AERMOD Concentration Predictions in the CBL

In its formulation of the vertical distribution for the CBL, AERMOD parts company with traditional Gaussian models such as ISC3. Since downdrafts are more prevalent in the CBL than updrafts, the observed vertical concentration distribution is not Gaussian. Figure 14 presents a schematic representation of an instantaneous plume in a convective boundary layer and its corresponding ensemble average. The base concentration prediction in AERMOD is representative of a one hour average. Notice that since a larger percentage of the instantaneous plume is effected by downdrafts, the ensemble average has a general downward trend. Since downdrafts are more prevalent the average velocity of the downdrafts is correspondingly weaker than the average updraft velocity to insure that mass is conserved.

 
Figure 14: Instantaneous and corresponding ensemble-averaged plume in the CBL 

The instantaneous plume is assumed to have a Gaussian concentration distribution about its randomly varying centerline. The mean or average concentration is found by summing the concentrations due to all of the random centerline displacements. This averaging process results in a skewed distribution which AERMOD represents as a bi-Gaussian p.d.f. (i.e., one for updrafts and the other for downdrafts). Figure 15 shows the superposition of the updraft and downdraft plumes..57

Figure 15: AERMOD’s pdf approach for plume dispersion in the CBL.  AERMOD approximates the skewed distribution by superimposing two Gaussian distributions, the updraft and downdraft distributions. 

The dispersion algorithms for the convective boundary layer (CBL) are based on Gifford's (1959) meandering plume concept in which a small "instantaneous" plume wanders due to the large eddies in a turbulent flow. The specific model form is a probability density function (p.d.f.) approach in which the distribution of the centerline displacement is computed from p_w and p_v , the w v p.d.f. of the random vertical (w) and lateral (v) velocities in the CBL, respectively. This approach is discussed in Misra (1982), Venkatram (1983) and Weil, et al. (1988). The total vertical displacement z_c of the plume centerline is based on the superposition of the displacements due to the  random and the plume rise as described in Weil et al. (1986, 1997). Thus, the AERMOD approach extends Gifford's model to account for plume rise. In addition, it includes a skewed distribution of z_c because p_w in the CBL is known to be skewed; however, the lateral plume displacement is assumed to be Gaussian. 

For material dispersing within a convective layer, the conceptual picture (see Figure 14) is a. plume embedded within a field of updrafts and downdrafts that are sufficiently large to displace the plume section within it. The p.d.f. of the plume centerline height z_c is found from the p.d.f. of (i.e., p_w ), as discussed in Weil (1988), and z_c is obtained by superposing the plume rise (h) and the displacement due to the random convective velocity (w):

where h_s is the stack height (corrected for stack tip downwash), u is the mean wind speed (a vertical average over the convective boundary layer) and x is the downwind distance. The h above includes source momentum and buoyancy effects as given by eq. (116) below (see Briggs, 1984).

A good approximation to the P_w in the CBL has been shown to be given by the superposition of w two Gaussian distributions (e.g., Baerentsen and Berkowicz ,1984; Weil, 1988) such that

where ₁ and ₂ are weighting coefficients for the two distributions (1=updrafts, 2=downdrafts) ₁ + ₂ = 1.  The w_i  and _i ( I =1,2) are the mean vertical velocity and standard deviation 1 2 I for each distribution and are assumed to be proportional to _w . A simple approach for finding , , ₁ , ₂ , _{1,  2}, as a function of _w and the vertical velocity skewness S = w³/ ³_w is given by Weil (1990, 1997).

In the p.d.f. approach used here (Weil et al., 1997), there are three primary sources that contribute to the modeled concentration field: 1) the “direct” or real source at the stack, 2) an “indirect” source that the model locates above the CBL top to account for the slow downward dispersion of buoyant plumes that “loft” or remain near, but below, z_i , and 3) a “penetrated source” that contains the portion of plume material that has penetrated into the stable layer above z_i . The direct source describes the dispersion of plume material that reaches the ground directly from the source via downdrafts. The indirect source is included to treat the first interaction of the “updraft” plume with the elevated inversion - that is, for plume sections that initially rise to the CBL top in updrafts and return to the ground via downdrafts. Image sources are added to treat.59 the subsequent plume interactions with the ground and inversion and to satisfy the zero-flux conditions at z = 0 and at z = z_i . This source plays the same role as the first image source above z_i in the standard Gaussian model, but differs in the treatment of plume buoyancy. For the indirect source, a modified reflection approach is adopted in which the vertical velocity is reflected at z = z_i , but an “indirect” source plume rise h_i is added to delay the downward dispersion of plume material from the CBL top. This is intended to mimic the lofting behavior. The penetrated source is included to account for material that initially penetrates the elevated inversion but subsequently can reenter the CBL via turbulent mixing of the plume and eventual reentrainment into the CBL. Figure 16 illustrates this three plume approach; a fundamental feature of AERMOD’s convective model.

Figure 16: AERMOD’s Three Plume Treatment of the CBL

The total concentration in the CBL for the horizontal plume state is.

where:

C_c {x_r , y_r , z_r } = Total concentration in CBL

C_d {x_r , y_r , z_r } = Direct Source concentration contribution

C_r {x_r , y_r , z_r } = Indirect Source concentration contribution

C_p {x_r , y_r , z_r } = Penetrated Source concentration contribution .

The total concentration for the terrain responding state has the form of eq. (66) with z_r replaced by z_p. 

In considering penetration, the fraction of the plume mass that remains in the CBL (f_p), is calculated as follows:

where h_h = z_i - h_s , and h_eq is the equilibrium plume rise in a stable environment (see h is Berkowicz et al., 1986) are calculated as follows

6 The AMS/EPA Regulatory Model AERMOD

6.1 General Structure of AERMOD Including Terrain
6.2 AERMOD Concentration Predictions in the CBL
      6.2.1 DIRECT SOURCE CONTRIBUTION TO CONCENTRATION  
              CALCULATIONS IN THE CBL
      6.2.2 INDIRECT SOURCE CONTRIBUTION TO CONCENTRATION  
              CALCULATIONS IN THE CBL
      6.2.3 PENETRATED SOURCE CONTRIBUTION TO CONCENTRATION 
              CALCULATIONS IN THE CBL
6.3 Concentrations in the SBL Calculated by AERMOD 
6.4 Estimation of Dispersion Coefficients 
      6.4.1 AMBIENT TURBULENCE FOR USE IN CALCULATING DISPERSION
      6.4.2 BUOYANCY INDUCED DISPERSION (BID) COMPONENT OF _ AND _
              y z
      6.4.3 COMPONENT OF DISPERSION COEFFICIENTS DUE TO 
              DOWNWASH
6.5 Plume Rise Calculations in AERMOD
      6.5.1 PLUME RISE IN THE CBL 
      6.5.2 PLUME RISE IN THE SBL
6.6 Source Characterization 
6.7 Adjustments for the Urban Boundary Layer

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