Gaussian diffusion formulas

TABLE 18.2 Point Source Gaussian Diffusion Formulas... 

Problems arise to get informations about the diffusion coeffients Ky and Kz. If equation (3.4) is interpreted as Gaussian distribution, a lot of available dispersion data can be taken into consideration because they are expressed in terms of standard deviations of the concentration distribution. Though there is no theoretical justification the Gaussian plume formula is converted to the K-theory expression by the transformation /11/... 

However, we must keep in mind the limitations of this approach, especially the transfer of consistent sets of dispersion parameters to the propagation of air pollution in the vicinity of a source. The Gaussian plume formula should be used only for those downwind distances for which the empirical diffusion coefficients have been determined by standard diffusion experiments. Because we are interested in emissions near ground level and immissions nearby the source, we use those diffusion parameters which are based on the classification of Klug /12/ and Turner /13/. The parameters are expressible as power functions,... 

V. Point Source Diffusion Formulas Based on a Gaussian Distribution. 233... 

The presumption of a Gaussian distribution for the mean concentration from a point source, although demonstrated only in the case of stationary, homogeneous turbulence, has been made widely and, in fact, is the basis for many of the atmospheric diffusion formulas in common use. Based on the developments of Section IV, we present in this section the Gaussian point source diffusion formulas that have been used for practical calculations. 

Dispersion Models Based on Inert Pollutants. Atmospheric spreading of inert gaseous contaminant that is not absorbed at the ground has been described by the various Gaussian plume formulas. Many of the equations for concentration estimates originated with the work of Sutton (3). Subsequent applications of the formulas for point and line sources state the Gaussian plume as an assumption, but it has been rigorously shown to be an approximate solution to the transport equation with a constant diffusion coefficient and with certain boundary conditions (4). These restrictive conditions occur only for certain special situations in the atmosphere thus, these approximate solutions must be applied carefully. 

Up to this point in this chapter we have developed the common theories of turbulent diffusion in a purely formal manner. We have done this so that the relationship of the approximate models for turbulent diffusion, such as the K theory and the Gaussian formulas, to the basic underlying theory is clearly evident. When such relationships are clear, the limitations inherent in each model can be appreciated. We have in a few cases applied the models obtained to the prediction of the mean concentration resulting from an instantaneous or continuous source in idealized stationary, homogeneous turbulence. In Section 18.7.1 we explore further the physical processes responsible for the dispersion of a puff or plume of material. Section 18.7.2 can be omitted on a first reading of this chapter that section goes more deeply into the statistical properties of atmospheric dispersion, such as the variances a (r), which are needed in the actual use of the Gaussian dispersion formulas. 

We have seen that under certain idealized conditions the mean concentration of a species emitted from a point source has a Gaussian distribution. This fact, although strictly true only in the case of stationary, homogeneous turbulence, serves as the basis for a large class of atmospheric diffusion formulas in common use. The collection of Gaussian-based formulas is sufficiently important in practical application that we devote a portion of this chapter to them. The focus of these formulas is the expression for the mean concentration of a species emitted from a continuous, elevated point source, the so-called Gaussian plume equation. 

SUMMARY OF GAUSSIAN POINT SOURCE DIFFUSION FORMULAS 923... 

Fig.4.19 Tseif(Q) obtained for a all the protons in PVE empty MD simulations,/ /// NSE, /=0.55) and b the main chain (filled circle, /=0.66) and the side group hydrogens (empty circle, /=0.51), both from the MDS. Dotted lines are expected Q-dependence from the Gaussian approximation in each case. Solid lines are description in terms of the anomalous jump diffusion model. Insets Chemical formula of PVE (a) and distribution functions obtained for the jump distances (b)...

Fig. 2 Positional detection and mean-square displacement (MSD). (a) The x, y-coordinates of a particle at a certain time point are derived from its diffraction limited spot by fitting a 2D-Gaussian function to its intensity profile. In this way, a positional accuracy far below the optical resolution is obtained, (b) The figure depicts a simplified scheme for the analysis of a trajectory and the corresponding plot of the time dependency of the MSD. The average of all steps within the trajectory for each time-lag At, with At = z, At = 2z,... (where z = acquisition time interval from frame to frame) gives a point in the plot of MSD = f(t). (c) The time dependence of the MSD allows the classification of several modes of motion by evaluating the best fit of the MSD plot to one of the four formulas. A linear plot indicates normal diffusion and can be described by = ADAt (D = diffusion coefficient). A quadratic dependence of on At indicates directed motion and can be fitted by = v2At2 + ADAt (v = mean velocity). An asymptotic behavior for larger At with = [1 - exp (—AA2DAt/)] indicates confined diffusion. Anomalous diffusion is indicated by a best fit with = ADAf and a < 1 (sub-diffusive)...

Formula (3.328) shows that the diffusion is anomalous, that is a power low dependence on time exists. When a = 1, Eq. (550) becomes Gaussian thus normal diffusion. All the above results are obtained assuming that the Maxwillian distribution of velocities is reached instantaneously by the ensemble of Brownian particles. In other words, the inertia of the particles is ignored. 

In Gaussian theory of diffusion (the most commonly used), x is a function of wind speed (inverse proportionality), the values of the standard deviation of the Gaussian distribution, the height of release and the position of a point in space with reference to the release point. The relevant formulae and diagrams, valid within several tens of kilometres from the release point, are given in Section 6-4. For simplified evaluations that usefully support quick decisions, they are not strictly necessary. For now it is sufficient to know that the formulae and diagrams... 

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