Dissipation scales and anomalous sinks in steady two-dimensional turbulence

Abstract

In previous papers I have argued that the \emph{fusion rules hypothesis}, which was originally introduced by L'vov and Procaccia in the context of the problem of three-dimensional turbulence, can be used to gain a deeper insight in understanding the enstrophy cascade and inverse energy cascade of two-dimensional turbulence. In the present paper we show that the fusion rules hypothesis, combined with \emph{non-perturbative locality}, itself a consequence of the fusion rules hypothesis, dictates the location of the boundary separating the inertial range from the dissipation range. In so doing, the hypothesis that there may be an anomalous enstrophy sink at small scales and an anomalous energy sink at large scales emerges as a consequence of the fusion rules hypothesis. More broadly, we illustrate the significance of viewing inertial ranges as multi-dimensional regions where the fully unfused generalized structure functions of the velocity field are self-similar, by considering, in this paper, the simplified projection of such regions in a two-dimensional space, involving a small scale $r$ and a large scale $R$, which we call, in this paper, the $(r, R)$-plane. We see, for example, that the logarithmic correction in the enstrophy cascade, under standard molecular dissipation, plays an essential role in inflating the inertial range in the $(r, R)$ plane to ensure the possibility of local interactions. We have also seen that increasingly higher orders of hyperdiffusion at large scales or hypodiffusion at small scales make the predicted sink anomalies more resilient to possible violations of the fusion rules hypothesis.