The Batchelor–Howells–Townsend spectrum: Three-dimensional case

Given a velocity field u(x,t), we consider the evolution of passive tracer θ governed by ∂tθ+u⋅∇θ=Δθ+g with time-independent source g(x). When u is small in some sense, Batchelor, Howells and Townsend (Batchelor et al., 1959) predicted that spectrum scales as |θk|2∝|k|−4|uk|2. Following our recent work for two-dimensional case, this paper prove BHT59 scaling does hold probabilistically, asymptotically large wavenumbers enough random synthetic three-dimensional incompressible fields u(x,t). We also relaxed assumptions on source, allowing finite variances both full power latter.