Anomalous Scaling of the Passive Scalar

We establish anomalous inertial range scaling of the structure functions for a model of homogeneous, isotropic advection of a passive scalar by a random velocity field. The velocity statistics is taken as Gaussian with decorrelation in time and velocity differences scaling as ${\left|\mathrm{x}\right|}^{\frac{\kappa }{2}}$ in space, with $0\le \kappa <2\mathrm{}$. The scalar is driven by a random forcing acting on spatial scale $L$. The structure functions for the scalar are well defined as the diffusivity is taken to zero and acquire anomalous scaling for large $L$. The anomalous exponent is calculated explicitly for the fourth structure function and for small $\kappa$.