The basic equations - Severe Convective Storms and Tornadoes: Observations and Dynamics - page 66

resulting equation over a horizontal area of radius R, at arbitrary height z, we find
that

R 2 w Þ 2 Rw ¼ d = dz ð ð 2 : 181 Þ

Note that the partial derivatives become ordinary derivatives because R and w are
symmetrical about r ¼ 0 and radial variations are identical for each variable. This
equation expresses the physical process of entrainment of mass through the (outer)
boundary of the plume as being proportional to the increase with height of the
vertical mass flux through the plume. The steady-state version of the inviscid ver-
tical equation of motion (2.7) is used and the vertical perturbation pressure
gradient is neglected, so that

u @ w =@ x þ w @ w =@ z ¼ B ð 2 : 182 Þ

We integrate (2.182) over a volume between two heights (z to z þD z), making use

of the Boussinesq continuity equation (2.35) and the divergence theorem to
express the volume integral in terms of a surface flux. We use the boundary
condition that w ¼ 0 at the sides of the plume. For infinitesimally small D z

R 2 w 2 R 2 B d = dz ð Þ¼ ð 2 : 183 Þ

The steady-state version of the adiabatic thermodynamic equation (2.161) is

z þ wN 2

u @ B =@ r þ w @ B =@ ¼ 0 ð 2 : 184 Þ

It is integrated over the same volume that (2.182) was integrated and, after having
used the Boussinesq continuity equation (2.35) and the divergence theorem, it is
found, since (wB) at the edge of the plume vanishes, that for infinitesimally small
D z

R 2 wB Þ¼ R 2 wN 2

d = dz ð ð 2 : 185 Þ

Factoring out the s, we end up with the following three, simultaneous, highly

nonlinear equations in w, B, and R:

dz ð R 2 w Þ¼ 2R d = w ð 2 : 186 Þ

dz ð R 2 w 2 Þ¼ R 2 B d = ð 2 : 187 Þ

z ð R 2 wB Þ¼ R 2 wN 2

= ð 2 :

188 Þ d

When the environment is dry adiabatic, N 2 ¼ 0 and from (2.188) we find that

R 2 wB ¼ constant. The reader is referred to other textbooks for exact solutions to
the variables for this case and for the case of an unstable environment (N 2

0).
The much more interesting case is that of a stably stratified environment

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(N 2 0). In an influential paper by Morton, Taylor, and Turner in 1956 (often

referred to as MTT; this might be the first case in meteorology when a triplet of
authors are abbreviated as three letters; RKW, which is used in a subsequent
chapter, might be the second), equations (2.186)-(2.188) were derived for the case
in which the mean vertical velocity and mean buoyancy follow a more realistic

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