 95165 Eyink, G. L.
 Exact results on stationary turbulence in 2D: Consequences
of vorticity conservation
(140K, LaTex)
Mar 25, 95

Abstract ,
Paper (src),
View paper
(auto. generated ps),
Index
of related papers

Abstract. We establish a series of exact results for a model of stationary
turbulence in twodimensions: the incompressible NavierStokes equation
with stochastic force whitenoise 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 blowup of mean energy as $\sim \ell_0^2{{\varepsilon}\over
{\nu}}$ for $\nu\rightarrow 0$ when there is no IRdissipation at the large
lengthscale $\ell_0;$ (2) with an additional IRdissipation, 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 steadystate and the
forward cascade of the higherorder 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.
 Files:
95165.src(
desc ,
95165.tex )