The basic equations - Severe Convective Storms and Tornadoes: Observations and Dynamics - page 68

Figure 2.14. Variations of mean radius, mean buoyancy, and a measure of upward vertical
mass flux as a function of height in the MTT steady-state plume in a uniform, stably stratified
environment for a Gaussian profile in mean vertical velocity and mean buoyancy: r is the radius
of the plume, w is vertical velocity, B is buoyancy, and z is the height, all expressed in non-
dimensional units (adapted from Morton et al., 1956).

top of the plume actually overshoots the equilibrium level (where the buoyancy
first drops to zero). The reader is again reminded of the limitations of this simple
model: not only is it steady and dry, but in addition the vertical perturbation
pressure gradient is ignored, which in nature should act to reduce vertical accelera-
tion. Finally, this model is valid for an environment in which there is no vertical
shear and, in fact, no flow at all outside the plume. Yet, despite the alarming
extent to which the atmosphere has been grossly oversimplified, the governing
equations (2.195)-(2.197) are still highly nonlinear. Similar equations have been
used to study chaos theory. The reader should be getting the feeling that, while
such a simple model is so mathematically complex, an even more realistic model
must be even more complex.
MTT in 1956 also derived and solved the equations for an entraining thermal
in a stably stratified environment. The three governing equations, analogous to
(2.189)-(2.191) for the plume model, are

R 3 R 2

=

dt ð

3

Þ¼ 4 ð 2 :

200 Þ

d w

R 3 w Þ¼ R 3 B

4 4

d = dt ð

3 3

ð 2 : 201 Þ

R 3 B Þ¼ R 3 WN 2

4 4

d = dt ð

3 3

ð 2 : 202 Þ

For N 2

¼ a positive constant, the behavior of the thermal when it begins with
zero radius, vertical velocity, and constant buoyancy at the outset is summarized
(but not shown in a figure) as follows: