Applied Mathematics and Plasma Physics

Evelyn Lunasin, Susan Kurien, Mark A. Taylor and Edriss S. Titi, "A study of the two-dimensional Navier-Stokes-*α* model
for two-dimensional turbulence", *J. of Turb.*, vol. 8, 2007

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 *k α* >> 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 the
dynamics of the conserved enstrophy. We measure the scaling of the
energy spectra from high-resolution simulations of the
two-dimensional Navier-Stokes-*α* model, in the limit as *α*
tends to infinity. The energy spectrum of the smoothed velocity
field scales as *k ^{-7}* in the direct enstrophy cascade regime,
consistent with dynamics dominated by the timescale associated
with the rough velocity field. We are thus able to deduce that the
dynamics of the dominant cascading conserved quantity, namely the
enstrophy of the rough velocity, governs the scaling of all
derived statistical quantities.

@article{lunasin-2007-Navier-Stokes-alpha,

author = {Evelyn Lunasin and Susan Kurien and Mark A. Taylor and Edriss S. Titi},

title = {A study of the two-dimensional Navier-Stokes-
$\alpha$
model
for two-dimensional turbulence},

year = {2007},

urlpdf = {http://math.lanl.gov/~skurien/papers/LunKurTayTiti_2dalpha_jot2007.pdf},

journal = {J. of Turb.},

volume = {8}

}