During 1991-1994, the four year period covered by this report, there has been a
shift in emphasis away from one-dimensional (radial) mantle viscosity models and
toward an understanding of the three-dimensional (radial and lateral) structure
of mantle viscosity. Our understanding of mantle viscosity comes primarily from
two sources: studying the creep properties of mantle minerals under appropriate
pressures and temperatures and using theoretical flow models to predict surface
observables, such as: plate velocities, gravity, geoid, and heat flow.
Laboratory measurements of deformation indicate that the rheology of upper
mantle minerals such as olivine ((Mg,Fe)
SiO
) is a strong function of
temperature, grain size, and stress [cf., Karato and Wu, 1993]. The
deformation of minerals under mantle conditions generally follows a flow law of
the form
where 
is the deformation rate, 
is
the deviatoric stress, 
is the shear modulus, d is the
grain size of the rock, Q is the activation energy for the deformation
mechanism, T is the temperature in Kelvin, R is the gas constant and A is a
constant. Viscosity is defined as
therefore, deformation is directly related to viscosity. Substituting equation (1) into equation (2), an effective viscosity,
can be defined. For temperature changes of 100 degrees K, the viscosity changes by an order of magnitude at constant stress [cf., Karato and Wu, 1993]. Changes of deviatoric stress by a factor of 2 change the viscosity by an order of magnitude [cf., Karato and Wu, 1993]. Other factors, such as partial pressure of oxygen and water content may also have important effects.
Two creep mechanisms are likely to dominate in the mantle; diffusional flow
(corresponding to n=1 in Equation (1)) and power-law creep (corresponding to
n>1 in Equation (1)). A rheology with a linear stress, deformation-rate creep
mechanism, such as diffusional flow, is referred to as a Newtonian rheology.
The question of which mechanism dominates in the mantle depends on the average
grain size of the mantle minerals. Ito and Sato [1991] found a significant
grain-size reduction after the transformation of (Mg Fe)
SiO
into (Mg
Fe)SiO
perovskite plus (Mg,Fe)O magnesiowüstite, and suggest this could
weaken the rheology of the uppermost part of the lower mantle. From
Equation (3) it is clear that reducing grain size, d, leads to a reduction in
all other things being held equal. Based on a study of
polycrystaline CaTiO
perovskite, a low pressure analog for the (Mg
Fe)SiO
perovskite in the lower mantle, Karato and Li [1992] find that
when the perovskite transforms from the orthorombic structure to the tetragonal
structure, creep is enhanced while the stress exponent remains unchanged. This
suggests that grain-size reduction occurs as a result of the phase
transformation in the analog system. Karato and Li further suggest that
the lower mantle may be controlled by diffusion creep. It should be noted that
the strain rates achieved in the lab (typically 10
--10
s
)
are much larger than those predicted in the lithosphere and mantle
(
s
).
Because of the difficulties in interpreting and applying laboratory creep measurements to mantle conditions, models of mantle viscosity based on large-scale geophysical observations continue to play an important role. Viscosity models deduced from these observations are not unique, however, and require knowledge of models for the surface load for post-glacial rebound studies [cf., Tushingham and Peltier, 1991] or internal density anomalies for geoid and plate velocity studies. Both ice sheet models and internal density models have their own associated uncertainties adding to the complexity of the problem. In addition, the theoretical models are often simplified to keep them tractable; commonly, a linear rheology that varies only with depth (i.e., n=1 and T constant in Equation (1)), is assumed. One of the most interesting new results from studies of radial mantle viscosity structure is the analysis of the radial resolution of the models. Once the resolution of the models are considered, it appears that some of the differences between viscosity models based on different observations may reflect the lack of resolving power in the data sets.
Until recently, little was known about the effect of lateral viscosity variations on surface observables. It is crucial to understand the effects of lateral viscosity variations, because the majority of mantle viscosity studies have assumed that the long wavelength flow is unaffected by lateral variations in viscosity. In the upper mantle, the lateral variation in viscosity could exceed the radial variation in viscosity. Consider the following argument. The viscosity of olivine increases by one order of magnitude for every decrease in temperature of 100 degrees. Based on simple thermal models, the center of a subducted slab is 500-800 degrees colder than the mantle surrounding the slab. Therefore, it would not be unreasonable to expect 5-8 orders of magnitude variation in viscosity from the center of the slab to the ambient mantle. This greatly exceeds even the most extreme radial viscosity models.
Due to the limitations of space, this review will focus on the results of theoretical flow models. Readers who want to follow the laboratory results in greater depth should consult Karato and Wu [1993] and Poirier [1994]. I have organized the review into two parts, one focusing on the resolution of radial viscosity models and the other focusing on the effects of lateral variations in viscosity. Other recent reviews on mantle viscosity include those by Peltier [1989], Hager [1991], and King [1994a].