  We develop a fundamental approach to a turbulent constitutive law for the 2D
inverse cascade, based upon a convergent multi-scale gradient (MSG) expansion.
To first order in gradients we find that the turbulent stress generated by
small-scale eddies is proportional not to strain but instead to `skew-strain,'
i.e. the strain tensor rotated by $45^\circ.$ The skew-strain from a given
scale of motion makes no contribution to energy flux across eddies at that
scale, so that the inverse cascade cannot be strongly scale-local. We show that
this conclusion extends a result of Kraichnan for spectral transfer and is due
to absence of vortex-stretching in 2D. This `weakly local' mechanism of inverse
cascade requires a relative rotation between the principal directions of strain
at different scales and we argue for this using both the dynamical equations of
motion and also a heuristic model of `thinning' of small-scale vortices by an
imposed large-scale strain. Carrying out our expansion to second-order in
gradients, we find two additional terms in the stress that can contribute to
energy cascade. The first is a Newtonian stress with an `eddy-viscosity' due to
differential strain-rotation, and the second is a tensile stress exerted along
vorticity contour-lines. The latter was anticipated by Kraichnan for a very
special model situation of small-scale vortex wave-packets in a uniform strain
field. We prove a proportionality in 2D between the mean rates of differential
strain-rotation and of vorticity-gradient stretching, analogous to a similar
relation of Betchov for 3D. According to this result the second-order stresses
will also contribute to inverse cascade when, as is plausible, vorticity
contour-lines lengthen on average by turbulent advection.
