Diffusion arrest of Gaussian packet compression (${\ell }_B\sim \sqrt{\kappa /\lambda }$)

For a chaotic incompressible planar flow with Lyapunov exponent pair $\lambda ,-\lambda$, an initially broad Gaussian scalar packet in a linear incompressible flow stretches and compresses exponentially while diffusion is negligible. Its compressed variance stops decreasing around $\kappa /\lambda$, the Batchelor scalar microscale balance, while the larger variance continues to grow. In a constant principal strain, the normalized covariance eigenvalues obey ${\dot{j}}_{\pm }=\pm 2\lambda j_{\pm }+2\kappa$, proving that the compressed one approaches $\kappa /\lambda$. Random finite-correlation strain instead gives a fluctuating compressed width; stationarity and integrability assumptions are required to turn this into exponential statistics of the packet amplitude.

After diffusion arrest of Gaussian packet compression, define the logarithmic area increase $A_t=\frac{1}{2}\log [\det J(t)/\det J(0)]$. It is nonnegative. Assume $A_t/t$ has a large deviation principle with convex rate function $\mathcal{I}$, minimized at $\lambda >0$. The exact amplitude relation $f(t)=f(0)e^{-A_t}$ and the Laplace principle for probability measures give the displayed moment decay rate. The rate function refers to area, so rare width fluctuations are included rather than discarded without justification. For differentiable strictly convex $\mathcal{I}$, the interior minimizer satisfies ${\mathcal{I}}^{\prime }(s_{\mu })=-\mu$. If ${\mu }_{\ast }=-{\mathcal{I}}^{\prime }(0)$ is finite, then ${\gamma }_{\mu }=\mathcal{I}(0)$ for $\mu \ge {\mu }_{\ast }$: rare nearly unstretched packets dominate high moments. Below the threshold, $d({\gamma }_{\mu }/\mu )/d\mu =-\mathcal{I}(s_{\mu })/{\mu }^2<0$. These statements concern a positive Gaussian packet with flow-only averaging; a homogeneous signed random initial scalar requires a different averaging kernel. Arbitrary random $C(t)$ alone does not imply this statistical model.