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.