A Lagrangian
stochastic (LS) model calculates the paths of a large number of individual
particles as they travel with the local wind field. Rather than calculating the wind field exactly, its statistical
properties (mean and variances) are prescribed and the values acting on an
individual particle at any time are selected from a Gaussian distribution of
random deviates. The basic assumption
made is that the particles have only limited memory of their previous state.
If the ground is
perfectly reflective, i.e. no deposition of the tracer occurs, particle
velocity is reversed and its position reflected whenever it passes below an
imaginary plane, typically located at surface roughness length Zo. To calculate concentration at a point in
space, the total residence time of particles crossing through a volume of space
surrounding the point is determined.
LS models have several advantages over
their Gaussian and Eulerian counterparts, as well as at least one
disadvantage. LS models are more
physically valid than Gaussian models, which do not incorporate wind shear or
inhomogeneity of the turbulence field, and which can be shown to be equivalent
to K-theory models with constant diffusivity.
They do not require artificial diffusivity, as do the Eulerian models
for convective transport. They can be
released from an arbitrarily complex source region, by approximating the source
as a collection of evenly distributed point sources; alternatively, they may be
released from the sensor and be allowed to travel backward in time
to determine what their source region might have been . Both deposition of
tracer to the underlying surface and other time-dependent phenomenon such as
radioactive decay can be handled in a natural way.
Its disadvantages
The most significant
disadvantage of LS models is their computation time requirements, which can be
several orders of magnitude larger than those required to solve algebraically
reduced Gaussian models or even Eulerian models. Because the calculation of concentration involves an averaging of
particle residence time over a volume of space, through which many of the
particles released in an experiment may not pass, a large number of particles
may be required to generate an estimate of concentration with sufficient
accuracy. Further, if the collector
volume is increased to capture more particles, the concentration will represent
an average over a larger volume and be less representative of the target
point. To circumvent this problem in
the backward-in-time mode used for area sources, WindTrax automatically generates touchdown catalogs that can be
used whenever the atmospheric conditions are similar.