We study a simple 2-D model of secular mantle cooling by convection in which the viscosity depends exponentially on the temperature. To understand the effects of weak zones near the sinking and rising areas, their position and viscosity are imposed. The heat transport efficiency is measured by introducing local Rayleigh (Ral) and Nusselt (Nul) numbers defined at each thermal boundary layer. We find that the Nul-Ral relationship for the bottom boundary layer is similar to that of the steady-state convection with constant viscosity in a rectangular box. The existence of weak zones predicts a higher Nul for the top thermal boundary layer, suggesting that this will enhance the heat transport of the top layer. If we change Ral of the top layer into the higher of the two Rals, which is usually that of bottom layer, the resultant Nul-Ral approaches that of steady-state constant viscosity convection. This may imply that the heat transport of convection with secular cooling may be controlled by the boundary layer which has higher Ral. We also suggest that the commonly used parametrization, in which the convecting medium is treated as having a constant viscosity equal to that of its isothermal core, may overestimate the heat transport efficiency. ¿ 1997 American Geophysical Union