Navier-Stokes-alpha model: LES equations with nonlinear dispersion

We present a framework for discussing LES equations with nonlinear dispersion. In this framework, we discuss the properties of the nonlinearly dispersive Navier-Stokes-alpha (NS−α) model of incompressible fluid turbulence — also called the viscous CamassaHolm equations in the literature — in comparison with the corresponding properties of large eddy simulation (LES) equations obtained via the approximate-inverse approach. In this comparison, we identify the spatially filtered NS−α equations with a class of generalized LES similarity models. Applying a certain approximate inverse to this class of LES models restores the Kelvin circulation theorem for the defiltered velocity and shows that the NS−α model describes the dynamics of the defiltered velocity for this class of generalized LES similarity models. We also show that the subgrid scale forces in the NS−α model transform covariantly under Galilean transformations and under a change to a uniformly rotating reference frame. Finally, we discuss in the spectral formulation how the NS−α model retains the local interactions among the large scales ; retains the nonlocal sweeping effects of large scales on small scales ; yet attenuates the local interactions of the small scales amongst themselves.