The basic equations - Severe Convective Storms and Tornadoes: Observations and Dynamics - page 63

2.8.1 Similarity models of plumes and thermals

Since plumes are of fundamental importance, we will look at them in some detail
and refer the reader to other texts (e.g., Emanuel, 1994 and Houze, 1993) for more
complete analyses of thermals and starting plumes. We will, for relative simplicity,
consider dry plumes maintained by a source of buoyancy in a homogeneous fluid.
Air motion is assumed to be turbulent and independent of molecular diffusion.
Suppose that F is the rate at which buoyancy is supplied by a point source. Then
the flux of buoyancy

F buoyancy velocity area ð 2 : 163 Þ

Since buoyancy has units of m s 2 (see (2.19)), velocity has units of m s 1 , and
area has units of m 2 , F must have units of m 4 s 3 . To see how the mean vertical
velocity (w), the mean buoyancy (B), and the mean radius of the plume (R)vary
with height, we use similarity theory and dimensional analysis. We will assume first
that the mean vertical velocity and buoyancy are functions of F and z, the height
above the point heat source. This assumption makes sense physically, as the
stronger the buoyancy flux, the stronger the vertical velocity should be; also, the
farther away one gets from the buoyancy source, the weaker the vertical velocity
should be.
So, we will assume that

w ¼ C 1 F a z b ð 2 : 164 Þ

where C 1 is a positive dimensionless constant. To be dimensionally correct, we
know that the units of m and s must match up on the LHS and RHS of (2.164).
It is necessary then that

4a þ b ¼ 1 ð 2 : 165 Þ

and

3a ¼ 1 ð 2 : 166 Þ

It follows that

w ¼ C 1 F 1 = 3 z 1 = 3

ð 2 : 167 Þ

which is consistent with physical intuition. Similarly, it can be shown that

buoyancy

B ¼ C 2 F 2 = 3 z 5 = 3

ð 2 : 168 Þ

where C 2 is a positive constant. Equation (2.168) is also consistent with physical
intuition: buoyancy decreases away from the source and is a function of buoyancy
flux. Of note is the finding that buoyancy decreases with height more rapidly than
vertical velocity. This finding follows even without considering the governing
equations. Finally, we find that the radius of the plume

R ¼ C 3 z ð 2 : 169 Þ

where C 3 is positive constant. In other words, the radius of the plume increases
linearly with height above the source.