  We study the fine-scale statistics of temperature and its derivatives in
turbulent Rayleigh-Benard convection. Direct numerical simulations are carried
out in a cylindrical cell with unit aspect ratio filled with a fluid with
Prandtl number equal to 0.7 for Rayleigh numbers between 10^7 and 10^9. The
probability density function of the temperature or its fluctuations is found to
be always non-Gaussian. The asymmetry and strength of deviations from the
Gaussian distribution are quantified as a function of the cell height. The
deviations of the temperature fluctuations from the local isotropy, as measured
by the skewness of the vertical derivative of the temperature fluctuations,
decrease in the bulk, but increase in the thermal boundary layer for growing
Rayleigh number, respectively. Similar to the passive scalar mixing, the
probability density function of the thermal dissipation rate deviates
significantly from a log-normal distribution. The distribution is fitted well
by a stretched exponential form. The tails become more extended with increasing
Rayleigh number which displays an increasing degree of small-scale
intermittency of the thermal dissipation field for both the bulk and the
thermal boundary layer. We find that the thermal dissipation rate due to the
temperature fluctuations is not only dominant in the bulk of the convection
cell, but also yields a significant contribution to the total thermal
dissipation in the thermal boundary layer. This is in contrast to the ansatz
used in scaling theories and can explain the differences in the scaling of the
total thermal dissipation rate with respect to the Rayleigh number.
