Jökull - 01.12.1973, Síða 21

A One Dimensionaí Convection Model:
Application to an Internally Heated Two-phase Mantle
ROBERT P. LOWELL1 AHD GUNNAR BÖÐVARSSON
SCHOOL OF OCEANOGRAPHY
OREGON STATE UNIVERSITY, CORVALLIS, OREGON 97331
abstract
We simplify the well known Rayleigh con-
vection model by neglecting horizontal heat
conduction, using an average flow velocity, and
assuming the cell size on physical grounds. For a
homogeneous Newtonian fluid layer, the result-
lng one dimensional ‘strip-model’ yields results
similar to those obtained by the classical linear-
iiation method.
Ry applying the strip model in the case of
an internally heated fluid with two phases, we
obtain a new critical Rayleigh number which
ls a function of the transition parameters, the
physical properties of the fluid, and the depth
°f the fluid layer. We treat both normal and
abnormal transitions, that is transitions with
either a positive or negative slope Clapeyron
curves respectively. For the olivine-spinel and
spinel-oxides transitions which are thought to
occur in the upper mantle, the new critical
number deviates little from the critical number
for a homogeneous fluid. Hence the known
phase transitions appear to be of minor import-
ance with regard to the existence of mantle
convection.
The one dimensional model can also be ap-
plied in the study of convection phenomena
tnvolving other complex geophysical systems.
INTRODUCTION
Although the plate tectonics model of litho-
spheric motion is now a widely accepted theory,
little is known about the physical causes o£
1) Now at School of Geophysical Sciences,
Georgia Institute of Technology, Atlanta,
Georgia 30332.
this global phenomenon. The available observa-
tional data provide only a kinematic, empirical
description of the surface motion and yield few
clues as to the driving mechanism involved.
Thermal convection in the mantle has general-
ly been invoked as the underlying cause (Hess,
1962; Runcorn, 1962, 1969), but there is no
agreement as to details of the convection
system. The simple Rayleigh model o£ cellular
convection in a homogeneous, incompressible
Newtonian fluid layer heated from below has
frequently provided a convenient basis for dis-
cussions of mantle convection. This model is
clearly an oversimplification, since the mantle
cannot be described by a homogeneous New-
tonian fluid. Recently, other more realistic
variants of the Rayleigh model have been sug-
gested (Allan, et al., 1967; Tozer, 1965; Els-
asser, 1969; Turcotte and Oxburgh, 1967; Kno-
poff, 1970; and Morgan, 1971). The results
obtained to date indicate that thermal convec-
tion is a possible driving mechanism. However,
a number of difficulties still have to be over-
come before the thermal convection theory can
be accepted as the final solution of the plate
motion problem. The most serious physical
problems stem from (1) lack of reliable data
on the thermal properties of the mantle materi-
al, particúlarly the radiative component of the
thermal conductivity; (2) considerable uncer-
tainty as to the rheological properties; and (3)
complications due to internal heating and in-
homogeneities within the mantle. Of particular
interest are the results of recent seismic in-
vestigations (Johnson, 1967; Archambeau et al.,
1969) which have indicated discontinuities in
the mantle at the depth of approximately 400
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