Abstract This paper contains two parts. In the first part, a new set of diagnostic equations is derived for the third-order moments for a buoyancy-driven flow, by exact inversion of the prognostic equations for the third-order moment equations in the stationary case. The third-order moments exhibit a universal structure: they all are a linear combination of the derivatives of all the second-order moments, w2, wθ, θ2, and q2. Each term of the sum contains a turbulent diffusivity Dt, which also exhibits a universal structure of the form Dt = aνt + bwθ. Since the sign of the convective flux changes depending on stable or unstable stratification, Dt varies according to the type of stratification. Here νt ∼ wl (l is a mixing length and w is an rms velocity) represents the “mechanical” part, while the “buoyancy” part is represented by the convective flux wθ. The quantities a and b are functions of the variable (Nτ)2, where N2 = gα∂Θ/∂z and τ is the turbulence time scale. The new expressions for the third-order ...