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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 $\omega_0\in C^{k+s},\,\,\,
k\in \bZ^+_0,00$
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
$1
0.$ 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 $00$! 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,\,\,
0From 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^{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 $s1/3.$
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, $-10.$ 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 $\zeta_p0$
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|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 $f_N\equiv \omega_N^{p-2}
\chi_{\{\omega_N2,$ 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 $11.$ 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 $01.$ 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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\end{document}
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