A generative diffusion framework for physically consistent 3D turbulence

Abstract Synthesizing fully developed three-dimensional turbulence remains a long-standing problem in fluid mechanics and an open challenge for generative modeling. This difficulty arises from extreme dimensionality, multiscale fluctuations, strong intermittency, and the need to satisfy exact physical constraints, including incompressibility and prescribed mass and momentum fluxes under given boundary conditions. We propose a physics-constrained diffusion model in which some a priori constraints can be incorporated directly into the generative dynamics. Using rotating turbulence as a paradigmatic system with key geophysical applications, we assess the proposed framework. Here, we show that it enables stable and statistically faithful single-time synthesis of inertial-range three-dimensional turbulent velocity fields, accurately reproducing anisotropic energy spectra, intermittent statistics, and physical constraints, whereas standard denoising diffusion probabilistic models exhibit multiscale statistical deviations, violations of physical consistency, and substantially slower training convergence, thereby pointing to broader implications for generative modeling of high-dimensional, physically constrained complex systems.