Lagrangian Ellipsoid Diagnostics in Rough Two-Dimensional Synthetic Flows: Aspect-Ratio Saturation and Reduced Modeling

We develop and test a Lagrangian methodology for extracting finite-cloud geometry and reduced dynamics from particle trajectories. A cloud in a rough, incompressible, two-dimensional synthetic flow is represented by a mass/gyration ellipse describing the particle-weighted bulk and a minimum-area enclosing ellipse describing the outer envelope and its supporting particles. The homogeneous Gaussian--H"older velocity has finite Ornstein--Uhlenbeck memory, and the moving centroid is followed explicitly.
The principal finding is unexpected: despite continual deformation by a non-smooth velocity field, the normalized aspect ratios of both ellipses reach broad, order-one distributions that are approximately stationary when conditioned on cloud scale. The mass ellipse is more elongated, whereas the enclosing ellipse spans a larger radial envelope and has the more scale-stable conditional shape distribution.
To understand this saturation, we compare two finite-cloud descriptions of the velocity gradient. A gradient averaged over the enclosing ellipse predicts more aligned stretching than the particles experience. The resulting gradient correction supplies most of the negative contribution offsetting aligned stretching, while a smaller correction is specific to representing the evolving outer cloud by an enclosing ellipse.
Finally, we demonstrate a data-to-model workflow. The measured time series is transformed to intrinsic variables, a constrained hierarchy of finite-lag stochastic models is fitted, and the models are compared on held-out realizations. The fitted coefficients and equations are specific to this synthetic experiment. The transferable contribution is the methodology: paired cloud geometries, finite-cloud coarse graining, intrinsic variables, transparent correction decomposition, and held-out validation for future simulations and experiments.