A Turbulent Constitutive Law for the Two-Dimensional Inverse Energy Cascade

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.Comment: 24 pages, 1 figur

Electracy: the Internet as fifth estate

This account of an experimental approach to apparatus invention is offered as an introduction to apparatus study in general (grammatology), and electracy as the digital apparatus in particular. The approach is that of the Florida Research Ensemble (FRE), working through the EmerAgency, a virtual consultancy developed to translate Arts & Letters pedagogy into institutional consulting. The term 'electracy' (modeled on 'literacy,' a portmanteau of 'electricity' and Derrida's 'trace') was adopted to clarify that digital technologies are not reducible to a 'media literacy,' but include, besides technological innovations, inventions in the dimensions of institution formation and related skill sets, and identity behaviors individual and collective (ethics and politics). Current members of the FRE include Gregory Ulmer (University of Florida), John Craig Freeman (Emerson College), Barbara Jo Revelle (University of Florida), Jack Stenner (University of Florida), Jan Holmevik (Clemson University)