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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 σz = 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 σz is replaced by a constant times zi:

Therefore a limit is placed on each term involving σz 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 σz = ax(1 + bx)1/2 corresponds to the term in Equation (1-62), σz is capped at for this P(x,zd) as well.  Note that these caps to σz 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 σz/zi = 1.6.  In Equation (1-62), the corresponding ratios are σz/zi = 1.4, 1.6, and 1.9.

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