It is found by an a priori test that the counter-gradient diffusion of the turbulent heat flux observed in a non-premixed swirling flame can be quantitatively explained by a new turbulence model of first-order closure type. The new model has been derived by applying the two-scale direct-interaction approximation to the low-Mach-number equations with the assumption that the variation of density is relatively small. It can be applied to general turbulent flows with the density variation, including the case where the Boussinesq approximation can not hold. The model expression of turbulent heat flux involves the term related to the mean pressure gradient in addition to the usual term of gradient diffusion type. This expression can also reproduce the vertical turbulent heat flux of the natural convection along a heated vertical plate, which can not be explained by the standard turbulence model of gradient diffusion type.