Concluding remarks and discussions

We have investigated the horizontal cell sizes of mantle convection induced by continental plates, especially those of larger horizontal sizes than considered in previous numerical studies and laboratory experiments.
Numerical calculations reveal a Rayleigh number dependency of the horizontal scale of convection cells, with an upwelling under the plate and downwelling outside the plate. When the Rayleigh number is small, an upwelling develops beneath the margin of the plate, while when the Rayleigh number is large the upwelling occurs beneath the plate’s center.
By applying boundary-layer theory to the thermal convection beneath the plate, we have theoretically estimated the difference in average temperatures below and outside the plate. We have confirmed the theory by comparing the results with those from the numerical calculations. Further, by comparing the velocity amplitude induced by the horizontal temperature difference with that of the intrinsic thermal convection induced by vertical temperature differences, we have obtained the maximum horizontal size of the convection cell as a function of the Rayleigh number. The analysis reveals that convection cells with larger horizontal scales emerge for larger Rayleigh numbers, a finding consistent with the results of the numerical experiments.
The theory presented in this paper cannot predict the optimum horizontal scale of convection cells. It addresses whether the amplitude of convection induced by the horizontal temperature difference for a given horizontal scale can exceed that of the intrinsic convection, but we cannot predict the horizontal scale of convection cells which actually emerge at scales below the line shown in Figure 10 (u_h > u,u'). In order to find the optimal horizontal scale theoretically, we would need to develop a simple model, similar to a loop model (Guillou and Jaupart 1995) or a more sophisticated boundary-layer theory (Turcotte and Schubert 1982), to describe the dynamical balance and a convection cell extending beyond the plate margin. Even when using such models, it is necessary to apply some selection rules to determine the preferred horizontal scale, such as those producing structures corresponding to the most efficient heat transport.
In estimating the horizontal temperature differences, we have considered an idealistic situation in which convection occurs separately under the plate and outside it. This implies that our estimates of convective velocities are maximum values. Once convection is induced by a horizontal temperature difference, it will transport heat and reduce that temperature difference from its initial value. This acts to weaken the convective motion, and the amplitude of the convective velocity will be smaller than our estimate. The line in Figure 10 shifts downwards and the maximum horizontal scale of the convection scale decreases. However, this factor does not change the prediction that convection cells of larger horizontal scales, corresponding to the region above the line in Figure 10, cannot emerge.
Our theoretical estimates suggest that an upwelling of the convection cell extending beyond the margin of a supercontinent can be generated beneath its center in the case of whole-mantle convection. However, in the case of upper mantle convection, the upwelling takes place beneath the plate’s margins.
It has been suggested that an upwelling in the mantle exists under the eastern part of the Asian continent (Miyashiro, 1986; Tatsumi, 1990). If we interpret this in terms of our model, the location of the upwelling corresponds to that of a convection cell of aspect ratio 10 extending beyond the margin of the continental plate. However, since it takes too long a time for an upwelling to fully develop and because the expected amplitude of the convective velocity is small, it is difficult to fully explain the generation of this upwelling by the mechanism presented here.
The analytical expression obtained by the theoretical discussion presented here describes characteristics of convection depending on the Rayleigh number, horizontal dimension and thickness of the plate, and is expected to illustrate the qualitative features of convection of the previous studies. For example, if we apply the condition used by Gurnis (1988) and Lowman and Jarvis (1993) (Ra=10⁵, L=1-3) to Figure 10, it is inferred that the upwelling takes place beneath the center of the plate, a prediction consistent with their numerical results. However we cannot compare the horizontal extent of the convection cells because their plates are displaced by the convective flow underneath and the extent of the convection cells cannot be clearly identified. Lowman and Jarvis (1996) investigated the effects of a horizontally extensive plate on whole and upper mantle convection. Their results showed that an upwelling emerges under the center of the plate in the case of whole mantle convection, whereas such an upwelling could not be found for upper mantle convection. This is consistent with the present discussion, however, we have better compare both results more carefully since their model has a rigidly mobile layer at the surface modelling oceanic plates.
Our design of the numerical calculations and theoretical analysis are highly simplistic compared with real convection in the earth's mantle, and there are a number of limitations that future work will need to address. However, we believe that our results provide fundamental physical insights into actual mantle convection and continental breakup, and will help to interpret the results of more realistic numerical simulations of mantle convection.
In future models, a rigid condition should be specified at the boundary between the plate and the fluid. Under this condition, the convective velocity below the plate is smaller than for the free-slip condition used here. Further, since the effective thermal boundary-layer beneath the plate becomes thicker and the temperature below the plate increases, convection cells with larger horizontal scales may be formed. The effects of internal heating by radioactive elements should also be considered. Since radioactive elements are concentrated in the continental plates, they may warm the fluid under the plate more than considered to date and cause the emergence of larger convection cells. Finally, future models should address more complicated factors such as mobile continental plates, temperature-dependent viscosity, phase changes, and so on.