= Diffusion arrest of Gaussian packet compression
{title2=$\ell_B\sim\sqrt{\kappa/\lambda}$}
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 $\dot j_\pm=\pm2\lambda j_\pm+2\kappa$, 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.
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