Inversion of two-dimensional numerical convection experiments for a fluid with a strongly temperature-dependent viscosity

2-D thermal convection numerical experiments are conducted for a fluid with an infinite Prandtl number, a strongly temperature-dependent viscosity, and isothermal horizontal boundaries. The core Rayleigh number (Ra-(0) over bar), determined with the temperature of the well-mixed interior, is in the range 5 x 10(5) < Ra ((0) over bar)< 2 x 10(7), and the ratio of the top to the bottom viscosity (Delta mu) can be as large as Delta mu = 10(6). Different convective regimes are possible, depending on the values of Ra-(0) over bar and Delta mu. This paper focuses on the conductive-lid regime, in which convection is confined to a sublayer. First, a least-squares fit of more than 40 numerical experiments suggests that the temperature difference across the lower thermal boundary layer (Delta T-1) depends mostly on the viscous temperature scale (Delta T-v) defined by Davaille & Jaupart (1993), and slightly on the temperature difference across the fluid layer (Delta T): Delta T-1 = 1.43 Delta T-v - 0.03 Delta T. Second, a generalized non-linear inversion of the data does not support the assumptions that the temperature difference across the upper boundary layer is proportional to Delta T-v, and that isoviscous scaling laws can be used for describing heat Bur through the convective sublayer. Third, a generalized non-linear inversion of the data is carried out in order to avoid any assumptions on the parameters. This leads to the following heat flux scaling law: Nu = 3.8(Delta T-v/Delta T)1.63 Ra-(0) over bar(0.258), where the Nusselt number (Nu) is the non-dimensional hear flux. This scaling law is different from that proposed by previous studies. It reproduces the data at better than 1 per cent and fits the results of previous numerical experiments very well (e.g. Christensen 1984). Finally, a thermal boundary layer analysis is performed. For a fluid heated from below, the upper and lower thermal boundary layers interact with one another, inducing a thermal erosion of the conductive lid. This study suggests that the dynamics of convection is driven by the instability of the lower thermal boundary layer. Therefore, an alternative way to determine the heat flux is to use the value of the lower thermal boundary layer Rayleigh number (Ra-delta). This value is not independent of Ra-(0) over bar, unlike the case for an isoviscous fluid. A least-squares fit of the data leads to Ra-delta = 0.28Ra((0) over bar)(0.21). This law provides a very convenient way to model the thermal evolution of planetary mantles.