  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.
