# AERMOD Tech Guide

## 6.5.2 Plume Rise in the SBL

Plume rise in the SBL is taken from Weil (1990), which is modified by using an iterative approach which is similar to that found in Perry, et al. (1989). When a plume rises in an atmosphere with a positive potential temperature gradient, plume buoyancy decreases because the ambient potential temperature increases as the plume rises; thus, plume buoyancy with respect to the surroundings decreases. The plume rise equations have to be modified to account for this. This modification (the reader should refer to Weil (1988b) for details) produces the following plume rise formula which is used by AERMOD.

The velocity, uP , and N are evaluated initially at stack height . Once plume rise has been p computed from these stack top values, subsequent plume rise estimates are made, iteratively, by averaging the uP and N values at stack top with these at . Equation **(120)** applies only when the plume is still rising. The distance at which the stable plume reaches its maximum rise is given by the following expression:

Upon substituting eq. **(121)** for x in eq. **(120)** the maximum final rise of the stable plume

hs{ xf } reduces to:

As with eq. **(120)**, the velocity, up , and N in eqs. **(122)** are evaluated initially at stack height and P then iteratively.

When the atmosphere is close to neutral, the Brunt Vaisala frequency, N, is close to zero, and eq. **(120)** can predict an unrealistically large plume rise. Under, these circumstances, we assume that plume rise is limited by atmospheric turbulence. This happens when the rate of plume rise under neutral conditions is comparable to w . Under these conditions (neutral limit) the plume rise can be calculated from:

Ln is calculated as:

Also, when the wind speed in near zero (calm conditions) unrealistically large plume rise estimates would result from applying eq. **(120)**. Under calm, stable atmospheric conditions we calculate plume rise from:

By applying each of the above limits the final plume rise equation under stable conditions becomes:

i.e., the minimum value from eqs. **(120)**, **(122)**, **(123)** or **(125)**; see for example Hanna and Paine (1989) . In addition AERMOD prevents the stable plume rise from exceeding the rise expected during neutral or convective conditions (i.e. hs eq. **(126)** is not to exceed the rise calculated s from eq. **(116)**). Note, for situations when Fb = 0 no rise is calculated in stable conditions.

Therefore, the distance dependent height of the plume in the SBL is given by the following expression: