  Under the limit of infinite Prandtl number, we derive analytical expressions
for the large-scale quantities, e.g., P\'{e}clet 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 $E_u(k)\sim
k^{-13/3}$, which is in a very good agreement with our numerical results. The
entropy spectrum $E_\theta(k)$ however exhibits dual branches consisting of
$k^{-2}$ and $k^0$ spectra; the $k^{-2}$ branch corresponds to the Fourier
modes $\hat{\theta}(0,0,2n)$, which are approximately $-1/(2n \pi)$. The
scaling relations for Prandtl number beyond $10^2$ match with those for
infinite Prandtl number.
