Under the limit of infinite Prandtl number, we derive analytical expressions for the large-scale quantities, e.g., Péclet number Pe, Nusselt number Nu, and rms value of the temperature fluctuations ${\theta }_{\mathrm{rms}}$. We complement the analytical work with direct numerical simulations, and show that $\mathrm{Nu}\sim {\mathrm{Ra}}^{\gamma }$ with $\gamma \approx (0.30–0.32)$, $\mathrm{Pe}\sim {\mathrm{Ra}}^{\eta }$ with $\eta \approx (0.57–0.61)$, and ${\theta }_{\mathrm{rms}}\sim \mathrm{const}$. The Nusselt number is observed to be an intricate function of $\mathrm{Pe}$, ${\theta }_{\mathrm{rms}}$, and a correlation function between the vertical velocity and temperature. Using the scaling of large-scale fields, we show that the energy spectrum $E_u\left(k\right)\sim k^{-13/3}$, which is in a very good agreement with our numerical results. The entropy spectrum $E_{\theta }\left(k\right)$, however, exhibits dual branches consisting of $k^{-2}$ and $k^0$ spectra; the $k^{-2}$ branch corresponds to the Fourier modes $\widehat{\theta }(0,0,2n)$, which are approximately $-1/\left(2n\pi \right)$. The scaling relations for Prandtl number beyond ${10}^2$ match with those for infinite Prandtl number.

• Received 30 September 2013
• Revised 3 December 2013

DOI:https://doi.org/10.1103/PhysRevE.89.023006