### Abstract:

Two thermal convection problems of geophysical interest are
examined, theoretically. First, convection in the earth's mantle is
treated on the basis of a one-dimensional 'strip model'. This model
results from further simplification of the well known 'Rayleigh model'.
For homogeneous, Newtonian fluids, the strip model yields results
similar to those obtained by the Rayleigh method.
The strip model is used to determine the critical Rayleigh
number for convection in an internally heated two-phase fluid. The
critical number depends on the parameters of the phase transition,
the physical properties of the fluid, and the depth of the fluid layer.
Depending on these factors, a univariant phase transformation may
either enhance or hinder convective instability. For the olivine-spinel
and spinel-oxides transitions in (MgFe)₂SiO₄ which are thought
to take place in the upper mantle, it is shown that the critical Rayleigh number is altered only slightly from the critical number for convection
in a fluid with one phase. This result holds both for convection in the
entire mantle or convection restricted to the upper mantle. Hence the
phase changes are of minor importance regarding the existence of
mantle convection in general.
A method for estimating the order of magnitude of the displacement
of the phase surface as a function of Rayleigh number is outlined
for a fluid with only one phase transition. The strip model is also
used to treat convection in non-Newtonian fluids obeying a power law
rheological equation. If the mantle is governed by a flow law of this
type, it appears that convection can take place. Lastly, the procedure
for applying the strip model to fluids with variable viscosity and
thermal conductivity is outlined.
The second convection problem concerns some aspects of convection
of fluids in thin vertical fractures in the crust. A steady
state model is developed to estimate the magnitude of the mass flow
as a function of fracture thickness. It is shown that fractures of the
order of a millimeter thick or greater can carry a measurable convective
flow. A time dependent model is used to estimate the rate of
decay of the mass flow with time. The results indicate that in fractures
of the order of a centimeter thick, a measurable decrease of
the mass flow takes place after a period of the order of a day. This
rapid decay rate suggests that the principal effect of sea water convection in extensive fracture systems which are expected on mid-ocean
ridge crests is to cool a volume of crustal rock in the vicinity
of the fractures. Circulation of sea water in vertical fractures in the
upper crust may provide an explanation of 1) the relatively low conductive
heat flow measured at some locations on ocean ridge axes
and 2) the very 'noisy' data obtained in the axial zone.