Macroscopic Fluctuation Theory applied to turbulence

One of the theoretical challenges in the study of turbulence is to develop an effective, coarse-grained model of its behaviour on a macroscopic scale. Turbulent flows are out-of-equilibrium systems that exhibit stochastic behaviour due to small-scale perturbations affecting macroscopic dynamics within a finite amount of time. This property, known as the spontaneous stochasticity of the incompressible Navier–Stokes equation, challenges traditional coarse-graining approaches because unresolved scales cannot simply be neglected. In this work, these scales are modelled as vanishing, momentum-conserving, multiplicative noise. The resulting stochastic Navier–Stokes equation is analysed using Macroscopic Fluctuation Theory (MFT), a statistical framework for non-equilibrium systems governed by stochastic hydrodynamics. The results and insights of this theory are discussed in the context of Langevin equations. Strategies are presented for evaluating the probability distribution of observables to leading order in the vanishing-noise limit by integrating the instanton equations. The solutions to these equations represent the most likely paths leading to a given rare event. The main contribution of this work is the derivation of the MFT action and instanton equations for the adopted stochastic Navier–Stokes equation. Some insights are presented for the case of isotropic, locally correlated noise, with a particular focus on evaluating the stationary measure and the emergence of Lagrangian phase transitions. These results demonstrate the potential of MFT to provide a parameter-free statistical description of turbulence; however, further work is required to extract practical predictions from the derived model, both analytically and numerically.