  We review the properties of the nonlinearly dispersive Navier-Stokes-alpha
(NS-alpha) model of incompressible fluid turbulence -- also called the viscous
Camassa-Holm equations and the LANS equations in the literature. We first
re-derive the NS-alpha 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-alpha model to roll off as k^{-3} for k \alpha > 1 in three
dimensions, instead of continuing along the slower Kolmogorov scaling law,
k^{-5/3}, that it follows for k \alpha < 1. This rolloff at higher wavenumbers
shortens the inertial range for the NS-alpha model and thereby makes it more
computable. We also explain how the NS-alpha 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-alpha
model and its inviscid limit (the Euler-alpha model).
