An ensemble of Gaussian fields with multifractal statistics for turbulence

Obtaining accurate field statistics continues to be one of the major challenges in turbulence theory and modelling. From the various existing modelling approaches, multifractal models have been successful in capturing intermittency in velocity gradient and increment distributions. Moreover, superstatistical models from non-equilibrium statistical mechanics have shown the capacity to model probability density functions (p.d.f.s) of various statistical turbulent quantities as ensembles of simpler stochastic processes. Here, we present an approach that generates field statistics in the form of a characteristic functional by promoting a model for multifractal increment statistics to an ensemble of Gaussian fields. By carefully designing the correlation function and the corresponding weight of each subensemble, we are able to define a functional that exhibits multifractal two-point inertial-range and dissipation-range statistics, and that blends into realistic large-scale behaviour. Additionally, the method is capable of producing multifractal statistics with any of the widely used singularity spectra. We characterise the fidelity of our approach through comparisons to literature results from direct numerical simulations. Overall, our framework thereby bridges between three different perspectives: superstatistics, multifractals and functional approaches to turbulence.