Jökull - 01.12.1973, Page 22

and 650 km. On the basis of high-pressure
laboratory data, Anderson (1967) and Ring-
wood and Major (1970) come to the conclusion
tliat the upper discontinuity probably repres-
ents a solid-solid phase transition in (MgFe)2-
•SiOj from an olivine to a spinel structure.
Moreover, Anderson (1967) suggests that the
lower discontinuity may represent a collapse of
the spinel structure to a phase composed of
MgO, FeO, and SÍO2. The implications of
these transitions cannot be ignored in any dis-
cussion of mantle convection. It should be not-
ed that whereas the olivine-spinel transition is
a normal one with a positive slope Clapeyron
curve, the lower transition is believed to be
anormal with a negative Clapeyron curve.
Phase transitions within the convecting fluid
considerably complicate the mathematics of the
Rayleigh convection model. Additional non-
linear terms are introduced and the stability
problem becomes more involved, in particular
in the case of an internally heated fluid. Schu-
bert et al. (1970), Schubert and Turcotte (1971),
and Busse and Schubert (1971) have shown that
an approximate solution to the stability pro-
blem for a layer of a normal two-phase fluid,
heated from below, can be obtained on the
basis of the well-known method of lineariza-
tion which has been applied in most problems
involving Rayleigh convection. On the other
hand, although a comprehensive discussion of
the Rayleigh model involving a homogeneous
internally heated fluid has been given by
Roberts (1967), the case of the internally heat-
ed two-phase fluid has not been treated in the
literature. Moreover, the implications of anorm-
al phase transitions have not been discussed.
The purpose of the present paper is to dis-
cuss the stability problem for layers of internally
heated normal or anormal two-phase fluids on
the basis of a much simplified physical model
which furnishes useful results relevant to the
mantle convection problem. This is achieved
by the introduction of a one-dimensional ‘strip-
model’ approximation for the convective flow.
As a result of the mathematical simplifications
obtained, this model can be applied to furnish
useful stability criteria in many cases involving
convective phenomena in complex geophysical
systems where the classical Rayleigh method
fails. In particular, it opens up the possibility
20 JÖKULL 23. ÁR
of investigating cases involving non-homogene-
ous, non-Newtonian fluids. Moreover, finite
amplitude convective flows can be studied by
similar methods. We will begin by pointing
out the physical relationship between the ‘strip-
model’ and the Rayleigh model by showing
that the ‘strip-model’ method furnishes good
results in two cases of Rayleigh convection
which have been solved by the classical method.
THE STRIP MODEL - DESCRIPTION
AND BASIC EQUATIONS
Since the strip model to be introduced below
is a further simplification of the Rayleigh
model, it is important to first point out the
many physical approximations which the Ray-
leigh model itself entails. Consider a horizontal
layer of an incompressible Newtonian fluid of
thickness h, zero surface temperature and a
constant bottom temperature To- Let the z-axis
be vertical, z be the unit vertical vector, and
let /3 = To/h. The linearized perturbation
equations of the Rayleigh model may be written
(.Jeffreys, 1926, 1928):
Vp' — vpV2 U + agpT'z = 0 (1)
uz/3 = aV2T' (2)
V- u = 0 (3)
where p' and T' represent the perturbation
pressure and temperature respectively. More-
over p denotes the kinematic viscosity, p
the density, a. the thermal expansivity, and a
the thermal diffusivity. The perturbation velo-
city is u = (ux, uy, uz), and the acceleration of
gravity is g. The non-linear terms which are
being neglected are the convective acceleration
p(“ ‘ V/u (4)
second order heat transport terms
u • VT' (5)
and the viscous dissipation of heat