  This article is devoted to the mathematical study of a new
Navier-Stokes-alpha model with a nonlinear filter equation. For a given
indicator function, this filter equation was first considered by W. Layton, G.
Rebholz, and C. Trenchea to select eddies for damping based on the
understanding of how nonlinearity acts in real flow problems. Numerically, this
nonlinear filter equation was applied to the nonlinear term in the
Navier-Stokes equations to provide a precise analysis of numerical diffusion
and error estimates. Mathematically, the resulting alpha-model is described by
a doubly nonlinear parabolic-elliptic coupled system. We therefore undertake
the first theoretical study of this system by considering periodic boundary
conditions in the spatial variable. Specifically, we address the existence and
uniqueness of weak Leray-type solutions, their rigorous convergence to weak
Leray solutions of the classical Navier-Stokes equations, and their long-time
dynamics through the concept of the global attractor and some upper bounds for
its fractal dimension. Handling the nonlinear filter equation together with the
well-known nonlinear transport term makes certain estimates delicate,
particularly when deriving upper bounds on the fractal dimension. For the
latter, we adapt techniques developed for hyperbolic-type equations.
