Jökull - 01.12.1973, Qupperneq 36

TABLE 3.
Critical Rayleigh numbers for a two-phase mantle-wide layer (based on (mks. units)).
h = 3 X 10«; A = 10-12; a = 2 X 10-6; c = 103; g _ 10;
« = 2x10-5; pí = 10-3
Transition Critical Rayleigh No. W. Phase Change Critical Rayleigh No. W/O Phase Change % Difference
Mg2Si04 -> Mg2Si04 forsterite -» spinel 1830 2000 -8
Mg2Si04 -» 2MgO + Si02 spinel -» oxides 2690 2000 + 34
Fe2Si04 -» Fe2Si04 fayalite -» spinel 1880 2000 -6
Fe2Si04 -» 2FeO + Si02 spinel -» oxides 2350 2000 + 18
For the range of viscosity 1015 to 1022 m2/sec,
the Rayleigh number for the entire mantle falls
in the range
103 5? R ^ 10io
Therefore for mantle-wide convection, the vis-
cosity is the dominant factor as to the dynamic
state of the mantle. The known phase transi-
tions are of only minor importance.
Next consider convection in the upper mantle
only. Because h is smaller, the critical number
is more affected by the phase transitions. Sup-
pose, for example, that convection occurs in
the range 100—700 km. Then the ratio di/h is
0.5 and 0.9 respectively. We have calculated
the critical number for this case with the as-
sumption that /3^ = 10-3 °C/m despite the smal-
ler value of h. This appears reasonable since
the heat production in the upper mantle is
probably greater than 10—i2 w/kg. The results
of these calculations are listed in Table 4.
Table 4 shows that the results for a 600 km
layer are essentially the same as for convection
in the entire mantle, though the phase transi-
tion effects tend to be larger. The spinel-oxides
transition in Mg2SiC>4 raises the critical number
by a factor of 2.
For viscosity in the range 10ið to 3 X 1017
m2/sec, the Rayleigh number falls in the range
3 x 104 ^ R ^ 107
If the Rayleigh number is actually of the order
of 3 X 104, the effect of the spinel-oxides transi-
tion may be to restrict convection to depths
less than 650 km.
We wish to point out that the negative terms
in the denominator of (58) are less than unity
for the mantle transition parameters. More-
over, had we used £ = 0.28, which appears to
be the appropriate value for an internally heat-
ed homogeneous fluid, the results would be
essentially the same.
CONCLUDING REMARRS
We have developed a simple, one-dimension-
al model which is suitable for treating convec-
tion in complex geophysical systems. As an
example, we have examined the important pro-
blem of convection in an internally heated
34 JÖKULL 23. ÁR