A passive scalar is a quantity that is transported by the flow but does not back react on it. This scheme can be used to describe real scalar transport in the case where the back reaction of the scalar on the velocity field is negligible. Scalar structures are bound to decay in the absence of an external source of fluctuations. If an external pumping is considered, at scales smaller than the injection lenghtscale, a flux of scalar fluctuations develops. In this range of scales the passive scalar phenomenology strongly resembles that of hydrodynamic turbulence, hence the expression “scalar turbulence”. On the contrary, thermal equilibrium is expected at scales larger than the injection scale, where no upscale flux is present. Surprisingly enough, this is not the general case.

In the isotropic case, the violation of equilibrium occurs at the level of the multipoint statistics. This has been firstly obtained in the Kraichnan model for passive scalar advection by Falkovich and Fouxon - PRL 2005. We generalized this result to realistic flows through direct numerical simulations of a passive scalar advected by a two-dimensional turbulent flow in the inverse cascade. The breakdown of Gibbs equilibrium is here related to the presence of long-range correlations between two evolving scalar blobs within the turbulent flow.

The violation of equilibrium expectation is even more dramatic for the anisotropic case, since it occurs already at the level of the two-point statistics. We obtained the result first analytically in the framework of the Kraichnan model. The Lagrangian interpretation of the result suggested that the same result may hold for realistic flows as well. We performed a numerical analysis of the problem that showed consistency with the theoretical results. From a single scalar snapshot one can detect elongated structures much larger than the scalar injection scale (see figure below). This means that a small degree of anisotropy introduced at small scales propagates toward large scales where it eventually dominates the structure of the concentration field.