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\begin{document}
\title{Exact Results on Stationary Turbulence in 2D:\\
       Consequences of Vorticity Conservation}
\author{Gregory L. Eyink\\{\em Department of Mathematics, Building No. 89}\\
        {\em University of Arizona, Tucson, AZ 85721}}
\date{ }
\maketitle
\begin{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.

\end{abstract}

\newpage 

\section{Introduction}

An essential difference between the incompressible 
Euler equations in 
2D and 3D is that the vorticity in each fluid 
element is conserved in 
the former but not the latter. This conservation 
is expressed by the fact 
that the vorticity field in 2D satisfies the equation
\be D_t\omega(\br,t)=0, \lb{1} \ee
where $D_t=\partial_t+\bv(\br,t)\bdot\grad$ is the convective 
or material 
derivative. It follows that each of the integrals
\be \om^{(n)}(t)={{1}\over{n!}}\int_\Lambda d^2\br\,\,
\omega^n(\br,t), \lb{2} \ee
for integers $n>0$ is an invariant of the dynamics, in 
addition to the energy integral
\be E(t)={{1}\over{2}}\int_\Lambda d^2\br\,\,v^2(\br,t). 
\lb{3} \ee
These statements are true in 2D not only for smooth, 
classical solutions,
but also for a wide class of {\em weak solutions}, relevant 
to the description
of turbulence. For example, the weak solutions constructed 
by Yudovich 
\cite{1} for the initial data $\omega_0\in L^\infty(\Lambda)$ 
conserve each of the above integrals. 

However, it has long been recognized that the
presence of an additional quadratic invariant, 
the {\em enstrophy} $\Omega(t)
=\om^{(2)}(t)$, changes drastically the nature of 2D 
turbulence in comparison 
to the 3D case: see the works of Lee \cite{2} and 
Fjortoft \cite{3}. 
In the late 1960's it was suggested by Kraichnan \cite{4}, 
Leith \cite{5},
and Batchelor \cite{6} that this extra constraint 
in 2D results in two 
distinct inertial ranges which can exist 
simultaneously: an energy-
transferring range in which energy is passed down 
to smaller wavenumbers 
and an enstrophy-transferring range in which enstrophy 
is passed up to larger
wavenumbers. To maintain such a steady state, energy 
and enstrophy need to 
be fed into the fluid at an intermediate range. More 
recently, such steady-states 
driven by random external forces have been rigorously 
constructed by Vishik 
and Fursikov: see Theorem XI.2.1 in \cite{7}. 
Nevertheless, much of the conventional
picture of ``dual cascades'' of energy and enstrophy 
remains conjectural. 
Furthermore, a new attempt by Polyakov \cite{8,9} 
to construct stationary
statistical solutions of the Euler equations 
from the hierarchy of 
correlation functions in suitable 2D conformal 
field theories has prompted
some questions \cite{10} concerning the role of the 
higher invariants in Eq.(\ref{2}) for $n\geq 3.$ 

Our goal in this work is to derive---either {\em a priori} 
or under suitable hypotheses
---exact results in the theory of 2D turbulence. We have 
in mind two audiences. On the 
one hand, the paper should be helpful to fluid mechanicians, 
statistical physicists and engineers 
working in this area in order to provide rigorous results 
as constraints and checks on physical 
theory-building. On the other hand, the paper should be 
useful for mathematicians 
interested in 2D turbulence in order to provide a mathematical 
statement of known results
and open problems in the area, which are generally formulated 
less precisely. In view 
of these objectives, many of the results of this 
paper are ``old,'' as the conclusions
have long been believed but without careful proofs. 
In such cases, we have found
that the traditional arguments can often be converted 
to proofs without much difficulty. However,
the paper also contains several new results, including, 
in particular, exact bounds on 
scaling exponents of vorticity. To make the paper readable 
to both audiences we have 
adopted a compromise between the styles of the different 
communities. We do not state
results as theorems in the text and many of technical 
details of the proofs are 
relegated to three long Appendices, for readers interested 
in those details. Hopefully
mathematicians reading the paper will be able to formulate 
precise theorem statements 
for themselves from the textual discussion and the technical 
appendices. 

The precise contents of this work are as follows: 
in Section 2 we introduce the balance equations for energy 
and the vorticity invariants
in 2D, which form the basis of most of our analysis. 
We discuss the relation between 
regularity of the 2D Euler solutions and the validity of 
the naive conservation laws.
We also introduce here the models with random stirring 
forces, whose stationary measures
are our main focus in this work. In Section 3 we prove 
results on the cascade directions
of energy and enstrophy in these steady states: that is, 
we establish the existence of
scale ranges of constant mean flux, as well as the signs 
and magnitudes of the fluxes. 
These results in general require hypotheses on bounded 
means of energy and enstrophy 

\newpage

\noindent as viscosity tends to zero, which are 
interesting open problems. Finally, we discuss here
the fluxes of the higher-order vorticity invariants, 
and establish their cascade direction
as toward the ultraviolet, appealing to stronger (but still 
reasonable) assumptions 
of finite means. In Section 4 we use the results of the 
previous sections to derive 
rigorous bounds on the scaling exponents of the vorticity 
difference in the 2D enstrophy
cascade range. These bounds all appear to be new, except 
for the energy spectrum bound
which was previously observed by Sulem and Frisch (who, 
however, never published a complete
argument.) Our bounds establish a precise connection between 
``anomalous scaling'' in
2D, or corrections to the 1967 Kraichnan scaling theory, 
and the existence of ``negative H\"{o}lder singularities'' 
of the vorticity. Although it follows easily from a 
heuristic 
``multifractal model'' of 2D turbulence, the connection 
does not seem to have been pointed out before.

Even more than the 3D energy inertial range, the 2D 
cascades are a ``theoreticians'
turbulence.'' From a practical point of view the subject 
has interest mostly from the 
possibility that certain geophysical flows, such as 
atmospheres on scales of 100-10,000 km,
are predominantly two-dimensional in nature, from the 
similiarity to certain problems in 
plasma physics, and from the relation to the rapid 
rotation limit of 3D turbulence, which,
by consideration of the Taylor-Proudman theorem, 
ought to behave as two-dimensional
for small Rossby number. From the purely theoretical 
point of view the 2D steady-state 
is a useful model on which to test ideas for the 3D case, 
since it is generally more amenable 
to exact analysis but is still a strongly nonlinear, 
statistically nonequilibrium system 
with highly nontrivial behavior. It is mostly this latter 
point of view which motivates our work here.

\section{Vorticity Balance and Transfer Equations}

\noindent {\em (i) Balance Equations for Free Evolution}

We here consider the balance equations governing the 
local conservation of the vorticity invariants 
in space and in scale. To introduce the concepts in 
the simplest context, we discuss first
free evolution, i.e. equations without any external forcing. 
Thus, our starting point is the 
2D Euler equations in the ``vorticity formulation'' 
\be \partial_t\omega(\br,t)+\grad\bdot(\bv(\br,t)
\omega(\br,t))=0, \lb{4} \ee
in which the velocity field is recovered instantaneously 
from the vorticity field via
the integral equation
\be \bv(\br,t)=\int_\Lambda d^2\br'\,\,\bB_\Lambda(\br-\br')
\omega(\br',t). \lb{5} \ee
Here $\bB_\Lambda(\br)\equiv \grad\btimes(\bE_3
D_\Lambda(\br))$ and $D_\Lambda(\br)$
is the Green function of the laplacian in the 2D domain 
$\Lambda$ with Dirichlet b.c. To avoid 
technically-involved discussions of the boundary conditions, 
we shall actually consider here
always $\Lambda=\bT^2,$ the 2D torus of sidewidth $\ell_0,$ 
and adopt periodic b.c. There are, in our opinion,
more substantial differences between free and forced turbulent 
flow in 2D than there are in 3D.
This is mostly connected with the fact that the 2D enstrophy 
cascade is ``non-accelerated,'' so that 
no finite-time singularities occur, and ``non-local,'' 
so that the statistics of the largest 
eddies play a more substantial role. We hope to treat 
elsewhere in more detail the subject of 
enstrophy cascade in freely-decaying 2D turbulence. 
Here we shall just discuss those aspects that 
will be relevant to our main topic, the forced steady-states.

To define a precise notion of ``scale of motion'' we use 
the {\em filtering approach} 
which is common in the turbulence modelling method of 
large-eddy simulation and already exploited by us
for an exact analysis of energy transfer in the 
incompressible Euler
equations for arbitrary dimension \cite{11,12}. 
The basic idea 
of the filtering technique is to define a ``large-scale'' 
vorticity field $\ol_\ell(\br,t)$
and a ``small-scale'' field $\os_\ell(\br,t)$ instantaneously 
by the equations
\be \ol_\ell(\br,t)=\int_\Lambda d^2\br'\,\,G_\ell(\br-\br')
\omega(\br',t) \lb{6} \ee
and 
\be \os_\ell(\br,t)=\omega(\br,t)-\ol_\ell(\br,t). \lb{7} \ee
The ``filter function'' $G_\ell(\br)=\ell^{-d}G(\ell^{-1}\br)$ 
is assumed to 
be smooth with rapid decay, both in physical and in Fourier 
space. The parameter
$\ell$ fixes the arbitrary length in the division of the field 
into ``large-scale''
and ``small-scale'' components. If the filter is convoluted 
with the equation of motion, Eq.(\ref{4}),
an equation for the large-scale vorticity field is obtained 
of the form
\be \partial_t\ol_\ell(\br,t)+\grad\bdot(\vl_\ell(\br,t)
\ol_\ell(\br,t)+\bsigma_\ell(\br,t))=0. \lb{8} \ee
The large-scale velocity field is defined here in the same 
manner by $\vl_\ell=G_\ell*\bv.$
Furthermore
\be \bsigma_\ell(\br,t)\equiv \overline{(\bv\omega)}_\ell
(\br,t)-\vl_\ell(\br,t)\ol_\ell(\br,t) \lb{9} \ee
plays the same role as the usual small-scale stress tensor 
$\btau(\br,t)$ in the analogous equation 
for the large-scale velocity. Thus, we refer to it as the 
{\em vortical stress.} Observe that 
$\sigma_i=\en_{jk}\partial_j\tau_{ik}$ in terms of the usual 
stress, with $\en_{ij}$ the 
antisymmetric Levi-Civita tensor in 2D. 

The {\em enstrophy} integral
\be \Omega(t)={{1}\over{2}}\int_\Lambda d^2\br\,\,\omega^2(\br,t) 
\lb{10} \ee
is formally conserved by the dynamics Eq.(\ref{4}). From the 
Eq.(\ref{8}) for the 
large-scale vorticity it is straightforward to derive by the 
standard methods of 
nonequilibrium thermodynamics a {\em local balance equation} for 
its large-scale density
\be h_\ell(\br,t)\equiv {{1}\over{2}}\ol_\ell^2(\br,t). 
\lb{11} \ee 
It has the form 
\be \partial_t h_\ell(\br,t)+\grad\bdot\bK_\ell(\br,t)=
-Z_\ell(\br,t) \lb{12} \ee
in which the current
\be \bK_\ell(\br,t)\equiv h_\ell(\br,t)\vl_\ell(\br,t)+
\ol_\ell(\br,t)\bsigma_\ell(\br,t) \lb{13} \ee
represents space-transport of the large-scale enstrophy, 
and the {\em enstrophy flux} 
\be Z_\ell(\br,t)\equiv -\grad\ol_\ell(\br,t)\bdot
\bsigma_\ell(\br,t) \lb{14} \ee
represents the enstrophy transfer to the small-scale 
modes. Note that
the large-scale {\em vorticity-gradient} $\xil_\ell
\equiv\grad\ol_\ell$ enters into 
the enstrophy transfer process in two different ways: 
the term $-\vs\bdot\grad\ol$
in the equation for $\os$ produces a growth in that 
quantity at the expense of the 
large-scale vorticity $\ol,$ and the term 
$-\overline{\xi}_j\cdot\partial_i\overline{v}_j$
in the equation for $\overline{\xi}_i$ itself produces a 
steepening of the large-scale vortex-gradient 
(hence a redistribution of the vorticity to higher 
wavenumbers) by nonlinear stretching 
due to the strain at that same scale. Both of these 
processes contribute to the dynamical 
development of the enstrophy flux $Z_\ell.$ 

An identical analysis can be made of the balance 
for the local densities
\be h^{(n)}_\ell(\br,t)={{1}\over{n!}}\ol^n_\ell
(\br,t) \lb{15} \ee
of the contribution to the invariants $\om^{(n)}$ 
in the large-scale modes. 
(Observe that $h^{(n)}_\ell$ has wavenumber support 
inside a spectral radius 
$\sim n/\ell.$) By a similar calculation as before it 
follows that
\be \partial_t h_\ell^{(n)}(\br,t)+\grad\bdot\bK_\ell^{(n)}
(\br,t)=-Z_\ell^{(n)}(\br,t) \lb{16} \ee
with
\be \bK_\ell^{(n)}(\br,t)\equiv h_\ell^{(n)}(\br,t)\vl_\ell(\br,t)
+h_\ell^{(n-1)}(\br,t)\bsigma_\ell(\br,t) \lb{17} \ee
and
\be Z_\ell^{(n)}(\br,t)\equiv h_\ell^{(n-2)}
(\br,t)Z_\ell(\br,t). \lb{18} \ee
The latter represents the flux of the invariant quantity 
$\om^{(n)}$ from the large-scale to the 
small-scale modes. It is of some interest that for $n>2$ 
it is simply proportional to the enstrophy flux itself.

\noindent {\em Vorticity conservation and regularity 
of initial data}

Although each of the invariants $\om^{(n)}$ is {\em formally} 
conserved by the 2D Euler equations, they need 
not be conserved for general weak solutions. In the 
terminology of Polyakov \cite{8,9} these naive
conservation laws may be vitiated by ``anomalies.'' This 
corresponds exactly to the traditional 
picture of the ultraviolet enstrophy cascade in \cite{4,5,6}, 
which posits some nonvanishing flux 
of enstrophy escaping to infinite wavenumber. The analogous 
possibility for failure of naive energy 
conservation was already noted by Onsager \cite{14}, who pointed 
out that a minimal degree of ``irregularity''
is required of the velocity field for this to occur: in fact, 
the velocity field must have H\"{o}lder 
index $h\leq {{1}\over{3}},$ for otherwise the flux will vanish 
asymptotically and energy will be conserved. 
A corresponding statement holds in the case of 2D enstrophy 
conservation \cite{13}: the H\"{o}lder index
of the {\em vorticity field} must take a value $h\leq 0,$ 
or otherwise enstrophy is conserved. 

The proof of this conservation statement is very similar 
to that for energy conservation \cite{15,11}. 
Indeed, an expression exists for the ``vortical stress vector'' 
$\bsigma$ quite similar to one discovered 
by Constantin et al. \cite{15} for the ordinary stress: 
\be \bsigma(\br,t)=[\Delta\omega(\br,t)\Delta\bv(\br,t)]_\ell-
[\Delta\omega(\br,t)]_\ell[\Delta\bv(\br,t)]_\ell. \lb{19} \ee
Here $\Delta_\bs f(\br)\equiv f(\br)-f(\br-\bs)$ denotes 
backward-difference and $[f]_\ell=
\int d^2\bs\,\,G_\ell(\bs)f(\bs)$ is the average over the 
separation-vector $\bs$ with respect to the filter
function. Since
\be \Delta_\bl\omega(\br,t)=O(\ell^h) \lb{20} \ee
implies
\be \Delta_\bl\bv(\br,t)=O(\ell) \lb{21} \ee
and 
\be \grad\ol_\ell(\br,t)=O(\ell^{h-1}), \lb{22} \ee
it follows directly from the expression Eq.(\ref{14}) for the 
enstrophy flux that
\be Z_\ell(\br,t)=O(\ell^{2h}). \lb{23} \ee
Hence, $Z_\ell\rightarrow 0$ for $\ell\rightarrow 0$ when $h>0,$ 
and enstrophy is conserved.
For that matter, it follows from the like expressions Eq.(\ref{18}) 
that $Z_\ell^{(n)}=O(\ell^{2h})$
and the higher-order vorticity invariants are conserved as well 
when $h>0.$ This H\"{o}lder criterion 
for enstrophy conservation is actually contained in Yudovich' 
more general result \cite{1}, since for $h>0$ 
it obviously holds that $C^h(\Lambda)\subset L^\infty(\Lambda).$

These results allow some classification of decaying turbulence 
in 2D according to the regularity of the 
initial data. It is known from the work of Wolibner \cite{15a}, 
Yudovich \cite{1} and Kato \cite{15b}, that 
a unique solution of the Euler equations exists globally 
in time for initial data $\omega_0\in C^{k+s},\,\,\,
k\in \bZ^+_0,0<s<1.$ Furthermore, the solution remains 
H\"{o}lder continuous at any finite time $t$ with positive 
exponent $h(t)=k+\exp[-C\|\omega_0\|_\infty t]s.$ Therefore, 
it follows that, regardless of the size of the 
Reynolds number initially, there will be no turbulence in 2D---
in the sense of a long enstrophy cascade range---
if the initial data $\omega_0\in C^{k+s}$ with $k\geq 1.$ This 
might be called initial data of Type O. On the other 
hand, if $\omega_0\in C^s,\,\,\,0<s<1$---which we may refer to 
as Type A---then turbulent cascades may develop 
over time. Although fluxes of enstrophy must vanish asymptotically 
for $\ell\rightarrow 0$ at any finite time $t>0$ 
starting from such data, nevertheless mean constant fluxes of 
these invariants may occur over growing ranges of 
$\ell$ as $t\rightarrow\infty.$ This accords very well with the 
traditional picture of the 2D enstrophy cascade, which 
postulates a ``non-accelerated cascade'' in which constant fluxes 
develop at time $t$ down to an exponentially 
small, but non-zero, length-scale $\sim \ell_0 e^{-C\overline
{\sigma}t}$ \cite{15c,15d}. The situation for 
$\omega\in L^{\infty}$ is the borderline case, but Yudovich' 
results show that it is essentially similar and
can still be considered Type A. 

However, it seems possible that the enstrophy conservation could 
be violated in finite time (or immediately) for less 
regular initial data, which we refer to as Type B. This phenomenon 
would be the 2D analogue of the Onsager conjecture 
on existence of energy-dissipative solutions of Euler in 3D. Weak 
solutions of 2D Euler have been constructed under
less restrictive assumptions on the initial data than those assumed 
by Yudovich. In fact, existence has now been
proved within successively larger classes: for $\omega\in L^2$ by 
Vishik and Komech \cite{40}, for $\omega\in L^p$ with any 
$1<p<\infty$ by DiPerna and Majda \cite{41}, for $\omega\in L\log L$ 
by Chae \cite{15e}, and, most generally, for 
$\omega\in L^1$ by Delort \cite{15f}. Nothing is known about uniqueness 
of solutions in these classes, nor about 
conservation properties. We conjecture that enstrophy conservation 
can be violated at finite time for suitable initial 
data $\omega_0\in L^p$ with any $p<\infty.$ In the case $\omega\in L^2$ 
it was argued by Sulem and Frisch \cite{34} that 
enstrophy will be conserved if the energy spectrum obeys a suitable 
decay estimate $E(k)=O\left(k^{-({{11}\over{3}}+\en)}
\right)$ for any $\en>0.$ We shall derive this result below in 
the context of forced, steady-states. However, it 
remains possible that enstrophy conservation may fail in the class 
$\omega\in L^2$ with spectrum decaying less steeply.

The previous results on weak solutions do not cover all the cases of 
physical interest. For example, a common 
situation considered for numerical simulation is decaying turbulence 
with initial data chosen from a Gaussian random 
ensemble of initial vorticity fields $\omega_0$ with the Kraichnan 
$k^{-3}$ energy spectrum: see
Farge et al. \cite{28,29}, Benzi and Vergassola \cite{30}. In that 
case, the initial vorticity fields
have {\em infinite enstrophy} with finite probability, and 
$\omega_0\in L^2$ fails. In this case, as well as others,
it is more natural to consider $\omega\in B^{s,\infty}_p,$ the 
{\em Besov spaces}, with suitable $s$ and $p.$ 
These spaces will be very important for our later analyses, but here 
we just remark that they correspond to 
classes of functions with H\"{o}lder index $s$ in the space 
$L^p$-mean sense. It is a consequence of Theorem 4 in 
\cite{36} for the case $p=2$ that the individual realizations of a 
homogeneous ensemble of random vorticity fields
with enstrophy spectrum $\Omega(k)=O\left(k^{-(1+2s)}\right)$ will 
belong to $B^{s-,\infty}_2$ with probability
one. Therefore, with the Kraichnan spectrum, for which $s=0,$ 
it will be true that $\omega_0\in B^{0-,\infty}_2.$ This
space is ``marginally worse'' than $L^2$ and existence within 
this class is not given by any of the previous
theorems. 

Recently, Shnirelman \cite{42a} has established solvability 
of the Euler equations for $\omega\in B^{s,\infty}_2$
with $s>2.$ Actually, solutions with such large values of $s$
 are classical and Shnirelman's results are only 
a slight improvement of the well-known results of Ebin and
 Marsden \cite{15g} and others on solvability with 
$\omega\in H^{s},\,\,\,s>2.$ The result on enstrophy conservation
 of Sulem and Frisch \cite{34} is best 
stated in terms of the Besov spaces, and implies that $\omega
\in B^{s,\infty}_2,\,\,\,s>{{1}\over{3}}$ 
is a sufficient condition for enstrophy conservation to hold. 
Clearly, the Besov index in Shnirelman's construction
is too large to be of interest for 2D enstrophy cascades, but 
we conjecture that Euler solutions in fact exist globally 
in $B^{0,\infty}_2.$  This conjecture has a very natural 
physical interpretation that a bound of the enstrophy spectrum 
by Kraichnan's spectrum, $\Omega(k)=O\left(k^{-1}\right),$ 
will be dynamically preserved in time. It would be very natural 
to look for enstrophy dissipation at finite time within this class. 

There is an issue raised in the works \cite{29,30} which shall 
be important in our discussion later of the steady-states.
Both of these works reported that ``negative exponents'' occur
due to ``cusps'' in vortex cores, with $h\approx -1/2$ 
the most typical value and $h_{\mn}\approx -1.$ Since there 
seemed to be no internal mechanism in 2D for generation of 
such singularities from regular initial data, it was conjectured 
in \cite{29,30} that the ``cusps'' observed 
in those works were present initially. Actually, a homogeneous 
{\em Gaussian} random field with enstrophy spectrum 
$\sim k^{-(1+2s)},$ from which initial data were selected in 
\cite{29,30} for $s=0,$ have for {\em all} $s\in \bR$ 
realizations $\omega\in B^{s-,\infty}_{\infty}$ a.s. and not 
merely $\omega\in B^{s-,\infty}_{2}$ as given by Theorem 4 
of \cite{36}. This is a simple generalization to nonpositive 
$s\in \bR$ of the well-known Wiener-L\'{e}vy theorem, which 
states that the minimum H\"{o}lder singularity for realizations 
of such a Gaussian ensemble is $h_{\mn}=s-$ a.s. 
(G. Eyink, unpublished). \footnote{Because of this fact, our 
classification of initial data into ``Type A'' and 
``Type B'' for vorticity fields chosen at random from Gaussian 
ensembles coincides with that used by She et al. 
(for velocity fields) in the study of decaying Burgers 
turbulence \cite{42aa}.} Therefore, the initial data of \cite{29,30}
are distributional, but with no singularities as severe as $h=-1/2$ 
and the ``cusps'' observed, if real and not 
numerical artefacts,  must have been produced in the course of the 
dynamical evolution. Such a phenomenon would not
be dissimilar to what happens for $h>0.$ As discussed above, the 
exponent in that case can decay, but at most to 
$[\![h]\!],$ the greatest integer less than $h$ \cite{15a,1,15b}.  
Similarly, one might suppose that if the initial data 
$\omega\in B^{0,\infty}_\infty(\bT^2),$ then a solution exists 
globally in $B^{-1,\infty}_\infty(\bT^2).$ 
Again, the exponent may deteriorate in magnitude by 1. This 
behavior would be consistent with our conjecture
that a solution exists globally in $B^{0,\infty}_2(\bT^2),$ 
because $B^{0,\infty}_2(\bT^2)\subset 
B^{-1,\infty}_\infty(\bT^2)$ in 2D as a consequence of Besov 
space embedding theorems (see Appendix III).

\noindent {\em Remark on nonlocality}

A few remarks are in order regarding the issue of ``locality'' 
of the enstrophy cascade. It is
possible to make an analysis of the contributions to the 
enstrophy flux from wavevector
triads in distinct octave bands, analogous to that made 
for energy transfer in \cite{16,12}.
Except for a single class of contributions, all triads 
yield an enstrophy flux $Z_{\ell_K}$ for 
$\ell_K=2^{-K}$ which is actually $O(\ell_K^{3h}).$ This 
is the magnitude of the flux 
contributed by the ``local triads'' with all wavevectors 
from octave bands near the $K$th.  
However, the nonlocal class of terms like $\omega_N
(\bv_M\bdot\grad)\omega_L$ with $N,L\approx K$
and $M\ll K$ contributes at the order
\be Z^{{\rm nloc.}}_{\ell_K}\sim 2^{K(1-2h)}2^{-M(1+h)}
{{2^M}\over{2^K}}=2^{-hM}2^{-2hK}. \lb{24} \ee
Note that $\bv_M\sim 2^{-(1+h)M}$ since the velocity field 
is in $C^{1+h}$ when $\omega\in C^h$ 
and that the factor $2^M/2^K$ comes from cancellations due 
to detailed conservation, as described 
in \cite{16,12}. This class of contributions is dominated 
by the triads with $M\approx 0,$ i.e. by the 
largest-scale modes, and gives the leading term in the flux 
$\sim \ell_K^{2h}.$ Hence, the enstrophy cascade
is {\em infrared-dominated} and nonlocal when $0<h<1.$ 
\footnote{Notice that our rigorous analysis is entirely in 
terms of {\em upper bounds}. Therefore, we have not really 
established nonlocal domination of the cascade 
unless we can show that these upper bounds are achieved. 
We expect this to be true but have no proof.}

The preceding analysis was made in Fourier language, but the 
same result may be obtained in physical space. 
In fact, the estimate $\sim \ell^{2h}$ obtained in Eq.(\ref{23}) 
is of the same magnitude, and its origin is 
similar. Locality is violated because $\Delta_\bl\bv$ is not 
a small-scale quantity when $\bv\in C^{1+h}$ 
and $h>0$! The proper small-scale quantity in this circumstance 
is instead
\be \bv(\br+\bl)-\bv(\br)-(\bl\bdot\grad)\bv(\br)=
O(\ell^{1+h}), \lb{25} \ee
requiring an additional subtraction $\sim O(\ell).$ The 
subtracted term still appears in $\Delta_\bl\bv(\br),$ 
which is dominated by the large-scales. For this reason 
the ``vortical stress'' $\bsigma$ in 2D
is not a truly small-scale quantity when $\omega\in C^h,\,\,
0<h<1,$ but instead gets its dominant contribution 
from the infrared. The infrared domination of enstrophy 
transfer was already remarked by Kraichnan in \cite{4} 
from his ``K41-style'' theory. Here it is demonstrated 
from the fluid equations themselves. 

\noindent {\em (ii) Statistical Steady States}

Up until this point we have considered only the 
formally-conservative inertial dynamics. However,
to produce stationary cascade states there must be added 
to the equations both external forces, 
to provide an input of energy and enstrophy, and viscous 
damping, to provide a source of dissipation. It is 
particularly convenient to produce the turbulence with a 
zero-mean, Gaussian random force, white-noise in time: 
\be \langle f_i(\br,t)f_j(\br',t')\rangle= 2F_{ij}(\br-\br')
\delta(t-t'). \lb{29} \ee
Here $F_{ij}(\br-\br')$ is the integral kernel of the covariance 
operator, a positive, trace-class operator on 
$L^2(\bT^d).$ In this case it is possible to construct in 
finite domain $\Lambda$ statistical solutions 
of the Navier-Stokes dynamics: see the monograph of Vishik 
and Fursikov \cite{7}, Theorems X.1.1 and X.1.2.
Those authors demonstrated also for this case the existence 
of stationary statistical solutions with finite
mean energy and enstrophy, by explicit construction through 
a rigorous application of the Bogolyubov-Krylov 
averaging method (Theorem XI.2.1 of \cite{7}.) Aside from 
the possibility of rigorous proofs of existence,
the most convenient aspect of the white-noise force case 
is the fact that energy and enstrophy inputs are 
entirely external control parameters without any dependence 
upon the detailed statistics of the fluid. 
This was first demonstrated formally by Novikov \cite{19}. 
The precise statement of his result is that 
\be \ven\equiv \nu\langle|\grad\bv(\br,t)|^2\rangle=F_{ii}(\bz). 
\lb{30} \ee
This is essentially an energy balance relation, expressing 
equality of mean dissipation on the left with 
mean energy input by the force on the right. Novikov's result 
has been rigorously verified by Vishik and Fursikov 
in 2D, but in 3D only an inequality was established:  
\be \ven\equiv \nu\langle|\grad\bv(\br,t)|^2\rangle\leq F_{ii}(\bz). 
\lb{31} \ee
(see Theorem XI.2.2 of \cite{7}). There is a real physical issue 
involved here, since singularities might 
provide an extra sink of energy in addition to regular viscous 
dissipation if they can occur for small but
positive viscosity. This is ruled out in 2D but not so far in 3D. 
Unless this can be done, we do not 
expect that the balance relation with equality, Eq.(\ref{30}), 
can be proved in 3D. At least it would 
need to be shown that such ``dissipative singularities'' 
occurred with zero probability with respect to
the steady-state measure (see Appendix I). 

In this section we discuss the steady-state balance equations 
for the vorticity-invariants
$\om^{(n)},\,\,n\geq 1$ in 2D, analogous to Novikov's energy 
balance relation Eq.(\ref{30}). These can be 
demonstrated to hold in 2D as exact equalities by the methods 
of Vishik and Fursikov, but we postpone 
the rigorous proofs to Appendix I of this paper. Here we shall 
simply give the formal arguments in the 
manner of Novikov's work. It may be useful to recall here the 
basic elements of his derivation of the 
energy balance, before extending it to the vorticity invariants. 
The starting point is the randomly-forced
Navier-Stokes system 
\be \partial_t\bv+(\bv\bdot\grad)\bv=-\grad p+\nu\nabla^2\bv+\bF, 
\lb{32} \ee
where $\bF$ is the random force defined above. Actually, let us 
for the moment consider a slight generalization
to a force with the more general covariance
\be \langle f_i(\br,t)f_j(\br',t')\rangle= 2F_{ij}(\br-\br',t-t'), 
\lb{33} \ee
no longer delta-correlated in time but still space-time homogeneous. 
Furthermore, we still assume its 
statistics to be {\em Gaussian} with zero mean. If one assumes the 
existence of a stationary measure
for this system with regular realizations, then there is a balance 
of viscous dissipation with energy input
by the force, as: 
\be \nu\langle|\grad\bv(\br,t)|^2\rangle=\langle\bv(\br,t)\bdot
\bF(\br,t)\rangle. \lb{34} \ee
However, the Gaussian integration-by-parts identity may be used 
to rewrite the input term as 
\be \langle\bv(\br,t)\bdot\bF(\br,t)\rangle= 2\int_{-\infty}^t dt'
\int_\Lambda d^2\br'\,\,
F_{ij}(\br-\br',t-t')G_{ij}(\br,t;\br',t'), \lb{35} \ee
where
\be G_{ij}(\br,t;\br',t')=\langle
{{\delta v_i(\br,t)}\over{\delta f_j(\br',t')}}\rangle \lb{36} \ee
is the {\em mean response function} to the stochastic force. 
\footnote{It is interesting to observe that the previous 
relation, which is exact for a Gaussian force, holds for a 
general stochastic driving force 
in the DIA approximation: see \cite{20}.} What is special to the 
case of delta-correlation in time is that
only the time-diagonal part of the individual response function 
appears, which satisfies
\begin{eqnarray}
\widehat{G}_{ij}(\br,t;\br',t) & \equiv & {{\delta v_i(\br,t)}
\over{\delta f_j(\br',t)}} \cr
                             \,& =      & {{1}\over{2}}
\delta_{ij}\delta^2(\br-\br'), \lb{37}
\end{eqnarray}
in every realization. Notice that this is the response to the 
{\em random} force and the result 
differs by a factor of ${{1}\over{2}}$ from the familiar result 
for infinitesimal response to a
deterministic force. A full justification of this prescription 
involves the use of the Ito stochastic
calculus (see \cite{7}). Inserting Eq.(\ref{37}) into Eq.(\ref{35}) 
with the delta-correlation 
Eq.(\ref{29}), the balance equation in its original form, Eq.(\ref{30}), results.

We now extend these considerations to vorticity conservation. 
By taking the curl of the Eq.(\ref{32}),
we derive the equation for the vorticity  
\be \partial_t\omega+(\bv\bdot\grad)\omega=\nu\nabla^2\omega+q, 
\lb{38} \ee
with 
\be q=\bE_3\bdot(\grad\btimes\bF). \lb{39} \ee
If we assume for simplicity isotropic forcing, i.e. $F_{ij}=
\delta_{ij}F$ in Eq.(\ref{33}), then
it follows easily that
\be \langle q(\br,t)q(\br',t')\rangle= 2Q(\br-\br',t-t'), 
\lb{40} \ee
with $Q=-\nabla^2F$ and the random force $q$ is still Gaussian. 
>From the Eq.(\ref{38}) we derive 
for $n\geq 2$ the local balance relation for individual 
solutions:
\be \partial_th^{(n)}+\grad\bdot(h^{(n)}\bv-\nu h^{(n-1)}
\grad\omega)=-\nu h^{(n-2)}|\grad\omega|^2+h^{(n-1)}q, 
\lb{41} \ee
with $h^{(n)}\equiv {{1}\over{n!}}\omega^n.$ In the steady
state we therefore find that
\be \eta^{(n)}\equiv\nu\langle h^{(n-2)}|\grad\omega|^2\rangle
=\langle h^{(n-1)}q\rangle. \lb{42} \ee
This is the average balance equation for dissipation of the 
$n$th vorticity invariant $\om^{(n)},$ and its
input by the force $q$ for the general case. 

However, with the Gaussian integration-by-parts identity we 
can derive in the same 
manner as before
\be \langle h^{(n-1)}(\br,t)q(\br,t)\rangle =2\int_{-\infty}^t 
dt'\int_\Lambda d^2\br'\,\,
                        Q(\br-\br',t-t') \langle h^{(n-2)}(\br,t)
\widehat{H}(\br,t;\br',t')\rangle, \lb{43} \ee
in which
\be \widehat{H}(\br,t;\br',t')={{\delta \omega(\br,t)}\over
{\delta q(\br',t')}} \lb{44} \ee
is the individual response operator of the vorticity to the 
stochastic force $q.$ For the particular
case $n=2,$ with $\eta=\eta^{(2)},$ we derive the result
\be \eta=2\int_{-\infty}^t dt'\int_\Lambda d^2\br'\,\,
Q(\br-\br',t-t')H(\br,t;\br',t'), \lb{45} \ee
analogous to Eqs.(\ref{34}),(\ref{35}), with $H(\br,t;\br',t')
\equiv \langle \widehat{H}(\br,t;\br',t')\rangle.$
In particular, for {\em white-noise} force in time we see that
\be \eta=Q(\bz) \lb{46} \ee
and is also independent of the fluid statistics. As we shall see, 
this fact has profound consequences. 
Another remarkable feature of this case is that 
\be \eta^{(n)}=\langle h^{(n-2)}\rangle\eta \lb{47} \ee
for $n\geq 2,$ so that all of the higher-order inputs are 
proportional to the input of enstrophy. 

For our discussion in the next section we shall need a slight 
generalization of the previous results
which we mention without a detailed derivation. It can be given 
as above. We consider the local-balance
of vorticity invariants in the large-scales, Eq.(\ref{12}), with 
the addition of viscous dissipation and
random stirring. The balance equation now becomes
\be \partial_t h_\ell^{(n)}(\br,t)+\grad\bdot\bK_\ell^{(n)}(\br,t)
                             =-Z_\ell^{(n)}(\br,t)-D_\ell^{(n)}
(\br,t)+h^{(n)}_\ell(\br,t)q_\ell(\br,t) \lb{48} \ee
in which the current
\be \bK_\ell^{(n)}(\br,t)\equiv h_\ell^{(n)}(\br,t)\vl_\ell(\br,t)
+h_\ell^{(n-1)}(\br,t)\bsigma_\ell(\br,t)
                             -\nu h^{(n-1)}_\ell(\br,t)
\grad\omega_\ell(\br,t), \lb{49} \ee
and 
\be D_\ell^{(n)}(\br,t)=\nu h^{(n-2)}_\ell(\br,t)
|\grad\omega_\ell(\br,t)|^2, \lb{50} \ee
is the dissipation of the $n$th invariant in the 
large-scales only. Note that $Z_\ell^{(n)}$ is the 
flux given in Eq.(\ref{18}). If we now define
\be \eta^{(n)}_\ell\equiv\langle h^{(n-1)}_\ell q_\ell\rangle 
\lb{51} \ee
to be the {\em input} of the $nth$ invariant into the 
length-scales $>\ell,$ then we obtain the balance equation
\be \eta^{(n)}_\ell=\langle Z_\ell^{(n)}\rangle+\langle 
D_\ell^{(n)}\rangle. \lb{52} \ee

\noindent {\em Remark on a curious independence property}

We just wish to point out here a property of the force 
white-noise in time, which follows from 
Eq.(\ref{47}) and the general result Eq.(\ref{42}). 
In combination they yield, for each $n\geq 0,$
\be \langle \omega^n(\br)\cdot|\grad\omega(\br)|^2\rangle
=\langle\omega^n(\br)\rangle
                            \langle|\grad\omega(\br)|^2
\rangle. \lb{53} \ee
Observe that this implies, under some technical assumptions, 
that the vorticity
at a point, $\omega(\br),$ and the magnitude of its gradient 
at that same point, $\xi(\br)=|\grad\omega(\br)|,$
are statistically independent! Some implications this 
curious fact will be discussed later.

\section{Cascade Directions of Energy and Enstrophy}

\noindent {\em Preliminaries}

We consider here a somewhat more general situation than 
before, in which we have an 
infrared dissipation in some range $[k_0,k_e]$ above 
the lowest wavenumber $k_0,$
and also ultraviolet dissipation in the range $[k_v,
\infty].$ More precisely, the 
dynamical equation in Fourier space is taken to be
\be \partial_t a_i(\bk)+\sum_{\bp j,\bq k}\,B_{\bk i,
\bp j,\bq k}a_j(\bp)a_k(\bq)=
               f_i(\bk)-\alpha_{(r)}k^{-r}\theta_{[k_0,k_e]}
(k)a_i(\bk)
                       -\nu_{(s)}k^{s}\theta_{[k_v,\infty]}(k)
a_i(\bk). \lb{54} \ee
Note that $B_{\bk i,\bp j,\bq k}={{i}\over{2}}(k_jP_{ik}(\bk)
+k_iP_{jk}(\bk))\delta_{\bk,\bp+\bq}$ is the 
usual coefficient of the Euler nonlinearity. The force $f_i(\bk)$ 
is assumed to have 
support in a range $[{{1}\over{2}}k_f,2k_f].$ We consider 
an arbitrary $s$th-order 
hyperviscosity with coefficient $\nu_{(s)},$ acting in the 
ultraviolet above some
wavenumber $k_v.$ Of course, even without the cutoff 
$k_v$ this dissipation would
be substantial only for large $k$ when $s$ was large 
enough. However, it will be
useful in our discussion below to put in the cutoff 
wavenumber. Likewise, we consider 
an arbitrary ``hypoviscosity'' with coefficient 
$\alpha_{(r)}$ acting in the infrared
below some wavenumber $k_e.$ Similar dissipation 
mechanisms are often argued to occur 
at the largest length-scales in atmospheric flows. 
Special cases of this model have been 
the subject of a number of recent high Reynolds 
number simulations \cite{21,22,23}. 

The primary issue we address in this section is the 
support of the energy spectrum and the 
direction of cascades of energy and enstrophy in the 
steady-states of these models. There is a 
simple heuristic argument for the cascade directions 
which is part of the ``folklore'' of the subject:
in essence it goes back to Kraichnan \cite{4}. If the 
energy input $\ven$ and enstrophy input 
$\eta$ are divided into infrared and ultraviolet fluxes, as 
\be \ven=\ven_{ir}+\ven_{uv}, \lb{55} \ee
and
\be \eta=\eta_{ir}+\eta_{uv}, \lb{56} \ee
then it is possible to {\em define} two wavenumbers in 
some sense characteristic of
the infrared and and ultraviolet ranges, as
\be k_{ir}^2\equiv \eta_{ir}/\ven_{ir}, \lb{57} \ee 
and 
\be k_{uv}^2\equiv \eta_{uv}/\ven_{uv}. \lb{58} \ee
Of course, it is also true that
\be \eta=k_f^2\ven \lb{59} \ee
up to an immaterial constant of order one. Solving these 
relations yields
\be {{\ven_{uv}}\over{\ven_{ir}}}={{k_f^2-k_{ir}^2}
\over{k_{uv}^2-k_f^2}}, \lb{60} \ee
and
\be {{\eta_{uv}}\over{\eta_{ir}}}={{k_{uv}^2(k_f^2-k_{ir}^2)}
\over{k_{ir}^2(k_{uv}^2-k_f^2)}}. \lb{61} \ee
A particular limit of interest is $k_{uv}\rightarrow \infty,$ 
in which case $\ven_{uv}/\ven_{ir}\rightarrow 0$
and $\eta_{uv}/\eta_{ir}\rightarrow (k_f/k_{ir})^2-1.$ Hence, 
in that limit all of the energy flows to the 
low wavenumbers, i.e. there is an {\em inverse cascade} of energy. 
In that same limit, a fraction $(k_{ir}/k_f)^2\eta$
of the enstrophy input goes to the low wavenumbers and a fraction 
$\left(1-\left({{k_{ir}}\over{k_f}}\right)^2\right)\eta$ 
goes to high wavenumbers. If the limit $k_{ir}\ll k_f$ is 
considered subsequently, then there is predominantly
a {\em direct cascade} of enstrophy. As we shall show in 
detail below, this argument is basically correct
and leads to the proper directions and magnitudes of transfer. 
Its weakness is that it does not tell how 
the characteristic wavenumbers $k_{ir},k_{uv}$ depend upon 
physical parameters of the model (such as viscosity
$\nu,$ etc.) Furthermore, it makes implicit assumptions on 
the existence of intervals of constant flux, finiteness
of energy, etc. It is one of our purposes below to give, 
as much as is possible, an {\em a priori} treatment of the problem
and, if additional assumptions are required, to make them explicit. 

\noindent {\em Notation: moments of the energy spectrum}

Although somewhat clumsy because of the generality of our model, 
we use the following notations:

\be E_{(p)}\equiv \int_{k_0}^\infty\,k^p E(k)dk, \lb{62} \ee

\be E_{(p)}^{<\ell}\equiv \int_{2\pi/\ell}^\infty\,k^p E(k)dk,  
\lb{63} \ee

\be E_{(p)}^{>\ell}\equiv \int_{k_0}^{2\pi/\ell}\,k^p E(k)dk,  
\lb{63a} \ee

\be E_{(p)}^{(\ell_1,\ell_2)}\equiv\int_{2\pi/\ell_1}^{2\pi/\ell_2}
\,k^p E(k)dk,  \lb{64} \ee
if $\ell_1>\ell_2,$ and otherwise $\equiv 0.$

Similar abbreviations will be used later for moments of the 
enstrophy spectrum.

\noindent {\em Energy and enstrophy balance equations}

Equations for energy balance in the scales $>\ell$ are derived, 
formally by the methods used above,
or, rigorously as in Appendix I:

For $\ell_e>\ell>\ell_f:$
\be \alpha_{(r)}E^{>\ell_e}_{(-r)}+\nu_{(s)}E_{(s)}^{(\ell_v,\ell)}
+\langle\Pi_\ell\rangle=0.  \lb{65} \ee
For $2^{-1}\ell_f>\ell:$
\be \alpha_{(r)}E^{>\ell_e}_{(-r)}+\nu_{(s)}E_{(s)}^{(\ell_v,\ell)}
+\langle\Pi_\ell\rangle=\ven.  \lb{66} \ee

In the same way, equations are derived for enstrophy balance 
in the scales $>\ell$: 

For $\ell_e>\ell>\ell_f:$
\be \alpha_{(r)}E^{>\ell_e}_{(-r+2)}+\nu_{(s)}
E_{(s+2)}^{(\ell_v,\ell)}+\langle Z_\ell\rangle=0.  \lb{67} \ee
For $2^{-1}\ell_f>\ell:$
\be \alpha_{(r)}E^{>\ell_e}_{(-r+2)}+\nu_{(s)}
E_{(s+2)}^{(\ell_v,\ell)}+\langle Z_\ell\rangle=\eta.  \lb{68} \ee

\noindent {\bf I. Case Without Infrared Dissipation}

\noindent {\em Divergence of energy as viscosity tends to zero}

Here we take also $k_v=k_0.$ This was the model constructed 
by Vishik and Fursikov for $s=2$ \cite{7}. 
They noted that $\nu\langle|\grad\bv|^2\rangle=F(\bz)$ is 
independent of viscosity $\nu=\nu_{(2)},$ and 
interpreted that result as verifying Kolmogorov's hypothesis 
of dissipation non-vanishing with viscosity 
going to zero. However, this is a misinterpretation. 
Kolmogorov's hypothesis was for the 3D case where 
there is an ultraviolet energy cascade. On the contrary, 
in the 2D case there will be an inverse 
cascade. Hence, energy will remain constant as 
$\nu\rightarrow 0$ in the unforced case (decaying
turbulence) and the mean energy will {\em diverge} 
in the randomly forced case as $\nu\rightarrow 0.$
We present a proof of this result (which was found 
by the author and Z.-S. She):  
 
In this case we have
\be \ven=\nu E_{(2)},  \lb{69} \ee
and 
\be \eta=\nu E_{(4)}.  \lb{70} \ee
By the Cauchy-Schwartz inequality,
\be E_{(2)}\leq \sqrt{E_{(4)}E_{(0)}}.  \lb{71} \ee
Thus,
\be \ven\leq \sqrt{\nu\eta E_{(0)}},  \lb{72} \ee
or,
\be E_{(0)}\geq {{\ven^{2}}\over{\nu\eta}}.  \lb{73} \ee
Hence, {\em energy must diverge like} $\nu^{-1}$ {\em as} 
$\nu\rightarrow 0.$ Observe from $\eta=\ven/\ell_f^2$ that
\be E_{(0)}\geq \ell_f^2{{\ven}\over{\nu}}.  \lb{74} \ee

A similar {\em upper bound} is obtained from 
\be \ven\geq\nu k_0^2E_{(0)},  \lb{75} \ee
so that 
\be E_{(0)}\leq \ell_0^2{{\ven}\over{\nu}}.  \lb{76} \ee 
This shows, by the way, that the energy is finite for finite 
viscosity. 
These bounds are consistent with the picture that most of 
the dissipation of energy, and also of 
enstrophy, occurs in the large-scales, at wavenumbers $\leq k_f,$ 
so that
\be \ven\sim\nu k_f^2E_{(0)},\,\,\,\,\,\eta\sim \nu k_f^4E_{(0)}.  
\lb{77} \ee
The amplitude of the velocity fluctuations is forced to rise 
to a high enough level in the low wavenumbers
to achieve energy balance with the input from the force.

\noindent {\em Localization of energy}

As a simple application of the Chebychev inequality, note that
\begin{eqnarray}
\int_K^\infty E(k)dk & \leq & \int_{k_0}^\infty 
\left({{k}\over{K}}\right)^4E(k)dk \cr
                   \,& = & {{\eta/\nu}\over{K^4}} \cr
                   \,& = & ({\rm const.}){{E_{(0)}
\over{(K \ell_f)^4}}}.  \lb{78}
\end{eqnarray}

\noindent {\em Flux directions}

Define 
\be \ven_{uv}(\ell)=\langle\Pi_\ell\rangle,  \lb{79} \ee
for $\ell<2^{-1}\ell_f.$ By energy balance,
\begin{eqnarray}
\ven_{uv}(\ell) & = & \ven-\nu E_{(2)}^{>\ell} \cr
              \,& = & \nu E_{(2)}^{<\ell}.  \lb{80}
\end{eqnarray}
Since $E_{(2)}^{<\ell}\leq \ell^2\cdot E_{(4)},$ by Chebychev 
again, 
\begin{eqnarray}
\ven_{uv}(\ell) & \leq & \ell^2\cdot\eta \cr
             \, & =    & \left({{\ell}\over{\ell_f}}\right)^2\ven.  
\lb{81}
\end{eqnarray}
Note the similarity to Fjortoft's old result \cite{3} on the 
amount of energy
able to reach small length-scales in freely decaying turbulence.

We may also define the UV enstrophy flux as
\be \eta_{uv}(\ell)=\langle Z_\ell\rangle  \lb{82} \ee
for $\ell<2^{-1}\ell_f.$ For this enstrophy flux the same argument 
yields 
\begin{eqnarray}
\eta_{uv}(\ell) & = & \nu E_{(4)}^{(<\ell)} \cr
              \,& = & \eta\cdot\left({{E_{(4)}^{<\ell}}
\over{E_{(4)}}}\right).  \lb{83}
\end{eqnarray}
It seems possible that a finite fraction of the enstrophy 
input $\eta$ goes
to the high wavenumbers.

\noindent {\bf II. Case With Infrared Dissipation}

\noindent {\em Finiteness of Energy}

Since we have seen that energy {\em must} diverge for 
vanishing viscosity without some 
low-wavenumber dissipation, it seems impossible that 
any $\nu\rightarrow 0$ limit of that
state can exist. Therefore, we consider here the 
general model of Eq.(\ref{54}) (with $s=2.$)
Although we believe it to be true, so far we have not been 
able to prove that energy remains finite even if the 
IR dissipation is added! For some of our arguments below 
we will use that fact. Therefore, we state it as a 
\begin{Hyp}
$E_{(0)}=O(1)$ as $\nu\rightarrow 0.$ 
\end{Hyp}
Some notation we will also use below is
\be \ven_e\equiv \alpha_{(r)}E_{(-r)}^{>\ell_e}  \lb{84} \ee
for the IR energy dissipation, and
\be \eta_e\equiv \alpha_{(r)}E_{(-r+2)}^{>\ell_e}  \lb{85} \ee
for the IR enstrophy dissipation, and likewise
\be \ven_v\equiv \nu E_{(2)}^{<\ell_v}  \lb{86} \ee
for the UV energy dissipation, and
\be \eta_v\equiv \alpha_{(r)}E_{(-r+2)}^{<\ell_v}  \lb{87} \ee
for the UV enstrophy dissipation. Now we have the balance 
relations
\be \ven=\ven_e+\ven_v,  \lb{88} \ee
and
\be \eta=\eta_e+\eta_v.  \lb{89} \ee
For the most part we shall consider the case $\ell_v=\ell_0,$ 
but consider the more general situation 
at one point later on. 

\noindent {\em Asymptotic locations of energy and enstrophy 
dissipation}

It is possible here to use the same method of argument as before. 
Thus
\begin{eqnarray}
0\leq \ven-\ven_e=\ven_v & =    & \nu E_{(2)} \cr
                      \, & \leq & \sqrt{\nu^2E_{(4)}E_{(0)}} \cr
                      \, & =    & \sqrt{\nu\eta E_{(0)}}.  \lb{90} 
\end{eqnarray}
>From this we can draw several conclusions. First,
\be \ven_v=O\left(\sqrt{\nu E_{(0)}}\right).  \lb{91} \ee
and
\be \ven_e=\ven +O\left(\sqrt{\nu E_{(0)}}\right),  \lb{92} \ee
so that asymptotically all of the energy dissipation is infrared 
for $\nu\rightarrow 0,$
under our hypothesis. Note actually that we need only the 
weaker result $E_{(0)}=o(1/\nu)$ 
to draw that conclusion. Secondly, 
\be \Omega=E_{(2)}=O\left(\sqrt{{{E_{(0)}\over{\nu}}}}\right).  
\lb{93} \ee
Thus, enstrophy may diverge as $\nu\rightarrow 0,$ but at most 
as $\nu^{-1/2}$ if the 
hypothesis is true. Finally, since $\eta_e=k_e^2\ven_e$ 
(up to an unimportant constant factor),
it follows from Eq.(\ref{92}) also that 
\be \eta_e =\left({{k_e}\over{k_f}}\right)^2\cdot\eta
+O\left(\sqrt{\nu E_{(0)}}\right).  \lb{94} \ee
Consequently, it follows also that
\be \eta_v =\left[1-\left({{k_e}\over{k_f}}\right)^2\right]
\eta+O\left(\sqrt{\nu E_{(0)}}\right).  \lb{95} \ee
Thus $\eta_v\approx\eta$ when $k_f\gg k_e$ and $\eta_v\approx 0$ 
when $k_f\approx k_e.$

\noindent {\em Fluxes of energy and enstrophy}

Now set 
\be \ven_{ir}(\ell)\equiv -\langle\Pi_\ell\rangle  \lb{96} \ee
for $\ell>2\ell_f,$ and
\be \ven_{uv}(\ell)\equiv +\langle\Pi_\ell\rangle \lb{97} \ee
for $\ell<2^{-1}\ell_f.$ Thus,
\be \ven_{ir}(\ell)=\ven_e+\nu E_{(2)}^{>\ell}. \lb{98} \ee
and
\be \ven_{uv}(\ell)=\ven-\ven_e-\nu E_{(2)}^{>\ell}. \lb{99} \ee
Note that 
\be E_{(2)}^{(>\ell)}\leq {{({\rm const.})}\over{\ell^2}}E_{(0)}. 
\lb{100} \ee
Along with our previous result for $\ven_e$ in Eq.(\ref{92})
,this gives 
\be \ven_{ir}(\ell)= \ven+O\left(\sqrt{\nu E_{(0)}},
{{\nu E_{(0)}}\over{\ell^2}}\right), \lb{101} \ee
and 
\be \ven_{uv}(\ell)=O\left(\sqrt{\nu E_{(0)}},{{\nu E_{(0)}}
\over{\ell^2}}\right). \lb{102} \ee
Under our hypothesis, or assuming even $E_{(0)}=o(1/\nu),$ 
we see that in the limit 
$\nu\rightarrow 0$ that $\ven_{ir}(\ell)=\ven,\,\,\,
\ell_e>\ell>2\ell_f,$ and $\ven_{uv}(\ell)=0,
\,\,\,\ell<2^{-1}\ell_f.$ Notice that this argument 
{\em proves} the constancy of flux over the 
relevant ranges. 

For enstrophy we can likewise set
\be \eta_{ir}(\ell)\equiv -\langle Z_\ell\rangle \lb{103} \ee
for $\ell>2\ell_f,$ and
\be \eta_{uv}(\ell)\equiv +\langle Z_\ell\rangle \lb{104} \ee
for $\ell<2^{-1}\ell_f.$ Thus,
\be \eta_{ir}(\ell)=\eta_e+\nu E_{(4)}^{>\ell}. \lb{105} \ee
and
\be \eta_{uv}(\ell)=\eta-\eta_e-\nu E_{(4)}^{>\ell}. \lb{106} \ee
Also,
\be E_{(4)}^{(>\ell)}\leq {{({\rm const.})}\over{\ell^4}}E_{(0)}. 
\lb{107} \ee
We conclude finally that
\be \eta_{ir}(\ell)=\left({{k_e}\over{k_f}}\right)^2\cdot\eta
                      +O\left(\sqrt{\nu E_{(0)}},
{{\nu E_{(0)}}\over{\ell^4}}\right), \lb{108} \ee
and 
\be \eta_{uv}(\ell)=\left[1-\left({{k_e}\over{k_f}}\right)^2
\right]\eta
                      +O\left(\sqrt{\nu E_{(0)}},{{\nu E_{(0)}}
\over{\ell^4}}\right). \lb{109} \ee
In the limit $\nu\rightarrow 0$ we recover the results 
of the traditional argument, with
$k_{ir}\equiv k_e.$ Note that we can get $\eta_{uv}\rightarrow 
\eta$ by taking $k_f\gg k_e$
and $\eta_{uv}\rightarrow 0$ by taking $k_f\approx k_e.$

\noindent {\em Extent of the enstrophy inertial range}

As before, we have proved that the enstrophy flux is really 
finite over the relevant UV range:
\be \eta_{uv}(\ell)\approx \eta_v, \lb{110} \ee
for 
\be 2^{-1}\ell_f\geq \ell\gg \ell_d\equiv {{\nu^{1/4} 
E_{(0)}^{1/4}}\over{\eta^{1/4}}}. \lb{111} \ee
In contrast, the classical Kraichnan dissipation length is
\be \ell_d^{{\rm Kr}}={{\nu^{1/2}}\over{\eta^{1/6}}}. 
\lb{112} \ee
Since $E_{(0)}\gg \nu\cdot\eta^{1/3}$ in the limit 
$\nu\rightarrow 0,$ it follows 
that $\ell_d\gg\ell_d^{{\rm Kr}}.$ Thus, this argument 
does not establish constant 
flux down to the classical dissipation scale.

In this context, it may be noted that it is possible to 
define a dissipation wavenumber 
intrinsically for the steady-state, by
\be \kappa_d^2\equiv {{E_{(4)}}\over{E_{(2)}}}. \lb{113} \ee
Many of our previous conclusions would follow if it 
were possible to prove 
\begin{Hyp}
$\lim_{\nu\rightarrow 0}\kappa_d=\infty.$ 
\end{Hyp}
Indeed, by its definition,
\begin{eqnarray}
\ven_v & =   &{{\eta_v}\over{\kappa_d^2}} \cr
       &\leq &{{\eta}\over{\kappa_d^2}}, \lb{114} 
\end{eqnarray}
so that $\ven_v\rightarrow 0$ would follow under this 
hypothesis as $\nu\rightarrow 0.$
Of course, in this case $\ven_e\rightarrow\ven.$ Furthermore, 
since $\eta_e=k_e^2\ven_e,$
it would then follow that $\eta_e\rightarrow (k_e/k_f)^2\eta$ 
for $\nu\rightarrow 0.$
As a consequence, $\eta_v\rightarrow [1-(k_e/k_f)^2]\eta$ 
in that same limit. These 
conclusions all involve dissipation. Note that the corresponding 
statements about fluxes
would still require the finite-energy assumption (Hypothesis 1). 
Unfortunately, Hypothesis 2 does not 
seem at the moment to be susceptible to proof. In \cite{24} 
some similar wavenumbers 
are studied. However, for those wavenumbers it is proved that 
$k_d^{{\rm Kr}}$ is an
{\em upper bound} in 2D, which does not suffice for our purpose. 
One situation where the 
results can be obtained cheaply is by keeping the cutoff 
$k_v$ in the model, and taking 
it by hand a function of $\nu$ which goes to infinity as 
$\nu\rightarrow 0.$ Since 
\be \ven_v\leq {{\eta}\over{k_v^2}}  \lb{115} \ee
in this circumstance, we get the desired conclusions.   	

Both of the previous two hypotheses follow from the even stronger 
one:
\begin{Hyp}
$\Omega_{(0)}=O(1)$ as $\nu\rightarrow 0.$ 
\end{Hyp}
Here we introduce the notation $\Omega_{(p)}=E_{(p+2)}$ for 
the $p$th moments of the enstrophy spectrum.
Obviously, it implies Hypothesis 1 on finiteness of energy. 
Note that the Hypothesis 3 is true when the 
energy spectrum is steeper than the classical Kraichnan spectrum 
$E(k)\sim k^{-3}$ but invalid at that spectrum. 
However, it implies an even stronger result than Hypothesis 2, 
namely, that the wavenumber $\kappa_d$ 
introduced there equals the classical Kraichnan wavenumber:
\be \kappa_d=\kappa_d^{{\rm Kr}}. \lb{116} \ee
In fact, an alternative way of writing $\kappa_d$ is as 
\be \kappa_d^2={{\Omega_{(2)}}\over{\Omega_{(0)}}}\sim 
{{\eta/\nu}\over{\Omega_{(0)}}}, \lb{117} \ee
with $\eta_{uv}\sim \eta.$ However, if Hypothesis 3 holds 
then 
\be \Omega_{(0)}\propto \eta^{2/3}, \lb{118} \ee
as $\nu\rightarrow 0.$ This is a consequence of dimensional 
analysis, since, if the 
zero-viscosity limit exists, then all statistical quantities 
must be functions only 
of $\eta,$ with units of $({\rm time})^3,$ and the model 
wavenumbers $k_0,k_f.$ Because
$\Omega$ has units of $({\rm time})^2,$ it must then be 
$\eta^{2/3}$ times some function
of the fixed wavenumber ratio $k_0/k_f.$ Hence, 
$\kappa_d^2\sim\eta^{1/3}/\nu,$ or
\be \kappa_d\sim {{\eta^{1/6}}\over{\nu^{1/2}}}
=\kappa_d^{{\rm Kr}}. \lb{119} \ee
In particular, the dissipation wavenumber diverges 
to infinity as $\nu\rightarrow 0.$

With the stronger Hypothesis 3 it is not hard, in 
fact, to show that the enstrophy
flux will be constant up to the Kraichnan wavenumber 
$\kappa_d^{{\rm Kr}}.$ This validates 
the significance of the latter as the dissipation wavenumber. 
In fact, with the Hypothesis 3,
our earlier estimates on the energy and enstrophy dissipation, 
Eqs. (\ref{91}),(\ref{92}),(\ref{94}),
(\ref{95}), may be improved to
\be \ven_v=\nu \Omega_{(0)}, \lb{120} \ee
\be \ven_e=\ven-\nu \Omega_{(0)},  \lb{121} \ee
\be \eta_e =\left({{k_e}\over{k_f}}\right)^2\cdot\eta
+O\left(\nu \Omega_{(0)}\right), \lb{122} \ee
and
\be \eta_v =\left[1-\left({{k_e}\over{k_f}}\right)^2\right]\eta
+O\left(\nu \Omega_{(0)}\right).  \lb{123} \ee
Likewise, the estimates of the fluxes in Eqs.(\ref{101}),
(\ref{102}),(\ref{108}), and (\ref{109}), may 
be improved under the stronger hypothesis to give
\begin{eqnarray}
\ven_{uv}(\ell) & = & \nu\Omega^{<\ell}_{(0)} \cr  
              \,& = & O\left(\nu \Omega_{(0)}\right), 
\lb{124} 
\end{eqnarray}
\begin{eqnarray}
\ven_{ir}(\ell) & = & \ven-\nu \Omega_{(0)}^{<\ell} \cr
              \,& = & \ven+O\left(\nu \Omega_{(0)}\right), 
\lb{125}
\end{eqnarray}
\begin{eqnarray}
\eta_{ir}(\ell) & = & \eta_e+\nu\Omega^{>\ell}_{(2)} \cr 
              \,& = & \left({{k_e}\over{k_f}}\right)^2\cdot\eta 
                      +O\left(\nu \Omega_{(0)},{{\nu \Omega_{(0)}}
\over{\ell^2}}\right), \lb{126} 
\end{eqnarray}
and 
\begin{eqnarray}
\eta_{uv}(\ell) & = & \eta-\eta_e-\nu\Omega_{(2)}^{>\ell} \cr
                  \,& = & \left[1-\left({{k_e}\over{k_f}}
\right)^2\right]\eta
                           +O\left(\nu \Omega_{(0)},
{{\nu \Omega_{(0)}}\over{\ell^2}}\right). \lb{127} 
\end{eqnarray}
In particular, it follows from the last estimate that 
$\eta_{uv}(\ell)\approx \eta$ down to a lengthscale
$\ell_d$ which is determined by 
\be \eta\sim {{\nu\Omega_{(0)}}\over{\ell_d^2}}\sim 
{{\nu\eta^{2/3}}\over{\ell_d^2}}, \lb{128} \ee
whose solution yields
\be \ell_d\sim {{\nu^{1/2}}\over{\eta^{1/6}}}\equiv 
\ell_d^{{\rm Kr}}. \lb{129} \ee
Therefore, the ultraviolet end of the inertial range 
is demarcated by the wavenumber $\kappa_d^{{\rm Kr}}$ when
the Hypothesis 3 is valid. 

\noindent {\em Fluxes of the higher-order vorticity invariants}

Notice that our arguments for constancy of fluxes do not 
{\em necessarily} work for the 
higher-order vorticity invariants. In fact, as already noted 
by Falkovich and 
Hanany in \cite{10}, the inputs for those quantities need not 
be supported just in 
the spectral range $[2^{-1}k_f,2k_f].$ In fact, this can be 
easily seen from our 
expression Eq.(\ref{51}), since $q_\ell=q$ when 
$\ell<2^{-1}\ell_f$ but the same is
not true for $h_{\ell}^{(n-1)}.$ Thus, the input of these invariants 
can be spread over 
the entire inertial range. In the white-noise force case we can 
derive an exact expression
for the input into the length-scales $>\ell,$ generalizing our 
earlier expression 
for total input. It is 
\be \eta^{(n)}_\ell=\langle h^{(n-2)}_\ell\rangle\cdot\eta,  
\lb{130} \ee
when $\ell\leq \beta\cdot\ell_f,$ where $\beta$ is some 
fraction $<1$ that depends upon
the choice of filter function $G_\ell.$ In fact, by applying 
the Gaussian integration-by-parts
identity to Eq.(\ref{51}), we obtain the expression 
for the input
\be \eta^{(n)}_\ell=2\int_\Lambda d^2\br'\,\, Q(\br-\br') 
\langle h^{(n-2)}_\ell(\br,t)
{{\delta \ol_\ell(\br,t)}\over{\delta q(\br',t)}}\rangle, 
\lb{131} \ee
where from the definitions and the Ito rule
\be {{\delta \ol_\ell(\br,t)}\over{\delta q(\br',t)}}
={{1}\over{2}}G_\ell(\br-\br'). \lb{132} \ee
Hence, 
\be \eta^{(n)}_\ell=\langle Q,G_\ell\rangle_{L^2}\cdot
\langle h_\ell^{(n-2)}\rangle.  \lb{133} \ee
However, if the filter has a Fourier transform 
$\widehat{G}(\bk)\equiv 1$ in some neighborhood of 
$\bk=\bz,$ then for a small enough $\ell<\beta\cdot\ell_f$ 
it follows that $\langle Q,G_\ell\rangle_{L^2}=Q(\bz),$
and thus Eq.(\ref{130}) follows. 

This formula shows explicitly how the input is distributed 
over length-scales in the white-noise
case. As one application, let us observe that the inputs 
of all of the {\em even-order} 
invariants must diverge if there is an ultraviolet 
divergence of the enstrophy, or if
\begin{eqnarray}
{{1}\over{2}}\langle\ol_\ell^2\rangle & =       
& \int_{k_0}^\infty \,|\widehat{G}_\ell(k)|^2\Omega(k)dk \cr
\,& \approx & \int_{k_0}^{2\pi/\ell}\,\Omega(k)dk.  \lb{134}
\end{eqnarray}
diverges for $\ell\rightarrow 0$ (after the limit 
$\nu\rightarrow 0.$) In fact, by the previous formula
and the H\"{o}lder inequality
\begin{eqnarray}
\eta^{(2m)}_\ell & =    & {{1}\over{(2m-2)!}}
\langle\ol_\ell^{2(m-1)}\rangle\cdot\eta \cr
              \, & \geq & {{1}\over{(2m-2)!}}
\langle\ol_\ell^2\rangle^{m-1}\cdot\eta.  \lb{135}
\end{eqnarray} 
If the effects of viscosity can be ignored, then 
the flux of these invariants must get increasingly 
large at small-scales (or else the flux in the 
infrared direction must be infinite) when the enstrophy
itself diverges.

This is the scenario proposed by Falkovich and Hanany \cite{10}, 
who considered energy spectra 
of the $k^{-3}$-form with log-corrections 
\be E(k)\sim\eta^{2/3}k^{-3}\ln^{-s}(k\ell_f)  \lb{136} \ee
(as proposed much earlier by Kraichnan \cite{25} with the 
precise value $s={{1}\over{3}}.$)
It is easy to calculate for these spectra that 
\be {{1}\over{2}}\langle\ol_\ell^2\rangle\sim \ln^{1-s}
\left({{\ell_f}\over{\ell}}\right). \lb{137} \ee
Therefore, the proposal of \cite{10} that
\be \langle Z^{(2m)}_\ell\rangle\sim \ln^{(m-1)(1-s)}
\left({{\ell_f}\over{\ell}}\right)  \lb{138} \ee
appears consistent for the white-noise force case. 
Our rigorous argument with the H\"{o}lder 
inequality shows that the ``reducible contribution''
---claimed in \cite{10} to be dominant---is at least 
a lower-bound. 

However, if $\langle \omega^2\rangle<\infty,$ then it is 
possible to have constant-flux inertial ranges
of the higher-order invariants. In fact, let us consider 
for $n>2$ a more general
\begin{Hyp}
$\Omega^{(n-2)}=O(1)$ as $\nu\rightarrow 0.$
\end{Hyp}
Under this assumption the inputs of the higher invariants
 all have limits
\be \lim_{\ell\rightarrow 0}\eta^{(n)}_\ell=\Omega^{(n-2)}
\cdot\eta<\infty, \lb{139} \ee
and thus are restricted to essentially a finite wavenumber 
range around $k_f.$ It seems most
likely that these invariants will behave as the enstrophy 
and develop {\em ultraviolet cascades.}
This, in fact, follows if we make an additional
\begin{Hyp}
$\Omega^{(2n-2)}=O(1)$ as first $\nu\rightarrow 0$ and then 
${{k_e}\over{k_f}}\rightarrow 0.$
\end{Hyp}
Indeed, it is easy to derive a bound for the infrared 
dissipation $\eta^{(n)}_e$ of the $n$th invariant: 
\be \eta_e^{(n)}=O\left(\sqrt{\alpha_{(r)}k_e^{-r}
\Omega^{(2n-2)}\eta_e}\right), \lb{140} \ee
as a direct consequence of its definition 
\begin{eqnarray}
\eta_e^{(n)} & = & \alpha_{(r)}\langle h^{(n-1)}
(-\bigtriangleup)^{-r/2}P^{<k_e}\omega\rangle \cr
               \,& = & \alpha_{(r)}\langle 
(-\bigtriangleup)^{-r/4}P^{<k_e}h^{(n-1)}
\cdot (-\bigtriangleup)^{-r/4}P^{<k_e}\omega\rangle, \lb{141} 
\end{eqnarray}
and the Cauchy-Schwartz inequality. In conjunction with 
Eq.(\ref{122}) and the Hypothesis 5, this yields
\begin{eqnarray}
\eta_e^{(n)} & = & O\left({{k_e}\over{k_f}}\sqrt{\alpha_{(r)}k_e^{-r}
\Omega^{(2n-2)}\eta}\right) \cr 
           \,& \rightarrow & 0, \lb{142} 
\end{eqnarray}
as $\nu\rightarrow 0,(k_e/k_f)\rightarrow 0$ in that order. 
Therefore, all of the higher-order 
vorticity-invariants must cascade to high wavenumbers. This 
conclusion supports the argument of Falkovich and 
Hanany \cite{10} against the conformal ``minimal models'' 
of Polyakov \cite{8,9}. As pointed out in \cite{10},
the constraints of constant fluxes of all the higher-order 
invariants would be satisfied in Polyakov's scheme 
by fixing---for each one---the dimension of a primary field. 
This could not be done in the minimal models with a finite,
closed algebra of primaries. Polyakov has suggested \cite{9} 
that the difficulty may be avoided if the 
higher-order invariants all cascade to low-wavenumbers. 
However, it is easy to check that his 
conformal solutions have all $\Omega^{(n)}_{{\rm conf}}
<\infty$ for $n\geq 2,$ and thus the previous argument
implies non-vanishing ultraviolet fluxes of the higher-order 
invariants. \footnote{However, it is not clear to us 
that Polyakov takes into account correctly the contribution 
of ``negative dimension'' operators to the $n$th order
correlator of the vorticity. This might lead to spectra 
$\Omega^{(n)}(k)$ of the higher invariants 
with slower decay at high-wavenumber than the enstrophy 
spectrum $\Omega(k)=\Omega^{(2)}(k).$} The only way 
that we can see for the minimal models to survive as viable 
descriptions of 2D turbulence is to appeal
to a non-conformal effect: namely, the contribution 
$\Omega^{(n)}_{{\rm cond}}$ of the ``infrared condensates'' 
must diverge for all $n>2$ as $\nu\rightarrow 0,(k_e/k_f)
\rightarrow 0.$ However, we see no likely 
reason for this to occur and, in the absence of concrete 
matching conditions between the conformal
and nonconformal ranges, it is a difficult question to 
investigate in Polyakov's approach. 

Our Hypothesis 5 implies not only that there will be ultraviolet 
cascades of the higher-order 
invariants but also it implies that the corresponding dissipation 
wavenumbers $\kappa^{(2m)}_d,$
defined as
\be [\kappa^{(2m)}_d]^2\equiv {{\langle h^{(2m-2)}|\grad\omega|^2
\rangle}\over{\langle h^{(2m)}\rangle}}, \lb{143} \ee
all coincide with the Kraichnan wavenumber $\kappa_d^{{\rm Kr}}.$ 
(Note that the definition only makes sense
{\em a priori} for even $n=2m$ when both sides are guaranteed 
to be positive.) This follows as a consequence 
of the curious ``independence'' property of the vorticity and 
its gradient magnitude, noted at the end of 
Section 2{\em (ii)}. In fact, appealing to Eq.(\ref{53}), 
it follows that 
\begin{eqnarray}
[\kappa^{(2m)}_d]^2 & = &{{\langle h^{(2m-2)}\rangle\langle 
|\grad\omega|^2\rangle}\over{\langle h^{(2m)}\rangle}}, \cr
                \,& = & {{\Omega^{(2m-2)}}\over{\Omega^{(2m)}}}
\cdot{{\eta}\over{\nu}}.   \lb{144} 
\end{eqnarray}
By the same reasoning which led to Eq.(\ref{118}), it follows 
under Hypothesis 5 that
\be {{\Omega^{(2m)}}\over{\Omega^{(2m-2)}}}\sim \eta^{2/3}, 
\lb{145} \ee
and thus
\be \kappa_d^{(2m)}\sim C^{(2m)}{{\eta^{1/6}}\over{\nu^{1/2}}}
=C^{(2m)}\kappa_d^{{\rm Kr}}. \lb{146} \ee
The ``independence'' property is special to the case of 
Gaussian force, white-noise in time and it is 
not clear that the result Eq.(\ref{146}) is more general than that.

\section{Scaling Indices in the Ultraviolet Enstrophy Range}

We present here exact bounds on scaling exponents of the vorticity 
field in forced, stationary turbulence 
in 2D. All of the estimates are based upon the condition of constant 
mean enstrophy flux $\eta$ at small scales,
which was established in the last section under various reasonable 
hypotheses. As observed by Paladin and Vulpiani 
\cite{27}, the velocity field is a poor indicator of small-scale 
structure in 2D and numerical studies have instead 
properly focused on the vorticity field \cite{28,29,30}. 
Therefore, the scaling laws we consider
are those for the {\em vorticity structure functions}, 
presumed in the form:
\be \langle |\Delta_\bl\omega|^p\rangle\sim \ell^{\zeta_p}. 
\lb{151} \ee
Here ``$\sim$'' is interpreted to mean that the logarithm 
of the LHS divided by $\log \ell$ goes 
to the limit $\zeta_p$ as $\ell \rightarrow 0.$ This type 
of ``multiscaling'' behavior is indicative
of intermittency in the vorticity distribution. We are not 
aware of any numerical work which directly
verifies such scaling laws in 2D steady-states. (The papers 
\cite{28,29,30} make some related analyses
of vorticity in freely-decaying 2D turbulence via wavelets, 
which is discussed further below.) 

We shall actually assume these scaling laws in just the weak 
sense of big-$O$ bounds:
\be \langle |\Delta_\bl\omega|^p\rangle=O(\ell^{\zeta_p}). 
\lb{152} \ee
More properly, we take $\zeta_p$ to be the supremum of the 
$\zeta$'s for which Eq.(\ref{152}) holds 
for each fixed $p$ value, $p\geq 1.$ The average $\langle
\cdot\rangle$ could be interpreted either 
as a space-average over the torus $\bT^2,$ or as an ensemble 
average. The two averages ought to be 
distinguished. To make clear the difference, we shall use 
$z_p$ for the (maximal) exponent 
in the estimate Eq.(\ref{152}) for space-averages. If this 
is in fact considered 
to be a space average, then it just a so-called ``Besov 
condition,'' the defining criterion 
that $\omega\in B^{s,\infty}_p,\,\,\,s={{z_p}\over{p}},$ one 
of the scale of so-called {\em Besov 
spaces} (a standard reference is \cite{35}.) Note that 
Eq.(\ref{152}) for space-averages
characterizes $\omega\in B^{s,\infty}_p$ only if $s>0,$ 
whereas for $s\leq 0$ it is just a sufficient condition. 
For further discussion, see Appendix III. We shall make 
use of some simple parts of the theory of Besov 
spaces in our proofs below. Of course, to appeal to the constant 
flux condition, it is the ensemble-average 
that ought to be considered. However, it is a consequence of 
Theorem 4 in \cite{36} that Eq.(\ref{152})
for ensemble-averages implies that the same estimate holds 
almost surely for space-averages with
$\zeta_p$ replaced by $z=\zeta_p-\en,$ for any $\en>0.$ In 
other words, the realizations 
$\omega\in B^{\sigma_p-\en,\infty}_p$ with probability one, 
for $\sigma_p=\zeta_p/p.$ \footnote{Under the additional 
condition that $\langle \|\omega\|_{B^{s_p(1-\en),\infty}_p}
\rangle<\infty$ for all $\en>0$ it can even be 
seen that $\sigma_p={\rm ess.inf}_\omega s_p(\omega),$ 
in which $s_p(\omega)$ is the maximal
index $s$ in the realization $\omega$ of the ensemble.} 
This will suffice for our purposes.

We shall generally consider below the idealized 
inertial-range of infinite extent, obtained as
the $\nu\rightarrow 0$ limit of the steady-states 
discussed in Section 2 (ii). A cautionary remark 
ought to be made that this limit has never been established 
to exist, and it could be false.
The only cases that we know for which the existence of the 
zero-viscosity limit has been 
established are that of freely-decaying, homogeneous 
turbulence in infinite Euclidean space $\bR^d$
for $d=3$ or $d=2$ (\cite{7},Theorem VIII.3.1) and that of 
evolving turbulence on the $2D$ periodic domain $\bT^2$
with suitable deterministic forces and with random initial 
data of finite mean enstrophy 
(\cite{7},Theorem VIII.4.2). Therefore, we shall give below 
some consideration of another formulation
of the results, in which the big-$O$ bounds Eq.(\ref{152}) 
are assumed valid only for $\nu>0$ but with 
constants that are independent of $\nu.$ This allows us 
to make statements about scaling exponents 
that are valid for the mathematically constructed (and more 
physically realistic) ensembles with $\nu>0.$

\noindent {\em Kolmogorov Relation and Exact Bounds}

One important exact result is a scaling law analogous to the 
von Karman-Howarth-Kolmogorov relation
in 3D. The result in 2D, derived in detail in Appendix II, is:
\be \eta=-{{1}\over{4}}\grad_\bl\bdot\langle [\Delta_\bl \bv]
[\Delta_\bl\omega]^2\rangle. \lb{153} \ee
The RHS is a ``physical-space enstrophy flux,'' $\eta(\bl),$ 
defined by the time-derivative of the vorticity
2-point correlation under the inertial dynamics (cf. \cite{9}, 
Eq.(20)) and the relation is simply an 
expression of the constant ultraviolet enstrophy flux $\eta$ 
for the $\nu\rightarrow 0$ limit. 
The physical-space flux is related to the usual Fourier-space 
flux $Z(k)$ via 
\be Z(k)={{k}\over{2\pi}}\int_{\bR^2}d^2\bl\,\,{{J_1(k\ell)}
\over{\ell}}\eta(\bl). \lb{154} \ee 
To make our arguments simpler, we will assume also homogeneity 
and isotropy, which follow from 
the analogous properties for the force. In that case, 
Eq.(\ref{153}) simplifies to 
\be \langle [\Delta_\bl v_{|\!|}(\br)][\Delta_\bl
\omega(\br)]^2\rangle=
    -2\eta\ell, \lb{155} \ee
for $\ell<\ell_f,$ with $\Delta_\bl v_{|\!|}(\br)=\hat{\bl}\bdot
\Delta_\bl\bv(\br).$ This is 
analogous to the Kolmogorov ``4/5-law'' in 3D for the 3rd-order 
velocity structure function.

We first use this relation to derive a bound on the exponent $x$ 
in the scaling law for the energy spectrum 
$E(k)\sim k^{-x}.$ Of course, this is related to the enstrophy 
spectral exponent $y$ in the law $\Omega(k)\sim k^{-y}$ 
and to the vorticity structure-function exponent $\zeta_2,$ as
\be x=2+y=3+\zeta_2. \lb{156} \ee
We note that there are a number of predictions for the exponent 
$x.$ The original 1967 theory of Kraichnan 
gave $x=3$ \cite{4}, modified later with a logarithmic correction 
$E(k)\sim k^{-3}\log^{-1/3}(k\ell_f)$ \cite{25} 
(see also \cite{31}). Saffman proposed instead $x=4$ \cite{32}, 
Moffatt $x=11/3$ \cite{33}, and, most recently, Polyakov 
has considered an infinite number of possible spectra with $x>3$ 
corresponding to solutions of various 2D conformal
field theories \cite{8,9}. An exact bound was proposed in 1975 
by Sulem and Frisch that $x\leq 11/3$ \cite{34}.
\footnote{It should be noted that the original Sulem-Frisch paper 
gave the estimate $x\leq 4$ as a rigorous bound 
and only a heuristic argument for $x\leq 11/3.$ Later, the authors 
suggested that the sharper bound 
should hold rigorously as well (U. Frisch, private communication, 
1992).} Interestingly, this yields Moffatt's spectrum 
as the upper limit. In particular, the Saffman prediction $x=4$ 
is ruled out as an ultraviolet spectrum in the 
enstrophy range. 

Here we shall give a simple derivation of the Sulem-Frisch bound,
\be \zeta_2\leq {{2}\over{3}}, \lb{157} \ee
based upon Eq.(\ref{155}) and the fundamental Besov space embedding 
theorem \cite{35}. We first give the proofs 
taking the average as over space, and afterward we shall formulate 
the argument for the (more proper) ensemble-average. 
Thus, we assume that the exponent $z_2>2/3,$ or $s_2>1/3.$
This implies that $\bv\in B^{1+s,\infty}_2$ for each $s<s_2$ 
(see Appendix III). In particular, it holds for an $s>1/3.$ 
However, the fundamental embedding theorem (see Section 2.7.1, 
\cite{35} and Appendix III) states that
\be B^{s,\infty}_p(\bT^d)\subset B^{s',\infty}_{p'}(\bT^d), 
\lb{158} \ee
(continuous embedding) with
\be s-{{d}\over{p}}=s'-{{d}\over{p'}}. \lb{159} \ee
Applying this for $d=2$ with $p=2$ and $p'=\infty,$ and noting 
that $B^{s,\infty}_\infty=C^s$ (the space of H\"{o}lder
functions of index $s$) it follows that $\bv\in C^s(\bT^2),$ 
or $\|\Delta_\bl\bv\|_\infty=O(\ell^s).$
Then, applying the H\"{o}lder inequality
\begin{eqnarray}  
|\langle [\Delta_\bl v_{|\!|}(\br)][\Delta_\bl\omega(\br)]^2
\rangle| 
             & \leq & \|\Delta_\bl \bv \|_\infty\cdot
\|\Delta_\bl\omega\|_2^2 \cr
          \, &   =  & O\left(\ell^{3s}\right).  \lb{160}
\end{eqnarray}
However, Eq.(\ref{160}) leads to a contradiction when $s>1/3$ 
since its LHS is $\sim \ell$ by Eq.(\ref{155}) but, 
by the above argument, is $O\left(\ell^{3s}\right).$ Thus, we 
conclude that $z_2\leq 2/3.$

To transpose the argument to ensemble averages---for which the 
constant flux condition was proved under reasonable
conditions in Section 3.II---we appeal to Theorem 4 of \cite{36}. 
This states that Eq.(\ref{152}) for 
ensemble averages implies that for any $s<\sigma_p,$
\be \|\Delta_\bl\omega\|_{L^p}=O(\ell^s)\,\,\,\,\,\,\,\,\,\,
\,\,\,\,\,{\rm a.s.}\lb{160a} \ee
In that case, the previous argument may be applied to infer 
that if $\zeta_2>2/3,$ then  
\be \lim_{\ell\rightarrow 0}{{1}\over{\ell}}\int_{\bT^2}d^2
\br\,\,[\Delta_\bl v_{|\!|}(\br)][\Delta_\bl\omega(\br)]^2=0
\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{\rm a.s.} \lb{160b} \ee
This is not consistent with Eq.(\ref{155}) as long as the quantity
\be {\cal Z}_\ell\equiv {{1}\over{\ell}}\int_{\bT^2}d^2\br\,\,
[\Delta_\bl v_{|\!|}(\br)][\Delta_\bl\omega(\br)]^2 
\lb{160c} \ee
is ``uniformly integrable,'' or when 
\be \lim_{A\rightarrow\infty}\sup_{\ell>0}\langle |{\cal Z}_\ell
|{\boldmath 1}_{\{|{\cal Z}_\ell|>A\}}\rangle=0. 
\lb{160d} \ee
In other words, if the contributions of the tails of the 
distribution to the average of $|{\cal Z}_\ell|$ 
go to zero uniformly in $\ell,$ then 
\be \lim_{\ell\rightarrow 0}{{1}\over{\ell}}\langle[\Delta_\bl 
v_{|\!|}][\Delta_\bl\omega]^2\rangle
    =\lim_{\ell\rightarrow 0}\langle{\cal Z}_\ell\rangle=0, 
\lb{160e} \ee
contradicting Eq.(\ref{155}). 

One can expect the assumption Eq.(\ref{160d}) to be true, 
since the quantity ${\cal Z}_\ell$
has $\ell$-independent expectation $-2\eta,$ providing uniformity 
in $\ell,$ and, since it is a space-average, making 
unlikely any ``intermittency'' as $\ell\rightarrow 0.$ In other 
words, it is possible in principle that 
${\cal Z}_\ell$ has a small probability $\sim \ell$ to take on a 
very large magnitude $\sim \eta\cdot\ell^{-1}$ for each 
$\ell>0$ and a large probability $\sim 1-\ell$ to be zero. That 
would be consistent with both $\langle {\cal Z}_\ell\rangle
=-2\eta$ and $\lim_{\ell\rightarrow 0}{\cal Z}_\ell=0\,\,\,{\rm a.s.}$ 
(the last as a consequence of the Borel-Cantelli 
lemma) but it would violate the uniform integrability assumption. 
No contradiction could then be derived. However, we do 
not expect a space-average quantity such as ${\cal Z}_\ell$ to show 
such strong intermittency as $\ell\rightarrow 0,$ 
in which the constant mean would be achieved by large, rare 
fluctuations. Therefore, the uniform integrability assumption 
Eq.(\ref{160d}) seems to us reasonable.

Another rather different condition would also suffice for the proof, 
namely 
\be \langle\|\omega\|_{B^{s,\infty}_2}^3\rangle<\infty. \lb{160f} \ee
This means that the constants in the big-$O$ estimate Eq.(\ref{152}), 
as a space-average, can be chosen for each 
realization $\omega$ so that the ensemble-average of their cube is 
finite. This also seems reasonable. A careful
examination of the argument in the previous paragraph shows that
\be |{\cal Z}_\ell|\leq ({\rm const.})\|\omega\|_{B^{s,\infty}_2}^3. 
\lb{160g} \ee
See Appendix III. Since the latter is assumed integrable, the 
dominated convergence theorem would yield
Eq.(\ref{160e}) and lead to a contradiction with constant flux. 
Thus, our bound  
$\zeta_2\leq 2/3,$ Eq.(\ref{157}), holds for the ensemble-average 
exponent under quite reasonable assumptions.

Bounds can obtained in the same way for $\zeta_p$ with $p=3$ and 
higher. The simplest such estimate is 
\be \zeta_p\leq 0, \lb{164} \ee
for every $p\geq 3.$  We shall give proofs of this assuming averages 
as over space. The extension to ensemble-averages 
can be made in the same way as before, assuming uniform integrability 
of ${\cal Z}_\ell,$ Eq.(\ref{160d}), 
or finite means of constants, such as 
\be \langle \|\omega\|_{B^{s,\infty}_p}^3\rangle<\infty. 
\lb{160h} \ee
Therefore, we do not explicitly consider the extension. For 
transparency of notation we shall use the 
more conventional ensemble-average notation, with the 
understanding that the argument is really made 
first for space-averages and then extended as above. 

To obtain the estimate first on $\zeta_3,$ we use a condition 
on the velocity,
\be \|\Delta_\bl^2\bv\|_3=O\left(\ell^{\sigma_3+1}\right), 
\lb{161} \ee
equivalent to Eq.(\ref{151}) for $p=3$ with $\zeta_3=
3\sigma_3>0.$ The proof of this equivalence requires some potential
theory estimates which are discussed in Appendix III. Note that 
the second-difference is necessary when $\sigma_3>0,$
since then $\|\Delta_\bl\bv\|_3=O(\ell).$ However, we show that, 
in fact, this assumption is not consistent with
constant flux and that
\be \zeta_3\leq 0. \lb{162} \ee
For the proof we just observe that by a different application of 
the H\"{o}lder inequality  
\begin{eqnarray}  
|\langle [\Delta_\bl v_{|\!|}(\br)][\Delta_\bl\omega(\br)]^2\rangle|
             & \leq & \|\Delta_\bl\bv\|_3\cdot
\|\Delta_\bl\omega_\ell\|_3^2 \cr
          \, &   =  & O\left(\ell^{2\sigma_3+1}\right). \lb{163}
\end{eqnarray}
Again, this contradicts Eq.(\ref{155}) if $\sigma_3>0$ and yields 
the bound Eq.(\ref{162}). Since the H\"{o}lder 
inequality also implies that the combination $\sigma_p={{\zeta_p}
\over{p}}$ is non-increasing in $p,$ 
it follows at once that $\zeta_p\leq 0$ for every $p\geq 3.$ 
\footnote{Incidentally, this gives another
proof of the bound $\zeta_2\leq 2/3,$ in conjunction with the 
Besov space embedding theorem. Indeed, 
$B^{s,\infty}_2\subset B^{s',\infty}_3$ with $s'=s-{{1}\over{3}},$ 
so that $s_3>0$ if $s_2>{{1}\over{3}}.$}

\noindent {\em Refined Similarity Hypothesis and Improved Bounds} 

The same results as above can be obtained using the condition
\be \langle Z_\ell(\br)\rangle=\eta, \lb{165} \ee
in terms of the local flux variable introduced in Section 2 (i). 
For example, assuming $\zeta_3=3\sigma_3>0,$ 
we can estimate in the same way with the H\"{o}lder inequality that  
\begin{eqnarray}  
\|Z_\ell\|_1 & \leq & ({\rm const.})\|\grad\ol_\ell\|_3\cdot
\|[\Delta\bv]_\ell\|_3\cdot\|[\Delta\omega]_\ell\|_3 \cr
          \, &   =  & O\left(\ell^{\sigma_3-1}\cdot\ell\cdot
\ell^{\sigma_3}\right)=O\left(\ell^{2\sigma_3}\right), \lb{166}
\end{eqnarray}
which gives Eq.(\ref{162}). Note that the estimate 
$\|\grad\ol_\ell\|_3=O(\ell^{\sigma_3-1})$ follows from the 
representation 
\be \grad\ol_\ell(\br)= -\int d^2\bl\,\,\grad G_\ell(\bl)
\Delta_\bl\omega(\br) \lb{166a} \ee
and the assumed bound on $\|\Delta_\bl\omega\|_3.$ 

However, these estimates can now be further improved. The 
basic idea is that
a scaling relation should exist between the {\em local 
enstrophy flux} variable $Z_\ell(\br)$ 
and the vorticity difference at the same point $\br$:
\be Z_\ell(\br)\sim {{[\Delta_\ell\bv(\br)][\Delta_\ell
\omega(\br)]^2}\over{\ell}}. \lb{167} \ee
This is an analogue of the {\em refined similarity 
hypothesis} (RSH) in 3D which---in the version 
of Kraichnan \cite{37}---states that local energy flux 
scales as $\Pi_\ell(\br)\sim [\Delta_\ell v(\br)]^3/\ell$ 
in terms of the velocity difference at the same point. 
The Eq.(\ref{167}) will imply relations between 
corresponding scaling exponents of the vorticity differences 
and the enstrophy flux, in the same way as the ordinary RSH 
in 3D. More precisely, if we assume 
\be \langle |Z_\ell|^p\rangle=O(\ell^{\tau_p}), \lb{168} \ee
then there follow heuristically from Eq.(\ref{167}) relations 
between the exponents $\zeta_p$ and $\tau_p:$
\be \zeta_p=\tau_{p/3}. \lb{168a} \ee
These relations are proved here as inequalities and yield 
our main estimates on exponents. 
The proofs given below follow closely methods used in our 
discussion of the 3D RSH in \cite{10}.
However, unlike there, they are formulated directly in 
terms of ensemble-averages. Thus 
a statistical hypothesis is required, either a moment 
condition of the form
\be \langle\|\omega\|_{B^{s,\infty}_p}^p\rangle<\infty, 
\lb{168aa} \ee
for all $s<\sigma_p,$ or else uniform integrability of 
flux, when $\zeta_p<0.$ \footnote{We have already seen 
that $\zeta_p\leq 0$ is required for $p\geq 3.$ If the 
minimum H\"{o}lder singularity of the 
vorticity is negative, $-1<h_\mn<0,$ then $\zeta_p<0$ 
for large enough $p$: see \cite{13}.} 

One useful observation for the proof is that the assumption 
Eq.(\ref{152}) can be reformulated using $L^p$-spaces 
based upon joint space \& ensemble-averaging. In other words, 
we define an $L^p$-norm on random functions of the 
torus, as 
\be |\!|\!|f|\!|\!|_p\equiv \left[{{1}\over{|\Lambda|}}
\int_\Lambda d^2\br 
                                          \langle|f(\br)|^p
\rangle\right]^{1/p}. \lb{168b} \ee
Appealing to statistical homogeneity of the ensemble, we 
can write the condition Eq.(\ref{152}) as
\be |\!|\!|\Delta_\bl\omega|\!|\!|_p=O\left(\ell^{\sigma_p}
\right). \lb{168c} \ee
The next key point is that for $1<p<\infty,$ with the usual 
space-$L^p$ norms,
\be \|\Delta_\bl\bv\|_p\leq C\cdot \ell^{1+s}\cdot\|\omega
\|_{B^{s,\infty}_p} \lb{168d} \ee
with a constant $C$ which is {\em independent of} $\omega$ 
whenever
\be \|\Delta_\bl\omega\|_p=O\left(\ell^{s}\right) 
\lb{168dd} \ee
with $-1<s<0.$ This result is shown in Appendix III 
to be true
as a consequence of the Calder\'{o}n-Zygmund inequality 
and a theorem on characterization of Besov spaces 
by differences. Note that it is {\em not} true when 
$s\geq 0.$\footnote{The corresponding statement
then is that $\|\Delta_\bl\bv\|_p\leq C\cdot \ell\cdot
\|\omega\|_{B^{s,\infty}_p},$ which 
was used to establish our previous results.} Furthermore, 
Eq.(\ref{168d}) will hold a.s. for the 
realizations of the ensemble by a generalization of Theorem 4 
of \cite{36} (Appendix III). Note that 
Theorem 4 does not apply when $\sigma_p<0.$ Taking the $p$th 
power 
of Eq.(\ref{168d}), ensemble-averaging, and taking finally the 
$p$th root, we obtain the inequality
\be |\!|\!|\Delta_\bl\bv|\!|\!|_p\leq 
                       C\cdot \ell^{1+s}\cdot \langle 
\|\omega\|^p_{B^{s,\infty}_p}\rangle^{1/p} \lb{168ddd} \ee
for the joint space\& ensemble norm. In conjunction with 
Eq.(\ref{168c}) we then derive that
\be |\!|\!|\Delta_\bl\bv|\!|\!|_p=O\left(\ell^{s+1}\right). 
\lb{169} \ee
In that case, we can use again the H\"{o}lder inequality to 
derive for every $s<\sigma_{3p},$ 
\begin{eqnarray}  
|\!|\!|Z_\ell|\!|\!|_p & \leq & |\!|\!|\grad\ol_\ell|\!|\!|_{3p}
\cdot|\!|\!|[\Delta\bv]_\ell|\!|\!|_{3p}
\cdot|\!|\!|[\Delta\omega]_\ell|\!|\!|_{3p} \cr
                    \, &   =  & O\left(\ell^{s-1}
\cdot\ell^{s+1}\cdot\ell^{s}\right)
                                =O\left(\ell^{3s}\right).
\lb{170}
\end{eqnarray}
However, using Eq.(\ref{168}) for $\ell\rightarrow 0$ with 
the {\em maximal} choice of $\tau_p,$ we obtain 
\be \zeta_p\leq \tau_{p/3} \lb{171}. \ee
This is our main estimate. Note from $\|Z_\ell\|_1\leq\|Z_\ell\|_p$ 
for $p\geq 1$ that 
\be \tau_p\leq 0. \lb{172} \ee
Hence, Eq.(\ref{171}) can improve upon Eq.(\ref{164}). The 
exponent $\tau_p<0$ if the {\em local} enstrophy flux is 
increasingly intermittent as $\ell\rightarrow 0.$ We emphasize 
that the local flux is distinct from the global quantity
obtained by space-averaging, which is presumably {\em not} 
intermittent (see above). 

\noindent {\em A Formulation Valid For Nonzero Viscosity}

All of the previous results on exponent bounds have assumed the 
existence of a zero-viscosity limit 
for the stationary ensemble. However, it is worthwhile to 
formulate versions of these results which 
are valid at finite $\nu>0.$ The way to reformulate the premise 
Eq.(\ref{152}) of the argument in that context is that
\be \langle |\Delta_\bl\omega|^p\rangle \leq C_{\sigma,p}\ell^\zeta,
\,\,\,\sigma=\zeta/p, \lb{172a} \ee
with a {\em viscosity-independent} constant $C_{\sigma,p}.$ Of course, 
at positive viscosity, the vorticity is
presumably analytic in 2D. Therefore, Eq.(\ref{172a}) will hold with 
an arbitrarily large $\sigma,$ but with
a constant which in general increases as $\nu\rightarrow 0.$ Therefore, 
our assumption will be that for each fixed $p\geq 1$
Eq.(\ref{172a}) holds for some real $\sigma$ with a viscosity-independent 
constant $C_{\sigma,p}.$ We then take 
$\sigma_p$ to be the supremum of the $\sigma$'s for which this is true. 
In that case, we can still show that
the previous derived bounds on the exponents $\sigma_p$ are valid. 
To see this, recall that the enstrophy 
flux was shown in Section 3.II to be constant down to a length-scale 
$\ell_d$ which {\em vanishes} as $\nu\rightarrow 0,$
if either of two reasonable hypotheses is satisfied. Under the 
assumption of uniformly bounded mean energy (Hypothesis 1)
it was shown that $\ell_d\sim \nu^{1/4}$ and under the stronger 
assumption of uniformly bounded mean enstrophy (Hypothesis 3)
it was shown that $\ell_d\sim \nu^{1/2}.$ In either case, we may consider 
a length-scale $\lambda$ intermediate between
$\ell_0$ and $\ell_d,$ for example
\be \lambda\equiv\sqrt{\ell_0\ell_d}. \lb{172b} \ee
It then follows from the results of Section 3.II that
\be \lim_{\nu\rightarrow 0}\langle Z_\lambda\rangle=\eta. \lb{172c} \ee
However, if we assume that Eq.(\ref{152}) holds for $p=3$ with 
$\sigma_3>0$---just to name one case---then
we obtain the bound
\be |\langle Z_\lambda\rangle|=O(\lambda^{\zeta_3}), \lb{172d} \ee
and that is inconsistent with Eq.(\ref{172c}). Hence, we still 
conclude under our modified assumptions that 
$\zeta_p\leq 0$ for $p\geq 3.$

It is not completely clear, however, that this reformulation avoids the 
assumption that the $\nu\rightarrow 0$ 
limit exists. Notice that the best constants in the big-$O$ bounds of 
Eq.(\ref{152}) are one choice of 
a seminorm for the Besov spaces, 
\be ||\omega||_{B^{s,\infty}_p}^*\equiv\sup_{\ell>0}{{\|\Delta_\bl
\omega\|_{L^p}}\over{\ell^s}}, \lb{152a} \ee
which provides along with the $L^p$-norms a Banach space norm for 
$B^{s,\infty}_p,\,\,\,s>0,p\geq 1:$
\be \|\omega\|_{B^{s,\infty}_p}\equiv \|\omega\|_{L^p}
+||\omega||_{B^{s,\infty}_p}^*. \lb{152b} \ee
Consider the case $p=2$ (since we showed above that it is not 
possible that $s>0$ for $p\geq 3$). In that case,
$C_s=||\omega||_{B^{s,\infty}_2}^*$ is the best constant that 
can be chosen in every realization for 
the enstrophy-spectrum decay estimate
\be \Omega(k)\leq C_s\cdot k^{-(1+2s)}. \lb{152c} \ee
If this constant $C_s$ has finite ensemble-average 
$\langle C_s\rangle_\nu<\infty$ and also
$\langle\omega^2\rangle_\nu<\infty,$ with as well {\em viscosity-independent} 
upper bounds on the averages, then 
\be \sup_{\nu>0}\langle\|\omega\|_{B^{s,\infty}_p}\rangle_\nu<\infty. 
\lb{152d} \ee
That is enough to infer from the Prokhorov theorem the existence 
(at least along a subsequence) of a $\nu\rightarrow 0$ 
limit supported on $B^{s-\en,\infty}_2.$ \footnote{Since $B^{s,\infty}_2$ 
is not separable, this is not 
immediate. However, $B^{s,\infty}_2$ is compactly imbedded in the 
separable Banach space $H^{s-\en}_2$
as a consequence of the continuous imbedding $B^{s,\infty}_2
\subset H^{s-(\en/2)}$ and the Rellich compactness lemma.
This is enough to infer that the measures $\mus_\nu,\,\,\,
\nu>0$ are tight as distributions on $H^{s-\en}_2;$
see Theorem II.3.1 of \cite{7}. The continuous imbedding 
$H^s\subset B^{s,\infty}_2$ for all real $s$
concludes the argument.} Therefore, a strong enough form 
of the assumption on viscosity-independent bounds
implies the existence of a zero-viscosity limiting ensemble.

\noindent {\em Multifractal Model and Negative Exponents of 
Vorticity}

An interpretation of ``multiscaling'' laws of the form of 
Eq.(\ref{151}), with $\zeta_p$ a nonlinear function of $p,$
was proposed by Parisi and Frisch in \cite{38}. Their 
hypothesis was that the scaling could be attributed to 
the function having local H\"{o}lder exponents $h$ on sets 
$S(h)$ of ``fractal dimension'' $D(h).$ A simple
heuristic argument led to 
\be \zeta_p=\inf_{h\geq h_\mn} [ph+(d-D(h))]. \lb{173} \ee
It was shown by Jaffard \cite{39} that if the scaling holds 
in just big-O sense with a space-average (for even 
a {\em single} $p$), then at least a part of the ``multifractal 
hypothesis'' is correct, namely, the function is 
locally H\"{o}lder continuous and the bound holds
\be D_H(h)\leq ph+(d-\zeta_p) \lb{174} \ee
for the Hausdorff dimension of the set $S(h).$ In \cite{36} 
an extension is made of these results to ensemble-averages
and to ``negative H\"{o}lder exponents,'' which may occur 
if $\zeta_p<d/p.$ It was also shown in \cite{36} that
Eq.(\ref{173}) holds when $D(h)$ is taken to be a certain 
``box-counting dimension spectrum'' (which is {\em not}
the dimension of a physical-space set such as $S(h)$ in general.) 
These results are unrestricted in dimension
and thus hold for the vorticity in $d=2.$ It was already proposed 
by Benzi and Vergassola in \cite{30} that the  
``multifractal model'' should hold in 2D for the vorticity field. 
They also made a very preliminary attempt to 
determine $D(h)$ from numerical data on freely decaying 
turbulence in 2D with initial data chosen from a 
Gaussian random field of energy spectrum $k^{-3}.$ An interesting 
conclusion of their analysis---discussed already 
above---was that ``negative exponents'' $h\approx -1/2$ occur due 
to ``cusps'' in vortex cores, confirming earlier work 
of Farge and Holschneider \cite{29}. 

The conditions considered by those authors are different than ours, 
since we discuss forced steady-states rather
than free decay. However, from our previous results, we can 
establish a condition for steady-states requiring the 
``negative exponents'' to occur there. Namely, we can show that 
such singularities are necessary for ``intermittency'' 
in the 2D enstrophy cascade. It is easy to see that $h_\mn>0$ 
can never be consistent with the enstrophy flux condition. 
However, it seems possible that $h_\mn=0.$ Supposing that is true, 
the formula Eq.(\ref{173}) shows that $\zeta_p\geq 0$ 
for $p\geq 0,$ and, in conjunction with Eq.(\ref{164}),
\be \zeta_p=0 \lb{175} \ee
for $p\geq 3.$ Hence, the 1967 theory of Kraichnan \cite{1} must be 
exact for $p\geq 3$---up to possible logarithmic 
corrections---if $h_\mn=0$! This result can be extended to the case 
$p=2$ if a somewhat stronger assumption is made,
that is, we can show that as well
\be \zeta_2=0. \lb{176} \ee
It was another theorem of Yudovitch \cite{1} that, if the initial 
vorticity is bounded, then at any later time  
under Euler evolution, $\|\Delta_\bl\bv\|_\infty\leq c(\Lambda)\cdot
\|\omega_0\|_\infty \ell(1+\log\ell).$ 
Consider the following (strong) form of this as an assumption for 
the steady-state:
\begin{eqnarray}
|\!|\!|\Delta_\bl\bv|\!|\!|_\infty & \equiv & {\rm ess.}
\,{\rm sup}_{\{\br\in\Lambda,\bv\}}|\Delta_\bl\bv(\br)| \cr
                      \, & =      & O(\ell(1+\log\ell)), \lb{177}
\end{eqnarray}
where the supremum is over all realizations of the statistical 
ensemble and points of the space domain $\Lambda.$
Notice that the assumption just means that the individual velocity 
fields are {\em quasi-Lipschitz} 
uniformly over the ensemble and space. 
In fact, we obtain then
\begin{eqnarray}  
|\!|\!|Z_\ell|\!|\!|_1 & \leq & ({\rm const.})|\!|\!|\grad\ol_\ell
|\!|\!|_{2}\cdot|\!|\!|[\Delta\bv]_\ell|\!|\!|_\infty
                                 \cdot|\!|\!|[
\Delta\omega]_\ell|\!|\!|_2 \cr
          \, &   =  & O\left(\ell^{s_2-1}\cdot\ell\log\ell\cdot
\ell^{s_2}\right)=O\left(\ell^{2s_2}\log\ell\right),
\lb{178}
\end{eqnarray}
which yields Eq.(\ref{176}). In this case, we obtain also the 
$k^{-3}$ energy spectrum as exact, up to possible logarithmic
terms. It may seem surprising that ``monoscaling'' is exact 
in 2D for $h_\mn=0,$ since dynamically one should expect 
that H\"{o}lder exponents of the vorticity $h>0$ will occur 
at least at some points in space. However, it is important 
to appreciate that the 1967 scaling is consistent with a 
{\em nontrivial} spectrum of singularities $D(h)$ over $h\geq 0.$ 
It follows from Eq.(\ref{173}) that the condition $\zeta_p=0$ 
for $p\geq 0$ is equivalent to the condition 
that $D(0)=2.$ This still allows $D(h)\neq 0$ for $h>0.$ 

For the zero-viscosity limit of the forced steady-state it seems 
even more probable that such negative H\"{o}lder 
singularities may form, due to the effects of the random forcing 
building up over time. If, on the other hand, the 
hypothesis $h_\mn=0$ is correct, it requires a universality in 
the ultraviolet range of 2D turbulence, which is not 
strongly confirmed by simulations. Although one simulation by 
V. Borue \cite{14} has confirmed the Kraichnan theory, a 
plethora of energy spectra, generally steeper than $k^{-3},$ 
have been observed (see references in \cite{14}.) Without any 
definitive explanation of this fact, we will just suggest that 
the results may be due to a limited range of Reynolds 
numbers and a consequent contamination of scaling from the 
infrared range. In fact, large-scale, coherent
vortex structures are commonly observed in such simulations 
which have very steeply-decaying spectra
in Fourier space. This would also explain why spectra much 
steeper than allowed by the bound $x\leq 11/3$
are commonly observed, and the lack of universality in such 
scaling, since the vortex structures depend upon the 
details of forcing in the large-scales. It should be clear 
that the energy spectrum itself is not the quantity most 
sensitive to the possible presence of ``negative H\"{o}lder 
singularities.'' To clear up the issue whether $h_\mn=0$ 
or not it would be far preferable to measure $\langle
(\Delta_\bl\omega)^p\rangle$ for $p\geq 3,$ the higher order 
vorticity structure functions.

\section{Appendices}

\noindent {\bf Appendix I: Rigorous Proof of the Steady-State 
Balance Equations} 
 
We give here the proof of the balance equations in the 
steady-state with white-noise 
force in time. The proof of the energy-balance equation, 
Eq.(\ref{30}), was already given 
as Theorem XI.2.2 of Vishik and Fursikov \cite{7}. We shall 
discuss therefore the 
corresponding equations for the vorticity invariants, 
Eq.(\ref{47}) in the text. 
In the case $n=2,$ corresponding to the enstrophy balance, 
we will outline a complete 
proof. However, for $n>2$ we cannot give a complete proof. 
Instead we shall show that 
the balance equations for (even) $n>2$ follow from expected 
regularity of the steady-state 
measure. As we discuss below, the regularity properties assumed 
should follow
by a generalization of the standard energy-estimates to the 
higher-order vorticity 
invariants. The difficulty in the argument arises from the fact 
that simple Fourier-Galerkin 
truncations of the 2D Euler dynamics preserve only the quadratic 
invariants, energy and enstrophy,
and not the higher invariants (which was already observed some 
time ago by Kraichnan \cite{42b}.) 
We emphasize again that the balance equations do require some 
proof, since they might be false 
if singularities were to occur at a viscosity which is positive 
but small enough. Of course, 
that is not expected to occur in 2D.

As in the proof of Theorem XI.2.2 in \cite{7}, our main tool is 
the {\em stationary Kolmogorov equation}
\be \int \overline{\mu}(d\omega)\,\exp(i\langle\omega,\varphi\rangle)
K(\omega,\varphi)=0, \lb{I1} \ee
in which $\overline{\mu}$ is the stationary measure, $\varphi
\in H^s,\,\,s>2$ and 
\be K(\omega,\varphi)=i\langle \nu\bigtriangleup\omega-
B(\bv,\omega),\varphi\rangle-
                                 \langle\varphi,H\varphi\rangle. 
\lb{I2} \ee
Note in the last equation that $B(\bv,\omega)=(\bv\bdot\grad)\omega$ 
and $\bv=({\rm rot})^{-1}\omega.$ 
This may be proved in the same manner as Eq.(XI.2.7) of \cite{7}, 
along with the additional {\em a priori} estimate for $\alpha\geq 0,$
\be \mus\left(\|\omega\|_{L^2}^{2\alpha}\|\grad\omega\|_{L^2}^2\right)
\leq C_\alpha<\infty. \lb{I3} \ee
Although this result is not stated in \cite{7}, it follows very 
directly from the methods 
developed there. Hence, we shall simply sketch here the proof. 
Time-dependent statistical solutions 
are obtained for 2D Navier-Stokes with force white-noise in 
time in Theorem XI.3.1 of \cite{7}.
However, the {\em a priori} bounds (XI.3.9) derived there 
are not sufficient to discuss the steady-state,
because they are exponential in time. Instead, the bounds
\be E\left(\|\omega(t)\|_{L^2}^{2+2\alpha}+\nu\int_0^{t}ds
\,\,\|\omega(s)\|_{L^2}^{2\alpha}
                       \|\grad\omega(s)\|_{L^2}^2\right)
\leq A_\alpha+B_\alpha\cdot t \lb{I4} \ee
are required. These can be proved by exactly the same 
argument used in \cite{7} to prove Theorem X.4.1, 
but with vorticity substituting for velocity. It then 
follows by the Bogolyubov-Krylov averaging
method used to prove Theorem XI.2.2 that a stationary 
statistical solution exists in 2D for each $\nu>0,$
which obeys the estimate Eq.(\ref{I3}). Note that the 
uniqueness of the solutions $\mu(t,\cdot)$ to the
time-dependent Kolmogorov equation is proved in 2D under 
certain conditions (Theorem XI.4.2 of \cite{7}),
but the uniqueness of the steady-state $\mus$ seems to 
be open. We expect it to be true.
The Kolmogorov equation Eq.(\ref{I1}) can also be directly 
obtained from Eq.(XI.2.7) of \cite{7} by setting 
\be \nu_i=-\en_{ij}\partial_j\varphi=-\partial_i^\top\varphi 
\lb{I5}. \ee
It is valid then for all $\varphi\in H^s,\,\,s>3,$ which 
suffices just as well for our purposes.

We give next the proof of the enstrophy balance equation,
\be \mus(\nu\|\grad\omega\|^2_{L^2})= H(\bz), \lb{I6} \ee
following closely the proof of the energy balance in \cite{7}. 
As there, we take 
\be \varphi(\br)=\varphi_\bk\cdot e^{i\bk\bdot\br}+\varphi_\bk^*
\cdot e^{-i\bk\bdot\br} \lb{I7} \ee
in the Kolmogorov equation Eq.(\ref{I2}). Differentiating twice 
with respect to $\varphi_\bk,\varphi_\bk^*,$
we obtain 
\be \int \mus(d\omega)\,\exp(i\langle\omega,\varphi\rangle)
                \left[|\wo_\bk|^2K+2i{\rm Re}\left(\wo^*_\bk
{{\partial K}\over{\partial\varphi_\bk}}\right)
                +{{\partial^2 K}\over{\partial\varphi_\bk^*
\partial\varphi_\bk}}\right]=0. \lb{I8} \ee
Summing over all wavenumbers $\bk$ in the sphere of radius 
$2^N,$ we obtain 
\be \int \mus(d\omega)\,\left[-2\langle \nu\bigtriangleup
\omega-B(\bv,\omega),P_N\omega\rangle
                                             - 2H_N\right]=0, 
\lb{I9} \ee
where $P_N$ is the projection onto the space spanned by 
the eigenmodes $\exp(i\bk\bdot\br)$ with $|\bk|\leq 2^N$
and $H_N$ is the partial trace ${\rm Tr}(P_NH)=H_N(\bz).$ 
Recall that for any $\omega\in H^1$ the following 
statements are true: $\omega_N\equiv P_N\omega\rightarrow\omega$ 
strongly in $H^1;$ 
\be |\langle \nu\bigtriangleup\omega-B(\bv,\omega),\omega_N
\rangle|\leq 
                     C\cdot (1+\|\omega\|_{L^2})\|\omega\|_{H^1}
\|\omega_N\|_{H^1}; \lb{I10} \ee
and, $\langle B(\bv,\omega),\omega\rangle=0.$ Since $\omega\in H^1$ 
a.s. with respect to $\mus,$
it therefore follows that 
\be \langle \nu\bigtriangleup\omega-B(\bv,\omega),P_N\omega\rangle
\rightarrow -\nu\|\grad\omega\|_{L^2}
            \,\,\,\,\,\,\,\,\,\,\,\,\,\,\mus-{\rm a.s.}. \lb{I11} \ee
On the other hand, it follows as well from Eq.(\ref{I10}) that 
the function in
the above limit for each $N$ is bounded by $C\cdot (1+\|\omega\|_{L^2})
\|\omega\|_{H^1}^2$ and the 
latter is integrable by the {\em a priori} estimate Eq.(\ref{I3}). 
It therefore follows by the Lebesgue 
dominated convergence theorem upon taking the limit $N\rightarrow 
+\infty$ in Eq.(\ref{I9}) that 
\be \mus(2\nu\|\grad\omega\|^2-2H(\bz))=0, \lb{I12}. \ee
This is equivalent to Eq.(\ref{I6}), as claimed. 

We would like to follow the same pattern of argument to 
prove the higher-order balance equations
\be \mus(\nu\omega^{p-2}\|\grad\omega\|^2)= \mus(\omega^{p-2})
H(\bz) \lb{I13} \ee
for even integers $p>2$ (or Eq.(\ref{47}), $n=p.$) For this 
we shall employ the following {\em assumed estimates}:
\be \mus\left(\|\omega\|_{L^p}^{(1+\alpha)p}\right)\leq 
C_p<\infty, \lb{I14} \ee
and 
\be \mus\left(\|\omega\|_{L^p}^{\alpha p}\cdot\int 
\omega^{p-2}|\grad\omega|^2 \right)\leq C_p<\infty 
\lb{I15} \ee
for $\alpha\geq 0$ and for even $p>2,$ as well as
\be \mus\left(\|\omega\|_{L^p}^{\alpha p}\|\grad\omega
\|_{L^2}^2 \right)\leq C_p<\infty. \lb{I15x} \ee
We shall afterward discuss the motivation for adopting 
these particular hypotheses by making 
generalized energy-estimates. (Note the third has a 
somewhat different status, but is still quadratic in 
$\grad\omega.$) Here we shall use these conditions to 
rigorously demonstrate the balance equations Eq.(\ref{I13}). 
\footnote{It is worth observing that a similar discussion
 can be made for the 3D energy balance. For example, the 
energy-balance Eq.(\ref{30}) would follow in 3D by the 
argument of \cite{7} if the {\em a priori} estimate could be 
established that $\int \mus(d\bv)\,\|\grad\bv\|^{5/2}
\|\bv\|^{1/2}<\infty,$ improving Eq.(XI.2.8). This would be 
consistent with violation of the energy equality (Ito formula) 
for a set of individual realizations with zero probability.}
The first step is to set $\varphi(\br)=\sum_{i=1}^p\varphi_{\bk_i} 
e^{i{\bk_i}\bdot\br}$ in the stationary Kolmogorov 
equation Eq.(\ref{I1}), apply the differential operator 
${{1}\over{p!}}(\partial^p/\partial\varphi_{\bk_1}^*\cdots
\partial\varphi_{\bk_p}^*),$ and then take all $\varphi_\bk=0.$ The result is 
\begin{eqnarray}
\, & & \int\mus(d\omega)\,\left[{{i^p}\over{p!}}\sum_{i=1}^p
\wo_{\bk_1}\cdots\underline{\wo_{\bk_i}}\cdots\wo_{\bk_p}
            \times\left(-\nu k_i^2\wo_{\bk_i}-\langle 
e^{i\bk_i\bdot\br},B(\bv,\omega)\rangle\right)\right. \cr
\, & & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,
\,\,\,\,\,\,\left.+{{i^p}\over{p!}}\sum_{i\neq j=1}^p
             \wo_{\bk_1}\cdots\underline{\wo_{\bk_i}}\cdots
\underline{\wo_{\bk_j}}\cdots\wo_{\bk_p}
             \times \widehat{H}(\bk_i)\delta_{\bk_i+\bk_j,\bz}
\right]=0, \lb{I16} 
\end{eqnarray}
where underlining indicates omission of those terms. 

The next step is to multiply through the preceding equation 
by the factor $\prod_{i=1}^p\whi(\bk_i/2^N),$ 
where $\phi$ is a smooth function on $\bR^2$ to be chosen, 
and then to sum over all $\bk_i,\,i=1,...,p$ with
$|\bk_i|\leq 2^N$ for each $i$ and $\bk_1+\cdots+\bk_p=\bz.$ 
The reason for introducing the smoothing
factor is that we wish to appeal to $L^p$-convergence and, 
as is well-known, Fourier series on $\bT^d$
do not converge in $L^p$-norm for $d>1$ (e.g. \cite{45},
Section VII.4). On the other hand, if the function 
$\phi$ considered above satisfies the modest properties that
\be |\phi(\br)|\leq A\cdot (1+r)^{-(d+\en)}, \lb{I16a} \ee
and 
\be |\whi(\bk)|\leq A\cdot (1+k)^{-(d+\en)}, \lb{I16b} \ee
for some constant $A<\infty$ and $\en>0,$ and also 
\be \whi(\bz)=\int d^d\br\,\,\phi(\br)=1, \lb{I16c} \ee
then the {\em generalized means}
\be (S_\en f)(\br)\equiv \sum_\bk \whi(\en\bk)\widehat{f}_\bk 
e^{i\bk\bdot\br}, \lb{I17} \ee
have the properties that $S_\en $ is bounded on $L^p(\bT^d)$ 
and that $S_\en f\rightarrow f$ in strong $L^p$-sense 
as $\en \rightarrow 0.$  See Section VII.2 of \cite{45}. 
Let us denote $S_N\equiv S_{2^{-N}}$ and 
$\omega_N=S_N\omega,$ etc. Then the result of the 
aforementioned operations is the equation
\begin{eqnarray}
\, & & \int\mus(d\omega)\,\left[{{1}\over{(p-1)!}}
\int_{\bT^2}d^2\br\,\,\omega_N^{p-1}(\br)
                                \left(-\nu\bigtriangleup
\omega_N(\br)+S_NB(\bv,\omega)\right)\right] \cr
\, & & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,
\,\,\,\,\,\,\,\,
       =\int\mus(d\omega)\,\left[{{1}\over{(p-2)!}}
\int_{\bT^2}d^2\br\,\,\omega_N^{p-2}(\br)\right]\cdot H_N. \lb{I18} 
\end{eqnarray}
We study now the convergence as $N\rightarrow\infty$ of each term.

It follows from the first estimate Eq.(\ref{I14}) that $\omega\in 
L^p$ for each even $p\geq 2$ a.s. with 
respect to $\mus.$ By the $L^p$-boundedness of $S_N,$
\be \int_{\bT^2}d^2\br\,\,\omega_N^{p-2}(\br)\leq C\cdot 
\int_{\bT^2}d^2\br\,\,\omega^{p-2}(\br) \lb{I19} \ee
for even $p>2$ and the righthand side is $\mus$-integrable by 
Eq.(\ref{I14}). Thus we may apply the Lebesgue theorem
to infer for the last term that
\be \lim_{N\rightarrow\infty}\mus\left[\int_{\bT^2}d^2\br\,\,
\omega_N^{p-2}(\br)\right]
            =\mus\left[\int_{\bT^2}d^2\br\,\,\omega^{p-2}(\br)
\right]. \lb{I20} \ee

Similar considerations can be made for the middle term. In 
fact, for all integers $p\geq 1$
\begin{eqnarray}
\left|\langle\omega_N^p,S_N B(\bv,\omega)\rangle\right|
               & \leq & \|\omega_N^p\|_{L^2}\|S_N 
B(\bv,\omega)\|_{L^2} \cr
             \,& \leq & \|S_N\omega\|_{L^{2p}}^p
\|B(\bv,\omega)\|_{L^2} \cr
             \,& \leq & \|\omega\|_{L^{2p}}^p\left
[1+\int v^2|\grad\omega|^2\right], \lb{I21} 
\end{eqnarray}
The last function is $\mus$-integrable by the rigorous 
estimate Eq.(\ref{I3}). To see this, we observe that 
the velocity may be represented by the line-integral
\be \bv(\br)=\bv(\br')+\int_0^{|\br-\br'|}ds\,(\hat{\bn}
\bdot\grad)\bv, \lb{I21a} \ee
where $\hat{\bn}=(\br-\br')/|\br-\br'|,$ valid when 
$|\grad\bv|\in L^1.$ We can average this equation over the 
torus $\bT^2$ with respect to $\br'$ and use the fact that
\be \int \bv=\bz \,\,\,\,\,\,\,\,\,\,\,\,\,\,\mus-{\rm a.s.}, 
\lb{I21b} \ee
when the force spectrum has support bounded away from the origin. 
In that case, 
\be \bv(\br)={{1}\over{\ell^2_0}}\int_{\bT^2}d^2\br'
\int_0^{|\br-\br'|}ds\,(\hat{\bn}\bdot\grad)\bv, \lb{I21c} \ee
and
\begin{eqnarray}
|\bv(\br)| & \leq & C(\ell_0)\cdot \|\grad\bv\|_{L^1} \cr
        \, & \leq & C'(\ell_0)\cdot \|\grad\bv\|_{L^2} \cr
        \, & \leq & C''(\ell_0)\cdot \|\omega\|_{L^2}, \lb{I22d} 
\end{eqnarray}
in which the dependence of the constants on the finite box-size 
$\ell_0$ is made explicit. Note the last line 
is a consequence of the well-known Calder\'{o}n-Zygmund 
inequality \cite{46}. Thus,
\be \int v^2|\grad\omega|^2\leq C'''(\ell_0)\cdot\|\omega\|_{L^2}^2
\|\grad\omega\|_{L^2}^2 \lb{I22e} \ee
and the righthand side is $\mus$-integrable by Eq.(\ref{I3}), 
as claimed. Furthermore,
since $\omega_N\rightarrow \omega$ in $L^{2p},$ also 
$\omega_N^p\rightarrow \omega^p$ in $L^2.$ 
Because $S_NB(\bv,\omega)\rightarrow B(\bv,\omega)$ 
in $L^2$ as well, we conclude that 
\begin{eqnarray}
\lim_{N\rightarrow 0}\langle\omega_N^p,S_N B(\bv,\omega)
\rangle & = & \langle\omega^p,B(\bv,\omega)\rangle \cr
\, & = & 0 \,\,\,\,\,\,\,\,\,\,\,\,\,\,\mus-{\rm a.s.} 
\lb{I22} 
\end{eqnarray}
for all integers $p\geq 1$. Therefore, the Lebesgue theorem 
allows us to conclude that
\be \lim_{N\rightarrow 0}\mus\left[\langle\omega_N^{p-1},S_N 
B(\bv,\omega)\rangle\right]=0. \lb{I23} \ee

However, we must treat the first term in Eq.(\ref{I18}) 
differently, since $\omega_N\rightarrow\omega$ in $H^1$
and thus $|\grad\omega_N|^2\rightarrow|\grad\omega|^2$ only 
in $L^1.$ Let us set
\be I_p[\omega]=\int \omega^{p-2}|\grad\omega|^2 \lb{I26}, \ee
and then, for some $R>0,$ define also
\be I_{p,R}[\omega]=\int_{\{|\omega|<R\}}\omega^{p-2}
|\grad\omega|^2, \lb{I25} \ee
and 
\be \overline{I}_{p,R}[\omega]=\int_{\{|\omega|\geq R\}}
\omega^{p-2}|\grad\omega|^2 \lb{I26a} \ee
so that $I_p[\omega]=I_{p,R}[\omega]+\overline{I}_{p,R}
[\omega].$ We shall show that it is enough to establish that 
\be \lim_{N\rightarrow\infty}\mus\left(I_{p,R}[\omega_N]
\right)=\mus\left(I_{p,R}[\omega]\right) \lb{I24} \ee 
in order to take the limit $N\rightarrow\infty$ in the first term. 

In fact, we shall show that by choosing $R$ sufficiently large, 
independent of $N,$ we may guarantee that
\be \left|\mus\left(I_p[\omega_N]\right)-\mus\left(I_{p,R}[\omega_N]
\right)\right|<\en \lb{I28} \ee
and 
\be \left|\mus\left(I_p[\omega]\right)-\mus\left(I_{p,R}
[\omega]\right)\right|<\en \lb{I30} \ee
for any preselected $\en>0.$ To prove this, we observe 
first that for even $p\geq 2,$
\begin{eqnarray}
\mus\left(\overline{I}_{p,R}[\omega_N]\right) & \leq & 
{{1}\over{R^2}}
\mus\left(\int \omega^{p}_N|\grad\omega_N|^2\right) \cr
\, & \leq & {{1}\over{R^2}}\mus\left(\|\omega\|_{L^p}^{p}
\|\grad\omega\|_{L^2}^2\right) \cr
                                \, & \leq & {{C_p}\over{R^2}}
<\infty. \lb{I31} 
\end{eqnarray}
The last line follows from the validity of the third assumed 
estimate, Eq.(\ref{I15x}), for arbitrary even $p.$ 
The middle line is a consequence of the fact that, for a suitable 
choice of $\phi,$ and for $f,g\geq 0,$
\be \int f_N g_N\leq \|f\|_{L^1}\|g\|_{L^1} \lb{I31aa} \ee
for each $N\geq 1.$  It is enough to take a real-valued $\whi\in 
C^\infty$ with compact support
and with also $\phi$ real and positive. It is easy to construct 
examples with all these properties. 
In that case, the square $|\whi|^2$ has the same properties and 
defines another operator $T_N$ (through multiplication of the
Fourier coefficients by $|\whi_N(\bk)|^2$) with the same properties 
as $S_N.$ Thus,
\begin{eqnarray}
\int d^2\br\,\,f_N(\br) g_N(\br) & =    & \sum_{\bk}|\whi_N(\bk)|^2
\widehat{f}_\bk^*\widehat{g}_\bk \cr
                              \, & =    & \|T_N(f*g)\|_{L^1} \cr
                              \, & \leq & \|f*g\|_{L^1} \cr
                              \, & \leq & \|f\|_{L^1} \|g\|_{L^1}. 
\lb{I31bb}
\end{eqnarray}
Next we use Jensen's inequality to obtain $\omega_N^p\leq S_N
(\omega^p)$ for $p\geq 1$ and $|\grad\omega_N|^2=
|S_N(\grad\omega)|^2\leq S_N\left(|\grad\omega|^2\right).$ In 
that case, taking $f=\omega^p$ with even $p\geq 2$ 
and $g=|\grad\omega|^2$ in Eq.(\ref{I31bb}), we find that
\be \int \omega_N^p|\grad\omega_N|^2\leq \|\omega^p\|_{L^1}
\|(\grad\omega)^2\|_{L^1}= 
                                    \|\omega\|_{L^p}^{p}
\|\grad\omega\|_{L^2}^2 \lb{I31cc} \ee
just as required in Eq.(\ref{I31}). The consequence of this 
equation, is, in short, that 
\be \left|\mus\left(I_p[\omega_N]\right)-\mus\left(I_{p,R}
[\omega_N]\right)\right| < {{C_p}\over{R^2}}, \lb{I31c} \ee
and $R$ may be chosen so large that $C_p/R^2<\en,$ giving 
Eq.(\ref{I28}). On the other hand, 
\be \lim_{R\rightarrow \infty}\mus\left(I_{p,R}[\omega]\right)
=\mus\left(I_p[\omega]\right) \lb{I29} \ee
by the monotone convergence theorem. Since $\mus(I_p[\omega])
<\infty,$ we may pick $R$ large enough that also 
Eq.(\ref{I30}) is valid. 

Therefore, as claimed, we must only show that
\be \lim_{N\rightarrow\infty}\mus\left(I_{p,R}[\omega_N]\right)
=\mus\left(I_{p,R}[\omega]\right). \lb{I31d} \ee
Since 
\be I_{p,R}[\omega_N]\leq R^{p-2}\|\grad\omega_N\|^2_{L^2}
\leq R^{p-2}\|\grad\omega\|^2_{L^2} \lb{I32} \ee
and the righthand side is $\mus$-integrable, it is enough 
to prove that
\be \lim_{N\rightarrow\infty}I_{p,R}[\omega_N]=I_{p,R}[\omega]
\,\,\,\,\,\,\,\,\,\,\,\,\,\,\mus-{\rm a.s.} \lb{I31e} \ee
However, we see that the functions $f_N\equiv \omega_N^{p-2}
\chi_{\{\omega_N<R\}},f\equiv \omega^{p-2}\chi_{\{\omega<R\}}$ 
and $g_N\equiv |\grad\omega_N|^2,g\equiv |\grad\omega|^2$ satisfy 
the following properties $\mus$-a.s.: 
({\em i}) $|f_N|\leq R^{p-2}<\infty,$  ({\em ii}) $f_N(\br)
\rightarrow f(\br)$ pointwise a.e. with respect 
to Lebesgue measure,  and ({\em iii}) $g_N\rightarrow g$ 
in $L^1.$ This is enough to conclude by a triangle argument that
$I_{p,R}[\omega_N]=\int f_Ng_N\rightarrow \int fg=I_{p,R}[\omega],$ 
as required. We conclude that
\be \lim_{N\rightarrow \infty}\mus\left(\int \omega_N^{p-2}
|\grad\omega_N|^2\right)
                            =\mus\left(\int \omega^{p-2}
|\grad\omega|^2\right), \lb{I33} \ee
which allows the limit to be taken in the first term of 
the Eq.(\ref{I18}). 

Therefore, from all of the preceding argument, we obtain 
from Eq.(\ref{I18}) in the limit $N\rightarrow 0$ that
\begin{eqnarray}
\, & & \int\mus(d\omega)\,\left[{{1}\over{(p-2)!}}\int_{\bT^2}
d^2\br\,\,\nu\omega^{p-2}(\br)|\grad\omega|^2\right] \cr
\, & & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,
       =\int\mus(d\omega)\,\left[{{1}\over{(p-2)!}}\int_{\bT^2}
d^2\br\,\,\omega^{p-2}(\br)\right]\cdot H, \lb{I34} 
\end{eqnarray}
for even $p>2,$ which are exactly the higher-order balance 
equations Eq.(\ref{I13}).

The motivation for the estimates Eqs.(\ref{I14}),(\ref{I15}) 
we have assumed is the following: Let 
again $\omega_N=P_N\omega$ and consider the Fourier-Galerkin 
approximation to the dynamics
\be \partial_t\omega_N+P_NB(\bv_N,\omega_N)=\nu\bigtriangleup
\omega_N+q_N. \lb{I35} \ee
It is unfortunately the case that 
\be \int \omega_N^{p-1}P_NB(\bv_N,\omega_N)\neq 0 \lb{I36} \ee
for $p>2.$ If we ignore this fact, assuming for the moment 
that $\|\omega_N(t)\|_{L^p}^p$ is conserved 
by the approximate dynamics, then the Ito formula gives for 
each $\alpha\geq 0,$ 
\begin{eqnarray}
d\|\omega_N(t)\|_{L^p}^{(\alpha+1)p} & = & {{-4\nu(\alpha+1)}
\over{q}}\|\omega_N(t)\|_{L^p}^{\alpha p}
                                           \int \omega_N^{p-2}(t)
|\grad\omega_N(t)|^2\cdot dt+dM_N(t) \cr
                                  \, &   & \,\,\,\,\,\,\,\,\,\,\,\, 
                                           +{{H_N}\over{2}}
\left[(\alpha+1)\alpha p^2\|\omega_N(t)\|_{L^p}^{(\alpha-1)p}
                                           \|\omega_N(t)
\|_{L^{2p-2}}^{2p-2}dt \right. \cr
                                  \, &   & \,\,\,\,\,\,\,\,\,
\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,
                                           \left. 
+(\alpha+1)p(p-1)\|\omega_N(t)\|_{L^p}^{\alpha p}
                                           \|\omega_N(t)
\|_{L^{p-2}}^{p-2}dt \right], \lb{I37} 
\end{eqnarray}
where
\be M_N(t)\equiv p\int_0^t\langle\omega_N^{p-1}(s),dq_N(s)
\rangle \lb{I38} \ee
is a martingale part. It is not too difficult to make this 
equation the basis of an inductive proof of the 
bounds Eqs.(\ref{I14}),(\ref{I15}). We do not give the 
argument here because the above equation is not really 
correct. Notice that
\be \langle\omega_{N-1}^{p-1},P_NB(\bv_{N-1},\omega_{N-1})
\rangle
    =\langle\omega_{N-1}^{p-1},B(\bv_{N-1},\omega_{N-1})
\rangle=0, \lb{I39} \ee
despite Eq.(\ref{I36}). Therefore, the violation of the 
higher-order conservation laws for the truncated dynamics 
is, in 
some sense, a ``boundary effect'' in Fourier space associated 
to the interactions near the wavenumber cutoff at $2^N.$
If some control of these boundary terms can be obtained, then 
the estimates Eqs.(\ref{I14}),(\ref{I15}) should
follow. It seems to us an interesting open mathematical problem 
to establish these estimates as {\em a priori}
bounds. More sophisticated approximations to 2D Euler may be 
useful here, such as the finite-mode dynamics based
upon $SU(N)$ Lie algebras which have $O(N)$ integrals of motion 
for $O(N^2)$ modes \cite{90}. However, we would be 
very surprised if the presumed bounds should not be valid and 
we regard the balance equations therefore as rather 
convincingly demonstrated. 

\noindent {\bf Appendix II: Kolmogorov Relation for the 2D 
Enstrophy Cascade}

We shall derive here the Kolmogorov-type relation for 2D, 
Eq.(\ref{155}). The method we 
use is patterned closely after U. Frisch's derivation of 
the ``4/5-law'' in Section 6.2
of \cite{42}. 

The first step is to define a {\em physical-space enstrophy 
flux} as
\be \eta(\bl)\equiv \left.-{{1}\over{2}}{{d}\over{dt}}\langle
\omega(\br,t)
                                        \omega(\br+\bl,t)\rangle
\right|_{{\rm Euler},t=0}, \lb{II1} \ee
where the time-derivatives are to be evaluated using the inviscid 
Euler equations. (Compare this 
definition with Polyakov's ``point-splitting'' method in \cite{9}.) 
A simple calculation, using 
statistical homogeneity and incompressibility, yields
\be \eta(\bl)=-{{1}\over{4}}\grad_\bl\bdot\langle\Delta_\bl\bv
(\Delta_\bl\omega)^2\rangle. \lb{II2} \ee
If we now consider the randomly-forced NS equation, Eq.(\ref{38}), 
\be \partial_t\omega+(\bv\bdot\grad)\omega=\nu\nabla^2\omega+q, 
\lb{II3} \ee
then we obtain {\em formally}, for the steady-state, the 
balance-equation
\be \eta(\bl)={{1}\over{2}}\langle\omega(\br)[q(\br+\bl)+q(\br-\bl)
]\rangle
               +\nu\nabla^2_\bl\langle\omega(\br)\omega(\br+\bl)
\rangle. \lb{II4} \ee
For a Gaussian force, white-noise in time, this becomes
\be \eta(\bl)={{1}\over{2}}(Q(\bl)+Q(-\bl))+\nu\nabla^2_\bl R(\bl)
, \lb{II5} \ee
where $Q(\br-\br')$ is the covariance function of $q$ and $R(\bl)
=\langle\omega(\br)\omega(\br+\bl)\rangle$
is the vorticity covariance function. We have sketched a formal 
derivation of the last 
balance equation, Eq.(\ref{II5}), but it should be possible to 
prove rigorously in 2D by the methods 
of Appendix I. 

Note that the viscous term will be negligible when $\nu$ is small 
and 
$\ell=|\bl|\gg \ell_d^{{\rm Kr}}.$ This can be easily checked to 
be true using arguments like
those in Section 3.II. In fact, for the case of isotropic forcing, 
this can be inferred from the 
well-known representation
\be R(\bl)=\int_0^\infty\,\,J_0(k\ell)\Omega(k)dk. \lb{II6} \ee
(See Theorem 2.5.2 of \cite{43}.) However, if the force spectrum 
has support only in a finite wavenumber 
range around $k_f,$ then 
\be Q(\bl)\approx Q(\bz) \lb{II7} \ee
when $\ell\ll \ell_f.$ We therefore obtain that 
\be -{{1}\over{4}}\grad_\bl\bdot\langle\Delta_\bl\bv
(\Delta_\bl\omega)^2\rangle\approx \eta \lb{II8} \ee
with $\eta\equiv Q(\bz),$ when $\ell_f\gg\ell\gg\ell_d^{{\rm Kr}}.$ 
Alternatively, we may consider 
idealized limits, first $\nu\rightarrow 0$ and subsequently 
$\ell\rightarrow 0,$ always assuming
that such limits exist. 

For completeness we note that this relation can be obtained 
in another way by relating 
the ``physical-space flux'' $\eta(\bl)$ to the usual spectral 
flux $Z_K,$ as
\be Z_K={{K}\over{2\pi}}\int d^2\bl {{J_1(K\ell)}
\over{\ell}}\eta(\bl). \lb{II9} \ee
By its definition
\begin{eqnarray}
Z_K & \equiv & \left.-{{1}\over{2}}{{d}\over{dt}}
\int_{|\bk|\leq K}{{d^2\bk}\over{(2\pi)^2}}
\langle|\widehat\omega(\bk)|^2\rangle\right|_{Euler} \cr
 \, & =      & \int_{|\bk|\leq K}{{d^2\bk}\over{(2\pi)^2}}
\int d^2\bl\,\,e^{i\bk\bdot\bl}\eta(\bl). \lb{II10} 
\end{eqnarray}
However,
\begin{eqnarray}
\int_{|\bk|\leq K}d^2\bk\,\,e^{i\bk\bdot\bl} & = & 
2\int_0^K kdk\int_0^\pi d\theta\,\,e^{ik\ell\cos\theta} \cr
                                          \, & = & 
2\pi\int_0^K\,J_0(k\ell)\,kdk. \lb{II11} 
\end{eqnarray}
We have used a common integral representation of the Bessel 
function $J_0$ (Eq.(9.1.18) of \cite{44}.)
It is also a well-known relation for the Bessel functions that
\be \int_0^z\,t^\nu J_{\nu-1}(t)dt=z^\nu J_\nu(z) \lb{II12} \ee
when ${\rm Re}\nu>0.$ (See Eq.(11.3.20) of \cite{44}.) Applying 
this for $\nu=1,z=K\ell,t=k\ell$ we obtain
Eq.(\ref{II9}) as claimed. It is then easy to see that in 
the range of scales where $\eta(\bl)\approx \hat{\eta},$
a constant, that also $Z_K\approx \hat{\eta}.$ We just use
\be \int_0^\infty J_\nu(t)dt=1 \lb{II13}, \ee
valid for ${\rm Re}\nu>-1$ (Eq.(11.4.17) of \cite{45}), in the 
case $\nu=1.$ Therefore, we see that 
we must have $\hat{\eta}=\eta,$ and we re-derive Eq.(\ref{II8}).

We obtain the special form in Eq.(\ref{155}), 
\be \langle [\Delta_\bl v_{|\!|}(\br)][\Delta_\bl\omega(\br)]^2
\rangle=
    -2\eta\ell, \lb{II14} \ee
under the condition of statistical isotropy. This can be easily 
guaranteed if the forcing is 
chosen to be isotropic.\footnote{More properly, this requires 
one to consider the infinite-volume setting.
In that case, it is easy to show for isotropic forcing that at 
least one homogeneous and isotropic stationary
statistical solution exists. Of course, it is not known whether 
the steady-state is unique. In particular,
it may be that there is a ``spontaneous-breaking'' of $SO(2)$ 
symmetry, so that the space-ergodic measures, or 
``pure phases,'' are not rotation-invariant. Barring this 
unlikely possibility, the symmetries of the stationary 
measure follow from those of the force.} In that case we may 
represent the vector function
\be \bA(\bl)\equiv \langle\Delta_\bl\bv(\Delta_\bl\omega)^2
\rangle \lb{II15} \ee
in terms of two functions $A_{|\!|}(\ell)$ and $A_{\perp}(\ell)$ 
depending only upon the magnitude $\ell=|\bl|,$ as 
\be A_i(\bl)=\hat{l}_i A_{|\!|}(\ell)+\en_{ij}\hat{l}_jA_{\perp}(\ell). 
\lb{II16} \ee
It is then easy to calculate that
\be \grad_\bl\bdot\bA={{1}\over{\ell}}A_{|\!|}(\ell)+{{dA_{|\!|}}
\over{d\ell}}(\ell). \lb{II17} \ee
Thus, we obtain from Eq.(\ref{II8}) that
\be {{dA_{|\!|}}\over{d\ell}}(\ell)+{{1}\over{\ell}}A_{|\!|}(\ell)
=-4\eta. \lb{II18} \ee
The only solution of this equation regular at $\ell=0$ is 
\be A_{|\!|}(\ell)=-2\eta\ell. \lb{II19} \ee
This is exactly Eq.(\ref{II14}). Observe that $A_{\perp}\equiv 0$ 
if the steady-state is invariant as well under 
space-reflection (i.e. the full invariance under the improper 
rotation group $O(2).$) In that case, we have even
\be \bA(\bl)= -2\eta\bl \lb{II20} \ee
as a vector identity. 

\noindent {\bf Appendix III: Potential Theory Estimates and Results 
for Besov Spaces in 2D}

We prove here the estimates from potential theory and the embedding 
theorems used in Section 4 to derive bounds 
on 2D scaling exponents. We shall consider these in essentially the 
order of their appearance in that section.

First, we give for completeness a derivation of the Besov space 
embedding theorem Eq.(\ref{158}),
since the proof is very elementary. The defining criterion for 
$f\in B^{s,q}_p(\bT^d)$ is that
the sequence
\be   (2^{sN}\|f*\varphi_N\|_p: N\geq 0)\in \ell^q, \lb{III1} \ee 
where $(\widehat{\varphi}_N:N\geq 0)$ is a smooth partition of 
unity with ${\rm supp}\widehat{\varphi}_N\subset 
[2^{N-1},2^{N+1}].$ If a function $g$ has its Fourier transform 
supported in the unit ball $\{\bk:|\bk|\leq 1\},$ 
then it can be written as $g=\phi*g,$ where $\phi$ is a function 
with $C^\infty$ Fourier transform $\widehat{\phi}$ 
supported in the larger ball $\{\bk:|\bk|\leq 2\}$ and $=1$ in 
the unit ball. For each $p\geq 1$ it follows by the 
H\"{o}lder inequality that
\be \|g\|_\infty=\|\phi*g\|_\infty\leq C_p \|g\|_p, \lb{III2} \ee
where $C_p=\|\phi\|_q,\,{{1}\over{p}}+{{1}\over{q}}=1.$ Therefore, 
for each $p'\geq p,$
\begin{eqnarray}
\|g\|_{p'} & \leq & \|g\|^{1-{{p}\over{p'}}}_\infty\cdot
\|g\|_p^{{{p}\over{p'}}} \cr
         \,& \leq & C \|g\|_p, \lb{III3} 
\end{eqnarray}
with $C=(C_p)^{1-{{p}\over{p'}}}.$ A simple scaling argument 
shows that if $g$
has instead its Fourier transform supported in the ball 
$\{\bk:|\bk|\leq 2^N\},$ then
\be \|g\|_{p'}\leq C\cdot 2^{N\left({{d}\over{p}}
-{{d}\over{p'}}\right)}\cdot \|g\|_p \lb{III4} \ee
for each $p'\geq p.$ If we apply this observation to the function 
$g=f*\varphi_N,$ then we see that for each $N\geq 0,$
\be 2^{s'N}\|f*\varphi_N\|_{p'}\leq C\cdot 2^{sN}\|f*\varphi_N\|_{p}, 
\lb{III5} \ee
when 
\be s'-{{d}\over{p'}}=s-{{d}\over{p}}. \lb{III6} \ee
This implies that $\|f\|_{B^{s',q}_{p'}}\leq C\|f\|_{B^{s,q}_{p}},$ 
concluding the proof.

We now discuss the relation between the two conditions 
$\|\Delta_\bl\omega\|_p=O(\ell^{s})$
and $\|\Delta_\bl^2\bv\|_p=O(\ell^{1+s}).$ Since $\omega
=\grad\btimes\bv,$ 
\be \bv=\bB*\omega, \lb{III7} \ee
as in Eq.(\ref{5}), where $\bB$ is the integral kernel of 
the inverse operator to ${\rm rot}=\grad\btimes\cdot.$
The relation therefore involves potential theory estimates, 
which are readily available in  the literature. We first 
observe that for any $s\in \bR,$ 
\be \|\Delta_\bl\omega\|_p=O(\ell^{s})\longrightarrow 
\|\Delta_\bl^2\bv\|_p=O(\ell^{1+s}). \lb{III8} \ee 
Here the derivative $\grad\bv$ is taken in the distribution 
sense, and a straightforward calculation gives
\be \grad\bv=P(\grad\bB)*\omega+ {{1}\over{2}}\ben\cdot\omega, 
\lb{III9} \ee
where the first term is a principal part convolution integral 
for the singular kernel $\grad\bB$ and
$\ben$ in the second term is the Levi-Civita matrix. Observe, 
furthermore, that the difference operator $\Delta_\bl$ 
commutes with convolution operators, so that 
\be \grad(\Delta_\bl\bv)=P(\grad\bB)*(\Delta_\bl\omega)+ 
{{1}\over{2}}\ben\cdot(\Delta_\bl\omega). \lb{III10} \ee
Applying the Calder\'{o}n-Zygmund inequality \cite{46} 
to the first term, it then follows that
\be \|\grad(\Delta_\bl\bv)\|_p\leq C\|\Delta_\bl\omega\|_p 
\lb{III11} \ee
for $1<p<\infty.$ However, if $\Delta_\bl f$ is represented 
by the line-integral
\be \Delta_\bl f(\br)=\int_0^\ell ds\,\,\hat{\bl}\cdot\grad 
f(\br+s\hat{\bl}), \lb{III12} \ee
valid for $|\grad f|\in L^1,$ then a simple application of 
Jensen's inequality gives in general that
\be \|\Delta_\bl f\|_p\leq \ell\cdot \|\grad f\|_p \lb{III13} \ee
for $p\geq 1.$ Applying this to $f=\Delta_\bl\bv,$ we obtain that 
\be \|\Delta_\bl^2\bv\|_p\leq C\cdot\ell\cdot
\|\Delta_\bl\omega\|_p=O\left(\ell^{1+s}\right), \lb{III14} \ee
as claimed. 

However, if also $0<s<1,$ then we actually have equivalence
\be \|\Delta_\bl\omega\|_p=O(\ell^{s})\longleftrightarrow 
\|\Delta_\bl^2\bv\|_p=O(\ell^{1+s}), \lb{III15} \ee 
for $p>1.$ In fact, if $\bv\in L^p,$ then 
\be \|\Delta_\bl^2\bv\|_p=O(\ell^{1+s})\longleftrightarrow 
\bv\in B^{1+s,\infty}_p
                                       \longleftrightarrow 
\grad\bv\in B^{s,\infty}_p.  \lb{III16} \ee 
The first equivalence follows from Theorem 2.5.12 in \cite{35} 
on characterization of Besov spaces by differences.  
The second equivalence is a well-known result for all $s\in \bR.$ 
For $0<s<1$ it follows in the $\longleftarrow$ 
direction from the characterization by differences applied to 
$\grad\bv,\,\,\,\,\|\Delta_\bl\grad\bv\|_p=O(\ell^s),$ 
and the inequality Eq.(\ref{III13}). It follows in the 
$\longrightarrow$ direction from the defining criterion 
Eq.(\ref{III1}) applied to $\bv,$ and the following consequence
\be \|\varphi_N*(\grad\bv)\|_p=\|\grad(\varphi_N*\bv)\|_p\leq 
C\cdot 2^N\cdot\|\varphi_N*\bv\|_p \lb{III17} \ee
of the Nikol'skij inequality (see Remark 1, 
Section 1.3.2 of \cite{35}). 

On the other hand, if $-1<s<0,$ then there is no longer 
equivalence but it is still true that 
\begin{eqnarray}
\|\Delta_\bl\omega\|_p=O(\ell^{s}) & \longrightarrow & 
\|\Delta_\bl^2\bv\|_p=O(\ell^{1+s}) \cr
                                \, & \longleftrightarrow & 
\|\Delta_\bl\bv\|_p=O(\ell^{1+s}) \cr
                                \, & \longrightarrow & 
\omega\in B^{s,\infty}_p. \lb{III18} 
\end{eqnarray}
for $p>1.$ The middle equivalence is another consequence 
of Theorem 2.5.12 in \cite{35} on characterization 
by differences, which states that both are equivalent to 
$\bv\in B^{1+s,\infty}_p$ when $0<1+s<1.$ 
To obtain the last $\longrightarrow$ implication, we recall 
$\bv\in B^{1+s,\infty}_p
\longleftrightarrow \grad\bv\in B^{s,\infty}_p$ and 
$\omega$ is, of course, just a component of $\grad\bv.$ 
We should emphasize that the Theorem 2.5.12 in \cite{35} 
on characterization by differences does not 
apply when $s\leq d\left({{1}\over{{\rm min}\{1,p\}}}-1\right)$ 
and, in particular, when $s\leq 0$ for 
$p\geq 1.$ Therefore, $\|\Delta_\bl\omega\|_p=O(\ell^{s})$ is only a sufficient condition for
$\omega\in B^{s,\infty}_p$ and not a necessary one when $s\leq 0.$ 

We next show that the condition 
\be \langle |\Delta_\bl\omega|^p\rangle=O(\ell^{\zeta_p}) 
\lb{III29} \ee
with $\zeta_p<0$ implies that $\bv\in B^{1+s,\infty}_p$ a.s. 
for each $s<\sigma_p$ and, therefore,
\be \|\Delta_\bl\bv\|_p\leq \ell^{1+s}\cdot 
\|\bv\|_{B^{s,\infty}_p}\,\,\,\,\,\,\,\,\,\,\,\,\,\,
\mus-{\rm a.s.} \lb{III30} \ee
Theorem 4 in \cite{36} does not apply when $\sigma_p<0.$ 
Instead we shall show directly that $\bv\in B^{1+s,\infty}_p$
by means of the following characterization. Define the 
{\em truncated-ball means}
\be ({\cal V}_\ell^M f)(\br)\equiv 
\overline{(\Delta_{\ell\cdot \bh}^Mf)(\br)}, \lb{III31} \ee 
where the line indicates a space-average over the 
``truncated-ball'' $\{\bh:1<|\bh|<2\}.$ Then it is
known that 
\be \|f\|^\#_{B^{s,\infty}_p}=\sup_{N\geq 0}\left(2^{sN}
\|{\cal V}_{2^{-N}}^M f\|_p\right) \lb{III32} \ee
defines a seminorm for the Besov space $B^{s,\infty}_p$ 
with any integer $M>s>0.$ See Sections 2.5.11-12
of \cite{35}. To employ this result, we first average 
the condition Eq.(\ref{III29}) over the truncated-ball:
\be \overline{\langle |\Delta_{\ell\cdot\bh}\omega|^p\rangle}
=O(\ell^{\zeta_p}). \lb{III33} \ee
This implies for each fixed $\ell,$ or for any fixed countable set 
of $\ell$'s, that 
\be \Delta_{\ell\cdot\bh}\omega\in L_p \,\,\,\,\,\,\,\,\,\,\,\,\,\,
                                             \mus-{\rm a.e.}\,\,\,
\omega,\lambda-{\rm a.e.}\,\,\,\bh, \lb{III34} \ee
where $\lambda$ is Lebesgue measure on the truncated-ball. We can 
then invoke Eq.(\ref{III14}) to obtain 
\be \|\Delta_{\ell\cdot\bh}^2\bv\|_p\leq C\cdot\ell\cdot
\|\Delta_{\ell\cdot\bh}\omega\|_p\,\,\,\,\,\,\,\,\,\,\,\,\,\,
                                             \mus-{\rm a.e.}
\,\,\,\omega,\lambda-{\rm a.e.}\,\,\,\bh. \lb{III35} \ee
This may be averaged jointly over the truncated-ball and 
the ensemble to obtain, with Eq.(\ref{III33}),
\be \overline{\langle\|\Delta_{\ell\cdot\bh}^2\bv\|_p\rangle}
=O(\ell^{1+\sigma_p}). \lb{III36} \ee
However, it follows from Jensen's inequality that 
\be \|{\cal V}_\ell^2\bv\|_p\leq 
\overline{\|\Delta_{\ell\cdot\bh}^2\bv\|_p} \lb{III37} \ee
and thus
\be \langle \|{\cal V}_\ell^2\bv\|_p\rangle \leq 
\overline{\langle\|\Delta_{\ell\cdot\bh}^2\bv\|_p\rangle} 
                                         = O(\ell^{1+\sigma_p}) 
\lb{III38} \ee
for each $\ell$ in a countable set ${\cal L}.$ Let us choose 
the latter set to be ${\cal L}=\{\ell_N=2^{-N}:N\geq 0\},$ 
or, in other words,  
\be \langle\|{\cal V}_{2^{-N}}^2\bv\|_p\rangle 
= O\left(2^{-N(1+\sigma_p)}\right) \lb{III39} \ee
for all $N\geq 0.$ Then the same Borel-Cantelli 
argument used in the proof of Theorem 4 of \cite{36} implies 
that
\be \|{\cal V}_{2^{-N}}^2\bv\|_p= O\left(2^{-N(1+s)}\right) 
\,\,\,\,\,\,\,\,\,\,\,\,\,\,
                                             \mus-{\rm a.s.}
\lb{III40} \ee
for each $s<\sigma_p.$ We conclude, by the characterization 
theorem, that $\bv\in B^{1+s,\infty}_p$ a.s. 
and that Eq.(\ref{III30}) holds.

The last result we needed in our arguments is the inequality
\be \|\bv\|_{B^{1+s,\infty}_p}\leq C\|\omega\|_{B^{s,\infty}_p} 
\lb{III41} \ee
for some constant $C>0,$ which was used in establishing 
sufficiency of the moment conditions. 
For Eq.(\ref{III41}) we may refer to the literature. Since 
Eq.(\ref{III9}) relates $\grad\bv$ to $\omega$
by a classical singular-integral operator $T,$ we may appeal 
to Theorem A of Lemarie \cite{91}:
\be \|T\omega\|_{B^{s,q}_p}\leq C' \|\omega\|_{B^{s,q}_p} 
\lb{III42} \ee
for $1>s>0$ and $p,q\in [1,+\infty].$ This is a result of 
Calder\'{o}n-Zygmund type in the Besov spaces.
See also \cite{92}. In our case, 
\be \|\grad\bv\|_{B^{s,\infty}_p}\leq C'' 
\|\omega\|_{B^{s,\infty}_p}. \lb{III43} \ee
>From this we conclude at once Eq.(\ref{III41}). 

\noindent {\bf Acknowledgements:} This paper had its beginnings 
in some conversations with Z.-S. She, 
who made many useful suggestions and criticisms. Several other 
UA colleagues---D. Levermore, S. Malham,
S. Nazarenko, J. Watkins, and X. Xin---provided either 
references or helpful remarks. U. Frisch and 
A. Shnirelman very kindly sent me their unpublished works. 
I am grateful to all of the above.

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