The basic equations - Severe Convective Storms and Tornadoes: Observations and Dynamics - page 71

noted ''M. Be´ nard does not appear to be acquainted with James Thomson's paper
''On a changing tesselated structure in certain liquids'' (Proc. Glasgow Phil. Soc.,
1881-2), where is described a like structure in much thicker layers of soapy water
cooling from the surface.'' Alas, Rayleigh has provided more evidence for Stigler's
Law of Eponymy. The essence of Lord Rayleigh's mathematical analysis is now
summarized.

2.9.1 Convection in a resting atmosphere without rotation
The Boussinesq equations are used. The equations of motion are expressed in
terms of density in the expression for the pressure gradient force, viscosity is now
included in all three components of the equations of motion, and thermal conduc-
tivity is included in the thermodynamic equation, which is now expressed in terms
of temperature, rather than potential temperature. The equations linearized about
a resting basic state, in which the base-state pressure varies only with height and
temperature varies linearly from the bottom plate to the top plate (separated by a
distance H) ( Figure 2.16 ) and is held fixed as a result of heating, may be expressed
as follows:

2

Þ u 0 ¼ 1

p 0

= @ =@ ð 2 :

203 Þ

ð@=@ t r x

Þv 0 ¼ 1 p 0

2

ð@=@ t r = @ =@ y ð 2 : 204 Þ

2

Þ w 0 ¼ 1 p 0 ð@=@ t r = @ =@ z þ B ð 2 : 205 Þ

2

Þ T 0 ¼ w 0

0

ð@=@ t r ð 2 : 206 Þ

u 0 x þ@v 0 w 0 @ =@ =@ y þ@ =@ z ¼ 0 ð 2 : 207 Þ

is the kinematic coecient of molecular viscosity; 5

where is the coecient of

0 is the mean temperature gradient between
the two parallel plates. It should be noted that when linearizing the equations, it
cannot be assumed that the perturbation wind velocities are small in magnitude
compared with the basic-state wind velocities, because the basic state is resting.
However, if the perturbation winds are expanded in a Taylor series expansion in
time, it is seen that products of perturbation terms may be neglected in compar-
ison with terms containing only one perturbation term for a very short period of
time after the initial time. Later on, the nonlinear terms cannot be neglected. Now
suppose the fluid between the two plates has the following characteristics:
¼ 1

thermal conductivity/diffusivity; and

= @=@ T ð 2 : 208 Þ

where is the coecient of thermal expansion and the constant

¼@=@ T ð 2 : 209 Þ

Now

ð T ð

T 0 ¼ T

0

T ¼

T @

T =@ d ¼ 1 = d ¼ = ð 2 : 210 Þ

av

5 The kinematic coecient of eddy viscosity if eddies are considered rather than molecules.