Multiscale turbulence synthesis: Validation in 2D hydrodynamics

Numerical simulations can follow the evolution of fluid motions through the intricacies of developed turbulence. However, they are rather costly to run, especially in 3D. In the past two decades, generative models have emerged that produce synthetic random flows at a computational cost equivalent to no more than a few time steps of a simulation. These simplified models qualitatively bear some characteristics of turbulent flows in specific contexts (incompressible 3D hydrodynamics or magnetohydrodynamics) but generally struggle with the synthesis of coherent structures. We aim to generate random fields (e.g. velocity, density, magnetic fields) with realistic physical properties for a large variety of governing partial differential equations and at a small cost relative to time-resolved simulations. We propose a set of simple steps applied to given sets of partial differential equations: we filter from large to small scales, derive a first order time evolution approximation from Gaussian random initial conditions during a prescribed coherence time, and finally sum over scales to generate the fields. We test the validity of our method in the simplest framework: 2D decaying incompressible hydrodynamical turbulence. We compare the results of 2D decaying simulations with snapshots of our synthetic turbulence. We first quantitatively assess the difference with standard statistical tools: power spectra, increments, and structure functions. These indicators can be reproduced by our method during up to about a third of the turnover timescale. We also consider recently developed scattering transforms statistics, which are able to efficiently characterise non-Gaussian structures. This reveals a more significant discrepancy; however, this can be bridged by bootstrapping. Finally, the number of Fourier transforms necessary for one synthesis scales logarithmically in the resolution, compared to linearly for time-resolved simulations. We have designed a multiscale turbulence synthesis (MuScaTS) method to efficiently short-circuit costly numerical simulations to produce realistic instantaneous fields.