A—Unstable or neutral conditions:
The buoyancy flux parameter, Fb, was written [5] as follows:
$$F_{b} = \, g \, v_{s} r_{s}^{2} \left( {T_{s} - \, T_{a} } \right)/T_{s}$$
(44)
For (hs < 305 m), g is the acceleration (ms−2), vs is the exit vertical speed (ms−1), rs is the exit radius (m), Ts is the gas exit temperature (K), and Ta is the ambient temperature (K) at hs [27].
The critical x* is given by:
$$x* = 2.16 \, F_{b}^{2/5} h_{s}^{3/5}$$
(45)
For x ≤ x*, we have:
$$\Delta h\left( x \right) = {\text{const}}. \, F_{b}^{1/3} \left( u \right)^{ - 1} x^{2/3}$$
(46)
Taking the constant equals 1.6 [6], one gets:
$$H \, = \, h_{s} + \, 1.6 \, F_{b}^{1/3} \left( u \right)^{ - 1} x^{2/3}$$
(47)
This equation is used if Ts > Ta (2/3 law) [27].
uh can be calculated at 10 m as follows:
$$u_{h} = u_{10} \;(H/10m)^{p}$$
(48)
where the parameter p is given from [20].
Substituting from (47) into (43), one gets:
$$H* = \frac{{ - c^{2} x^{2d} (b + d)}}{{2/3(\frac{{1.6F^{1/3} }}{{u_{h} }})x^{2/3} - d(\frac{{1.6F^{1/3} }}{{u_{h} }})x^{2/3} - h_{s} d}}$$
(49)
where H* is the maximum value.
In unstable condition, taking d = 1.17 ([8] H* becomes:
$$H* = \frac{{c^{2} x^{2d} (b + d)}}{{0.5(\frac{{1.6F^{1/3} }}{{u_{h} }})x^{2/3} + h_{s} d}}$$
(50)
Then,
$$H*^{2} = \left[ {\frac{{c^{2} x^{2d} (b + d)}}{{0.5(\frac{{1.6F^{1/3} }}{{u_{h} }})x^{2/3} + h_{s} d}}} \right]^{2}$$
(51)
The maximum value for \(\chi\)* has the form:
$$\chi^{*} \left( {x, \, 0, \, 0, \, H} \right) \, = \, [Q \, / \, (\pi \sigma_{y} \sigma_{z} u_{h} )] \, \exp \frac{ - 1}{{2\sigma_{z}^{2} }}\left[ {\frac{{c^{2} x^{2d} (b + d)}}{{0.5\left( {\frac{{1.6F^{1/3} }}{{u_{h} }}} \right)x^{2/3} + h_{s} d}}} \right]^{2}$$
(52)
In neutral condition, taking d = 0.95 [8] \(\chi\)* becomes:
$$\chi^{*} \left( {x, \, 0, \, 0, \, H} \right) \, = \, [Q \, / \, (\pi \sigma_{y} \sigma_{z} u_{h} )] \, \exp \frac{ - 1}{{2\sigma_{z}^{2} }}\left[ {\frac{{c^{2} x^{2d} (b + d)}}{{0.28\left( {\frac{{1.6F^{1/3} }}{{u_{h} }}} \right)x^{2/3} + h_{s} d}}} \right]^{2}$$
(53)
In stable stability, there are two methods for maximum concentration at z = 0 as follows:
(First)—In stable, where d = 0.67 [8] the maximum value for \(\chi\)* is:
$$\chi^{*} \left( {x, \, 0, \, 0, \, H} \right) \, = \, [Q \, / \, (\pi \sigma_{y} \sigma_{z} u_{h} )] \, \exp \left[ {\frac{{ - c^{2} x^{2d} (b + d)^{2} }}{{2d^{2} h_{s}^{2} }}} \right]$$
(54)
(Second)—In stable (E and F), the stability parameter is written as follows:
$$s = \frac{g}{{T_{a} }}\left( {\frac{\Delta \theta }{{\Delta Z}}} \right)$$
(55)
where \(\left( {\frac{\Delta \theta }{{\Delta Z}}} \right)\) = 0.02 K/m for E and \(\left( {\frac{\Delta \theta }{{\Delta Z}}} \right)\) = 0.035 K/m for F ([27].
\(\Delta h\) is written as follows:
$$\Delta h = 2.6\left( {\frac{{F_{b} }}{{u_{h} s}}} \right)^{1/3}$$
(56)
H* becomes:
$$H* = \left( {\frac{{c^{2} x^{2d} (b + d)}}{d}} \right)^{0.5}$$
(57)
Also, χ* has the form:
$$\chi^{*} \left( {x, \, 0, \, 0, \, H} \right) = [Q/(\pi \sigma_{y} \sigma_{z} u_{h} )] \, \exp \left\{ {\frac{ - (b + d)}{{2d^{{}} }}} \right\}$$
(58)