The set of time-dependent, single-mode, mean-field equations appropriate for mantle convection has been converted by means of the method of lines formulation to a set of non-linear ordinary differential equations. The resulting system has been integrated in time t to steady-state by means of differential/algebraic system F(t, y, dy dt) = 0, in which the solution vector y consists of the vertical velocity, the thermal perturbation and the mean temperature and the spatial operators are finite differenced by high order schemes (up to 4th order). This large differential system-up to 500 equations-can be solved conveniently and economically by employing variable step-size, variable order (up to 5th order) stiff method such as backward differentiation formulas. We have studied thermal convection of infinite Prandtl number fluids with constant viscosity for both base heated and constant internally heated configurations. Both stress-free and rigid boundaries have been examined for these two modes of heating. Solutions for Rayleigh numbers up to 104 times of the critical value have been obtained. Heat transfer coefficients in the relationship between the Nusselt and Rayleigh number of the form Nu = aRab are derived from the steady-state solutions. In general, the exponent b is ∼ 10% higher than the corresponding value obtained from asymptotic or numerical solutions of the full set of 2-dimensional equations. We have also examined the effects of time-dependent heating and different types of initial conditions on the subsequent thermal evolution. The role of time-dependent heating from radioactive sources is to lengthen substantially the period of time during which the system is still influenced by the initial conditions. This time of thermal equilibration is found to vary as cRa-d, where d = 0.24 and is related to the heat-transfer coefficient. The constant c depends on the decay time of the heat sources. For Ra between 106 and 107 this time of readjustment is one-tenth of the thermal diffusion time of the entire layer and is 0(109 y) for upper-mantle convection, thus suggesting that modeling thermal history in the Archean may depend on earlier primeval events of the Earth.
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We are very much indebted to Bob Kee and Linda Petzold for showing us the DAE method. We have also benefited from fruitful discussions with Uli Christensen, Peter Olson, Fritz Busse and Lute Fleitout. Prof. Enzo Boschi has provided continuous encouragement in this and other matters. This research has been supported by NSF. grants EAR-8117439 and EAR-8214094 and by Petroleum Research Foundation grant 13550-G2, administered by the American Chemical Society.