# Scalar Turbulence

The ability of efficiently mixing transported substances is one of the most distinctive properties of turbulence. For instance, it is turbulence (induced by the spoon) that allows cream to rapidly invade a cup of coffee, indeed if only molecular diffusion would be at play in the coffee at rest the same process would require many hours! Given the statistical complexity of a turbulent velocity field, it is natural to wonder about the resulting complexity in the statistical features of the transported concentration field of a substance (e.g. a fluorescent dye, as in the figure below on the right, the temperature o magnetic field in a star, etc.). Extensive experimental and numerical studies have indeed demonstrated that scalar substances transported by turbulent velocity fields share with the turbulent velocity field many common properties such as intermittency and anomalous scaling laws, with the associated strongly non Gaussian statistics. Therefore, naively one would conclude that as for turbulence the problem cannot be solved.

Fortunately, this is not the case. As recognized after an ispiring work due to the Dirac medalist R. Kraichnan, at least in some “theoretical circumstances”, when the velocity field statistics can be prescribed and mimic that of a true turbulent flow, it is possible to understand much of the scalar statistics and, in particular, to identifythe mechanism at the basis of intermittency. The main reason underlying the possibility to understand scalar turbulence is the fact that this is a linear problem. Indeed, unlike Navier-Stokes equations, which are non linear in the velocity field, the advection diffusion equation is linear in the scalar field. In a nutshell, we can say that the problem of scalar turbulence, after a suitable procedure of averaging of the velocity field statistcs can be reduced to a generalization of heat transport and thus can be solved.

However, this is possible only when the transported scalar field is passive, i.e. it does not influence (modify) the carrier flow, as e.g. the dye in the figure. The situation is completely different when the transported field has some feedback on the fluid as, e.g., temperature (acting on the fluid via bouyancy) or the magnetic potential (which in two dimensions is also a scalar field acting on the fluid via the Lorentz force). In this case we speak of active scalars and the problem is again fully nonlinear as the Navier-Stokes equations and we are back to all the difficulties of turbulence.

The research activity undertaken by us has focused on several aspects of turbulent scalar transport. In particular, active and passive scalar fields (whose evolution rules are the same) evolving in the same realization of a turbulent velocity field have been numerically and theoretically investigated in order to elucidate the differences and similarities between them. The problem has been brought back to the statistical properties of the trajectories of fluid elements and their correlation with the mechanism of excitation of the scalar field fluctuations. Extensive studies have shown that under different circumstances (different flow settings) active and passive scalars may or may not share statistical features. For instance, it has been found that there are cases in which the two fields blatantly differ not only in terms of statistics but also in their dynamical behavior as, e.g., illustrated in the movie below. For details on these studies see a short review[1].

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In the framework of passive transport it has been also investigated the Lagrangian (i.e. based on the properties of particle trajectories) origin of the formation of very strong and very weak fluctuations of the concetration field — i.e. of those structures which are dubbed plateaus or fronts in atmospheric science—. This was possible thanks to the development of a new algorithmic strategy allowing the back in time integration of the Navier-Stokes equation to be efficiently performed [2]. Moreover, the universality of passive scalar statistics with respect to the energy injection mechanism has been explored in comparison with fluid (and modeled) turbulence by considering power law forcing which stand at the basis of Renormalization Group approaches to turbulence [3], while an explanation has been proposed for some non-universal aspects of scalar statistics in the presence of a mean shear superimposed to turbulent flows [4].

**Relevant Publications**

*[1] Active and passive fields face to face *

A. Celani, M.Cencini, A. Mazzino and M. Vergassola

*New J. Phys. ***6 **, 72 (2004)

*[2] Going forth and back in time: a fast and parsimonious algorithm for mixed initial/final-value problems*

A. Celani, M. Cencini and A. Noullez

*Physica D ***195 **, 283 (2004)

*[3] Anomalous scaling and universality in hydrodynamic systems with power-law forcing *

L. Biferale, M. Cencini, A. Lanotte, M. Sbragaglia and F. Toschi

*New J. Phys. ***6**, 37 (2004)

*[4] Shear effects in passive scalars spectra *

A. Celani, M. Cencini, M. Vergassola, D. Vincenzi and E. Villermaux

*J. Fluid Mech. ***523 **, 99 (2005)