Inhomogeneous anisotropic passive scalars

We investigate the behaviour of the two-point correlation function in the context of passive scalars for non-homogeneous, non-isotropic forcing ensembles. Exact analytical computations can be carried out in the framework of the Kraichnan model for each anisotropic sector. We will focus our attention on the isotropic sector with isotropic forcing in order to obtain a description of the influence of purely inhomogeneous contributions. It is shown how the homogeneous solution is recovered at separations smaller than an intrinsic typical lengthscale induced by inhomogeneities, and how the different Fourier modes in the centre-of-mass variable recombine themselves to give a 'beating' ( superposition of power laws) described by Bessel functions. The pure power-law behaviour is restored even if the inhomogeneous excitation takes place at very small scales.