= Gaussian scalar packet in a linear incompressible flow
{c}
{title2=$\dot J=CJ+JC^T+2\kappa QI$}
For $\chi=f\exp(-\boldsymbol x^TB\boldsymbol x)$ in two dimensions, take $B$ positive definite, $\operatorname{tr}C=0$ 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 $\dot B=-C^TB-BC-4\kappa B^2$ and $\dot f/f=-2\kappa\operatorname{tr}B$, equivalently the displayed moment equation. If $\dot F=CF$, its solution is $J=F[J_0+2\kappa Q\int_0^tF^{-1}(s)F^{-T}(s)ds]F^T$, with $\det F=1$.
Back to article page
