The Vertical Term
The Vertical Term (V), which is included in Equation (1-1), accounts for the vertical distribution of the Gaussian plume. It includes the effects of source elevation, receptor elevation, plume rise (Section 1.1.4), limited mixing in the vertical, and the gravitational settling and dry deposition of particulates. In addition to the plume height, receptor height and mixing height, the computation of the Vertical Term requires the vertical dispersion parameter (z) described in Section 1.1.5.
The Vertical Term Without Dry Deposition.
In general, the effects on ambient concentrations of gravitational settling and dry deposition can be neglected for gaseous pollutants and small particulates (less than about 0.1 microns in diameter). The Vertical Term without deposition effects is then given by:
where:
The infinite series term in Equation (1-50) accounts for the effects of the restriction on vertical plume growth at the top of the mixing layer. As shown by Figure 1-3, the method of image sources is used to account for multiple reflections of the plume from the ground surface and at the top of the mixed layer. It should be noted that, if the effective stack height, he, exceeds the mixing height, zi, the plume is assumed to fully penetrate the elevated inversion and the ground-level concentration is set equal to zero.
Equation (1-50) assumes that the mixing height in rural and urban areas is known for all stability categories. As explained below, the meteorological preprocessor program uses mixing heights derived from twice-daily mixing heights calculated using the Holzworth (1972) procedures. The ISC models currently assume unlimited vertical mixing under stable conditions, and therefore delete the infinite series term in Equation (1-50) for the E and F stability categories.
The Vertical Term defined by Equation (1-50) changes the form of the vertical concentration distribution from Gaussian to rectangular (i.e., a uniform concentration within the surface mixing layer) at long downwind distances. Consequently, in order to reduce computational time without a loss of accuracy, Equation (1-50) is changed to the form:
at downwind distances where the z/zi ratio is greater than or equal to 1.6.
The meteorological preprocessor program, RAMMET, used by the ISC Short Term model uses an interpolation scheme to assign hourly rural and urban mixing heights on the basis of the early morning and afternoon mixing heights calculated using the Holzworth (1972) procedures. The procedures used to interpolate hourly mixing heights in urban and rural areas are illustrated in Figure 1-4, where:
Hm{max} =
maximum mixing height on a given day
Hm{min} = minimum mixing height on a given day
MN = midnight
SR = sunrise
SS = sunset
The interpolation procedures are functions of the stability category for the hour before sunrise. If the hour before sunrise is neutral, the mixing heights that apply are indicated by the dashed lines labeled neutral in Figure 1-4. If the hour before sunrise is stable, the mixing heights that apply are indicated by the dashed lines labeled stable. It should be pointed out that there is a discontinuity in the rural mixing height at sunrise if the preceding hour is stable. As explained above, because of uncertainties about the applicability of Holzworth mixing heights during periods of E and F stability, the ISC models ignore the interpolated mixing heights for E and F stability, and treat such cases as having unlimited vertical mixing.
The Vertical Term in Elevated Simple Terrain.
The ISC models make the following assumption about plume behavior in elevated simple terrain (i.e., terrain that exceeds the stack base elevation but is below the release height):
-
The plume axis remains at the plume stabilization height above mean sea level as it passes over elevated or depressed terrain.
-
The mixing height is terrain following.
-
The wind speed is a function of height above the surface (see Equation (1-6)).
Thus, a modified plume stabilization height he´ is substituted for the effective stack height he in the Vertical Term given by Equation (1-50). For example, the effective plume stabilization height at the point x, y is given by:
where:
zs
= height above mean sea level of the base of
the stack (m)
z(x,y) = height above mean
sea level of terrain at the receptor location (x,y) (m)
It should also be noted that, as recommended by EPA, the ISC models "truncate" terrain at stack height as follows: if the terrain height z - zs exceeds the source release height, hs, the elevation of the receptor is automatically "chopped off" at the physical release height. The user is cautioned that concentrations at these complex terrain receptors are subject to considerable uncertainty. Figure 1-5 illustrates the terrain-adjustment procedures used by the ISC models for simple elevated terrain. The vertical term used with the complex terrain algorithms in ISC is described in Section 1.5.6.
The Vertical Term With Dry Deposition.
Particulates are brought to the surface through the combined processes of turbulent diffusion and gravitational settling. Once near the surface, they may be removed from the atmosphere and deposited on the surface. This removal is modeled in terms of a deposition velocity (vd), which is described in Section 1.3.1, by assuming that the deposition flux of material to the surface is equal to the product vdd, where d is the airborne concentration just above the surface. As the plume of airborne particulates is transported downwind, such deposition near the surface reduces the concentration of particulates in the plume, and thereby alters the vertical distribution of the remaining particulates. Furthermore, the larger particles will also move steadily nearer the surface at a rate equal to their gravitational settling velocity (vg). As a result, the plume centerline height is reduced, and the vertical concentration distribution is no longer Gaussian.
A corrected source-depletion model developed by Horst (1983) is used to obtain a "vertical term" that incorporates both the gravitational settling of the plume and the removal of plume mass at the surface. These effects are incorporated as modifications to the Gaussian plume equation. First, gravitational settling is assumed to result in a "tilted plume", so that the effective plume height (he) in Equation
(1-50) is replaced by
where hv = (x/us)vg is the adjustment of the plume height due to gravitational settling. Then, a new vertical term (Vd) that includes the effects of dry deposition is defined as:
V(x,z,hed) is the vertical term in the absence of any deposition--it is just Equation (1-50), with the tilted plume approximation. FQ(x) is the fraction of material that remains in the plume at the downwind distance x (i.e., the mass that has not yet been deposited on the surface). This factor may be thought of as a source depletion factor, a ratio of the "current" mass emission rate to the original mass emission rate. P(x,z) is a vertical profile adjustment factor, which modifies the reflected Gaussian distribution of Equation
(1-50), so that the effects of dry deposition on near-surface concentrations can be simulated.
For large travel-times, hed in Equation (1-53) can become less than zero. However, the tilted plume approximation is not a valid approach in this region. Therefore, a minimum value of zero is imposed on hed. In effect, this limits the settling of the plume centerline, although the deposition velocity continues to account for gravitational settling near the surface. The effect of gravitational settling beyond the plume touchdown point (where hed = 0) is to modify the vertical structure of the plume, which is accounted for by modifying the vertical dispersion parameter (z).
The process of adjusting the vertical profile to reflect loss of plume mass near the surface is illustrated in Figures 1-6 and 1-7. At a distance far enough downwind that the plume size in the vertical has grown larger than the height of the plume, significant corrections to the concentration profile may be needed to represent the removal of material from the plume due to deposition. Figure 1-6 displays a depletion factor FQ, and the corresponding profile correction factor P(z) for a distance at which z is 1.5 times the plume height. The depletion factor is constant with height, whereas the profile correction shows that most of the material is lost from the lower portion of the plume. Figure 1-7 compares the vertical profile of concentration both with and without deposition and the corresponding depletion of material from the plume. The depleted plume profile is computed using Equation (1-54).
Both FQ(x) and P(x,z) depend on the size and density of the particles being modeled, because this effects the total deposition velocity (See Section 1.3.2). Therefore, for a given source of particulates, ISC allows multiple particle-size categories to be defined, with the maximum number of particle size categories controlled by a parameter statement in the model code (see Volume I). The user must provide the mass-mean particle diameter (microns), the particle density (g/cm3), and the mass fraction () for each category being modeled. If we denote the value of FQ(x) and P(x,z) for the nth particle-size category by FQn(x) and Pn(x,z) and substitute these in Equation (1-54), we see that a different value for the vertical term is obtained for each particle-size category, denoted as Vdn. Therefore, the total vertical term is given by the sum of the terms for each particle-size category, weighted by the respective mass-fractions:
FQ(x) is a function of the total deposition velocity (vd), V(x,zd,hed), and P(x,zd):
where zd is a height near the surface at which the deposition flux is calculated. The deposition reference height is calculated as the maximum of 1.0 meters and 20z0. This equation reflects the fact that the material removed from the plume by deposition is just the integral of the deposition flux over the distance that the plume has traveled. In ISC, this integral is evaluated numerically. For sources modeled in elevated or complex terrain, the user can input a terrain grid to the model, which is used to determine the terrain elevation at various distances along the plume path during the evaluation of the integral. If a terrain grid is not input by the user, then the model will linearly interpolate between the source elevation and the receptor elevation.
The profile correction factor P(x,z) is given by
where R(z,zd) is an atmospheric resistance to vertical transport that is derived from Briggs' formulas for z (Gifford, 1976). When the product vgR(z,zd) is of order 0.1 or less, the exponential function is approximated (for small argument) to simplify P(x,z):
This simplification is important, since the integral in Equation (1-57a) is evaluated numerically, whereas that in Equation (1-57b) is computed using analytical approximations.
The resistance R(z,zd) is obtained for the following functional forms of z defined by Briggs:
For this last form, the x(z) and x(zd) must be solved for z and zd (respectively) by finding the root of the implicit relation
The corresponding functions for P(x,zd) for the special case of Equation (1-57) are given by:
For the last form,
,
and
The added complexity of this last form arises because a simple analytical solution to Equation (1-57) could not be obtained for the urban class A and B. The integral in P(x,zd) for z = ax(1 + bx)1/2 listed above matches a numerical solution to within about 2% for zd = 1 m.
When vertical mixing is limited by zi, the profile correction factor P(x,zd) involves an integral from 0 to zi, rather than from 0 to infinity. Furthermore, V contains terms that simulate reflection from z = zi as well as z = 0 so that the profile correction factor, P(x,zd), becomes a function of mixing height, i.e, P(x,zd,zi). In the well-mixed limit, P(x,zd,zi) has the same form as P(x,zd) in Equation (1-60) but z is replaced by a constant times zi:
Therefore a limit is placed
on each term involving z in Equation (1-60) so that each term
does not exceed the corresponding term in zi. Similarly, since
the leading order term in P(x,zd) for z = ax(1 + bx)1/2
corresponds to the 
term in Equation (1-62), z is capped at 
for this P(x,zd) as well. Note that these caps to z
in Equation (1-60) are broadly consistent with the condition on the use of
the well-mixed limit on V in Equation (1-51) which uses a ratio z/zi
= 1.6. In Equation (1-62), the corresponding ratios are z/zi
= 1.4, 1.6, and 1.9.
In many applications, the removal of material from the plume may be extremely small, so that FQ(x) and P(x,z) are virtually unity. When this happens, the vertical term is virtually unchanged (Vd = V, see Equation (1-54)). The deposition flux can then be approximated as vd rather than vdd. The plume depletion calculations are optional, so that the added expense of computing FQ(x) and P(x,z) can be avoided. Not considering the effects of dry depletion results in conservative estimates of both concentration and deposition, since material deposited on the surface is not removed from the plume.
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