Some Aspects of Thermal Convection Theory for the Earth's Mantle

Abstract

Summry A serious obstacle to developing a theory of the Earth’s thermal history, allowing for the possibility of convection, is the non-linearity of the convection equations. In particular, the non-linearity arising from viscosity variation with temperature is very marked. Instead of treating viscosity either as an adjustable parameter or assigning it some value derived from indirect observation, one would like to know what value it would assume from a consideration of convection theory itself, since this might largely determine the course of thermal history. Although an analytical approach to this is impossible, if one uses a reasonable hypothesis derived from Btnard cell experiments, an answer to this problem for the Upper Mantle can be suggested. The purpose of this paper is to draw attention to a few of the major problems that confront one in setting up a realistic theory of convection for the Earth‘s mantle and wh.ich are to a considerable extent a feature of any planetary convection theory. The first of these is in the nature of a brief reminder about the possibility of ever creating a self-contained theory of large-scale motions in the Earth‘s mantle. Although we do not have at present any direct observational data on long-term movements in the mantle, we have sufficient circumstantial evidence from crustal geology, vulcanology and seismology to make us believe that mantle and crustal motions occur over a very large range of length scales. We therefore have considerable a priori reasons for thinking that a complete theory of motions in the Earth’s interior will have a great complexity, comparable, perhaps with that for the atmospheric motions. As in meteorology one tries to break the total problem up into manageable pieces and what is normally described as a ‘ Theory of Mantle Convection ’ has its parallel in the problem of the general atmospheric circulation. However, this division of the problem raises a number of formal difficulties, which have their origin in the nonlinearity of the appropriate equations of hydrodynamics. This nonlinearity means that there is interaction between motions of all length scales and that a statement of initial and boundary conditions does not suffice to specify a theory of the motions within a limited length scale range. In the case of the atmospheric motions, which are more or less accessible to observation, a theory of motions within a restricted range of length scales has some prospect of success because it can be supplemented with empirical obFervations on motions outside the range of interest. Such observations would indicate how and to what extent the coupling of motions occur. In the absence of such detailed observational data, which is the situation with regard to mantle theories, it is an insoluble problem how to describe the interaction of large- and small-scale motions. What is more, unless there turns out to be a fairly well defined gap in the power spectrum of internal motions separating the largest scale motions of interest and the rest, the prospect of obtaining an adequate