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,z

_{d}) for σ_{z}= ax(1 + bx)^{1/2}listed above matches a numerical solution to within about 2% for z_{d}= 1 m.When vertical mixing is limited by z

_{i}, the profile correction factor P(x,z_{d}) involves an integral from 0 to z_{i}, rather than from 0 to infinity. Furthermore, V contains terms that simulate reflection from z = z_{i}as well as z = 0 so that the profile correction factor, P(x,z_{d}), becomes a function of mixing height, i.e, P(x,z_{d},z_{i}). In the well-mixed limit, P(x,z_{d},z_{i}) has the same form as P(x,z_{d}) in Equation (1-60) but σ_{z}is replaced by a constant times z_{i}:Therefore a limit is placed on each term involving σ

_{z}in Equation (1-60) so that each term does not exceed the corresponding term in z_{i}. Similarly, since the leading order term in P(x,z_{d}) for σ_{z}= ax(1 + bx)^{1/2}corresponds to the term in Equation (1-62), σ_{z}is capped at for this P(x,z_{d}) as well. Note that these caps to σ_{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 σ_{z}/z_{i}= 1.6. In Equation (1-62), the corresponding ratios are σ_{z}/z_{i}= 1.4, 1.6, and 1.9.

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