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Inertial-diffusive range for a passive scalar advected by a white-in-time velocity field

It is shown analytically and by Monte Carlo simulations that a passive scalar with finite diffusivity, advected by a white-in-time velocity field with a power law spectrum ∝ k−1−ξ (0 < ξ < 2), has an inertial-diffusive range with a spectrum ∝ k−3−ξ. This is the analog of the Batchelor-Howells-Townsend (J. Fluid Mech., 5 (1959) 134) phenomenological derivation of the k−17/3 law for low-Schmidt-number passive-scalar dynamics in ordinary turbulence.