The basic equations - Severe Convective Storms and Tornadoes: Observations and Dynamics - page 64

The vertical mass flux in the plume
wR 2 z 1 = 3 z 2 ¼ z 5 = 3 ð 2 : 170 Þ

Since the vertical mass flux increases with height, there must be turbulent

entrainment of environmental air into the plume.
In a thermal, unlike a plume, time t is also a significant independent variable,
rather than just height z. For simplicity we will assume that the environment of
the thermal is neutrally stratified. We assume that the vertical velocity, buoyancy,
and radius of the spherical thermal bubble are functions of the volume-integrated
buoyancy of the thermal Q as

ððð BdV

Q ¼ ð 2 : 171 Þ

where V is the volume and the units of Q are m 4 s 2 .So
w ¼ Q a z b f ð r Þ ð 2 : 172 Þ

where r is the radius of the thermal bubble. To be dimensionally correct
4a þ b ¼ 1 ð 2 : 173 Þ

and

2a ¼ 1 ð 2 : 174 Þ

So

w ¼ Q 1 = 2 z 1 f ð r Þ ð 2 : 175 Þ

Similarly, it follows that

B ¼ Qz 3 f ð r Þ ð 2 : 176 Þ

and

R ¼ Cz ð 2 : 177 Þ

where C is a positive constant. To find the variation with time, we make use of the
relation between vertical velocity and height (w ¼ Dz Dt) and (2.175). Integrating

(2.175) from the ground (z ¼ 0) to an arbitrary height z from t ¼ 0 to an arbitrary
time t, we find that

=

2 Q 1 = 2 f ð r Þ t z 2

178 Þ
so that z t 1 = 2 and therefore w t 1 = 2 (and w z 1 ). In other words, vertical
velocity decreases with time as the thermal rises. It can similarly be shown that
buoyancy B t 3 = 2 (and B z 3 ) (i.e., buoyancy decreases with time as the
thermal rises). It decreases because it entrains neutrally buoyant environmental air
(as the volume of the thermal increases, it must be taking in outside air and
mixing with it). Since the mean radius of the thermal increases linearly with height
(2.177), the thermal traces out a conical cross section as it ascends, like a plume
(2.169).

= ð 2 :

2.8.2 The plume dynamical model

We now return to the idealized plume model, but this time we will use the
Boussinesq continuity equation, the vertical equation of motion, and the adiabatic