Yahia, Hussein, Schneider, Nicola, Bontemps, Sylvain, Bonne, Lars, Attuel, Guillaume, Dib, Sami, Ossenkopf, Volker, Turiel, Antonio, Zebadua, Augusto, Elia, Davide, Maji, Suman Kumar, Schmitt, François G, Robitaille, Jean-François, Geometry and Statistics in acquisition data (GeoStat), Inria Bordeaux - Sud-Ouest, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), I. Physikalisches Institut [Köln], Universität zu Köln = University of Cologne, FEMIS 2021, Laboratoire d'Astrophysique de Bordeaux [Pessac] (LAB), Université de Bordeaux (UB)-Institut national des sciences de l'Univers (INSU - CNRS)-Centre National de la Recherche Scientifique (CNRS)-Université de Bordeaux (UB)-Institut national des sciences de l'Univers (INSU - CNRS)-Centre National de la Recherche Scientifique (CNRS), Université de Bordeaux (UB)-Institut national des sciences de l'Univers (INSU - CNRS)-Centre National de la Recherche Scientifique (CNRS), Physikalisches Institut [Köln], Institute of Marine Sciences / Institut de Ciències del Mar [Barcelona] (ICM), Consejo Superior de Investigaciones Científicas [Madrid] (CSIC), Istituto Nazionale di Astrofisica (INAF), Indian Institute of Technology Patna (IITP), Laboratoire d’Océanologie et de Géosciences (LOG) - UMR 8187 (LOG), Institut national des sciences de l'Univers (INSU - CNRS)-Université du Littoral Côte d'Opale (ULCO)-Université de Lille-Centre National de la Recherche Scientifique (CNRS)-Institut de Recherche pour le Développement (IRD [France-Nord]), Institut de Planétologie et d'Astrophysique de Grenoble (IPAG), Centre National d'Études Spatiales [Toulouse] (CNES)-Observatoire des Sciences de l'Univers de Grenoble (OSUG ), Institut national des sciences de l'Univers (INSU - CNRS)-Université Savoie Mont Blanc (USMB [Université de Savoie] [Université de Chambéry])-Centre National de la Recherche Scientifique (CNRS)-Institut National de Recherche pour l’Agriculture, l’Alimentation et l’Environnement (INRAE)-Université Grenoble Alpes (UGA)-Météo-France -Institut national des sciences de l'Univers (INSU - CNRS)-Université Savoie Mont Blanc (USMB [Université de Savoie] [Université de Chambéry])-Centre National de la Recherche Scientifique (CNRS)-Institut National de Recherche pour l’Agriculture, l’Alimentation et l’Environnement (INRAE)-Université Grenoble Alpes (UGA)-Météo-France, This work is supported by the GENESIS project (GENeration and Evolution of Structure in the ISm), via the french ANR and the german DFG through grant numbers ANR-16-CE92-0035-01 and DFG1591/2-1. N. Schneider acknowledges support by the German Deutsche Forschungsgemeinschaft, DFG project number SFB 956. L. Bonne acknowledges support by the Région Nouvelle-Aquitaine, France., ANR-16-CE92-0035,GENESIS,GENeration et Evolution des Structures du milieu InterStellaire(2016), Universität zu Köln, Centre National de la Recherche Scientifique (CNRS)-Université du Littoral Côte d'Opale (ULCO)-Université de Lille-Institut national des sciences de l'Univers (INSU - CNRS), Institut national des sciences de l'Univers (INSU - CNRS)-Université Savoie Mont Blanc (USMB [Université de Savoie] [Université de Chambéry])-Centre National de la Recherche Scientifique (CNRS)-Institut National de Recherche pour l’Agriculture, l’Alimentation et l’Environnement (INRAE)-Université Grenoble Alpes (UGA)-Institut national des sciences de l'Univers (INSU - CNRS)-Université Savoie Mont Blanc (USMB [Université de Savoie] [Université de Chambéry])-Centre National de la Recherche Scientifique (CNRS)-Institut National de Recherche pour l’Agriculture, l’Alimentation et l’Environnement (INRAE)-Université Grenoble Alpes (UGA), Université du Littoral Côte d'Opale (ULCO), Université de Lille, Université Grenoble-Alpes, IPAG, PCMI, and PNPS
International audience; Observations of the interstellar medium (ISM) show a complex density and velocity structure, which is in part attributed to turbulence. Consequently, the multifractal formalism should be applied to observation maps of the ISM in order to characterize its turbulent and multiplicative cascade properties. However, the multifractal formalism, even in its more advanced and recent canonical versions, requires a large number of realizations of the system, which usually cannot be obtained in astronomy. We present a self-contained introduction to the multifractal formalism in a "microcanonical" version, which allows us, for the first time, to compute precise turbulence characteristic parameters from a single observational map without the need for averages in a grand ensemble of statistical observables (e.g., a temporal sequence of images). We compute the singularity exponents and the singularity spectrum for both observations and magnetohydrodynamic simulations, which include key parameters to describe turbulence in the ISM. For the observations we focus on the 250 µm Herschel map of the Musca filament. Scaling properties are investigated using spatial 2D structure functions, and we apply a two-point log-correlation magnitude analysis over various lines of the spatial observation, which is known to be directly related to the existence of a multiplicative cascade under precise conditions. It reveals a clear signature of a multiplicative cascade in Musca with an inertial range from 0.05 to 0.65 pc. We show that the proposed microcanonical approach provides singularity spectra that are truly scale invariant, as required to validate any method used to analyze multifractality. The obtained singularity spectrum of Musca, which is sufficiently precise for the first time, is clearly not as symmetric as usually observed in log-normal behavior. We claim that the singularity spectrum of the ISM toward Musca features a more log-Poisson shape. Since log-Poisson behavior is claimed to exist when dissipation is stronger for rare events in turbulent flows, in contrast to more homogeneous (in volume and time) dissipation events, we suggest that this deviation from log-normality could trace enhanced dissipation in rare events at small scales, which may explain, or is at least consistent with, the dominant filamentary structure in Musca. Moreover, we find that subregions in Musca tend to show different multifractal properties: While a few regions can be described by a log-normal model, other regions have singularity spectra better fitted by a log-Poisson model. This strongly suggests that different types of dynamics exist inside the Musca cloud. We note that this deviation from log-normality and these differences between subregions appear only after reducing noise features, using a sparse edge-aware algorithm, which have the tendency to "log-normalize" an observational map. Implications for the star formation process are discussed. Our study establishes fundamental tools that will be applied to other galactic clouds and simulations in forthcoming studies.