We consider turbulent convection in a horizontal fluid layer of depth h, which lies above a solid base with thickness h_b. The fluid parameters are the kinematic viscosity n, the thermal diffusivity k, which is taken to be comparable with n, the density r, the specific heat c_p and the expansion parameter b. The thermal diffusivity of the solid base is k_b. The buoyancy forces determined by the average heat flux rc_pF_q produce turbulence with a typical velocity w*=(g b F_q h)^1/3. It is well-known from numerical simulation but also from various observations that the turbulence in convective conditions consists of large-scale eddy motions or coherent structures. At very high Reynolds numbers with Re=w* h / n >10⁴ as occurs in the atmospheric boundary layer but also in other geophysical applications such as the ocean and earth's interior the surface is highly turbulent with a logarithmic velocity and temperature profile. An order of magnitude analysis shows that plumes can only develop if the surface flux is uniform, for example by radiant heat transfer or if the base is very thin (assuming constant heat flux below the base). In other cases puffs can form. To check our analysis we have conducted a DNS. At the bottom of the base a heat flux is applied so that the mean heat flux at the bottom of the fluid layer is rc_pF_q. Computations are carried out for two values of the Rayleigh number Ra=10⁴ and 10⁵ and with 0.1 < k_b / k < 10. The simulations are performed in a rectangular box with an aspect ratio of the fluid domain equal to 5:1. The depth of the conductive base is the same as the depth of the fluid (h_b=h). The resolution in the fluid layer is 40³ grid points for Ra=10⁴ and 100³ for Ra=10⁵. It is found from the numerical solutions that the largest temperature fluctuations occur for a small value of k_b / k and the smallest temperature fluctuations for a large value of k_b / k. The spatial scales of eddy structures in the lowest surface layer of the fluid layer become significantly smaller as k_b / k is reduced from 10 to 0.1.