Scaling Relations for a Randomly Advected Passive Scalar Field

A recent ansatz for dissipation terms gave anomalous inertial-range scaling exponents ( $\propto n^{1/2},n\to \infty$) for the $n$th-order structure functions of a passive scalar field advected by a random velocity field. Analysis of a series expansion for the conditional mean of a dissipation term suggests that the ansatz gives the only possible anomalous scaling. Anomaly of inertial-range scaling is supported by realizability inequalities on the dissipation field. Predictions for conditional means and structure functions are compared with simulations.