where xr is radial distance to the receptor. Although the form of the vertical distribution function

remains unchanged for the two plumes, its magnitude is based on downwind distance for the

coherent plume and radial distance for the random plume.

Once the two concentration limits (CCh - coherent plume; CR - random plume) have been

calculated, the total concentration for stable or convective conditions (Cc,s) is determined by

interpolation. Interpolation between the coherent and random plume concentrations is

accomplished by assuming that the total horizontal “energy” is distributed between the wind’s

mean and turbulent components. That is,

where F


2 is a measure of the total horizontal wind energy and F


2 is a measure of the random

component of the wind energy. Therefore, the ratio F

r2/ F


2 is an indicator of the importance of the

random component and can therefore be used to weight the two concentrations as done in eq.


The horizontal wind is composed of a mean component , and random components F

u u and


. Thus, a measure of the total horizontal wind “energy” (given that the alongwind and

crosswind fluctuations are assumed equal i.e., F

u = F

v), can be represented as


where . The random energy component is initially and becomes equal to F

h u (u ) 2 at v = ~2 ~2 1/2 2σ 2 2 ~σ


large travel times from the source when information on the mean wind at the source becomes

irrelevant to the predictions of the plume’s position. The evolution of the random component of

the horizontal wind energy can be expressed as


where Tr is a time scale (= 24 hrs) at which mean wind information at the source is no longer

correlated with the location of plume material at a downwind receptor. Analyses involving

autocorrelation of wind statistics (Brett and Tuller 1991) suggest that after a period of

approximately one complete diurnal cycle, plume transport is “randomized.” Equation (73)

shows that at small travel times, σ σ , while at large times (or distances) , r v

2 = 2 ~ 2 σ σ r v 2 = 2 ~ 2 + u 2

which is the total horizontal kinetic energy ( F

h2) of the fluid. Therefore, the relative contributions

of the coherent and random horizontal distribution functions (eq. (71)) are based on the fraction of

random energy contained in the system (i.e.,σ σ ). r h

2 2

The application of eq. (71) is relatively straight forward in the SBL. Since concentrations in

the SBL are represented as a single plume, Cs can be calculated directly from eq. (71). By

contrast for convective conditions the situation is complicated by the inclusion of plume

penetration. Since F


2 depends on the effective parameters (eq. (73)), the concentration weighting

factors found in eq. (71) will be different for the non-penetrated and penetrated plumes of the


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