Three statistical theories of turbulence are applied to thermal convection between infinite slippery plates in the limit of infinite Prandtl number. The range of Rayleigh number R investigated is 657 ≤ R ≤ 1.25 X 10 4. The theories, which are compared in a series of numerical calculations, are the direct interaction approximation, the quasilinear approximation, and the quasinormal approximation. The direct interaction approximation gives results for the evolution Nusselt number in good agreement with some simple exact solutions to the problem. The flow predicted by this method is very persistent. Turbulence first appears just at critical R and its intensity gradually increases with increasing R. The quasinormal approximation gives satisfactory results for R ≲ 2000, but some of the initial value problems lead to an unphysical negative temperature autocorrelation speotrum for larger R. In none of the initial value problems for R > 2 X 103 did the quasinormal procedure yield a sensible stationary state. The quasilinear approximation appears to upperbound the heat transport given by both the other approximations predicting about 8% larger heat transport than the direct interaction approximation.
