It is known that the action of a small-scale incompressible flow (having suitable symmetries) on a large-scale perturbation of small amplitude is ‘formally’ diffusive (Kraichnan 1976; Dubrulle & Frisch 1991). There are two essential assumptions. The first one is scale-separation: the ratio e between the typical length-scale of the basic flow and that of the perturbation is small. The second one is the absence of a large-scale AKA effect (Frisch et al 1987). If the basic flow is parity-invariant (i.e. has a center of symmetry), this condition is automatically satisfied. By ‘formally’ diffusive, we understand that, unlike the case of the eddy diffusivity for a passive scalar (Frisch 1989), the eddy viscosity tensor need not be positive definite. There are indeed examples of strongly anisotropic flows (e.g. the Kolmogorov flow), where some components of the tensor are negative, resulting in a large-scale instability (Meshalkin & Sinai 1961; Green 1974; Sivashinsky 1985; Sivashinsky & Yakhot 1985).

When the eddy viscosity tensor is isotropic, the equation for the perturbation reduces to an ordinary diffusion equation, with diffusion coefficient uE.