A—Unstable or neutral conditions:

The buoyancy flux parameter, F_{b,} was written [5] as follows:

$$F_{b} = \, g \, v_{s} r_{s}^{2} \left( {T_{s} - \, T_{a} } \right)/T_{s}$$

(44)

For (h_{s} < 305 m), g is the acceleration (ms^{−2}), v_{s} is the exit vertical speed (ms^{−1}), r_{s} is the exit radius (m), T_{s} is the gas exit temperature (K), and T_{a} is the ambient temperature (K) at h_{s} [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 T_{s} > T_{a} (2/3 law) [27].

u_{h} 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)