  The characterization of intermittency in turbulence has its roots in the K62
theory, and if no proper definition is to be found in the literature,
statistical properties of intermittency were studied and models were developed
in attempt to reproduce it. The first contribution of this work is to propose a
requirement list to be satisfied by models designed within the Lagrangian
framework. Multifractal stochastic processes are a natural choice to retrieve
multifractal properties of the dissipation. Among them, following the
proposition of \cite{Mandelbrot1968}, we investigate the Gaussian
Multiplicative Chaos formalism, which requires the construction of a
log-correlated stochastic process $X_t$. The fractional Gaussian noise of Hurst
parameter $H = 0$ is of great interest because it leads to a log-correlation
for the logarithm of the process.Inspired by the approximation of fractional
Brownian motion by an infinite weighted sum of correlated Ornstein-Uhlenbeck
processes, 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 process
with speed of mean reversion $x$ and $k$ is a kernel. A regularization of
$k(x)$ is required to ensure stationarity, finite variance and logarithmic
auto-correlation. A variety of regularizations are conceivable, and we show
that they lead to the aforementioned multifractal models.To simulate the
process, we eventually design a new approach relying on a limited number of
modes 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 can
retrieve the expected behavior with only one mode per decade, making this
strategy versatile and computationally attractive for simulating such
processes, while remaining within the proposed framework for a proper
description of intermittency.
