substituting these series into (17) and
(18). He arranged the right-hand sides of the resulting
equations in double-Fourier-series form, by replacing products
of trigonometric functions of x (or z) by
sums of trigonometric functions, and then equated coefficients
of similar functions of x and z. He then reduced
the resulting infinite system to a finite system by omitting
reference to all but a specified finite set of functions
of t, in the manner proposed by the writer (1960).
He then obtained time-dependent
solutions by numerical integration. In certain cases all
except three of the dependent variables eventually tended
to zero, and these three variables underwent irregular,
apparent1y nonperiodic fluctuations.
These same solutions would have
been obtained if the series had at the start been truncated
to include a total of three terms. Accordingly, in this
study we shall let
where X, Y, and Z are functions
of time alone. When expressions (23) and (24) are substituted
into (17) and (18), and trigonometric terms other than those
occurring in (23) and (24) are omitted, we obtain the equations
Here a dot denotes a derivative with
respect to the dimensionless time τ=π2H-2(1+a2)kt,
while σ=k-1υ is the Prandtl
number, r=Rc-1Ro, and
b=4(1+a2)-1. Except for multiplicative
constants, our variables X, Y, and Z are
the same as Saltzman's variables A, D, and G.
Equations (25), (26), and (27) are the convection equations
whose solutions we shall study.
In these equations X is
proportional to the intensity of the convective motion,
while Y is proportional to the temperature difference
between the ascending and descending currents, similar signs
of X and Y denoting that warm fluid is rising
and cold fluid is descending. The variab1e Z is proportional
to the distortion of the vertical temperature profile from
linearity, a positive value indicating that the strongest
gradients occur near the boundaries.
Equations (25)-(27) may give realistic
results when the Rayleigh number is slightly supercritical,
but their solutions cannot be expected to resemble those
of (17) and (18) when strong convection occurs, in view
of the extreme truncation.
6. Applications of linear theory
Although equations (25)-(27), as they stand, do not have the form
of (4), a number of linear transformations
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will convert them to this form. One of the simplest of
these is the transformation
Solutions of (25)-(27) therefore remain
bounded within a region R as r → ∞,
and the general results of Sections 2, 3 and 4 apply to
these equations.
The stability of a solution X(r),
Y(r), Z(r) may be formally investigated by considering
the behavior of small superposed perturbations xo(r).
yo(r), zo(r). Such perturbations
are temporarily governed by the linearized equations
Since the
coefficients in (29) vary with time, unless the basic state
X, Y, Z is a steady-state solution of (25)-(27),
a general solution of (29) is not feasible. However, the
variation of the volume Vo of a small
region in phase space, as each point in the region is displaced
in accordance with (25)-(27), is determined by the diagonal
sum of the matrix of coefficients; specifically
This
is perhaps most readily seen by visualizing the motion in
phase space as the flow of a fluid, whose divergence is
Hence each small volume shrinks to zero
as r → ∞, at a rate independent of X,
Y, and Z. This does not imply that each small
volume shrinks to a point; it may simply become flattened
into a surface. It follows that the volume of the region
initially enclosed by the surface S shrinks to zero
at this same rate, so that all trajectories ultimately become
confined to a specific subspace having zero volume. This
subspace contains all those trajectories which lie entirely
within R, and so contains all central trajectories.
Equations (25)-(27) possess the
steady-state solution X = Y = Z =
0, representing the state of no convection. With this
basic solution, the characteristic equation of the matrix
in (29) is
This
equation has three real roots when r>0; all are
negative when r< 1, but one is positive when r>1.
The criterion for the onset of convection is therefore r=
1, or Ro=Rc in agreement
with Rayleigh's result.
When r> 1, equations
(25)-(27) possess two additional steady-state solutions
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