Learning Scalar Gradient Dynamics with Normalizing Flows

Stochastic models of the Lagrangian evolution of velocity gradients, with closure schemes for the pressure Hessian and the viscous Laplacian, have shown tremendous success in reconstructing data from simulations of homogeneous, isotropic turbulence (HIT). Similar modelling approaches, however, have proven to be far less successful for predicting the behaviour of scalar gradient, with the models being highly susceptible to finite-time singularities (Zhang et al., JFM, 964, A39, 2023). This is due to the lack of a pressure-Hessian type term in the scalar gradient equation, which plays a key role in regularizing the growth of velocity gradients. Recently, Carbone et al. (PRL 133, 2024) developed a reduced-order deterministic model for the Lagrangian velocity gradient tensor that accurately reproduces the joint probability density function (PDF) of velocity gradients in HIT using a normalizing flow framework. They constructed a Liouville equation with the learned PDF as a stationary solution, deriving a dynamical system that reproduces trajectories similar to those obtained from simulations. A crucial feature of these models is that, by construction, they do not produce run-away trajectories, and therefore offers a promising alternative to scalar gradient modeling that does not suffer from unphysical finite-time singularities. We have extended this modeling framework to the dynamics of passive scalar gradients in HIT, and further aim to extend this for different Reynolds and Schmidt numbers.