The Kraichnan Model: A Short Introduction

The Kraichnan model provides one of the clearest solvable settings for passive-scalar transport in turbulence, but its standard formulations often presuppose familiarity with stochastic-process and field-theoretic methods. Here we develop the model from the passive-scalar advection--diffusion equation upward, emphasizing its physical content while introducing the required formalism as it becomes necessary. Instead of using the full Navier--Stokes velocity, with nonlinear dynamics and multi-scale statistics not known in closed form, the model prescribes the advecting velocity as a Gaussian random field that is white in time, correlated in space, and allowed to have a rough spatial increment covariance. It is therefore not a literal model of Navier--Stokes turbulence, but a controlled idealization designed to isolate scalar transport by a rough random flow. The central simplification is that Gaussianity and temporal delta-correlation allow mixed velocity--scalar averages to be eliminated exactly, producing a closed hierarchy for equal-time scalar correlation functions. The two-point equation becomes a diffusion equation in separation space, with molecular diffusion supplemented by a scale-dependent turbulent diffusivity determined by the velocity-increment covariance. This gives a direct route to Richardson-type dispersion, the scalar variance cascade, the dissipative anomaly, and the inertial-range scaling of the second-order scalar structure function, F²(r) = ⟨[c(x + r) − c(x)]²⟩ ∼ r^2−ξ . Higher-order equations show how a Gaussian advecting velocity can nevertheless generate non-Gaussian scalar statistics, intermittency, and anomalous scaling through zero modes and statistical conservation laws of multi-particle configurations. We also discuss the Lagrangian particle interpretation, smooth versus rough stochastic flows, slow modes and particle-cluster geometry, compressible and anisotropic extensions, and the relation to renormalization-group and operator-product-expansion descriptions. The emphasis throughout is on the physical meaning of closure, roughness, stochastic particle motion, scalar diffusion, and the connection between Kraichnan-type advection and high-Schmidt-number liquid diffusion.