Inducing intermittency in the inverse cascade of two dimensional turbulence by a fractal forcing

Abstract

We demonstrate that like in the forward cascade of three dimensional turbulence that displays intermittency (lack of self-similarity) due to the concentration of energy dissipation in a small set of fractal dimension less than three, the inverse cascade of two-dimensional turbulence can also display lack of self-similarity and intermittency if the energy injection is constrained in a fractal set of dimension less than two. A series of numerical simulations of two dimensional turbulence are examined, using different forcing functions of the same forcing length-scale but different fractal dimension $D$ that varies from the classical $D=2$ case to the point vortex case $D=0$. It is shown that as the fractal dimension of the forcing is decreased from $D=2$, the self-similarity is lost and intermittency appears, with the scaling of the different structure functions $\langle |\delta u_\|^p| \rangle\propto r^{\zeta_p}$ differs from the dimensional analysis prediction $\zeta_p=p/3$. The present model thus provides a unique example that intermittency is controlled and can thus shed light and provide test beds for multi-fractal models of turbulence.