A passive scalar (dye/temperature) advected by a turbulent velocity field undergoes stretching and folding. Scalar variance decays as ⟨θ²⟩ ~ t^(-n) while forming the characteristic lamellar structures of the Batchelor scale.
Batchelor scale: λ_B = (κ/σ)^(1/2). Below it, diffusion dominates. Kraichnan (1968): in random flow, scalar spectrum E_θ(k) ~ k^(-1) in the viscous-convective range. Rms scalar ~ exp(-σt/2) for uniform strain.