In this talk we will show the global well-posedness of the three
dimensional viscous Camassa-Holm equations, also known as the
Navier-Stokes-alpha model. The dimension of their global attractor
will be esitmated and shown to be comparable with the number of
degrees of freedom suggested by classical theory of turbulence. We
will present semi-rigorous arguments showing that up to a certain wave
number, in the inertial range, the translational energy power specturm
obeys the Kolmogorov power law for the energy decay of the three
dimensional turbulent flow. However for the rest the inertial range
the energy spectrum of this model obeys the Kraichnan power law for
the energy decay of the two dimensional turbulent follows. This
observation makes the Navier-Stokes-alpha model more computable than
the Navier-Stokes equations. Furthermore, we will show that by using
the Camassa-Holm equations Navier-Stokes-alpha model) as a closure
model to the Reynolds averaged equations of the Navier-Stokes one gets
very good agreement with empirical and numerical data of turbulent
flows in infinite pipes and channels.
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