Scalar variance cascade in turbulence: η_B = η·Sc^{-1/2}
Batchelor (1959): a passive scalar (temperature, dye) advected by turbulence forms filaments down to the Batchelor scale η_B = η·Sc^{-1/2}, where η is the Kolmogorov scale. For Sc>1 (e.g. temperature in water), scalar fluctuations persist to scales smaller than the velocity field. The spectrum E_θ(k) ~ k^{-1} in the viscous-convective range.