Abstract
We consider the problem of anomalous dissipation for passive scalars advected by an incompressible flow. We review known results on anomalous dissipation from the point of view of the analysis of partial differential equations, and present simple rigorous examples of scalars that admit a Batchelor-type energy spectrum and exhibit anomalous dissipation in the limit of zero scalar diffusivity. This article is part of the theme issue 'Scaling the turbulence edifice (part 1)'.
Highlights
We consider the motion of a passive scalar in an incompressible fluid
We are interested in anomalous dissipation, that is, the scalar dissipation rate does not vanish in the limit of zero scalar diffusivity
Anomalous dissipation has been observed in experiments as well as in numerical simulations of turbulent mixing
Summary
We consider the motion of a passive scalar in an incompressible fluid. We confine ourselves to two space dimensions, which impose more restrictions on the motion of the fluid, but simplify the analysis somewhat. The Kraichnan model [25] has played an important role in the study of anomalous diffusion and anomalous scaling (among the many references, we mention in particular [26,27,28,29,30], and the monograph [31]) In this model, the passive scalar is advected by a random flow, generated by a Gaussian-in-space, white-in-time vector field. The purpose of this work is to present simple deterministic examples of scalar fields and advecting flows, compatible with the Batchelor scaling (1.1), for which anomalous dissipation exists and is a purely diffusive effect The flows in these examples are exact solutions of the incompressible Euler or Navier–Stokes equations, with a certain symmetry compatible with that of the scalar.
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