We review the properties of the nonlinearly dispersive Navier-Stokes-alpha (NS-) model of incompressible fluid turbulence - also called the viscous Camassa-Holm equations in the literature. We first re-derive the NS- model by filtering the velocity of the fluid loop in Kelvin's circulation theorem for the Navier-Stokes equations. Then we show that this filtering causes the wavenumber spectrum of the translational kinetic energy for the NS- model to roll off as k-3 for k > 1 in three dimensions, instead of continuing along the slower Kolmogorov scaling law, k-5/3, that it follows for k < 1. This roll off at higher wavenumbers shortens the inertial range for the NS- model and thereby makes it more computable. We also explain how the NS- model is related to large eddy simulation (LES) turbulence modeling and to the stress tensor for second-grade fluids. We close by surveying recent results in the literature for the NS- model and its inviscid limit (the Euler- model). 2001 Published by Elsevier Science B.V.
