The basic equations - Severe Convective Storms and Tornadoes: Observations and Dynamics - page 73

Adding the resulting two equations to (2.214), we see that p is eliminated and the
following equation is the result:

2 2 w ¼ð@ 2

x 2

2

y 2 ð@=@ t r Þr =@ þ@ =@ Þ B ð 2 : 215 Þ

We now seek to eliminate B so that we can get one equation in terms of one
dependent variable, namely w. To do so, we apply the operator ð@=@

2

t r Þ to

2

x 2

2

y 2 (2.215) and apply the operator ð@ Þ to the thermodynamic equation

(2.213) and then subtract the latter from the former to get

=@ þ@ =@

2 2 2 w ¼ ð@ 2

x 2

2

y 2 ð@=@ t r Þð@=@ t r Þr =@ þ@ =@ Þ w ð 2 : 216 Þ

This is a sixth-order partial differential equation in terms of the vertical velocity w.
Three physical constants/parameters that describe the characteristics of the

problem are . The reader is reminded that the former two quantify the

molecular diffusion of momentum and heat, while the latter quantifies the tem-
perature gradient between the two plates and the characteristics of the fluid
between them.
Before we continue on in the solution of this equation, it is useful from the
standpoint of simplification to recombine these three parameters into two. The
procedure follows that of Willem Malkus and George Veronis in 1958: we rewrite
the equations by scaling each spatial variable by H as follows:
x ¼ Hx

, ,and

y ¼ Hy and z ¼ Hz ; ; ð 2 : 217 Þ

H is the typical space scale in units of space (e.g., meters) and it is assumed that
the horizontal and space scales are probably similar (isotropic); in addition, H is
the spacing between the two plates. The variables expressed without an asterisk
are non-dimensional and O ð 1 Þ . The time scale is estimated from the equations

of motion (2.203)-(2.205), assuming that the inertial acceleration is the same order
of magnitude as the acceleration due to viscosity:

H 2 1 = = ð 2 : 218 Þ

so that

t ¼ H 2 = ð 2 : 219 Þ

The resulting equation, which is of sixth order in w, is as follows:

2 2 2 w ¼ Ra ð@ 2

x 2

2

y 2 ð@=@ t r Þð@=@ t r Þr =@ þ@ =@ Þ w ð 2 : 220 Þ

where the Prandtl (pronounced without the ''t'') number
¼ = ð 2 : 221 Þ

and the Rayleigh number

H 4

0 H 4

Ra ¼ =ðÞ¼g =ðÞ ð 2 : 222 Þ

The Rayleigh number varies monotonically and linearly as the temperature
gradient between the two plates (

0 ), monotonically, and very rapidly as the space
scale, but inversely proportional to both the kinematic coe cient of molecular vis-
cosity and the coecient of thermal conductivity/diffusivity. The physical meaning
of the Rayleigh number is that when buoyancy is the same order of magnitude as
the viscous term in the vertical equation of motion (i.e., the flow is laminar), it