Tracer Turbulence: The Batchelor–Howells–Townsend Spectrum Revisited

Given a velocity field u ( x , t ), we consider the evolution of a passive tracer $$\theta $$ θ governed by $$\partial _t\theta + u\cdot \nabla \theta = \Delta \theta + g$$ ∂ t θ + u · ∇ θ = Δ θ + g with time-independent source g ( x ). When $$\Vert u\Vert $$ ‖ u ‖ is small in some sense, Batchelor, Howells and Townsend (J Fluid Mech 5:134, 1959) predicted that the tracer spectrum scales as $$|\theta _k|^2\propto |k|^{-4}|u_k|^2$$ | θ k | 2 ∝ | k | - 4 | u k | 2 . In this paper we prove that, for random synthetic two-dimensional incompressible velocity fields u ( x , t ) with given energy spectra, this scaling does indeed hold probabilistically, asymptotically almost surely for large | k | and small $$\Vert u\Vert $$ ‖ u ‖ . We also propose an asymptotic correction factor to the BHT scaling arising from the time-dependence of u .