Uncovering latent singularities from multifractal scaling laws in mixed asymptotic regime. Application to turbulence

In this paper we revisit an idea originally proposed by Mandelbrot about the possibility to observe negative dimensions'' in random multifractals. For that purpose, we define a new way to study scaling where the observation scale $\tau$ and the total sample length $L$ are respectively going to zero and to infinity. This mixed’’ asymptotic regime is parametrized by an exponent $\chi$ that corresponds to Mandelbrot supersampling exponent''. In order to study the scaling exponents in the mixed regime, we use a formalism introduced in the context of the physics of disordered systems relying upon traveling wave solutions of some non-linear iteration equation. Within our approach, we show that for random multiplicative cascade models, the parameter $\chi$ can be interpreted as a negative dimension and, as anticipated by Mandelbrot, allows one to uncover the hidden’’ negative part of the singularity spectrum, corresponding to ``latent’’ singularities. We illustrate our purpose on synthetic cascade models. When applied to turbulence data, this formalism allows us to distinguish two popular phenomenological models of dissipation intermittency: We show that the mixed scaling exponents agree with a log-normal model and not with log-Poisson statistics.