Advection-diffusion of a passive scalar θ by incompressible 2D velocity field. Watch the Batchelor spectrum k⁻¹ emerge at scales below the diffusion cutoff. Click to add dye blobs.
Advection-diffusion: ∂θ/∂t + u·∇θ = κ∇²θ. Batchelor (1959): for κ→0, scalar spectrum E_θ(k) ~ k⁻¹ in the viscous-convective range (scales smaller than Kolmogorov, larger than Batchelor). Obukhov-Corrsin spectrum for k