Broad theoretical arguments are proposed to show, formally, that the
magnitude G of the temperature gradients in turbulent thermal convection at
high Rayleigh numbers obeys the same advection-diffusion equation that governs
the temperature fluctuation T, except that the velocity field in the new
equation is substantially smoothed. This smoothed field leads to a -1 scaling
of the spectrum of G in the same range of scales for which the spectral
exponent of T lies between -7/5 and -5/3. This result is confirmed by
measurements in a confined container with cryogenic helium gas as the working
fluid for Rayleigh number Ra=1.5x10^{11}. Also confirmed is the logarithmic
form of the autocorrelation function of G. The anomalous scaling of
dissipation-like quantities of T and G are identical in the inertial range,
showing that the analogy between the two fields is quite deep.