Gaussian scalar packet in a linear incompressible flow ($\dot{J}=CJ+J{C}^T+2\kappa QI$)

For $\chi =f\exp (-{\mathbf{x}}^{T}B\mathbf{x})$ in two dimensions, take $B$ positive definite, $\mathrm{tr}C=0$ and positive conserved mass $Q=\int \chi dA$. The Gaussian integral gives $Q=\pi f/\sqrt{\det B}$, $J=(Q/2)B^{-1}$ and $f=Q^2/(2\pi \sqrt{\det J})$. Substitution in the advection-diffusion equation gives $\dot{B}=-C^{T}B-BC-4\kappa B^2$ and $\dot{f}/f=-2\kappa \mathrm{tr}B$, equivalently the displayed moment equation. If $\dot{F}=CF$, its solution is $J=F[J_0+2\kappa Q{\int }_0^t{F}^{-1}(s)F^{-T}(s)ds]F^T$, with $\det F=1$.