A Semi-analytical Model for Short-Range Near-Ground Continuous Dispersion

Abstract

We present a semi-analytical model for the dispersion of passive scalars from continuous ground sources up to distances of a few hundred metres. We attempt to cope with problems typical of analytical models, such as the correct representation of the near-ground concentration and lateral dispersion, while avoiding the use of any empirical parameters. A previous analysis of Prairie Grass Project (PGP) data has shown that the near-ground, cross-wind integrated concentration decreases as some power of the distance from the source that is, itself, distance dependent. As the conventional power-law model is incapable of reproducing this behaviour, we propose a model in which the vertical diffusivity depends on both the height as a power law, and on the distance from the source. For this equation, we construct an infinite-series formal solution, with the first term used as an approximation. A set of equations based on this approximation and on Monin–Obukhov similarity theory is proposed for the the vertical diffusivity, from which the cross-wind integrated concentration is derived analytically. We further construct a simple empirical model for the distance-dependent vertical diffusivity. For the plume lateral width, a Langevin stochastic model depending on the plume height is proposed, whose formal analytical solution is used to derive a set of equations for the cloud width, which are easily solved numerically, with the results validated against PGP data. We apply four statistical measures to evaluate the performance of the model, including the computation of the 95% confidence intervals, for which we find very good agreement. Implementation of this model is extremely simple and computationally efficient.

Appendices

Appendices

Content of the Electronic Supplementary Material

The Electronic Supplementary Material file contains detailed derivations of some of the results presented in the paper. Sect. S1 contains the derivation of the approximate solution to the advection-diffusion equation with power-law profiles of the horizontal velocity and the turbulent vertical diffusion coefficient in which the vertical diffusivity depends on the distance from the source. Section S2 examines the range of validity of this approximation. Section S3 specifies the boundary layer parametrization we use. Section S4 presents the confidence intervals for the statistical measures used to evaluate the performance of the different models computing the CWI concentration, Sects. 7 and 8, and the cloud lateral width Sect. 11. Section S5 contains a detailed derivation of the equation for the cloud lateral width used in Sect. 10.

PGP Meteorological Data

Following are the meteorological data prevailed in PGP runs considered above (Table 6).

Table 6 Meteorological parameters in the PGP dataset

$u_*$ and L for the stable conditions were taken from Wilson (1982) and for the unstable conditions from Wilson et al. (1981) except for runs 10, 27, 31, 47, 48, 51, 52 taken from Horst et al. (1979). The PGP meteorological data are 10-min and 20-min averages according to Barad (1958). For the sake of comparison we present power law and stability parameters derived for three characteristic stability conditions. Assume the wind speed at 10 m, $U_{10}$ and the roughness length $z_0$ are given, L is derived from Golder (1972) and Irwin (1979). $u_*$, a and m are derived from:

$$\begin{aligned} u_*=&\,\kappa U\Big /\Big [\ln {\big (\frac{z}{z_0}\big )}-\varPsi _m\big (\frac{z}{L}\big )\Big ] \end{aligned}$$

(50a)

$$\begin{aligned} m=&\, u_* \varPhi _m/\kappa U \end{aligned}$$

(50b)

$$\begin{aligned} ~a=&\,U/z^m \end{aligned}$$

(50c)

with $\varPhi _m$ given in Eq. 8 and $\varPsi _m$ in Eq. 9. Table 7 shows the derived parameters with $z_0=0.03\, \mathrm{m}$.

Table 7 MOST, power law and stability parameters for three classes; $z_0=0.03\, \mathrm{m}$