## Feasibility of mantle convection

In order to gain insight into the feasibility and nature of convection within a spherical, rotating Earth, it is convenient to assume that the mantle approximates a Newtonian fluid. Although this assumption may be erroneous, it does allow simple calculations to be made on the convective process.

The condition for the commencement of thermal convection is controlled by the magnitude of the dimen-sionless Rayleigh number (R), which is defined as the ratio of the driving buoyancy forces to the resisting effects of the viscous forces and thermal diffusion.

Figure 12.6 Frames from numerical models illustrating (a) convection in a layer heated from below and (b) convection in a layer heated internally and with no heat from below (from Davies, 1999. Copyright © Cambridge University Press, reproduced with permission).

Ra = aPpgdVkn where a is the coefficient of thermal expansion, P the superadiabatic temperature gradient (the gradient in excess of that expected to be associated with the increasing pressure), p the density of the fluid, g the acceleration due to gravity, d the thickness of the convecting fluid, k the thermal diffusivity (the ratio of the thermal conductivity to the product of density and specific heat at constant volume), and n the dynamic viscosity (Section 2.10.3). For convection in the mantle, the Ray-leigh number corresponding to the onset of convection is approximately 103. This corresponds to the minimum temperature gradient required for convection to occur. For the actual temperature gradient the Rayleigh number is of the order of 106 or greater. This implies very favorable conditions for convection and, as a consequence, thin boundary layers compared to the total layer thickness.

The nature of the flow in a convecting fluid can be judged by the magnitude of the Reynolds number (Re), which allows discrimination between laminar and turbulent flow. Re is defined:

Re= vd/v where v is the velocity of flow and V is the kinematic viscosity (the ratio of the dynamic viscosity, n, to density). Taking v = 200 mm a-1 = 6 X 10-9 ms-1, d = 3000 km = 3 X 106 m and V = 2 X 1017 m2 s-1, Re = 9 X 10-20. This very low value indicates that viscous forces dominate and hence the flow is laminar. The effect of the Earth's rotation on convection can be judged by the magnitude of the Taylor number (T), which is defined:

where w is the angular velocity of rotation. Putting w = 7.27 X 10-5 rad s-1 and other values as above, T ~ 4 X 10-17. A value of T less than unity implies no significant effect of rotation on convection and so the Earth's rotation should have no effect on the pattern of mantle convection.

The efficiency of convection is measured by the Nusselt number (Nu), which is the ratio of the total heat transferred to that transferred by thermal conduction alone. Elder (1965) computed experimentally the relationship between Nu and Ra. He found that at values of Ra appropriate to marginal convection Nu is unity and very little heat is transferred by convection. At Ra values 106 or greater, appropriate to the mantle, Nu is about 100, indicating the predominance of heat transfer by convection.

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