Density-variance dissipation rate ($\chi ={\kappa }_{\rho }\overline{|\nabla {\rho }^{\prime }{|}^2}$)

Molecular scalar diffusion removes density-fluctuation variance at the positive rate displayed here. With this convention the budget for one half of the variance has dissipation term $-\chi$; the full variance budget has $-2\chi$. In a stationary locally homogeneous shear flow the half-variance budget gives $\overline{{\rho }^{\prime }{w}^{\prime }}{\overline{\rho }}_z+\chi =0$. An integral-scale model writes $\chi =\overline{{\rho }^{\prime 2}}q/L_{\rho }$, absorbing any normalization constant into $L_{\rho }$.