The concentration of a gas dispersing in a turbulent atmosphere is a random variable depending on spatial coordinates and time. Thus, a model for predicting concentration at a given point in space and time can be based on statistical properties of a data set taken in the course of an experiment. The covariance matrix is formed with respect to all time points at which measurements of concentration were made and its eigenvalues and eigenvectors are computed. If the sequence of eigenvalues converges sufficiently rapidly, then only the first few eigenvalues with the largest proportion of variance are needed and the corresponding eigenvectors are the principal components or the empirical eigenfunctions. The original signal is then reconstructed in terms of the empirical orthogonal functions found. A dosage model is also proposed based on the concentration model. Principal components of the dosage covariance matrix are computed from the data and are then used to reproduce the observed dosages.

ANASTASIOS ANTYPAS
Department of Applied Mathematics
University of Sheffield
Hicks Building
Sheffield, S3 7RH, U.K.
STP98aa@sheffield.ac.uk
 
 

The exact evolution equation for the PDF of a dispersing scalar in a turbulent flow is well-known, being easily derivable from the standard advection-diffusion equation for the concentration of the scalar in any one realisation. As is inevitable for any equation governing any statistical property of a turbulent flow, this evolution equation is not closed. Based on (a) some published "toy" solutions, and (b) simple physical arguments, a new closure hypothesis for the so-called small-scale mixing term is proposed and
investigated. Links with other work in PDEs and turbulence will be discussed.

PHILIP CHATWIN
School of Applied Mathematics
Hicks Building
University of Sheffield
Sheffield, S3 7RH, UK
P.Chatwin@sheffield.ac.uk
 
 

The distribution function for concentrations of a scalar pollutant dispersing in the turbulent atmosphere has a finite domain which is bounded above and below. Three methods, based on extreme value statistics, are used to obtain estimates for the upper bound and to describe the tail behaviour of the distribution; all three methods are applied to concentration data obtained from experimental atmospheric releases. Quantile-quantile plots are used to assess the goodness of fit of the resulting estimates of the distribution, and also to compare the performance of the three methods. The predicted values for the upper bound are orders of magnitude less than the source concentration, illustrating that molecular diffusion has a large effect on the high concentrations. The models are sensitive to the lack of high concentration data due to the meandering of the plume, intermittently moving off and onto the sensor position.

RICK MUNRO
Department of Applied Mathematics
University of Sheffield
Hicks Building
Sheffield, S3 7RH, UK
r.j.munro@sheffield.ac.uk