The basic equations - Severe Convective Storms and Tornadoes: Observations and Dynamics - page 76

(2.238) to get the following equation for ! i :

Ra k 2 =ð k 2 þ n 2

2 2

i þð k 2

þ n 2

2 2 2

Þ¼f! Þ ½ð 1 Þ = 4 g ð 2 : 239 Þ

The LHS of (2.239) is always ! i

must be zero (i.e., if the Rayleigh number is greater than zero), and if the bottom
plate is maintained at a warmer temperature than the top plate, small-amplitude
oscillations are not possible. So, let us return to (2.238) and find out what values
of

> 0. However, the RHS is always < 0. Therefore

! r are allowed when ! i ¼ 0. We find that

½! r þ k 2 þ n 2

2

½! r þ k 2 þ n 2

2

ð k 2 þ n 2

2

Þ¼ Ra k 2 ð 2 : 240 Þ

It appears from (2.240) that 0 (i.e., that there could be

instability or stability), depending on what the values of Ra, n, k, and

! r can be either > 0or <

are. For

''marginal'' stability, ! r ¼ 0, so that

Ra ¼ð k 2

þ n 2

2 3

k 2

241 Þ
What is the smallest value of the Rayleigh number and what is the value of k at
this smallest value for n ¼ 1, which is the shortest vertical mode possible? If within
the expression for the Rayleigh number, only the horizontal temperature gradient
between the plates varies, then by cranking up the temperature gradient we finally
arrive at the ''critical'' Rayleigh number. (The n ¼ 0 case is ''degenerate'', so that
w ¼ 0 everywhere and is therefore not considered.) This minimum value of the
Rayleigh number is called the ''critical'' value of the Rayleigh number (Ra c )and
is found by differentiating (2.241) with respect to k and setting it to zero to find
that the critical horizontal wave number, which is
k c ¼

Þ = ð 2 :

2

242 Þ
The critical value of the Rayleigh number is found by substituting (2.242) into
(2.241), so that

= 2 ð 2 :

4

Ra c ¼ð 27 = 4 Þ 657 : 5 ð 2 : 243 Þ

When the critical Rayleigh number is exceeded, ! i ¼ 0, so that

''stationary overturning'' is possible. When the temperature gradient is turned up
so that the critical Rayleigh number is exceeded, the first mode to go unstable
(Ra c increases with increasing n; cf. (2.241)) is that described by (2.242), or since

! r > 0 and

p 2

k c ¼ 2 = L c ¼ = ð 2 : 244 Þ

where L c is the critical horizontal wavelength. Thus
L c ¼ 2 2

p

ð 2 : 245 Þ

Since x ¼ Hx and y ¼ Hy [ cf. (2.217)],
L c ¼ 2 p H

ð 2 : 246 Þ

which is a nice, simple result. Eventually, of course, as the size of the
perturbations grows, linear results no longer hold and matters become more com-
plicated. However, theory does dictate what the spacing of the first-appearing cells
must be.
Now, (2.246) tells us only what the critical horizontal wavelength is, but does