Plume fluctuations in surface layer canopies

Many significant agricultural and ecological processes depend on chemical plumes in surface layers of the atmosphere. Insect pheromone tracking is one example, and integrated pest management is an outcome of the science [4]. Plume properties are generally understood on the basis of the distribution of mean concentration. This can be predicted with simple Gaussian plume modelling in simple flows. In more complicated flows, like canopies, simple diffusion is inadequate and it is better to use Lagrangian particle models [5]. In addition, the mean is a poor instantaneous representation of a plume (from a signalling insect say) and turbulent fluctuations of chemical concentrations are important drivers of ecosystem responses. In fact the mean may always be below significant threshold detection levels and it is only isolated large fluctuations that force a response.

Practical applications of plume fluctuations are limited by the predictability in real world situations. Real-world effects of meteorology and canopy structure can be systematically integrated into Lagrangian particle models for mean-field calculations, and are essential in many flux footprint inferences in ecosystem exchange processes [5]. Fluctuating quantities are more problematic, but a new approach based on conditioned clusters of Lagrangian particles, embedded in the ensemble of trajectories governed by well understood ‘single’ particle models, allows for the systematic prediction.

We have developed a detailed single-particle Lagrangian model with new parameterisations for turbulence statistics emphasising the spatial structure of local mean dissipation of energy. Comparisons with standard surface-layer benchmarks, for mean concentrations in plumes, show good agreement. Within the ensemble of trajectories generated by the one-particle model we also consider coupled pair motions of particles (relative dispersion). A collection of such pair separations, evolve in time under the influence of net sampling of local dissipation rates. At a given time the local energy dissipation for each pair in the collection of separations forms a distribution from which collective local separation properties can be inferred. In particular, Lagrangian pair separation models can be constructed to incorporate the real-world controls on local separation processes. For the first time systematic theoretical estimates of plume fluctuations in non-idealised flows are possible.

The simplest encoding of surface layer dissipation relies on local similarity theory of separation velocities as functions of dissipation and separation alone and uses the moments

ε_W(t) = ∫_Wε(x)P₁(x,t)d³x = <ε>_W, <ε^2/3>_W, <ε^1/3>_W

where P₁(x,t) is the probability density function for marked plume particles at time t. The fractional moments of dissipation occur in simple inertial range models for the collective pair separation process. This simple stochastic model for pair separation uses the empirical result that pairs accelerate apart with quadratic dependence on the separation speed plus an additive random turbulent acceleration. The model is

du = (α<ε^2/3>_Wr^−1/3 + β<ε^1/3>_Wr^−2/3u + γr⁻¹u²)dt + (2C₀ε_W)^1/2dW

for the change in separation speed u, with the separation distance adjusting as dr=udt for small time increments dt. The term dW is standard white noise and C₀ is a constant for Lagrangian processes. Such models easily describe pair separation in idealised flow [2], but here the parameters <ε>_W, <ε^2/3>_W, <ε^1/3>_W which occur in the model, depend on surface-layer, stability and canopy properties. Thus the real-world characterisation exerts direct explicit control on the pair separation process and consequently on the resultant plume concentration fluctuations.

The coupling of one-particle Lagrangian dispersion with pair separation processes is common in a broad class of meandering plume models [6], but never with any real non-ideal dynamical control of the pair separation process. We instead consider plume integral properties [3], where fluctuations depend solely on the probability of pair separations, decoupled from the bulk motion of the plume. This decoupling focuses on in-plume fluctuations. The distribution of the plume integral fluctuation intensity across the plumes is easily parameterised, so that point distributions of fluctuations in plumes are also predicted. The main output is the fluctuation intensity of concentration fluctuations and local concentration structure functions in the plume, both indicators of the texture and variability of concentrations as functions of downwind distance in the plume. The main novelty here is to extend beyond idealised predictions of fluctuations targeted at benchmark wind tunnel studies [1].

An example systematic study undertaken is to track the influence of changing stability on concentration fluctuations. Flow over a prescribed surface and stability may be described in terms of Monin-Obukhov similarity theory, but with modified parameterisations of energy dissipation rate to better represent local behaviour very near the wall (the pair separation process is sensitive to dissipation rate so emphasis is played on its realism). Identical plume releases in increasing stability leads to a general increase in the intensity of plume fluctuations and a reduction in the length scales of variability.

An interest for applications of the model are Lure & Kill pest management strategies where an attractant plume penetrates into a canopy and lures out insect pests. The attractant power of the plume and the penetration of fluctuating structures into the canopy determine the effectiveness of the pest management. In the framework of the the present work we consider the transition using one particle Lagrangian models to penetrate into a canopy and construct the modified separation parameters <ε>_W, <ε^2/3>_W, <ε^1/3>_W using the one-particle ensemble. We are able to see a change to much higher dissipation for plumes within the canopy and consequently faster mixing and reduction of plume fluctuations.

Endless other practical scenarios can be modelled by this generic system, and further improvements based on refined models for local pair separation accelerations can be designed.

References

[1] Borgas, M.S. & Sawford, B.L., 1996. J. Fluid Mech., 324, 25-54.

[2] Borgas, M.S. & Yeung, P.K. 2004 J. Fluid Mech 503, pp. 125-160.

[3] Brown, R.J., & Sawford, B.L., 2001. Boundary Layer Meteorology, 97, pp 219-249.

[4] Cardé, R.T., Minks, A.K., 1995. Annu. Rev. Entomol., 40, pp. 559-585

[5] Leuning, R., Denmead, O.T., Miyata, A., Kim, J., 2000. Agricultural and Forest Meteorology, 96, pp. 293-314

[6] Luhar, A.K., Hibberd, M.F. & Borgas, M.S., 2000. Atmos. Environ., 34, 3599-3616.