Parametrization of heat transport by non-Newtonian convection

We study the heat transport efficiency of the 2-D steady convection in a box (aspect ratio = 1) with a power-law creep which may be important for mantle convection. Effects of pressure- (depth) and temperature-dependent creep are also studied. To analyse the heat transport, we use the local Rayleigh (Ra-1) and Nusselt (Nu(1)) numbers, which are defined by the values characterizing each thermal boundary layer except the length scale. The commonly used definition of the Rayleigh and Nusselt numbers is only useful when the top and bottom thermal boundary layers show similar behaviour. Our definition has the advantage of treating each thermal boundary layer separately. By choosing an appropriate temperature drop and viscosity, we find that the Ra-1-Nu(1) relation of non-Newtonian fluid is in good agreement with that of Newtonian convection with a constant viscosity for most cases. Generally, the viscosity weighted by the strain rate is an appropriate viscosity for the range of Rayleigh number (10(4) similar to 10(7)) studied. However, it appears that this weighting becomes unsuitable at higher Rayleigh numbers. Strongly temperature-dependent creep introduces a new temperature scale-viscous temperature-for the treatment of the top thermal boundary layer. We find that the viscous temperature should be calculated using an enthalpy reduced by a factor of about 0.4-0.5, which is consistent with previous work. The bottom boundary layer can be interpreted using the results of the constant-viscosity case. The introduction of the pressure dependence degrades the simplicity of Nu(1) and Ra-1. We cannot obtain satisfactory results for the Nu(1)-Ra-1 relation of the top thermal boundary layer when only the strongly depth-dependent viscosity is considered. This may happen because we underestimate the local viscosity because of the large vertical variation in viscosity. For cases which include both temperature and pressure dependences, a satisfactory result is obtained if we use the viscosity averaged within a boundary layer. Our studies generally support the idea of local destabilization of the thermal boundary layer. In other words, our studies show that the exponent of the Nusselt-Rayleigh number relation is around 1/3, if we define them appropriately.