Scalar determinants of heat and mass transport in compressible fluids

The advection-diffusion equations for heat and mass transport are examined to establish their scalar determinants for the compressible case. Thermodynamic analyses reveal that advection does not constitute “thermal contact” as denoted in the 0^th Law and that for compressible fluids its scalar determinant is not the temperature, but rather the potential temperature (Θ). The 1^st Law is applied to show that conservation of Θ requires including a pressure covariance term in the (micro-scale advection) definition of the turbulent heat flux. For gas constituent conservation, advection and Fickian diffusion are universally agreed to depend directly on gradients in “concentration” (c), a nonetheless ambiguous term. Depending upon author, c may represent a scalar intensity defined either as a dimensionless fraction (in ppm or g kg^-1) or as a density (in kg m^-3), with nontrivial differences for the gas phase. Analyses of atmospheric law, scalar conservation and similarity theory demonstrate that both advection and diffusion of scalars in compressible fluids depend on gradients, not in constituent density but rather in the dimensionless, relative concentration such as the molar fraction or mixing ratio.