have the following local well-posedness result: when $d\neq 2$, for any
initial data $u_{0}\in\dot{H}^{\sigma}$, there exists
$T=T(\|u\|_{\dot{H^{\sigma}}})$ such that (NLS) has a unique solution

$u\in C^{0}_{t}\dot{H}^{\sigma}_{x}\cap
L^{2}_{t}\dot{W}_{x}^{\sigma,2^{*}}([-T,T]\times\mathbb{R}^{d})=:Y_{d};$

when $d=2$, for any $u_{0}\in\dot{H}^{\sigma}$ and $r\in[2,\infty)$, there
exists $T=T(\|u\|_{\dot{H^{\sigma}}},r)$ such that (NLS) has a unique solution

$u\in C^{0}_{t}\dot{H}^{\sigma}_{x}\cap
L^{r^{\prime}}_{t}\dot{W}_{x}^{\sigma,r}([-T,T]\times\mathbb{R}^{2})=:Y_{2}.$

Moreover, $\|u\|_{Y_{d}}\lesssim\|u_{0}\|_{\dot{H}^{\sigma}}$. and the
solution map $u_{0}\mapsto u$ is Lipschitz.

###### Proof.

The proof follows standard Banach fixed point argument and the
$\dot{H}^{\sigma}$-subcritical nature of (NLS). For clarity, we only prove the
case $d\neq 2$. The $d=2$ case is no more than a substitution by non-endpoint
Strichartz norm.

Define

$\Psi_{u_{0}}(u):=e^{it\Delta}u_{0}+i\int_{0}^{t}e^{i(t-s)\Delta}(|u|^{p-1}u)(s)ds.$

We will show $\Psi_{u_{0}}$ is contractive in some suitable metric space.

Let $M:=\|u_{0}\|_{\dot{H}^{\sigma}}$. Define the closed ball

$X_{T,M}:=\left\\{u\in C^{0}_{t}\dot{H}^{\sigma}_{x}\cap
L^{2}_{t}\dot{W}_{x}^{\sigma,2^{*}}([-T,T]\times\mathbb{R}^{d}):\|u\|_{L^{\infty}_{t}\dot{H}^{\sigma}_{x}\cap
L^{2}_{t}\dot{W}_{x}^{\sigma,2^{*}}}\leq 2C_{0}M\right\\}$ (D.3)

and metric

$\mathfrak{d}_{T}(u,v):=\|u-v\|_{L^{q_{0}}_{t}L^{r_{0}}_{x}([-T,T]\times\mathbb{R}^{d})}$
(D.4)

where $C_{0}$ is the constant in the Strichartz estimate for $e^{it\Delta}$.
Here $r_{0}\in(2,2^{*})$ satisfies

$\frac{d}{2}-\frac{d}{r_{0}}=\frac{\frac{d}{2}-\sigma}{\frac{d}{2}-s_{c}}$
(D.5)

and $q_{0}=2\frac{\frac{d}{2}-s_{c}}{\frac{d}{2}-\sigma}$ such that
$(q_{0},r_{0})$ is Strichartz admissible. From
$s_{c}<\sigma\leq\alpha_{c}=\frac{d}{d+2-2s_{c}}$, we know
$\sigma\leq\mathrm{RHS\,\,of\,\,}\eqref{eqrsigma}<\min\left\\{1,\frac{d}{2}\right\\}$
and thus
$(\dot{H}^{\sigma}\cap\dot{W}^{\sigma,2^{*}})(\mathbb{R}^{d})\hookrightarrow
L^{r_{0}}(\mathbb{R}^{d})$. Consequently, $(X_{T,M},\mathfrak{d}_{T})$ is a
complete metric space for $T<\infty$.

We first check $\Psi_{u_{0}}:X_{T,M}\to X_{T,M}$. By Strichartz estimate,

$\displaystyle\|\Psi_{u_{0}}(u)\|_{L^{\infty}_{t}\dot{H}^{\sigma}_{x}\cap
L^{2}_{t}\dot{W}_{x}^{\sigma,2^{*}}}\leq
C_{0}\|u_{0}\|_{\dot{H}^{\sigma}}+C_{1}\left\|D^{\sigma}(|u|^{p-1}u)\right\|_{L_{t}^{q_{1}^{\prime}}L^{r_{1}^{\prime}}_{x}}.$

Then by fractional chain rule for $C^{1}$ nonlinearity (see [12]), we get

$\displaystyle\left\|D^{\sigma}(|u|^{p-1}u)\right\|_{L_{t}^{q_{1}^{\prime}}L^{r_{1}^{\prime}}_{x}}\lesssim\left\|\|u\|_{\dot{W}_{x}^{\sigma,r_{1}}}\|u\|_{L^{pr_{2}}_{x}}^{p-1}\right\|_{L^{q_{1}^{\prime}}_{t}}\lesssim\|u\|_{L^{q_{1}}_{t}\dot{W}^{\sigma,r_{1}}_{x}}^{p}T^{\beta}\lesssim
M^{p}T^{\beta}$

Thus the onto property of $\Psi_{u_{0}}$ is guaranteed by taking
$T\sim\|u_{0}\|_{\dot{H}^{\sigma}}^{-\frac{p-1}{\beta}}=M^{-\frac{p-1}{\beta}}$.

Next, we check that $\Psi_{u_{0}}$ is a contraction concerning the metric
$\mathfrak{d}_{T}$. Taking $\tilde{r}_{0}$ as
$\frac{1}{\tilde{r}_{0}}=\frac{1}{r_{0}^{\prime}}-\frac{1}{r_{0}}$, we then
have $\dot{H}^{\sigma}\hookrightarrow L^{\tilde{r}_{0}(p-1)}$. Thus

$\displaystyle\|\Psi_{u_{0}}(u)-\Psi_{u_{0}}(v)\|_{L^{q_{0}}_{t}L^{r_{0}}_{x}}\lesssim\||u|^{p-1}u-|v|^{p-1}v\|_{L^{q_{0}^{\prime}}_{t}L^{r_{0}^{\prime}}_{x}}$
$\displaystyle\lesssim$
$\displaystyle\left\|\|u-v\|_{L_{x}^{r_{0}}}\left(\|u\|_{L_{x}^{\tilde{r}_{0}(p-1)}}^{p-1}+\|v\|_{L_{x}^{\tilde{r}_{0}(p-1)}}^{p-1}\right)\right\|_{L^{q_{0}^{\prime}}_{t}}$
$\displaystyle\lesssim$
$\displaystyle\|u-v\|_{L^{q_{0}}_{t}L^{r_{0}}_{x}}\left(\|u\|_{L^{\infty}_{t}\dot{H}^{\sigma}_{x}}^{p-1}+\|v\|_{L^{\infty}_{t}\dot{H}^{\sigma}_{x}}^{p-1}\right)T^{\beta}\lesssim\|u-v\|_{L^{q_{0}}_{t}L^{r_{0}}_{x}}\|u_{0}\|_{\dot{H}^{\sigma}}^{p-1}T^{\beta}.$

Here the same $\beta$ comes from
$\frac{1}{q_{0}^{\prime}}-\frac{1}{q_{0}}=\frac{\sigma-
s_{c}}{\frac{d}{2}-s_{c}}=\beta$. By further shrinking $T$ by a constant if
necessary, we can conclude that $\Psi_{u_{0}}$ is a contraction on
$(X_{T,M},\mathfrak{d}_{T})$ for $d\neq 2$. We omit the standard arguments for
the rest statements in this proposition.

∎

Next, we derive a similar local-wellposedness result of the renormalized
system (1.14).

######  Proposition D.2.

Suppose $d\geq 1$ and $0<s_{c}<\min\left\\{1,\frac{d}{2}\right\\}$ and
consider the Cauchy problem (1.14). For any $s_{c}<\sigma\leq\alpha_{c}$, we
have the following local well-posedness result: when $d\neq 2$, for any
initial data $Z_{0}\in(\dot{H}^{\sigma})^{2}$, there exists
$T=T(\|Z_{0}\|_{(\dot{H^{\sigma}})^{2}})$ such that (1.14) has a unique
solution

$Z\in C^{0}_{t}([-T,T],(\dot{H}^{\sigma}_{x}(\mathbb{R}^{d}))^{2})\cap
L^{2}_{t}([-T,T],(\dot{W}_{x}^{\sigma,2^{*}}(\mathbb{R}^{d}))^{2})=:\tilde{Y}_{d};$

when $d=2$, for any $Z_{0}\in(\dot{H}^{\sigma})^{2}$ and $r\in[2,\infty)$,
there exists $T=T(\|Z_{0}\|_{(\dot{H^{\sigma}})^{2}},r)$ such that (NLS) has a
unique solution

$Z\in C^{0}_{t}([-T,T],(\dot{H}^{\sigma}_{x}(\mathbb{R}^{d}))^{2})\cap
L^{r^{\prime}}_{t}([-T,T],(\dot{W}_{x}^{\sigma,r}(\mathbb{R}^{d}))^{2})=:\tilde{Y}_{2}.$

Moreover, $\|Z\|_{\tilde{Y}_{d}}\lesssim\|Z_{0}\|_{(\dot{H}^{\sigma})^{2}}$
and the solution map $Z_{0}\mapsto Z$ is Lipschitz.

###### Proof.

This is still a contraction argument as above, where we borrow the onto
estimate and contractive estimate from Lemma 6.5 and Proposition D.1
respectively. Again, for simplicity, we only discuss $d\neq 2$ and $1<p\leq 2$
case.

We view the potential as a source term and write

$\displaystyle\Psi_{Z_{0}}(Z):=e^{i\tau({\mathcal{H}}_{b}-ibs_{c})}Z_{0}+i\int_{0}^{\tau}e^{i(\tau-\tau^{\prime})({\mathcal{H}}_{b}-ibs_{c})}(F-VZ)(\tau^{\prime})d\tau^{\prime}.$

We will show $\Psi_{Z_{0}}$ is a contraction mapping on a ball of radius
$C\|Z_{0}\|_{\dot{H}^{\sigma}}$ in $S([0,T]\times\mathbb{R}^{d})$ for
$T=T(\|Z_{0}\|_{\dot{H}^{\sigma}})$. Here $S$ is the Strichartz norm in
$\dot{H}^{\sigma}$ as (6.15). As in Proposition D.1, we take a distance given
by the weaker norm $L^{q_{0}}_{\tau}L^{r_{0}}_{y}$.

Since we only consider well-posedness local in time, we may assume $T_{0}\leq
1$. Thereafter, all exponential timewise weight can be neglected and
$e^{it({\mathcal{H}}_{b}-ibs_{c})}$ has the Strichartz estimates in both
$\dot{H}^{\sigma}$ and $L^{2}$.

Using the nonlinear estimate from Lemma C.1 and linear estimate Lemma B.1 for
potential, we get the estimate similar to (6.23) (with exponents borrowed
there)

$\displaystyle\|F\|_{\dot{W}^{\sigma,r_{1}^{\prime}}}$ $\displaystyle\lesssim$
$\displaystyle\|Z\|_{\dot{W}^{\sigma,r_{1}}\cap\dot{W}^{\sigma,r_{1}^{-}}}^{p},\quad\|VZ\|_{\dot{W}^{\sigma,r_{1}^{\prime}}}\lesssim\|Z\|_{\dot{W}^{\sigma,r_{1}}}.$

Like (6.24), the space-time estimate follows

$\displaystyle\|F-VZ\|_{L^{q_{1}^{\prime}}_{\tau}\dot{W}^{\sigma,r_{1}^{\prime}}_{y}}\lesssim(\|Z\|_{S}^{p}+\|Z\|_{S})T^{\beta}$

Now the Strichartz estimate for $e^{it({\mathcal{H}}_{b}-ibs_{c})}$ in
$\dot{H}^{\sigma}$ yields the onto estimate straightforwardly.

As for the contraction estimate, notice that

$\displaystyle|R(Z_{1},Q_{b})-R(Z_{1}^{\prime},Q_{b})|\lesssim|Z_{1}-Z_{1}^{\prime}|(|Z_{1}|^{p-1}+|Z_{1}^{\prime}|^{p-1}+|Q_{b}|^{p-1}).$

Then mimicking Proposition D.1, we get

$\displaystyle\|(F-VZ)-(F^{\prime}-VZ^{\prime})\|_{L^{q_{0}^{\prime}}_{\tau}L^{r_{0}^{\prime}}_{y}}\lesssim\|Z-Z^{\prime}\|_{L^{q_{0}}_{\tau}L^{r_{0}}_{y}}\left(\|Z\|_{L^{\infty}_{\tau}\dot{H}^{\sigma}_{y}}^{p-1}+\|Z^{\prime}\|_{L^{\infty}_{\tau}\dot{H}^{\sigma}_{y}}^{p-1}+\|Q_{b}\|_{\dot{H}^{\sigma}}^{p-1}\right)T^{\beta}$

which suffices to conclude the contraction estimate.

∎

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# Anisotropic characteristics of the Kraichnan direct cascade in two-
dimensional hydrodynamic turbulence

E.A. Kuznetsova,b,c and E.V. Sereshchenkoa,d,e<EMAIL_ADDRESS>a
Novosibirsk State University, 630090 Novosibirsk, Russia
b Lebedev Physical Institute, RAS, 119991 Moscow, Russia
c Landau Institute for Theoretical Physics, RAS, 119334 Moscow, Russia
d Khristianovich Institute of Theoretical and Applied Mechanics, SB RAS,
630090 Novosibirsk, Russia
e Far-Eastern Federal University, 690091 Vladivostok, Russia

###### Abstract

Statistical characteristics of the Kraichnan direct cascade for two-
dimensional hydrodynamic turbulence are numerically studied (with spatial
resolution $8192\times 8192$) in the presence of pumping and viscous-like
damping. It is shown that quasi-shocks of vorticity and their Fourier
partnerships in the form of jets introduce an essential influence in
turbulence leading to strong angular dependencies for correlation functions.
The energy distribution as a function of modulus $k$ for each angle in the
inertial interval has the Kraichnan behavior, $\sim k^{-4}$, and
simultaneously a strong dependence on angles. However, angle average provides
with a high accuracy the Kraichnan turbulence spectrum
$E_{k}=C_{K}\eta^{2/3}k^{-3}$ where $\eta$ is enstrophy flux and the Kraichnan
constant $C_{K}\simeq 1.3$, in correspondence with the previous simulations.
Familiar situation takes place for third-order velocity structure function
$S_{3}^{L}$ which, as for the isotropic turbulence, gives the same scaling
with respect to separation length $R$ and $\eta$, $S_{3}^{L}=C_{3}\eta R^{3}$,
but the mean over angles and time $\bar{C_{3}}$ differs from its isotropic
value.

PACS: 52.30.Cv, 47.65.+a, 52.35.Ra

##  I Introduction

Developed two-dimensional hydrodynamic turbulence, in contrast to three-
dimensional turbulence, in the inertial interval of scales possesses an
additional integral of motion — enstrophy that is a half integral of the
squared vorticity, $(1/2)\int_{s}\Omega^{2}d\mathbf{r}$. As was demonstrated
in 1967 by Kraichnan [1], the existence of this integral generates in the
inertial range its own Kolmogorov-type spectrum of turbulence,

$E(k)~{}\sim~{}k^{-3},$ (1)

(now called the Kraichnan spectrum). This spectrum corresponds to a constant
enstrophy flux toward the small-scale region. Simultaneously, according to
[1], a conventional Kolmogorov spectrum $E(k)~{}\sim~{}k^{-5/3}$ is formed at
large scales with a constant energy flux directed toward the region of small
values of $k$ (inverse cascade). Since the Kraichnan paper in 1967 there were
performed many numerical experiments (see [2] \- [9]; a more complete list in
[10]) testified in favor of existence of the Kraichnan spectrum. Already in
the first numerical experiments (see, e.g., [2]) there was observed the
emergence of sharp vorticity gradients (consistent with the high Reynolds
number). It corresponds to the formation of the vorticity jumps similar to
shock waves with thickness small compared with their length. Based on these
observations, Saffman [11] proposed another spectrum, $E(k)~{}\sim~{}k^{-4}$.
The Saffman spectrum was obtained under the assumption that the main
contribution to the spectrum comes from isotropically distributed vorticity
shocks. (It should be noted that in two-dimensional turbulence, the formation
of singular vorticity gradients in finite time is forbidden, in accordance
with the rigorous theorems [12]; therefore we will call such structures as
quasi-shocks.) On the other hand, it follows from simple considerations that
the spectrum with such shocks is expected rather to have the Kraichnan type
behavior than that predicted by Saffman. The Fourier amplitude $\Omega_{k}$
from one shock will be $\propto k^{-1}$, that immediately yields the spectrum
$E(k)~{}\sim~{}k^{-3}$. However, the situation is not too simple. If one
assumes that the characteristic length of the step $L\gg k^{-1}$, then the
energy distribution from one such step in the $k$-space has the form of a jet
with an apex angle of the order of $\left(kL\right)^{-1}$ . For freely decay
turbulence as was shown numerically in [10, 13, 14] the spectrum (1) arises
owing to vorticity quasi-shocks, which form a system of jets with weak and
strong overlapping in the $k$-space. In their turn, a reason of appearance of
quasi-shocks can be regarded due to the compressible features of the so-called
di-vorticity, $\mathbf{B=\nabla\times}\Omega\widehat{z}$. For inertial scales
this vector field is frozen-in-fluid and its value, in the general case,
according to [10, 13, 14, 15] can be changed due to the velocity component
$\mathbf{v_{n}}$, normal to the $\mathbf{B}$-line, for which
$\mbox{div}\,\mathbf{v_{n}}\neq 0$. This is the origin of compressibility for
continuously distributed di-vorticity lines and thereby appearance of quasi-
shocks in 2D. Besides, for freely decay turbulence numerically it was
established that (i) in each of such jets, the fall in $E(k)$ at large $k$ is
proportional to $~{}k^{-3}$, which after the angle averaging yields the
spectrum (1) ; (ii) the third-order velocity structure function
$S_{3}^{L}(\mathbf{R})=\langle\delta v_{\|}^{3}\rangle$ depends proportionally
to $R^{3}$ (where $\mathbf{R}=\mathbf{r}-\mathbf{r}^{\prime}$, $\delta v_{\|}$
the projection of the velocity difference onto the vector $\mathbf{R}$), in
complete agreement with the Kraichnan theory.

The main aim of this paper is to study numerically in which extent all the
above statistical properties observed in freely decay two-dimensional
turbulence remain for the direct (Kraichnan) cascade when the presence of both
pumping and viscous damping. Here we use the same numerical scheme reported in
our previous paper [14] with $8192\times 8192$ grid points and periodical
boundary conditions. In order to eliminate an influence of the inverse cascade
we introduce at very large scales (comparable with the box size) a big damping
that allows us to increase the inertial interval for the direct cascade. In
the inertial range we put the damping rate equal to zero and switch on the
viscous-like dissipation starting from $k=0.6k_{max}$.

The main results of this paper are as follows:

1\. The energy spectrum averaged over angles, as a function of modulars $k$,
with a high accuracy has the Kraichnan behavior: $E(k)=C_{K}\eta^{2/3}k^{-3}$
where $\eta$ is the enstrophy flux and $C_{K}$ the Kraichnan constant. In our
simulations $C_{K}\simeq 1.3$, in correspondence with the previous simulations
[16]-[19].

2\. The formation of $k^{-3}$-dependence happens for all rays in the $k$-space
starting from sufficiently earlier evolution times (about two-three
characteristic inverse maximal vorticity $\omega_{max}^{-1}$). However, the
angular dependence of energy distribution $\epsilon(\mathbf{k})$, in all our
numerical experiments, remains far from isotropic. In $k_{x}$-$k_{y}$ plane it
has a very pronounced jet behavior, typical for freely decay turbulence [10,
13, 14]. At small evolution times when flow is still not-turbulent quasi-
shocks and their respective jets are very visible. With time jets have a
tendency to mutual overlapping but significant angular fluctuations retain in
the energy distribution at the steady state.

3\. The jet structures reflect also in the behavior of third-order velocity
structure function $S_{3}^{(L)}(\mathbf{R})=\langle(\delta
v_{\|})^{3}\rangle$. Like for the isotropic direct cascade with constant
enstrophy flux [20]-[23] fitting shows that this correlation function behaves
linearly on both $\eta$ and $R^{3}$, $S_{3}^{(L)}(\mathbf{R})=C_{3}\eta
R^{3}$, but demonstrates a significant dependence for coefficient $C_{3}$ on
angles $\theta$, in correspondence with those observed for
$\epsilon(\mathbf{k})$. Average of $C_{3}$ over angles yields for mean value
$\bar{C_{3}}$ being far from the theoretical prediction for the isotropic
turbulence (see, e.g., review [24] and references therein).

##  II Physical model and numerical scheme

We consider the two-dimensional Navier-Stokes equation for an incompressible
flow $\mathbf{u(x},t)=(u_{x}(x,y,t),u_{y}(x,y,t))$ in the vorticity
formulation, $\Omega=\nabla\times\mathbf{u}$,

$\frac{\partial\Omega}{\partial
t}+(\mathbf{u}\nabla)\Omega=F+G\quad\quad\mbox{\rm
with}\quad\quad\emph{div}\text{ }\mathbf{u}=0$ (2)

supplemented by the periodic boundary conditions in $x$ and $y$ directions in
the square box with size $L=1$. Here $F$ is a function responsible for both
injection of the energy and its dissipation on large scales and $G$ for
enstrophy dissipation at large $k$.

To study the direct cascade we model $F$ so that to eliminate influence of the
energy condensation due to inverse cascade by introducing a strong dissipation
at $k<k_{p}$ where $k_{p}$ is the characteristic injection wave number. In our
numerics we used function $F$ in the form $F=\hat{\Gamma}\Omega$ where the
Fourier transform of operator $\hat{\Gamma}$

$\displaystyle\Gamma_{k}$ $\displaystyle=$ $\displaystyle
A\frac{(k^{2}-b^{2})(k^{2}-a^{2})}{k^{2}}\quad\text{at}\quad 0\leq k\leq
b,\quad$ (3) $\displaystyle\quad\Gamma_{k}$ $\displaystyle=$ $\displaystyle
0\quad\text{at}\quad k>b.$ (4)

Parameters $a$ and $b$ were chosen by such a way to get a rapid transition to
the stationary energy dissipation at small $k$.

Dissipation function $G=\hat{\gamma}\Omega$ was taken in the viscous-type
form:

$\displaystyle\gamma_{k}$ $\displaystyle=$ $\displaystyle 0\quad\mbox{\rm
at}\quad k\leq k_{c},\quad$ (5) $\displaystyle\quad\gamma_{k}$
$\displaystyle=$ $\displaystyle-\nu(k-k_{c})^{2}\quad\mbox{\rm at}\quad
k>k_{c},$ (6)

where viscous cut-off $k_{c}$ was taken as $0.6k_{\max}$ with $k_{\max}=4096$
being the maximal value of $k$. Coefficient $0.6$ in front of $k_{\max}$
provide to prevent aliasing.

Appropriate changes of boundaries $k=b$ and $k=k_{c}$ for zeroth values of
both $\Gamma_{k}$ and $\gamma_{k}$ allowed us to get a maximally possible
inertial interval for a given spatial resolution. Note that, in the interval
$b<k<k_{c}$, Eq. (2) transforms into the Euler equation. As it was shown
analytically and confirmed by numerical experiments [10, 13, 14, 15] for
freely decay turbulence there exists a tendency for breaking of the di-
vorticity lines. This is consequence of the representation analogous to that
for the 3D Euler equation, i.e. the vortex line representation [25, 26], and
thereby the compressible character of the di-vorticity lines. Just this
compressibility is a reason for appearance of sharp vorticity gradients in the
form of quasi-shocks and respectively jets in the $k$-space.

Equation (2) was solved numerically using pseudo-spectral Fourier method,
while integration on time was performed with the help of hybrid Runge-Kutta /
Crank-Nicholson third-order scheme. Derivatives was calculated in the spectral
space, whereas the nonlinear terms were determined on a computational grid in
the physical space. Convective term was approximated explicitly, and for
linear terms ($F$ and $G$) we used implicit scheme. The transformations from
the physical to the spectral space and back were performed through the Fast
Fourier Transform (FFT). Spatial resolution was $8192\times 8192$ points.
Simulations were performed (with the use of the NVIDIA CUDA technology) at the
Computer Center of the Novosibirsk State University.

Initial conditions were chosen familiar to those used in our previous paper
[14] as random sets of vortices with zero mean vorticity. The initial spectrum
was concentrated in the region comparable with the injection scale $k_{c}$. We
verified that variations in the initial conditions qualitatively did not
change behavior of the system and its statistical characteristics.

##  III Numerical results

First, we determined the parameters for pumping rate $\Gamma$ when turbulence
reached its stationary state. These are $A=0.004$, $a=3$, $b=6$, and
$\nu=1.5$. Choice of the first three parameters provided sufficiently rapid
transition to the stationary state for $k\leq 3$ and combination with the
fourth one to approaching stationary asymptotics for the mean (average over
angles) enstrophy flux.

Figure 1: Time evolution of total energy $E$ and total enstrophy $H$.

Figure 2:  Energy spectrum $E(k)$ at different instants of time.

Fig. 1 for the given parameters $A$, $a$, $b$, and $\nu$ show temporal
behavior of the total energy and the enstrophy. As seen total energy $E$ tends
to the constant value (stationary state) at earlier time than the total
enstrophy does. In our numerical experiments we found also how the energy
spectrum and both the vorticity and di-vorticity spatial distributions evolve
with time.

Fig. 2 shows the angularly averaged spectrum of turbulence $E(k)$ at different
instants of time. At times $t=220$ one can see the steady (compare with Fig.
1) spectrum with the Kraichnan behavior $E(k)\sim k^{-3}$ which is seen almost
on three decades. At the steady state ($t\gtrsim 200$) by calculating the
enstrophy flux $\eta$ as the integral $\int\gamma(k)|\Omega({\bf k})|^{2}d{\bf
k}$ we verified that the steady energy spectrum $E(k)$ with a high accuracy
coincides with the Kraichnan spectrum $E(k)=C_{K}\eta^{2/3}k^{-3}$ where
$C_{K}\simeq 1.3$ the Kraichnan constant, in correspondence with the previous
simulations [16]-[19].

As was shown [10, 13, 14], for the decay turbulence such a spectrum is formed
owing to the sharp vorticity gradients appearing due to breaking of di-
vorticity lines. Results presented in Figs. 3-5 also show a significant role
of this process on the formation of the Kraichnan spectrum. In Fig. 3 one can
see the spatial distributions of $\Omega$ at $t=100$, before reaching the
steady state, and at $t=220$, after its reaching.

Figure 3: Vorticity distributions at $t=100,220$.

Figure 4: Distribution of $|\mathbf{B}|$ at $t=100,220$.

These are typical distributions for $\Omega$, but they are less informative
than the corresponding distributions of di-vorticity $|\mathbf{B(r},t)|$ which
are presented in Fig. 4 (for the same time instants as in Fig. 3).

As is seen from these figures the di-vorticity value concentrates along lines
which form a whole network, between these lines $|\mathbf{B}|$ is much less.
This difference is most clearly seen in the dependence of $|\mathbf{B}|$ along
projection $y=0.5$ (see Fig. 5) where maxima of the di-vorticity exceed its
minimal values for 1-2 orders of magnitude.

Figure 5:  Distribution of $|\mathbf{B}|$ along line $y=0.5$ at $t=100,220$.

These maxima define positions of the vorticity quasi-shocks which in the
$k$-space, due to jets, are responsible for anisotropy in 2D energy
distribution $\epsilon(k_{x},k_{y})$. Fig. 6 shows 2D compensated spectrum
$k^{4}\epsilon(k_{x},k_{y})$ for the steady state where one can see a set of
jets with both strong and weak mutual overlapping.

Figure 6:  2D compensated spectrum $k^{4}\epsilon(k_{x},k_{y})$ at $t=220$.

Along each such jets $\epsilon\sim k^{-4}$ which after angular average leads
with a high accuracy to the Kraichnan spectrum.

However, anisotropy, due to quasi-shocks, plays a more significant role for
the higher moments of the velocity field than for the spectrum. Fig. 7 shows
the dependencies of the third-order velocity structure function
$S_{3}^{(L)}(\mathbf{R})=\langle(\delta v_{\|})^{3}\rangle$, depending on $R$
for various values of angles, where
$\mathbf{R}=\mathbf{r}-\mathbf{r}^{\prime}$ and $\delta v_{\|}$ the projection
of the velocity difference onto the vector $\mathbf{R}$. At each angle
$S_{3}^{(L)}(\mathbf{R})$ is close to the cubic parabola, i.e. $\propto
R^{3}$, with a linear dependence relative $\eta$. However, the average of the
third-order velocity structure function over angles only gives a significant
difference with the constant $C_{3,isotr}=1/8$ for the isotropic turbulence.
However, the angular averaging constant $C_{3}$ undergoes temporal
fluctuations: its maximal value sometimes reaches 5 (for this ¡¡steady¿¿
state!). Average over time within window $210\geq t\leq 465$ with
characteristic period $\simeq 17$ gives better correspondence:
$\bar{C}_{3}\simeq 2.4C_{3,isotr}$.

Figure 7:  Dependence of $S_{3}^{(L)}$ as function of $R$ at different angles.

##  IV Conclusion.

The main resume of this work is the fact that for the 2D turbulence, in the
presence of pumping and damping, quasi-shocks, originated because of
compressibility of di-vorticity lines, play very essential role in the direct
cascade. Quasi-shocks are key factors which define 2D turbulence anisotropy,
in particular, anisotropy of both spectrum and structure functions.
Surprisingly that averaging over angles of 2D energy distribution
$\epsilon(k_{x},k_{y})$ gives with a high accuracy the Kraichnan spectrum.
This result coincides with results of the previous numerical experiments
[16]-[19]. Note, the same averaging for the third-order velocity structure
function yields a value for constant $\bar{C_{3}}$ different from a pure
isotopic case (see, e.g., review [24] and references therein), in spite of the
correct dependence of $S_{3}^{(L)}$ on both enstrophy flux $\eta$ and
separation $R$. This is an open question which we are going to study in our
future work.

The authors thank A.N. Kudryavtsev and G.E. Falkovich for useful discussions.
This work was supported by the RSF (Grant No. 14-22-00174).

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$\frac{\absolutevalue{z+\lambda}}{\absolutevalue{z}}\leq
1+\frac{\absolutevalue{\lambda}}{\absolutevalue{z}}\leq 3$ and
$\absolutevalue{1-\eta
z}^{\frac{1}{\eta\lambda}}\leq\absolutevalue{1+\frac{1}{2}\eta\lambda}^{\frac{1}{\eta\lambda}}\leq\sqrt{e}$.
Hence, condition (C1) also holds for $\varphi_{\lambda}^{\mathrm{GD}}$.

As for condition (C2), we need to demonstrate that

$\sup_{z\in
D_{\lambda},\lambda\in(0,\varepsilon)}\absolutevalue{\frac{z+\lambda}{\lambda}\psi_{\lambda}^{\mathrm{GD}}(z)}=\sup_{z\in
D_{\lambda},\lambda\in(0,\varepsilon)}\absolutevalue{\frac{z+\lambda}{\lambda}(1-\eta
z)^{\frac{1}{\eta\lambda}}}<\infty$

for some constant $\varepsilon>0$. In fact, when $\absolutevalue{1-\eta z}\geq
1$,

$\absolutevalue{\frac{z+\lambda}{\lambda}(1-\eta
z)^{\frac{1}{\eta\lambda}}}\leq\frac{3}{2}\absolutevalue{1+\frac{1}{2}\eta\lambda}^{\frac{1}{\eta\lambda}}\leq\frac{3}{2}\sqrt{e}$

for all $z\in D_{\lambda}$. When $\absolutevalue{1-\eta z}<1$,

$\absolutevalue{\frac{z+\lambda}{\lambda}(1-\eta
z)^{\frac{1}{\eta\lambda}}}\leq 1+\absolutevalue{\frac{z}{\lambda}(1-\eta
z)^{\frac{1}{\eta\lambda}}}\leq 1+\absolutevalue{z(1-\eta
z)^{\frac{1}{\eta}}}\leq\frac{3}{2}+2\kappa^{2}$

for all $z\in D_{\lambda}$ and $\lambda\in(0,1)$. ∎

###  B.3 Source condition on the regression function

###### Proof of Example 3.1.

With Assumption 1 and $\sum_{k=1}^{m}d_{k}\asymp m^{\gamma}$, we have
$\mu_{m}\asymp m^{-\gamma\beta}$. Select $s=\frac{p}{\beta}$. Then, for all
$t<s$,

$\sum_{m=1}^{\infty}\frac{\bar{f}_{m}^{2}}{\mu_{m}^{t}}=O\left(\sum_{m=1}^{\infty}m^{-(\gamma
p+1)}m^{t\gamma\beta}\right)=O\left(\sum_{m=1}^{\infty}m^{-1-\gamma(p-\beta
t)}\right)<\infty.$

Hence, $f^{*}\in[\mathcal{H}]^{t}$ for all $t<s$.

On the other hand,

$\displaystyle\sum_{m:\mu_{m}\leq\lambda}\bar{f}_{m}^{2}\geq c\sum_{m\geq
C\lambda^{-(\gamma\beta)^{-1}}}m^{-\gamma
p-1}=\Omega(\lambda^{\frac{p}{\beta}})=\Omega(\lambda^{s}).$

Finally, if $f_{j}\asymp j^{-(p+1)/2}$, we have

$\bar{f}_{m}^{2}=\sum_{j=D_{m-1}}^{D_{m}}f_{j}^{2}\asymp\int_{(m-1)^{\gamma}}^{m^{\gamma}}x^{-p-1}dx\asymp
m^{-(\gamma p+1)}$

where $D_{m}\coloneqq\sum_{k=1}^{m}d_{k}\asymp m^{\gamma}$. ∎

###### Proof of Example 3.2.

Let $s=\frac{p}{q\beta}$. Then, with Assumption 1, for all $t<s$,

$\sum_{j=1}^{\infty}\lambda_{j}^{-t}\absolutevalue{f_{j}}^{2}=\sum_{l=1}^{\infty}\lambda_{j(l)}^{-t}\absolutevalue{f_{j(l)}}^{2}\leq
C\sum_{l=1}^{\infty}l^{tq\beta}l^{-(p+1)}\leq
C\sum_{l=1}^{\infty}l^{-1+(p-tq\beta)},$

where we recall that $j(l)\asymp l^{q}$ and $\absolutevalue{f_{j(l)}}\asymp
l^{-(p+1)/2}$, so $f^{*}\in[\mathcal{H}]^{t}$ for all $t<s$.

On the other hand,

$\displaystyle\sum_{j:\lambda_{j}\leq\lambda}\absolutevalue{f_{j}}^{2}=\sum_{l:\lambda_{j(l)}\leq\lambda}\absolutevalue{f_{j(l)}}^{2}\geq
c\sum_{l\geq
C\lambda^{-1/(q\beta)}}l^{-(p+1)}=\Omega(\lambda^{\frac{p}{q\beta}})=\Omega(\lambda^{s}).$

∎

###  B.4 Proof of Lemma 4.19

The proof follows the same idea as the proof of Lemma A.5 and Lemma A.10 in Li
et al. (2023), but we establish the results for spectral algorithms and also
improve the estimates by Lemma 4.2 using the regular RKHS property. The proof
will be divided into two cases: $t>1/\beta$ and $t\leq 1/\beta$. For the first
case, the regression function $f^{*}$ is $L^{\infty}$-bounded so that we can
directly apply the Bernstein inequality. For the second case, we have to apply
a truncation technique.

##### Proof for the case $t>1/\beta$.

By the inclusion relation of $[\mathcal{H}]^{t}$, we may further restrict that
$t\leq 2\tau$. We will use the Bernstein inequality in Lemma A.7. Let us
denote

$\displaystyle\xi(x)=T_{\lambda}^{\frac{1}{2}}(K_{x}f^{*}(x)-T_{x}f^{*}_{\lambda}).$
(89)

Then, we have

$\displaystyle
T_{\lambda}^{\frac{1}{2}}\left[\left(\tilde{g}_{X}-T_{X}f_{\lambda}^{*}\right)-\left(g-Tf_{\lambda}^{*}\right)\right]=\frac{1}{n}\sum_{i=1}^{n}\xi(x_{i})-\mathbb{E}_{x\sim\mu}\xi(x).$

Moreover, we have

$\displaystyle\mathbb{E}\norm{\xi(x)}_{\mathcal{H}}^{m}$
$\displaystyle=\mathbb{E}\norm{T_{\lambda}^{\frac{1}{2}}K_{x}(f^{*}(x)-f^{*}_{\lambda}(x))}_{\mathcal{H}}^{m}$
$\displaystyle\leq\mathbb{E}\left[\norm{T_{\lambda}^{\frac{1}{2}}k_{x}}_{\mathcal{H}}^{m}\cdot\mathbb{E}\big{(}\absolutevalue{f^{*}(x)-f^{*}_{\lambda}(x)}^{m}~{}\big{|}~{}x\big{)}\right].$
(90)

The first term in (B.4) is bounded through (37) and Lemma 4.3:

$\displaystyle\norm{T_{\lambda}^{\frac{1}{2}}k_{x}}_{\mathcal{H}}^{2}\leq
M\mathcal{N}_{1}(\lambda)\leq C\lambda^{-1/\beta}.$

For the second term, since $t>1/\beta$, using the embedding property
Proposition 2.2 and also Lemma 4.4, for $\alpha\in(1/\beta,t)$, we have

$\displaystyle\norm{f^{*}_{\lambda}-f^{*}}_{L^{\infty}}\leq
M_{\alpha}\norm{f^{*}_{\lambda}-f^{*}}_{[\mathcal{H}]^{\alpha}}\leq
M_{\alpha}F_{(t-\alpha)/2}\norm{f^{*}}_{[\mathcal{H}]^{t}}\lambda^{\frac{t-\alpha}{2}}\leq
CM_{\alpha}R\lambda^{\frac{t-\alpha}{2}}$

Moreover, Lemma 4.4 also implies

$\displaystyle\mathbb{E}\absolutevalue{f^{*}_{\lambda}(x)-f^{*}(x)}^{2}=\norm{f^{*}_{\lambda}(x)-f^{*}(x)}_{L^{2}}^{2}\leq
F_{t/2}^{2}\norm{f^{*}}_{[\mathcal{H}]^{t}}^{2}\lambda^{t}\leq
CR^{2}\lambda^{t}.$

Plugging in these estimations in (B.4), we get

(B.4)
$\displaystyle\leq(C\lambda^{-1/(2\beta)})^{m}\cdot\norm{f^{*}-f^{*}}_{L^{\infty}}^{m-2}\cdot\mathbb{E}\absolutevalue{f^{*}_{\lambda}(x)-f^{*}(x)}^{2}$
$\displaystyle\leq(C\lambda^{-1/(2\beta)})^{m}\cdot\left(CM_{\alpha}R\lambda^{\frac{t-\alpha}{2}}\right)^{m-2}\cdot
CR^{2}\lambda^{t}$
$\displaystyle\leq\frac{1}{2}m!\left(CR^{2}\lambda^{t-1/\beta}\right)\left(CM_{\alpha}R\lambda^{\frac{t-\alpha-1/\beta}{2}}\right)^{m-2}$

Consequently, applying Lemma A.7 with

$\displaystyle\sigma^{2}=CR^{2}\lambda^{t-1/\beta},\qquad
L=CM_{\alpha}R\lambda^{\frac{t-\alpha-1/\beta}{2}}.$

yields

$\displaystyle\norm{T_{\lambda}^{\frac{1}{2}}\left[\left(\tilde{g}_{X}-T_{X}f^{*}_{\lambda}\right)-\left(g^{*}-Tf^{*}_{\lambda}\right)\right]}_{\mathcal{H}}\leq
CR\sqrt{\frac{\lambda^{-1/\beta}}{n}}\left(1+M_{\alpha}\sqrt{\frac{\lambda^{-\alpha}}{n}}\right)\lambda^{t/2}.$

Since $\lambda=\Omega(n^{-\theta})$ for some $\theta<\beta$, choosing
$\alpha>1/\beta$ sufficiently close to $1/\beta$ yields the desired result.

##### Proof for the case $t\leq 1/\beta$.

For this case, we have to apply a truncation technique. To bound the extra
error terms caused by truncation, we have to use the following proposition
about the $L^{q}$ embedding of the RKHS (Zhang et al., 2023, Theorem 5).

######  Proposition B.6.

Under Assumption 2, for any $0<s\leq\alpha_{0}$ and $\alpha>\alpha_{0}$, we
have embedding

$\displaystyle[\mathcal{H}]^{s}\hookrightarrow
L^{q}(\mathcal{X},\differential\mu),\quad q=\frac{2\alpha}{\alpha-s}.$ (91)

Now, let us consider
$\Omega_{B}=\left\\{x\in\mathcal{X}:\absolutevalue{f^{*}(x)}\leq BR\right\\}$
and

$\displaystyle\bar{\xi}(x)=\xi(x)\bm{1}_{\Omega_{B}}(x)$

as $\xi$ is defined in (89), where the choice of $B$ will be determined later.
Then,

$\displaystyle\begin{aligned}
\norm{\frac{1}{n}\sum_{i=1}^{n}\xi(x_{i})-\mathbb{E}\xi(x)}_{\mathcal{H}}&\leq\norm{\frac{1}{n}\sum_{i=1}^{n}\bar{\xi}(x_{i})-\mathbb{E}\bar{\xi}(x)}_{\mathcal{H}}+\norm{\frac{1}{n}\sum_{i=1}^{n}\xi(x_{i})\bm{1}_{\Omega_{B}^{\complement}}(x_{i})}_{\mathcal{H}}\\\
&\quad+\norm{\mathbb{E}\xi(x)\bm{1}_{\Omega_{B}^{\complement}}(x)}_{\mathcal{H}}.\end{aligned}$
(92)

For the first term in (92), we can repeat the same argument in the first case
with the extra bound

$\displaystyle\norm{\bm{1}\left\\{x\in\Omega_{r}\right\\}(f^{*}_{\lambda}-f^{*})}_{L^{\infty}}$
$\displaystyle\leq\norm{f^{*}_{\lambda}}_{L^{\infty}}+\norm{\bm{1}\left\\{x\in\Omega_{r}\right\\}f^{*}}_{L^{\infty}}$
$\displaystyle\leq
M_{\alpha}\norm{f^{*}_{\lambda}}_{[\mathcal{H}]^{\alpha}}+BR$
$\displaystyle\leq M_{\alpha}R\lambda^{\frac{t-\alpha}{2}}+BR,$

where we apply Lemma 4.4 in the last inequality. Consequently, we get

$\displaystyle\norm{\frac{1}{n}\sum_{i=1}^{n}\bar{\xi}(x_{i})-\mathbb{E}\bar{\xi}(x)}_{\mathcal{H}}\leq
CR\sqrt{\frac{\lambda^{-1/\beta}}{n}}\left(1+\frac{M_{\alpha}\lambda^{-\alpha/2}+B\lambda^{-t/2}}{\sqrt{n}}\right)\lambda^{t/2},$
(93)

For the second term in (92), Proposition B.6 together with Markov’s inequality
yields

$\displaystyle\mathbb{P}_{\mu}(\Omega_{B}^{\complement})=\mathbb{P}_{x\sim\mu}\left\\{\absolutevalue{f^{*}(x)}>BR\right\\}\leq(BR)^{-q}\norm{f^{*}}_{L^{q}}^{q},$

where $q=\frac{2\alpha}{\alpha-s}$. Then,

$\displaystyle\mathbb{P}\left\\{x_{i}\in\Omega_{B},~{}i=1,\dots,n\right\\}=\left(1-\mathbb{P}_{\mu}(\Omega_{B}^{\complement})\right)^{n}\geq(1-(BR)^{-q}\norm{f^{*}}_{L^{q}}^{q})^{n}.$

Consequently, as long as

$\displaystyle B^{-q}=o\left(\frac{1}{n}\right),\mbox{\quad or
equivalently,\quad}B=\omega\left(n^{1/q}\right),$ (94)

we have $\mathbb{P}\left\\{x_{i}\in\Omega_{B},~{}i=1,\dots,n\right\\}\to 1$
and thus the second term in (92) vanishes with high probability when $n$ is
large enough.

For the third term in (92),

$\displaystyle\norm{\mathbb{E}\xi(x)\bm{1}_{\Omega_{B}^{\complement}}(x)}_{\mathcal{H}}$
$\displaystyle\leq\mathbb{E}\norm{\xi(x)\bm{1}_{\Omega_{B}^{\complement}}(x)}_{\mathcal{H}}$
$\displaystyle=\mathbb{E}\left[\bm{1}_{\Omega_{B}^{\complement}}(x)(f^{*}(x)-f^{*}_{\lambda}(x))\norm{\varphi_{\lambda}^{1/2}(T)k_{x}}_{\mathcal{H}}\right]$
$\displaystyle\stackrel{{\scriptstyle(a)}}{{\leq}}C\lambda^{-1/(2\beta)}\mathbb{E}\left[\bm{1}_{\Omega_{B}^{\complement}}(x)(f^{*}(x)-f^{*}_{\lambda}(x))\right]$
$\displaystyle\stackrel{{\scriptstyle(b)}}{{\leq}}C\lambda^{-1/(2\beta)}\norm{f^{*}-f^{*}_{\lambda}}_{L^{2}}\mathbb{P}(\Omega_{B}^{\complement})^{\frac{1}{2}}$
$\displaystyle\stackrel{{\scriptstyle(c)}}{{\leq}}C\lambda^{-1/(2\beta)}\lambda^{t/2}B^{-q/2}\norm{f^{*}}_{L^{q}}^{q/2}=C\norm{f^{*}}_{L^{q}}^{q/2}\lambda^{t/2}\sqrt{\lambda^{-1/\beta}B^{-q}},$
(95)

where in $(a)$ we apply (37) and in $(b)$ we apply Cauchy-Schwarz inequality.

Finally, since $\theta<\beta$, we can choose
$\alpha\in(\beta^{-1},\theta^{-1})$. Then we have

$\displaystyle 1-\theta/\beta>1-\alpha\theta>0,$

and we can choose some $l$ such that

$\displaystyle\frac{1}{q}=\frac{1}{2}(1-\frac{t}{\alpha})<l<\frac{1-\theta
t}{2}.$

Now we set $B=n^{l}$. Then (94) immediately holds. For the term (93), we have

$\displaystyle\sqrt{\frac{\lambda^{-1/\beta}}{n}}=O\left(n^{-(1-\theta/\beta)/2}\right),\quad\frac{\lambda^{-\alpha/2}}{\sqrt{n}}=O\left(n^{-(1-\alpha\theta)/2}\right),\quad\frac{B\lambda^{-t/2}}{\sqrt{n}}=O\left(n^{-(\frac{1-\theta
t}{2}-l)}\right);$

For the term (95), we have

$\displaystyle\sqrt{\lambda^{-1/\beta}B^{-q}}=O\left(n^{-(ql-\theta/\beta)/2}\right)=O\left(n^{-(1-\theta/\beta)/2}\right).$

Therefore, we prove the desired result since all the exponents are negative.