A study of the Navier-Stokes-α model for two-dimensional turbulence

The Navier–Stokes-α model of turbulence is a mollification of the Navier–Stokes equations, in which the vorticity is advected and stretched by a smoothed velocity field. The smoothing is performed by filtering the velocity field over spatial scales of size smaller than α. This is achieved by convolution with a kernel associated with Green's function of the Helmholtz operator scaled by a parameter α. The statistical properties of the smoothed velocity field are expected to match those of Navier–Stokes turbulence for scales larger than α, thus providing a more computable model for those scales. For wavenumbers k such that kα ≫ 1, corresponding to spatial scales smaller than α, there are three candidate power laws for the energy spectrum, corresponding to three possible characteristic time scales in the model equations: one from the smoothed field, the second from the rough field and the third from a special combination of the two. In two dimensions, the second time scale may be understood to characterize th...