Breaking gravity waves generate and maintain a background level of turbulence which is capable of producing substantial cooling and/or heating in the upper mesosphere and lower thermosphere. The net thermodynamic effect of breaking gravity waves is critically dependent on the eddy Prandtl number (Pr) applicable to mesospheric turbulence. When Pr~1, our calculations of the heat budget for the mesopause region imply that the globaly averaged eddy or turbulent diffusion coefficient cannot exceed 106 cm3 s-1. This upper limit on turbulent diffusion applies to both potential temperature transport and chemically inert tracer transport when radiative damping in negligible. For chemically active species, larger diffusion coefficients are permitted because the effective eddy diffusion coefficient is increased by an additive term L/2&ggr;2, where L is the chemical loss rate and is the vertical wave number. For Pr~4--6, the turbulent diffusion of momentum DM is sufficiently greater than the turbulent diffusion of heat DH that the conversion of gravity wave energy to heat with high efficiency nearly balances the divergence of the downward eddy heat flux in the wave breaking zone. Therefore the heat budget of the mesopause region would no longer provide a powerful and useful constraint on DH. If Pr exceeds 6 with high efficiency for energy conversion to heat, gravity waves would heat the mesosphere throughout the wave breaking region.