Multiscaling in passive scalar advection as stochastic shape dynamics

TL;DR

This paper models passive scalar advection using stochastic trajectory dynamics, revealing multiscaling as a geometric consequence and calculating anomalous exponents through perturbation theory.

Contribution

It introduces a stochastic shape dynamics framework for the Kraichnan model, providing new insights into multiscaling and calculating anomalous exponents perturbatively.

Findings

Perturbative calculation of third and fourth order anomalous exponents.

Fourth order results agree with previous studies.

Trajectories become nearly deterministic in high dimensions.

Abstract

The Kraichnan rapid advection model 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.