Jökull - 01.12.1973, Side 33

Therefore, for convection to be possíble, for a
givn parameter an upper limit is placed
on the latent heat. Alternatively, for a given
latent heat, (64) places a lower limit on
Since is mainly a function of the heat pro-
duction A, (64) sets a lower limit on A. For a
wide range of values of RM and R^q , Fig. 7
shows that the critical number for a two-phase
fluid is lower than that for a single phase fluid.
These results agree in general with those of
Schubert et al. (1970), Schubert and, Turcotte
(1971) and Busse and Schubert (1971) for the
simpler case of a fluid heated from below.
Fig. 8 shows the corresponding curves for a
transition with a negative slope. In this case
Rc is plotted against | R^p | f°r various values
°f j Rlt | from the equation
R>
2000
•|RA9|+|RM|(l-(4/3)|RAe |)
(65)
where |RM| and |RAq | represent the absolute
values of the expressions given by (61) and (62)
respectively. It is apparent from comparing
(63) with (65) that Fig. 7 and 8 are identical
except that RM and R^q have been inter-
changed. For a transition with negative slope
the condition
hence
I rAo I <1
Ap
Pl2dgh|v|
<2/3
(66)
(67)
Ís obtained. This inequality may be looked
upon as placing a limit on the parameter
Ap /1 y | of the phase transition.
The above results would be modified slightly
if the transition does not occur at the mid-plane
of the layer. More critical to the validity o£ the
preceding analysis, however, is the assumption
rnade above that \ and £ are the same for one
phase and two phase fluids. On physical
grounds, one would not expect the critical
wavelength to change substantially. Fig. 2 o£
Busse and Schubert (1971) indicates a slight in-
erease in the critical wavelength as the phase
transition parameter P increases (P corresponds
to the parameter R^q of the strip model).
When P becomes infinite, however, \c goes to
Fig. 8. Critical Rayleigh number Rc for con-
vection in an anormal two-phase fluid as a
function of |RAq | l°r various values |RM|.
zero. Since the critical number given by (58) is
not strongly dependent on \, changes is \
should not seriously affect the above results
unless RAq becomes infinite.
The parameter £ is also indeterminate on
the strip model, and clearly the appropriate
value is not the same for a fluid heated inter-
nally as for a fluid heated from below. These
two types of convective flow behave somewhat
differently (Tritton and Zarraga, 1967). On
physical grounds one would tend to select a £
value slightly in excess of 0.35, but we do not
anticipate that our assumption of £ = 0.35 will
result in a serious error despite the sensitivity
of Rc to changes in £. Lastly we point out that
\ and £ affect only the definition of RM and
the value of Rc at the vertical axis. The shape
of the curves in Figs. 7 and 8 would remain
essentially unmodified. Therefore the strip
model results appear to satisfactorily depict the
principal effects of phase transitions on convec-
tion.
APPLICATION TO THE MANTLE
We shall now apply the strip model formula
(58) to estimate the effects of the olivine-spinel
and the spinel-oxides phase transitions on
JÖKULL 23. ÁR 31