Geology Reference
In-Depth Information
Fig. 13.13
Critical Rayleigh number
Ra
cr
for the onset of Rayleigh-Bénard convection in a layer of thickness
H
,asa
function of the dimensionless parameter 2
H
/œ
satisfied the process of thermal equilibration is
too fast for allowing motion under the combined
effect of the buoyancy force and the viscous drag.
This suggests that a single parameter can be used
to measure the capacity of thermal anomalies
to be transported by buoyancy forces through a
fluid. In fact, using the expression (
13.141
)we
see that the condition for thermal convection,
L
/
a
1, can be expressed as follows:
Ricard
2007
). In this instance, the Rayleigh
number and the critical value are defined as
follows:
'¡H
3
gT
ǜ›
D
'H
3
gT
›
Ra
D
(13.146)
œ
2
2
3
4
2
H
2
œ
2
C
Ra
cr
D
(13.147)
4
2
H
2
'¡a
3
gT
ǜ›
D
'a
3
gT
›
Ra
D
>> 1 (13.145)
where œis a characteristic wavelength of the ther-
mal fluctuations. If
Ra
<
Ra
cr
, any fluctuation
will decay with time, whereas for
Ra
>
Ra
cr
perturbations will grow exponentially with time.
The critical Rayleigh number depends from the
dimensionless parameter 2
H
/œ as illustrated in
Fig.
13.13
. For any fluctuation wavelength œ,if
Ra
lies above the curve, then the correspond-
ing perturbation generates instability. Conversely,
thermal convection is freezed for all wavelengths
such that
Ra
<
Ra
cr
.
The curve of Fig.
13.13
shows that the stability
curve has an absolute minimum. It is easy to find
the value of 2
H
/œ such that the curve attains a
minimum. This is given by:
where is the kinematic viscosity and we
have suppressed the multiplicative constant
10/9
1. The dimensionless parameter
Ra
is
called
Rayleigh number
. It represents the relative
importance of the buoyancy forces with respect to
viscous drag and rate of heat diffusion. However,
the form (
13.145
) is not quite general to be useful,
because here
Ra
depends from the blob radius
a
.
Furthermore, it is still unclear how much greater
than one should be
Ra
to trigger free thermal
convection. In general, a linear stability analysis
of the Boussinesq equations for a fluid layer of
thickness
H
, heated from below and cooled from
above, shows that a minimum
critical Rayleigh
number
,
Ra
cr
, exists for the onset of thermal
convection (e.g., Turcotte and Schubert
2002
;
2 H
œ
D
p
2
(13.148)
