The basic equations - Severe Convective Storms and Tornadoes: Observations and Dynamics - page 72

Figure 2.16. Setup for Rayleigh-Be´ nard convection.

It then follows that

T 0 B ¼ g ð 2 : 211 Þ

We define

0

¼ g ¼ constant ð 2 : 212 Þ

(Note that this is not the same as the entrainment rate in (2.179).) From (2.211)

and (2.212), we can now express the thermodynamic equation (2.206) in terms of
buoyancy, rather than temperature, as

2

Þ B ¼ w 0

ð@=@ t r ð 2 : 213 Þ

So, we have five equations in five unknowns (u 0 , v 0 , w 0 , p 0 , and B). We now drop
the prime notation to simplify writing of the equations, where it is understood that
u, v , w, and p represent perturbation quantities.
Before we begin our analysis, we digress briefly to point out, for historical
reasons, that Barry Saltzman in 1962 reported on a study of Rayleigh-Be´ nard
convection that employed his simplifying assumption that motions are two dimen-
sional in the x-z-plane (i.e.,

y ¼ 0: that there are convective rolls). He formed
a horizontal vorticity equation from (2.203) and (2.205). The resulting vorticity
equation and thermodynamic equation, which is essentially (2.51), form a system
of two equations for the streamfunction in the x-z-plane (the Boussinesq continu-
ity equation states that the flow in the x-z-plane is non-divergent, so that a
streamfunction can be defined). Incidentally, Saltzman noted the similarity of his
approach to that of Joanne Malkus (Simpson) and G. Witt a few years earlier. Ed
Lorenz, in his famous and influential 1963 paper ''Deterministic nonperiodic flow''
in which he first described the essence of chaos theory, used these equations as the
basis for his study. So, it turns out that the equations that Rayleigh used to
describe Be´ nard convection were the prototypical nonlinear equations that were so
influential in the argument for limits to the predictability of weather systems.
Instead of forming a vorticity equation as did Saltzman et al., we now
eliminate p from (2.203) and (2.204) to form a horizontal divergence equation
and, using the equation of continuity, get the following:

@=@

2 2 w

z 2

2 p

x 2

2 p

y 2 ð@=@ t r Þ@ =@ ¼ 1 = @=@ z ð@ =@ þ@ =@ Þ ð 2 : 214 Þ

Thus, both u and v have been eliminated. We now differentiate the vertical
equation of motion twice with respect to x, and also twice with respect to y.