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.