Multiscaling in passive scalar advection as stochastic shape dynamics

The Kraichnan rapid advection model [Phys. Fluids 11, 945 (1968); Phys Rev. Lett. 72, 1016 (1994)] is recast as the stochastic dynamics of tracer trajectories. This framework replaces the random fields with a small set of stochastic ordinary differential equations. Multiscaling of correlation functions arises naturally as a consequence of the geometry described by the evolution of $N$ trajectories. Scaling exponents and scaling structures are interpreted as excited states of the evolution operator. The trajectories become nearly deterministic in high dimensions allowing for perturbation theory in this limit. We calculate perturbatively the anomalous exponent of the third- and fourth-order correlation functions. The fourth-order result agrees with previous calculations.