Lagrangian Ellipsoid Diagnostics for Stochastic Hydrodynamics: Source--Sink Modeling of Deforming Particle Clouds

We propose the Lowner--John deform-cloud scheme as a Lagrangian diagnostic for incompressible stochastic flows with an inertial range. A volume-filled particle cloud is released at the ultraviolet scale and summarized at each time by two objects: the inertia tensor of its minimum-volume enclosing ellipsoid and the velocity gradient coarse-grained over that ellipsoid. We test the scheme on a two-dimensional isotropic incompressible Gaussian--Holder finite-time-correlated velocity field with Kolmogorov exponent, generated spectrally with Ornstein--Uhlenbeck Fourier modes. The resulting empirical train shows a broadly fluctuating but statistically saturated ellipsoid aspect ratio, a clear scale dependence of the perceived gradient, and an approximately ordinary tensor-level strain--vorticity balance. We then formulate reduced modeling of the train as physics-informed generator identification. In intrinsic variables describing scale, aspect ratio, strain amplitude, vorticity, and strain--ellipsoid alignment, the aspect-ratio dynamics separates into an aligned-strain source and a Lowner--John residual. The final open-box closure models strain and vorticity as scale-dependent stochastic drivers, represents alignment by a stationary von--Mises bias, and closes the residual by a scale-dependent affine feedback. Thus the observed aspect-ratio saturation is not merely fitted; it is explained as a balance between persistent strain alignment and geometric relaxation of the enclosing ellipsoid. The construction provides a portable route from particle-cloud data to interpretable finite-dimensional stochastic dynamics for future turbulent-flow applications.