A turbulent constitutive law for the twodimensional inverse energy cascade
Abstract
The inverse energy cascade of twodimensional turbulence is often represented phenomenologically by a Newtonian stress strain relation with a ‘negative eddy viscosity’. Here we develop a fundamental approach to a turbulent constitutive law for the twodimensional inverse cascade, based upon a convergent multiscale gradient (MSG) expansion. To first order in gradients, we find that the turbulent stress generated by smallscale eddies is proportional not to strain but instead to ‘skewstrain,’ i.e. the strain tensor rotated by 45°. The skewstrain 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 scalelocal. We show that this conclusion extends a result of Kraichnan for spectral transfer and is due to absence of vortex stretching in two dimensions. 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 smallscale vortices by an imposed largescale strain. Carrying out our expansion to second order in gradients, we find two additional terms in the stress that can contribute to the energy cascade. The first is a Newtonian stress with an ‘eddyviscosity’ 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 smallscale vortex wavepackets in a uniform strain field. We prove a proportionality in two dimensions between the mean rates of differential strain rotation and of vorticitygradient stretching, analogous to a similar relation of Betchov for three dimensions. According to this result, the secondorder stresses will also contribute to inverse cascade when, as is plausible, vorticity contour lines lengthen, on average, by turbulent advection.
 Publication:

Journal of Fluid Mechanics
 Pub Date:
 2006
 DOI:
 10.1017/S0022112005007883
 arXiv:
 arXiv:nlin/0512023
 Bibcode:
 2006JFM...549..191E
 Keywords:

 Nonlinear Sciences  Chaotic Dynamics
 EPrint:
 24 pages, 1 figure