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The characterization of intermittency in turbulence has its roots in the K62theory, and if no proper definition is to be found in the literature,statistical properties of intermittency were studied and models were developedin attempt to reproduce it. The first contribution of this work is to propose arequirement list to be satisfied by models designed within the Lagrangianframework. Multifractal stochastic processes are a natural choice to retrievemultifractal properties of the dissipation. Among them, following theproposition of \cite{Mandelbrot1968}, we investigate the GaussianMultiplicative Chaos formalism, which requires the construction of alog-correlated stochastic process $X_t$. The fractional Gaussian noise of Hurstparameter $H = 0$ is of great interest because it leads to a log-correlationfor the logarithm of the process.Inspired by the approximation of fractionalBrownian motion by an infinite weighted sum of correlated Ornstein-Uhlenbeckprocesses, our second contribution is to propose a new stochastic model: $X_t =\int_0^\infty Y_t^x k(x) d x$, where $Y_t^x$ is an Ornstein-Uhlenbeck processwith speed of mean reversion $x$ and $k$ is a kernel. A regularization of$k(x)$ is required to ensure stationarity, finite variance and logarithmicauto-correlation. A variety of regularizations are conceivable, and we showthat they lead to the aforementioned multifractal models.To simulate theprocess, we eventually design a new approach relying on a limited number ofmodes for approximating the integral through a quadrature $X_t^N = \sum_{i=1}^N\omega_i Y_t^{x_i}$, using a conventional quadrature method. This method canretrieve the expected behavior with only one mode per decade, making thisstrategy versatile and computationally attractive for simulating suchprocesses, while remaining within the proposed framework for a properdescription of intermittency.

physical review. e

Revisiting the framework for intermittency in Lagrangian stochastic  models for turbulent flows: a way to an original and versatile numerical  approach