Dissipation Intermittency Increases Long-Distance Dispersal of Heavy Particles in the Canopy Sublayer

Abstract

The dispersion of heavy particles such as seeds within canopies is evaluated using Lagrangian stochastic trajectory models, laboratory, and field experiments. Inclusion of turbulent kinetic energy dissipation rate intermittency is shown to increase long-distance dispersal (LDD) by contributing to the intermittent ejection of particles to regions of high mean velocity outside the canopy volume. Model evaluation against controlled flume experiments, featuring a dense rod canopy, detailed flow measurements, and imaged trajectories of spherical particles, demonstrates that superimposing a terminal velocity on the fluid velocity is insufficient to determine the particle dispersal kernel. Modifying the trajectory model by adding dissipation intermittency is found to be significant for dispersal predictions along with the addition of inertial and crossing trajectories’ effects. Comparison with manual seed-release experiments in a forest using wind-dispersed seeds shows that the model captures most of the measured kernels when accepted uncertainties in plant area index and friction velocity are considered. Unlike the flume experiments, the model modifications for several wind-dispersed seeds have minor effects on short-distance dispersal. A large increase was predicted in LDD when including dissipation intermittency for the forest experiment. The main results suggest that fitting or calibrating models to the ‘main body’ of measured kernels may not offer extrapolating foresight to LDD predictions. As inertial effects were found mostly negligible in the field conditions here, the extended trajectory model requires specifying only the seed’s terminal velocity and a constant variance of the normalized dissipation rate. Therefore, the proposed modifications can be readily applied to classical trajectory models so as to improve LDD predictions.

Appendices

Appendix 1: The Estimation of the Constant $C_0$ for the LS Model

The variations in the value of $C_0$ associated with the Lagrangian structure function are large, unlike the Kolmogorov constant of the Eulerian structure function for inertial subrange scales. Estimates of $C_0$ range between 2 to 7, with $C_0$ used as an adjustment parameter in several instances (Poggi et al. 2008). Here, $C_0$ was not used as a fitting parameter and the goal here is to clarify how $C_0$ was determined. Here, $C_0$ was calculated by matching the effective diffusivity of the Lagrangian stochastic model $K_\mathrm{f}=\sigma _w^2 T_\mathrm{L}$ with the boundary-layer eddy diffusivity $K_m=ku_*z$, where k is the von Karman constant. Taking into account the relation $T_\mathrm{L}=2\sigma _w^2/(C_0 \epsilon _0)$, and a neutral boundary-layer dissipation rate $\epsilon _0=u_*^3/(kz)$,

$$\begin{aligned} C_0 = 2(\sigma _w/u_*)^4, \end{aligned}$$

(20)

where the ratio $\sigma _w/u_*$ is taken to be 1.25, which is the typical value used in atmospheric surface-layer flow (Sorbjan 1989), and matches the measured value above the canopy both for the flume and the field experiments (Fig. 10).This results in $C_0=4.88$, which is used in all calculations.

Appendix 2: Heavy Particles of the Flume Experiment

The $w_\mathrm{g}$ value for the heavy particles released in the flume was measured for a sample of 136 spheres. The observed probability distribution of $w_\mathrm{g}$ was computed from this sample, binning the results into 10 equal-sized intervals. The measured probability distribution was fitted to a Weibull probability distribution, given by Eq. 15, and shown in Fig. 9.

Fig. 9

Measured p.d.f. of terminal velocity $w_\mathrm{g}$ of the spherical particles in the flume experiment, compared with a fitted Weibull distribution and a synthetic sample of 1000 values from the Weibull distribution to illustrate uncertainty originating from a finite sampling size during the experiment

A sample of 1000 $w_\mathrm{g}$ values was synthetically drawn from the Weibull distribution and their histogram computed to illustrate uncertainty originating from a finite sampling size during the experiment (in which 1000 particles were released). The observed probability distribution, computed from only 136 samples, cannot be distinguished from the synthetically generated Weibull distribution. The mean of $w_\mathrm{g}$ ($=1.79 u_*$), shown in dashed vertical line in Fig. 9, is calculated based on the measurements, and equals to the mean of the approximated Weibull function.

Appendix 3: Eulerian Canopy-Flow Model

The model that is used to generate the Eulerian flow statistics for the forest LS runs is a closure model presented by Siqueira et al. (2012). The model assumes that the Eulerian turbulent flow field is stationary, planar-homogeneous with no subsidence, and fully developed so that an extensive inertial subrange exists in all Eulerian velocity spectra. The model solves a set of six coupled equations for $\overline{u}$, $\sigma _u$, $\sigma _v$, $\sigma _w$, $\overline{u^{\prime } w^{\prime }}$, and $\epsilon _0$. The model equations and model constants are provided in Siqueira et al. (2012). The boundary conditions that are used for solving the flow-model equations are conventional (e.g. Katul et al. 2004): zero velocity and zero momentum flux at the ground; and prescribed values for the velocity and all Reynolds-stress terms at the top boundary, provided by measurements. The equations are solved to the top measurement height available, and matched with a neutral boundary layer for the velocity and dissipation above that height. All Reynolds-stress terms above the solution are maintained constant. The flow model was tested for an oak-hickory-pine canopy experimental set-up with multiple flow measurements at nine heights (Baldocchi and Meyers 1988a, b; Meyers and Baldocchi 1991), and was found to give comparable results to other closure models (Katul et al. 2004). The $\textit{PAD}$ profile that is used in the LS forest runs was measured at the site of the release experiment, and is shown in Fig. 10a. The flow model was solved with this $\textit{PAD}$ profile and a canopy drag constant of 0.25 for two values of $\textit{PAI}=0.89$ and 1.4. Model results are shown in Fig. 10 together with the measured flow statistics from the flume experiment for comparison.

Fig. 10

Calculated profiles of the velocity statistics for the field experiment set-up, using the Eulerian closure flow model (lines), compared with the measured profiles in the flume experiment (symbols). The profiles were calculated with a measured $\textit{PAD}$ profile at the site of the release experiment, shown in (a), for $\textit{PAI}=0.89$ and 1.4