Eyink, G. L.
Exact results on stationary turbulence in 2D: Consequences
of vorticity conservation
(140K, LaTex)
ABSTRACT. We establish a series of exact results for a model of stationary
turbulence in two-dimensions: the incompressible Navier-Stokes equation
with stochastic force white-noise in time. Essentially all of our conclusions
follow from the simple consideration of the simultaneous conservation of energy
and enstrophy by the inertial dynamics. Our main results are as follows:
(1) we show the blow-up of mean energy as $\sim \ell_0^2{{\varepsilon}\over
{\nu}}$ for $\nu\rightarrow 0$ when there is no IR-dissipation at the large
length-scale $\ell_0;$ (2) with an additional IR-dissipation, we establish
the validity of the traditional cascade directions and magnitudes of flux
of energy and enstrophy for $\nu\rightarrow 0,$ assuming finite mean energy
in the limit; (3) we rigorously establish the balance equations
for the energy and vorticity invariants in the 2D steady-state and the
forward cascade of the higher-order vorticity invariants assuming finite
mean values;
(4) we derive exact inequalities for scaling exponents
in the 2D enstrophy range,
as follows: if $\langle|\Delta_\bl\omega|^p\rangle\sim
\ell^{\zeta_p},$ then
$\zeta_2\leq{{2}\over{3}}$ and $\zeta_p\leq 0$ for $p\geq 3.$
If the minimum
H\"{o}lder exponent of the vorticity $h_{\mn}<0,$ then we
establish a 2D analogue
of the refined similarity hypothesis which improves these bounds.
The most novel and interesting conclusion of this work is the
connection established between ``intermittency'' in 2D and ``negative
H\"{o}lder singularities'' of the vorticity: we show that the latter are
necessary for deviations from the 1967 Kraichnan scaling to occur.