two second-order species relative to a bulk reaction rate that does not include segregation effects (R k ) and relative to a bulk reaction rate that does include these effects (R ke ).

14 104 E. G. PATTON ET AL. second-order species are presented in Figure 4a. I s is maximized near the canopy top for both species. In the simple formulation we impose here, the magnitude of I s varies much like the magnitude of the isoprene variance ( I 2 ) but with opposite sign (Davis, 1992); therefore the discussion in Section will help to explain the vertical profile of I s. For a twofold increase in P OH, the intensity of segregation increases by factors of (14, 18, 22) at non-dimensional heights (0.5, 1.0, 2.0). Overall, I s increases through the domain by a factor of about twenty. This overall increase results from an increase in the volume integrated hydroxyl concentration, which in turn acts to increase the reaction rate and the anti-correlation between I and OH (Verver et al., 1997). The smaller increase inside the canopy versus that aloft can be attributed to the fact that it is in this region that the isoprene source is distributed and therefore OH in this region is rapidly diminished. Above the canopy, however, isoprene must be transported to the region, while undergoing rapid chemistry en route, which results in less isoprene aloft, or equivalently, a greater OH concentration aloft. A result of the increased intensity of segregation for the higher Damköhler number second-order species is seen through a much steeper vertical gradient, both positive and negative, through the depth of the domain for this species than for the firstorder species of similar Damköhler number (Figure 3). To quantify the turbulent modification of the reaction rate, we note that Equation (9) shows that the reaction rate has been modified by turbulence. Therefore we can define an effective reaction rate constant (k e = k I (1 + I s )). Following Molemaker and Vilà-Guerau de Arellano (1997) we can also define an effective Damköhler number for second-order species that includes the effect of segregation: D ae = t k ev OH V, (10) where k ev implies that I s for this calculation is defined from deviations from the volume- and time-mean, rather than from the horizontal- and time-mean. For the two second-order species simulated here, D ae attains values of 0.02 and 0.16 respectively. So, ignoring the turbulent chemical reactions in the case of the highest P OH second-order species would overestimate the bulk reaction rate by 9%. For more segregated species, this estimate would increase. Of interest, however, is the influence imposed by the canopy on the reaction rate. To investigate the horizontal- and time-averaged local reaction rates we introduce two quantities R k and R ke, which we define as: R k = k e OH k I OH V, and R ke = k e OH k ev OH V. (11) The numerators of both R k and R ke are calculated from the horizontal- and timeaveraged fields and are derivable directly from Equation (9). R k represents the reaction rate that is felt locally by the second-order species with respect to the

15 DECAYING SCALARS EMITTED BY A FOREST CANOPY 105 bulk reaction rate that is traditionally presumed for large-scale chemical models (i.e., ignoring the covariance between reacting species), while R ke is the locally felt reaction rate with respect to the effective bulk reaction rate which includes the effects of turbulent segregation. With respect to the technique used to estimate second-order reaction rates in traditional large-scale chemical models, the reaction rate felt by our D a =0.17 second-order species has been reduced between 0 z/h 2(R k in Figure 4b). In the upper-most reaches of the canopy, the reduction of the reaction rate is 30%. A large component of the reduction in the reaction rate that is seen in the withincanopy airspace is the result of the distributed source of isoprene through the depth of the canopy, as is confirmed by the 5% reduction in the reaction rate for the D a = 0.02 species in the same region for which the segregation effect is small. However, we speculate that another component of the reduced reaction rate seen up to z/h = 2 results from an increased efficiency of isoprene transport from the within-canopy airspace by canopy-scale organized motions (see Section 3.4) that reduces the available OH. Somewhat fortuitously, above z/h = 2 the reaction rate for the same species would be well estimated by the bulk estimate suggesting that the modification by turbulence has been balanced by an increased availability of OH. Both sets of curves (R k and R ke ) are presented to illuminate the effects of ignoring segregation on a bulk reaction rate, even though the relationship between the two is precisely the ratio of the Damköhler numbers (D ae /D a ). Note that the calculation of both of these domain averaged parameters (the denominators of R k and R ke ) already include the reduced reaction rate within the canopy and therefore the percent decrease in reaction rate is not necessarily exact compared to an equivalent boundary layer that does not include plants. However, from these curves it is certainly apparent that in the vicinity of the vertically distributed source of isoprene the local reaction rate is markedly reduced TURBULENT SCALAR MIXING RATIO STATISTICS Scalar Mixing Ratio Variance Vertical profiles of scalar mixing ratio variance peak at the top of the canopy (z/h = 1) and decay rapidly with descent into the canopy (Figure 5a). Comparisons with Coppin et al. s (1986) measurements (Figure 6) of passive (conserved) scalar variances reveal similar structure through the depth of the canopy. Mechanical production of scalar variance near the canopy top is largely responsible for the variance peak at z/h = 1, since it is in this region that maximum scalar fluxes occur (Figure 5b) coincident with maximum scalar gradients (Figure 3). Due to the rigid boundaries at the top and bottom of the domain, the rapid attenuation of velocity fluctuations within the canopy, and the well-mixed nature of the scalar profiles above the canopy, the variances fall off away from the canopy top. Therefore, in these regions the scalar variances are largely a result of turbulence transport

16 106 E. G. PATTON ET AL. Figure 5. Vertical profiles of horizontal- and time-averaged (a) scalar mixing ratio variances for each of the first- and second-order species normalized by χ 2, and (b) vertical scalar mixing ratio flux for each of the first- and second-order species normalized by u χ. First-order species are lines only, while second-order species are lines and symbols. originating near the canopy top. This process will be discussed in more detail in Section The least first-order chemical modification to the variance profiles occurs near the canopy top (Figure 5a), where the most rapidly decaying first-order species variance is 10% that of the conserved species (compare this with ( 6%, 1% at z/h = (0.5, 2.0), and modification to the standard deviations (not shown) of (24, 31, 12)% at z/h = (0.5, 1.0, 2.0)). Since the variances peak near the canopy top, one would expect the greatest chemical effects on the variance to be coincident. However, it is in the vicinity of the canopy top that mechanical generation of scalar variance is a maximum, and where turbulent transport is most efficient, therefore scalar variance in this region is transported away from the canopy top region faster than the chemistry can operate. Comparing first- and second-order chemistry of similar Damköhler number (between D a = 0.12 and 0.23 for first-order and D a = 0.17 for second-order) reveals that the second-order scalar mixing ratio variance is markedly larger in magnitude throughout the domain than is the first-order species. First-order decay destroys variance, while turbulence interacting with second-order species produces competing mechanisms, one that destroys and one that produces variance. A more detailed analysis of the variance budget for both first- and second-order species will be presented in Section

17 DECAYING SCALARS EMITTED BY A FOREST CANOPY 107 Figure 6. Figure 2 from Coppin et al. (1986) Scalar Mixing Ratio Flux Within the canopy, profiles of vertical mixing ratio flux increase with height and peak at the canopy top (Figure 5b) Aloft, the profiles fall off linearly to zero at the top of the domain as dictated by the upper boundary condition. Typical field measurements of scalar fluxes show a nearly constant flux profile just above the canopy (e.g., Lee and Black, 1993). Gao et al. (1993) implemented a numerical method and boundary conditions that forced their flux profiles to be constant with height above the canopy. Due to our imposed upper boundary condition, our flux profiles are necessarily forced to zero at a height of only 60 m. However, the profiles above the canopy presented in Figure 5b are in reasonable agreement with the flux profiles taken during a wind-tunnel experiment presented by Coppin et al. (1986) (Figure 6). The negative vertical flux gradient above the canopy implies an increase in scalar mixing ratio through the region as scalar emitted through the depth of the canopy is transported aloft. Within the mid-levels of the canopy, it is significant that counter-gradient scalar fluxes are reproduced by the LES. Scalar transport through this region is typically non-local, which is why traditional down-gradient diffusion techniques are not viable for describing transport processes within a plant canopy of this sort. Canopy-scale structures thought to be generated by the canopyinduced elevated velocity shear layer are largely responsible for the transport of heat, momentum and scalars from regions aloft to regions deep within the canopy

18 108 E. G. PATTON ET AL. (Gao et al., 1989). The small negative fluxes witnessed below z/h = 0.3 suggest limited penetration of these structures to this depth. First-order chemistry has the largest effect on the profiles of the fluxes in the bottom half of the canopy (Figure 5b). Of importance is the height and magnitude of the switch from positive to negative fluxes. This shift largely occurs because, with increasing Damköhler number, the covariance between the scalar and temperature is destroyed in the lower reaches of the canopy, which is one of the key producers of scalar flux in this region (see Figure 12). Therefore any flux deep within the canopy is imported from the upper layers of the plants by turbulence. The smallest modification of the fluxes by the chemistry occurs at the canopy top (z/h = 1). At this height, the most rapidly decaying species is 36% that of the conserved species compared to 15% at z/h = 2. While this difference is somewhat clouded by the influence of the upper boundary condition, nevertheless, it is apparent that, as a whole, first-order chemistry is less effective at reducing fluxes than variances. This same chemical effect is also reflected in comparing fluxes and standard deviations (scalar perturbations to the same order) but the difference is not as substantial. Turbulence interacting with second-order species also produces flux-preserving contributions as exhibited by the fact that vertical flux of the second-order species D a = 0.17 is decidedly larger in magnitude than the first-order species of similar Damköhler number. See Section for further detail as to the mechanisms responsible for the features in the flux profiles presented here Scalar - Velocity Correlation Coefficient Profiles of the wχ correlation coefficient, R wχ = w χ + τ wχ /(σ w σ χ ), show the very different nature of the interaction between the velocity and scalar fields above and below the canopy (Figure 7). At the canopy top (and just within), the conserved scalar shows the correlation coefficient to peak at a value of nearly 0.8, while the first-order decay species all increase with increasing Damköhler number to a value of 0.94 for the D a = 1.38 species. This peak in the correlation coefficient has yet to be fully explained and is not seen over smooth surfaces, however, it is reasonable to say that it suggests that the turbulence in the canopy top region is especially efficient at transporting scalars (relative to the amount of turbulence present) and that there is a distinct difference in the structure of the turbulence within the canopy layers than is seen over smoother surfaces (Shaw et al., 1988; Brunet et al., 1994). This statement has recently been supported by Raupach et al. (1996) where they showed canopy turbulence to be more like a mixing layer than a surface layer. The value for the conserved scalar of 0.8 at the canopy-top is large compared with that measured in the field ( 0.4, e.g., Coppin, 1985) and likely results from the rigid upper boundary condition preventing the interaction with large-scale motions. This inconsistency, while dramatic, does not preclude an investigation into the influence of simple chemistry on correlation coefficients.

19 DECAYING SCALARS EMITTED BY A FOREST CANOPY 109 Figure 7. Vertical profiles of horizontal- and time-averaged (a) vertical velocity-scalar correlation coefficient (R wχ ) and (b) scalar eddy diffusivity (K χτ ) normalized by u h. First-order species are lines only, while second-order species are lines and symbols. The line labeled K is the expected inertial sublayer eddy diffusivity. The open squares are a corrected eddy diffusivity proposed by Hamba (1993), calculated from the conserved diffusivity and corrected for τ = 50 s first-order chemistry (= K χ /(1 + t /τ)). Within the canopy, the correlation coefficients for all scalars diminish rapidly with decreasing height and become negatively correlated below z/h = 0.3 for the conserved species and z/h = 0.5 for the most rapidly decaying species. The height of the change in sign and change in sign itself are consistent with the vertical scalar flux results (Figure 5b), since the flux is negative below these heights. The strong anti-correlation exhibited by the more rapidly decaying scalars in the subcanopy air space suggests that vertical scalar transport in the lower canopy region is inefficient. Coppin et al. (1986) also reported R wχ diminishing and becoming negative with descent into their wind-tunnel canopy. Above the canopy, due to the upper boundary condition, the correlation coefficients diminish with increasing z/h, however they are not identically zero at the upper boundary since we present the resolved plus subgrid contributions. Above a rough surface, observations have shown R wχ to be 0.35 to 0.5 for conserved scalars (Coppin, 1985); laboratory measurements (Coppin et al., 1986) show the above canopy R wχ to remain constant at between 0.5 and 0.6. In the region between 1.2 z/h 2, Figure 5 shows our numerical results to fall well within those measured by Coppin et al. (1986), confirming that our flow is similar to a laboratory flow. For the case presented here, chemistry tends to increase R wχ above the canopy. This

20 110 E. G. PATTON ET AL. result implies that compared with conserved scalars, with increasing Damköhler number, scalar fluctuations diminish with increasing height faster than do the fluxes which is confirmed by the discussion presented in Sections and Scalar Eddy Diffusivity It has been well established that first-order closure models that make use of eddy diffusivities to describe vertical transport are decidedly inaccurate when applied within a plant canopy where transport is largely counter-gradient. Nonetheless, due to complexities that arise in developing second-order models that contain chemistry, numerical modellers typically assume that an eddy diffusivity formulation is valid within the canopy and that the eddy diffusivity for chemically reactive scalars is equal to that for heat (e.g., Gao et al., 1993). Holtslag and Moeng (1991) derived an eddy diffusivity for scalars that retained turbulent transport terms in the scalar-flux equation in an attempt to include the effects of non-local transport. Since within canopy turbulence is dominated by counter-gradient transport, we follow the suggestions of Holtslag and Moeng (1991) and calculate the vertical profile of the scalar eddy diffusivity as: where K χτ = w χ τ χ τ z γ χτ (12) γ χτ = 2u2 χ h w 2. (13) Note that γ χτ is independent of reaction rate, but does depend on the scalar source strength. Although the assumptions presumed by Holtslag and Moeng (1991) when deriving γ χτ are not necessarily valid for canopy flows (namely a constant partitioning between turbulent transport and pressure destruction in the scalar flux budget, see Section 3.3.2), counter-gradient transport is a recognized feature of withincanopy turbulence (Denmead and Bradley, 1985). Therefore the inclusion of a term like γ χτ, which attempts to represent the effects of turbulent transport, is warranted. Within the upper reaches of the canopy, the eddy diffusivity for the conserved scalar is enhanced with respect to that expected in an inertial sublayer with a turbulent Prandtl number of unity (K ). Coppin et al. (1986) and others have shown that the scalar diffusivity should be larger (between 1.5 and 2 times) than K for the region between z/h =1andz/h = 2 as would be expected in the presence of a roughness sublayer. Our results however, do not have this feature. Although it is possible that the roughness sublayer in this simulation has been squeezed by the upper boundary limiting the eddy size and their diffusive abilities, it is also in K /(u h) = k/h(z d χ ),wherek is the von Kármán constant, u is the friction velocity, h is canopy height, and d χ is the displacement height (e.g., Raupach, 1979).

21 DECAYING SCALARS EMITTED BY A FOREST CANOPY 111 this region that one of the key assumptions in deriving γ χτ is violated. The ratio of turbulence transport to pressure destruction in the flux budget varies dramatically between these levels (Section 3.3.2). Therefore, it is possible that the reduction in the presented K χ in this region results from too small a γ χ. Nonetheless, Figure 7b provides the means to investigate the influence of chemistry on the magnitude and vertical distribution of K χτ compared with a conserved scalar. In general, increasing Damköhler number tends to reduce the ability of the turbulence to diffuse the scalar for a given gradient. Since the modification of the vertical mean scalar gradient is limited (Figure 3b), this effect must largely result from the chemical reduction of the scalar flux (Figure 5b). Below z/h =0.3,K χτ is near zero within the canopy for all decay rates. At z/h = 0.7, the diffusivity for the D a = 1.38 species is 28% smaller than K χ.however,atz/h =1.3, the diffusivity for the same species is 48% smaller than K χ, suggesting that eddy diffusivities are less modified by first-order chemistry in the vicinity of the source than at regions away from the source. This is likely a consequence of the fact that diffusivities are properties of both the flow and the source distribution. A comparison with a proposed first-order chemical modification to the conserved eddy diffusivity (Hamba, 1993) suggests that the proposed diffusivity does not capture the reduction in height of the maximum diffusivity and overestimates the diffusivity above z/h = Mean Source/Sink Height Following Brunet et al. (1994) we calculate the mean source/sink height for a particular quantity φ, by: d φ = h 0 z( w φ / z)dz h 0 ( w φ / z)dz, (14) where φ can be momentum, heat, or any particular scalar. Table II reveals that although the differences are not substantial, the source heights calculated from the current simulation all differ from that of momentum. The source height for heat is about 8% smaller than the height of mean momentum absorption. d φ for the conserved scalar is equal to that for heat, which suggests that the source distribution is more important in determining the mean source height than is the issue of whether the constituent is dynamically active versus passive. Of interest, however, is that the decaying scalars reveal a trend of increasing mean source height with increasing decay rate. This upward shift most likely results from a chemically-reduced ability of the turbulence to transport the scalar away from the peak in the canopy scalar source, increasing the importance of the source distribution in determining d φ. Because second-order chemistry has modified the vertical scalar flux profile less than that of a first-order decay species of similar Damköhler number, the mean source height for the second-order species (D a = 0.17) is lower than that for a less rapidly decaying first-order species (D a = 0.12).

22 112 E. G. PATTON ET AL. TABLE II Normalized mean source/sink heights (d φ /h) calculated via Equation (14). φ d φ /h Momentum Heat First-Order Decay Scalars D a = D a = D a = D a = D a = D a = Second-Order Decay Scalars D a = D a = It should also be noted that the mean source/sink height for momentum presented in Table II is lower than the rule-of-thumb estimate suggested by Kaimal and Finnigan (1994) of d/h = However, it is directly in line with measurements over a forest of similar leaf area index (Raupach et al., 1991) BUDGETS FOR SCALAR MIXING RATIO VARIANCE AND FLUX Scalar Variance Budgets Figure 5b reveals distinct differences in the variance profiles of scalars with different lifetimes. To explain the mechanisms responsible for these profiles, we turn to the variance budget to investigate the terms in the budget that are most affected by the existence of chemical loss. Scalar variance budgets within and above a plant canopy have not been discussed in the literature since the work of Coppin et al. (1986). In their work, they presented the equations in their full form taking into account the proper averaging procedures previously reported in Raupach and Shaw (1982). Numerous terms arise in their formulation due to small averaging volumes and short averaging times. For the work presented here, we ignore these dispersive flux terms on the basis that this work assumes horizontally homogeneous sources and sinks, so no time-averaged horizontal fluctuations can appear at plant scales. For the conserved and first-order decay species under quasi-steady, horizontally homogeneous conditions, the resolved scalar mixing ratio variance budget valid for the LES is of the form: χ 2 t = 0 (15)

23 DECAYING SCALARS EMITTED BY A FOREST CANOPY 113 Figure 8. Vertical profiles of all terms in the horizontal- and time-averaged mixing ratio variance budget (Equations (15) and (16)) normalized by χ 2 /t. Panel (a) is the conserved case (D a = 0.00), and panel (b) is the τ = 50s case (D a = 1.38). = 2 w χ χ z } {{ } P g w χ 2 z } {{ } T t τ xj 2 χ x j } {{ } D sgs 2k χ 2 }{{} C 1 where summation is implied over repeated indices and the subscripts are interchanged (1 x; 3 z). Using notation similar to that of Coppin et al. (1986), we define four terms that contribute to the production, transport and destruction of horizontally-averaged scalar variance: (1) gradient production (P g ), (2) turbulent transport (T t ), (3) destruction by SGS processes (D sgs ), interpreted in the LES context as dissipation of resolved-scale variance, and (4) chemical destruction (C 1 ). The most important differences to note between the budgets presented here and those in Coppin et al. (1986) are: (1) due to their inability to measure scalar dissipation, they were forced to calculate this term as a residual, while in the current simulations we are able to calculate the effects of SGS destruction the resolvedscale variance directly, and (2) the current simulations also include the destruction of scalar variance by the action of chemistry. For the conserved species, C 1 is identically zero and therefore does not contribute to the budget. Comparing terms P g and T t presented in Figure 8a to those measured by Coppin et al. (1986) (Figure 9a) shows that the current numerical results are in reasonable agreement. Gradient production generates scalar variance at the top of the canopy, while turbulence transport exports variance from the canopy top region to regions deep within the canopy and well above. Coppin et al. s (1986) measurements of P g suggest that P g does not contribute to the budget below the height of their planar source. In our case, however, P g acts as a sink of scalar variance in the sub-canopy

24 114 E. G. PATTON ET AL. Figure 9. Figure 8 from Coppin et al. (1986). region. This difference largely results from our vertically distributed scalar source. In addition, for our case T t transports 60% of the variance produced by P g to near the canopy top, where Coppin et al. s (1986) results show the ratio to be more like 50%. Coppin et al. (1986) suggest, however, that their measurements of T t are underestimated by their sensors by at least 10%, so both sets of results are in agreement. Since their dissipation term is calculated as a residual, the differences seen in T t can easily explain the fact that our D sgs is about two-thirds the magnitude of T t while they show dissipation to be three-halves T t at the canopy top. One other difference in D sgs is the shape of the profile within the canopy. Our D sgs profile smoothly decays to zero at the bottom of the domain, where their profile shows a minimum coincident with the height of their constant planar source. It is important to note that the dominant sink of variance for scalars existing in boundary layers not interacting with a vertically distributed drag force is typically the dissipation term, not the turbulent transport term (Moeng, 1984). Chemistry has little effect on the general profiles of each term as to the height at which each of the physical terms either contributes or destroys variance (Figures 8b and 10). However, chemistry does affect the magnitude and percent contribution of each term. Gradient production at the canopy top for the D a = 1.38 case is 26% that of the conserved case, and within the canopy, P g is now only weakly destructive. Turbulent transport now removes only 32% of the variance produced near the canopy top, and is no longer the dominant sink for variance in this region. However, T t continues to import variance to within- and above-canopy regions even if it is to a weaker degree and over a smaller height range. At the canopy top T t remains 7% larger than D sgs. Although gradient production at the canopy top is

25 DECAYING SCALARS EMITTED BY A FOREST CANOPY 115 Figure 10. Vertical profiles of the individual terms for all first-order decay species in the horizontaland time-averaged mixing ratio variance budget (Equation (15)) normalized by χ 2 /t. The terms are labeled following the definitions in Equation (15). reduced 74% by the 50-s time-scale chemistry, the SGS destruction term is reduced nearly 87%. The balance is now removed by chemistry (C 1 ), which is the dominant sink destroying 48% of the variance at the canopy top. First-order chemistry appears to reduce all three non-chemical terms in the scalar variance budget (Figure 10). Figure 3b shows that in general, the gradient of the first-order decay scalars is only weakly affected by first-order chemistry; therefore the chemical modification of the gradient production term in the vari-

26 116 E. G. PATTON ET AL. ance budget is largely due to a modification of the flux profile (Figure 5b). Since first-order chemistry is so effective at modifying the main variance producing mechanism, the turbulent transport term has less variance to redistribute and therefore must follow suit. The chemical term (C 1 ) is the product of the reaction rate (k) and the local scalar variance (Equation (15)). Even though P g decreases as the Damköhler number increases from smaller than one to one, T t is able to transport scalar variance away from the source region faster than the chemistry can act. Therefore, locally, k increases faster than C 2 decreases resulting in an increase in the magnitude of C 1 through this Damköhler range. Note that the elevated source of turbulence induced by presence of the plants provides the means for T t to transport scalar variance to the regions both above and within the canopy. Therefore, the magnitude of C 1 is proportionally larger in these regions compared to that at the canopy top than for species with a Damköhler number greater than one. For flows with a Damköhler number larger than one, P g continues to decrease with increasing reaction rate. However, for these species T t is unable to act as efficiently to redistribute variance before C 1 destroys it. C 1 can only destroy what P g creates and therefore the magnitude of C 1 now diminishes with increasing Damköhler number. The variance budget valid for second-order species is nearly identical to Equation (15), with the exception that the chemical destruction term contains multiple components: I 2 t =... 2k I ( OH I 2 + OH I I }{{}}{{} A B + OH I 2 }{{} } {{ } C 2 C ). (16) We call the sum of all three terms contributing chemically to the second-order species C 2. Although as a whole C 2 does not play a major role in the variance budget for our two second-order species (Figures 11a, 11b), a first-order species of similar Damköhler number reveals a notably smaller variance through the depth of the domain (Figure 5a). Figures 11c and 11d reveal an interesting interplay between the three terms that contribute to C 2. Term A is always a sink since the OH mixing ratio and the isoprene variance are both positive definite. In the formulation presented here, OH and I are always negatively correlated, thus Term B is always a variance source. Term C, while small in magnitude, acts to produce variance above the canopy, and to destroy it within. The presence of Terms B and C is in contrast with first-order decay species where chemistry acts solely to destroy mixing ratio variance. Thus, for similar Damköhler number, a second-order species exhibits larger variance than a first-order species. With increasing Damköhler number a shift occurs in the balance between Terms A and B. At the canopy top, a twofold increase in P OH results in a factor of nine

27 DECAYING SCALARS EMITTED BY A FOREST CANOPY 117 Figure 11. Vertical profiles of the chemical terms C 2 for both second-order decay species in the horizontal- and time-averaged mixing ratio variance budget (Equation (16)) normalized by χ 2 /t. The terms are labeled following the definitions in Equation (16). increase in Term A and only a factor of six increase in Term B (Figures 11c, 11d). This difference results in a net chemical contribution from C 2 that is opposite in sign, shifting from a net source of isoprene variance to a net sink. Inspection of Terms A and B in Equation (16) suggests that although a twofold increase in P OH produces a 5% decrease in isoprene variance (Figure 5a) and a factor of thirty-six increase in OH I (not shown), then the proportional change between Terms A and B must result because of a shift in the mean mixing ratio of the respective scalars. As the Damköhler number increases, the scalars move from an isoprene dominated regime to a hydroxyl dominated regime (confirmed by volumeintegrated time traces of the second-order species in Figure 2). Although the triple

28 118 E. G. PATTON ET AL. moment term (Term C) increases by a factor of (45, 26) at z/h = (0.75, 1.35), the overall contribution from this term still makes up a small percentage of C 2,and in fact is near zero at z/h =1.0whereC 2 is maximized. We conclude, therefore, that although turbulent interactions are important in mitigating chemical variance loss for second-order species compared to first-order species, the mean mixing ratio regime dominates the magnitude ratio of the contributing terms and therefore defines the sign of the overall chemical contribution to the second-order mixing ratio variance budget Scalar Flux Budgets In an attempt to explain the mechanisms responsible for the vertical scalar flux profiles (Figure 5b), we examine the vertical scalar flux budget. Scalar flux budgets within and above a plant canopy have not been discussed in the literature since the work of Coppin et al. (1986) and Raupach (1987). Again, we ignore the dispersive flux terms on the basis that our flow assumes horizontally homogeneous sources and sinks and thus no time-averaged horizontal fluctuations can appear at plant scales. Deriving the resolved scalar flux budget from the LES equations for the conserved and first-order decay species under quasi-steady, horizontally homogeneous conditions leads to w χ t = 0 = w 2 χ }{{ z } χ + g θ χ θ } {{} P b P g ( p ρ e ) z } {{ } D p w 2 χ z }{{} T t τ xj ( w x j τ zj + χ x j ) } {{ } D sgs + F z χ k w χ, (17) }{{}}{{} D cd C 1 where summation is implied over repeated subscripts and the subscripts are interchanged (1 x; 3 z). Using notation similar to that of Coppin et al. (1986), we define seven terms that contribute to the production, transport and destruction of horizontally-averaged vertical scalar flux: (1) gradient production (P g ), (2) buoyant production (P b ), (3) turbulent transport (T t ), (4) pressure destruction (D p ), (5) destruction by SGS processes (D sgs ), destruction by interaction with canopy drag (D cd ), and chemical destruction (C 1 ). Important differences to note between the budgets presented here and those in Coppin et al. (1986) are: (1) the presence of

29 DECAYING SCALARS EMITTED BY A FOREST CANOPY 119 Figure 12. Vertical profiles of the chemical terms in the horizontal- and time-averaged vertical scalar flux budget (17) normalized by (u χ )/t for (a) the conserved species (D a = 0.00) and (b) the 50 s first-order decay species (D a = 1.38). Note that the scales on the horizontal axis are different between the two figures. buoyancy forces (the results presented in Coppin et al. (1986) were for a neutrally stratified flow), (2) the ability to evaluate the pressure contributions, (3) the canopy drag term acting solely to convert resolved scale flux to SGS flux (i.e., acting as a scalar flux sink, rather than wake production), and (4) the destruction of vertical scalar flux by the action of chemistry. For the conserved species, C 1 is identically zero and therefore does not contribute to the budget. Comparison of Figure 12a to the measurements presented in Coppin et al. (1986) (Figure 9b), reveals that within and just above the canopy, P g and T t are in full agreement. Above z/h 1.7, however, the LES profiles cross the zero line from what would be expected from the measurements. This difference is undoubtedly a result of our upper boundary condition, and thus results above this height should be taken in that light. However, below this height, the scalar budgets are dominated by the presence of the plant canopy and are therefore considered reliable. Gradient production is the major source of flux near the canopy top, while in the sub-canopy layers, the flux is largely imported by turbulence. In regions where the mean scalar gradient is small (0.6 z/h 0.8 and z/h 1.3) buoyant forces act as the main contributing source of scalar flux as warm air brings high mixing ratio air from the scalar source region (the mid-reaches of the canopy) upwards. While We note that the wake production term that Coppin et al. (1986) refers to would appear as an equal magnitude and oppositely signed term (to D cd ) in the SGS scalar flux equation were we to formulate one. In this case, however, since the individual plant parts are actually SGS, interaction with the plant parts acts solely to convert resolved scale flux to subgrid scale.

30 120 E. G. PATTON ET AL. turbulence transports the flux away from the canopy top, pressure dominates the destruction of the flux at most levels. SGS destruction (transport) plays its largest role near the canopy top, where the fluctuations of both vertical velocity and scalar mixing ratio are maximized. Canopy drag imposes its greatest influence at a height consistent with the mean peak of leaf area density and scalar variability. The influence of first-order chemistry on the scalar flux budget can be seen by comparing Figures 12a and 12b. While the general shape of the profiles remains consistent between the conserved and decaying cases, the proportional contribution from the terms is decidedly modified. Most noticeable is the modification of the buoyant term, P b. In the conserved case, it contributes nearly 37% of the flux production at the canopy top, while for the decaying case, its contribution has fallen to 20% at the same height. Inspection of Equation (17) reveals that since the heating is identical for both cases, a reduction in this term is solely due to a reduction in the covariance between the scalar and temperature perturbations due to chemistry. Fitzjarrald and Lenschow (1983) suggested that P b could be affected by chemistry. They said, Because the covariance equation for potential temperature and a chemically reactive scalar does not contain a pressure correlation term, the only way that covariance can be lost is through molecular destruction. In our case, this statement is equivalent to saying that SGS processes are responsible for the reduction of P b. Gradient production continues to produce flux near the canopy top, but at a slightly reduced rate compared with the conserved case due to the slightly modified mean scalar mixing ratio gradient. Turbulent transport is now the dominant source of scalar flux in the sub-canopy region (z/h 0.7). The pressure destruction term is no longer as efficient at destroying flux as compared to the conserved case, suggesting a modification of the covariance between the scalar and the vertical pressure gradient. The destruction of the flux in this sub-canopy region is now dominated by P g. At the canopy top, pressure destruction (D p ) destroys 42% of the flux in the conserved case, while in the decaying case, D p only destroys 27% of the flux. First-order chemistry with a Damköhler number of 2.77 has only shifted the contribution T t at the canopy top from 43% to 37%, and the contribution from D sgs from 14% to 13%, therefore D p is most affected by decay chemistry. Chemistry (C 1 ) is a major player destroying 23% of the flux at the canopy top and nearly 50% above the canopy. Overall, only minimal changes are witnessed in the turbulent velocity interactions (P g and T t ), as would be expected since chemistry has no influence on the dynamics. (The differences that are present are due solely to chemical modification to the scalar component of these two terms.) Therefore, the reduction in the scalar flux for first-order decaying species seen in Figure 5b appears to be largely a result of chemical modification of the P b production term in the flux budget. Verver et al. (1997) attempt to quantify the influence of chemistry on a dynamically passive scalar by separating all the terms in the flux budget into those that contain a reaction

31 DECAYING SCALARS EMITTED BY A FOREST CANOPY 121 Figure 13. (a) Vertical profiles of the terms in the horizontal- and time-averaged vertical scalar flux budget valid for our two second order species, (a) D a = 0.02, and (b) D a = Note that with respect to Equation (17) Term C 1 has been replaced with Term C 2 (Equation (18)). (c) and (d) are the partitioning of the three components making up Term C 2 (Equation (18)) for each respective second-order species. All terms are normalized by (u χ )/t. Note that the scales on the horizontal axis are different between the two figures. rate coefficient (k) and those that do not. The results here suggest that this technique is inappropriate due to the drastic chemical modification of terms such as P b. Second-order species exhibit a larger flux than do first-order species of similar Damköhler number (Figure 5b). To explain this feature, we form an equation for the horizontal- and time-averaged vertical isoprene flux budget valid for our species. All terms appearing in the second-order species flux budget are identical to those in the first-order species budget (Equation (17)) with the exception of the

32 122 E. G. PATTON ET AL. chemistry term. The interaction between the turbulence, isoprene and the hydroxyl radical generates multiple chemistry terms that contribute to the simple isoprene flux budget. These terms are of the form: w I t =... k I ( OH w I + I w }{{} OH + w }{{} OH I ). }{{} (18) } A B {{ C } C 2 Inspection of Term A in (18) suggests that it should be necessarily a sink of scalar flux in the regions above z/h =0.3since OH is positive definite and above this region the flux of isoprene is positive. Figures 13c and 13d reveal that this is indeed the case. Since our previous discussions have shown that OH is negatively correlated with I, it follows that Term B should act opposingly to produce isoprene flux. The triple product (Term C), although small in magnitude, acts to destroy isoprene flux. All three terms are maximized near the canopy top and rapidly decay with descent into the canopy. This should be expected since it is in the upper reaches of the canopy that the scalar source is maximized (Figure 1) in concert with the strongest production of turbulent kinetic energy (e.g., Raupach and Thom, 1981). Comparing the chemical terms in the second-order species flux budget reveals that, with increasing Damköhler number, the balance shifts from C 2 producing flux to C 2 destroying flux (Figures 13c and 13d). At the canopy top, a twofold increase in P OH results in a factor of (10, 6, 37) increase in Terms (A, B, C). Although such a large increase in Term C occurs, the overall contribution from this term is still relatively small compared to the other two terms. Since Figure 5b shows the quantity w I decreases with increasing Damköhler number (which implies that w OH increases with the same Damköhler number change), the shift in C 2 from flux production to flux destruction results largely from a switch in the mean scalar mixing ratio regime from isoprene dominated to hydroxyl dominated. So, the overall influence of the chemistry terms in both the variance and flux budgets is largely moderated by the mean mixing ratio rather than by turbulent interactions. This statement should be modified somewhat since the triple moment term has increased its importance and acts in concert with Term A to destroy mixing ratio flux. Note that C 2 is maximized not at the canopy top, but rather at z/h =1.3,since for the case presented here it is in regions away from the isoprene source that have the highest hydroxyl mixing ratio and regions near the top of the canopy that have the highest isoprene flux INSTANTANEOUS FIELDS Field measurements of the instantaneous spatial distribution of dynamic variables and chemical species are extremely difficult due to the vast array of instruments required. These instruments are not only expensive, but in the case of dynamic

33 DECAYING SCALARS EMITTED BY A FOREST CANOPY 123 Figure 14. An instantaneous x, z slice of the deviation from the instantaneous horizontal mean of the D a = 0.17 second-order species normalized by χ overlaid with the coincident velocity vectors normalized by u. Contours with solid lines are positive, dotted lines are negative. In non-dimensional units the contours range from 1.5 to 1.5 with an interval of The maximum velocity vector corresponds to a non-dimensional magnitude of 4.2. The dashed-line is the top of the forest canopy. quantities, their presence would also alter the properties of the flow being measured. Researchers such as Gao et al. (1989) have resorted to time-height cross sections derived from tower measurements to gain insight into the spatial structure of turbulence within and above a forest canopy. The three-dimensional and timedependent nature of the LES output allows comparisons between simulated spatial fields and measured time-height cross sections. To illuminate the issue of segregation previously discussed in Section 3.1.2, Figure 14 presents an instantaneous x,z slice of the simplified isoprene deviation from the horizontal mean. The coincident instantaneous fluctuating velocity vectors are overlaid. Mean flow is from left to right. Rising motion brings isoprene-rich air from within the canopy layers to regions above. Subsequently, sinking motion brings isoprene depleted air to depths within the canopy. Recall that OH and I are necessarily negatively correlated in this study, and thus regions of low isoprene imply high OH. Note the relatively sharp gradients in the isoprene field at the upwind edge of the rising motion, and the downwind edge of the sinking motion (especially in the region 1.2 x/h 3.6). Previously witnessed in experimental time-height cross-sections, features such as this have been termed scalar microfronts and are characteristic of both convective plumes in shear flow and of scalar ramp patterns in the high shear region near the top of a forest (Gao et al., 1989). These microfronts are considered evidence of coherent motions within the flow and have been shown to be responsible for between 60