  Upper bounds on the heat transport and other quantities of interest in
Rayleigh-B\'enard convection are derived in previous work from constraints
resulting from the equations of time evolution for kinetic energy, the root
mean square of temperature, and the temperature averaged over horizontal
planes. Here, we investigate the effect of a new constraint derived from the
time evolution equation for the advective heat transport. This additional
constraint leads to improved bounds on the toroidal dissipation.
