Local Rayleigh and Nusselt numbers for Cartesian convection with temperature-dependent viscosity

We present a local treatment of the heat transport efficiency of the base-heated cartesian convection with a temperature-dependent viscosity, which allows us to give a clearer interpretation of Nusselt (Nu) - Rayleigh (Ra) numbers relation. The ''local'' Nusselt (Nu(l)) and Rayleigh (Ra-l) numbers are defined by the local values at each boundary layer except the length scale which is the total depth. We find that Nu(l) - Ra-l relation obtained for the convection with constant viscosity and free surfaces is nearly coincident with that of the bottom boundary layer. For top boundary layer, we can treat it as a constant viscosity convection with rigid surface for large enough viscosity contrast. This interpretation can be made by choosing an appropriate temperature drop and viscosity for the definitions of Nu(l) and Ra-l.