The basic equations - Severe Convective Storms and Tornadoes: Observations and Dynamics - page 67

Gaussian (radial) profile. For this profile the following similar set of equations
hold:

dz ð R 2 w Þ¼ 2R

d = w ð 2 : 189 Þ

dz ð R 2 w 2 Þ¼ 2R 2 B

d = ð 2 : 190 Þ

z ð R 2 wB Þ¼ 2R 2 wN 2

d = ð 2 : 191 Þ

These equations are similar to (2.186)-(2.188); the differences are that a factor of 2
appears in the equation of motion (second equation in the set) and the thermo-
dynamic equation (third equation in the set). The solution for these equations is
sought subject to the boundary conditions that the mean vertical velocity and
mean radius are zero at the ground and that the flux of buoyancy at the ground is
a specified constant. The variables are first changed as follows:

V Rw ð 2 : 192 Þ

U R 2 w ð 2 : 193 Þ

F R 2 wB ð 2 : 194 Þ

so that the three simultaneous differential equations become

dU = dz ¼ 2 V ð 2 : 195 Þ

dV 2 V 2 = dz ¼ 2FU = ð 2 : 196 Þ

dz ¼ 2UN 2 dF = ð 2 : 197 Þ

The boundary conditions are

V ¼ U ¼ 0

ð 2 :

198 Þ

at z ¼ 0, and

F ¼ð 2 =Þ F 0 ð 2 : 199 Þ

at z ¼ 0 and R ¼ 0, where F 0 is the flux of buoyancy over the point source of
heat. The salient characteristics of the solutions are as follows ( Figure 2.14 ):

a. The mean radius of the plume first increases linearly with height, but eventually
increases faster and faster with height.
b. Mean buoyancy is greatest at the ground and decreases with height until it
becomes negative. It is above the height at which buoyancy drops below zero
that the mean radius increases most rapidly with height.
c. A measure of mean upward vertical mass flux (R 2 w) decreases with height and
approaches zero rapidly near the height at which the mean radius increases
towards infinity quickly.

The physical interpretation of these results is that as stable environmental air is
entrained into the plume, the buoyancy of the plume decreases, vertical velocity
decreases, and the plume spreads out aloft into the characteristic anvil shape. The