  We consider the behaviour of a passive tracer $\theta$ governed by
$\partial_t\theta + u\cdot\nabla\theta = \Delta\theta + g$ in two space
dimensions with prescribed smooth random incompressible velocity $u(x,t)$ and
source $g(x)$. In 1959, Batchelor, Howells and Townsend (J.\ Fluid Mech.\
5:113) predicted that the tracer (power) spectrum should then scale as
$|\theta_k|^2\propto|k|^{-4}|u_k|^2$ for $|k|$ large depending on the velocity
$u$. For smaller $|k|$, Obukhov and Corrsin earlier predicted a different
spectral scaling. In this paper, we prove that the BHT scaling does indeed hold
probabilistically for sufficiently large $|k|$, asymptotically up to controlled
remainders, using only bounds on the smaller $|k|$ component.
