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.