# Vortex polarization and circulation statistics in isotropic turbulence

L. Moriconi1, R.M. Pereira2, and V.J. Valadão3111Corresponding author:
<EMAIL_ADDRESS>1Instituto de Física, Universidade Federal do
Rio de Janeiro, C.P. 68528, CEP: 21945-970, Rio de Janeiro, RJ, Brazil
2Instituto de Física, Universidade Federal Fluminense, 24210-346 Niterói, RJ,
Brazil  3Dipartimento di Fisica and INFN - Università degli Studi di Torino,
Via Pietro Giuria, 1, 10125 Torino TO, Italy.

###### Abstract

We carry out an in-depth analysis of a recently introduced vortex gas model of
homogeneous and isotropic turbulence. Direct numerical simulations are used to
provide a concrete physical interpretation of one of the model’s constituent
fields: the degree of vortex polarization. Our investigations shed light on
the complexity underlying vortex interactions and reveal, furthermore, that
despite some striking similarities, classical and quantum turbulence exhibit
distinct structural characteristics, even at inertial range scales. Crucially,
these differences arise due to correlations between the polarization and
circulation intensity within vortex clusters.

##  I Introduction

Turbulence, a fascinating and intricate phenomenon, is a natural state of
fluid motion that pervades our world. A deeper understanding of turbulence is
expected to lead to technological advancements in fields such as aircraft
turbulence prediction cat , turbulent combustion combust , and automotive
design auto . A milestone in this endeavor is due to Kolmogorov in 1941 (K41)
k41 , who, with a self-similar approach to the velocity structure function,
made predictions which found experimental evidence decades later obsk41 .
However, it is now widely recognized that the K41 theory does not offer a
definitive representation of turbulence. This limitation arises from
significant non-Gaussian deviations of turbulent fluctuations frisch , often
associated with the tendency exhibited by turbulence to self-organize into
thin elongated vortex structures carrying an expressive amount of kinetic
energy kaneda .

In spite of its central importance in fluid dynamics, velocity circulation was
historically overlooked in the context of turbulence research. It was more
than 50 years after the publication of the K41 theory that the circulation
variable was first explored by Migdal migdal1  in the early 1990s. In the past
decade, a remarkable improvement in hardware and software platforms enabled
the exploration of high Reynolds number direct numerical simulations (DNS) of
the Navier-Stokes equations. This has naturally renewed interest in
circulation and the analysis of its statistical properties iyer1 ; iyer2 ,
which in turn encouraged progress both from the theoretical migdal2  and the
modeling apol1 ; mori1 ; mori2 ; mori3 ; apol2 ; mori4  points of view. The
apparent simplicity of this variable also pushed forward the computational and
experimental analyses of circulation in other systems such as quantum muller1
; muller2  and quasi-two-dimensional turbulence zhu ; muller3 .

A promising phenomenological model of circulation statistics, referred to as
vortex gas model (VGM) apol1 , combines a mathematically formal model of the
cascade nature of turbulence through the Gaussian multiplicative chaos (GMC)
vargas  with the structural view of turbulence, seen as an entangled gas of
vortex filaments. The resulting statistical model not only accurately
reproduces the general statistical behavior of velocity circulation observed
in DNS, but motivated the discovery of new phenomena, as, for instance, a
statistical repulsion between vortex fluid structures at small scales mori3 .
In the context of probability distribution functions (PDFs), VGM gives a
fundamental interpretation of the behavior of extreme far tails through the
breakdown of multifractality mori2 ; mori4 , and offers a straightforward
explanation of the multifractal behavior of circulation in terms of a maximum
vortex packing fraction inside a fixed contour $\mathcal{C}$ where the
circulation is computed thesisvic .

The VGM is described in terms of two statistically independent fields. One
field entails the GMC framework needed to model intermittent energy
fluctuations, akin to the Obukhov-Kolmogorov modeling (OK62) O62 ; K62 , and
its physical role has been scrutinized in mori4 . The other field is self-
similar, targeting the K41 scaling exponents. The present study aims to
validate, from statistical analyses of DNS data, this self-similar modeling
field of the VGM. By adapting the approach of cluster summation in quantum
turbulence muller2  and investigating the spatial correlation function of
detected structures on DNS data, we unveil clear distinctions between
classical and quantum turbulence. Additionally, our findings align with the
GMC model and an analogous $4/5$-law for circulation statistics.

This paper is organized as follows. Sec. II outlines the main ideas of the VGM
and its consequences to the cluster summation which are additionally
strengthened in Sec. III through the analysis of high Reynolds number DNS
data. In Sec. IV we summarize our findings, in contrast to the current
understanding of cluster summation in the context of quantum turbulence.

##  II Vortex Gas Model and Cluster summation

The VGM was first introduced in apol1  and further developed in mori1 ; mori3
; mori4 , where circulation is accounted for at planar domains by the use of
Green’s theorem, as in

$\Gamma_{r}=\int_{\mathcal{D}_{r}}\\!\bm{\omega}\,d^{2}x\,,$ (1)

where $\mathcal{D}_{r}$ is the surface delimited by some contour of typical
linear size $r$ and $\bm{\omega}$ is the vorticity. A sum of dilute, quasi-
homogeneously distributed point vortices composes the statistical framework of
the model. Moreover, at inertial range scales, the statistical properties of
$\Gamma_{r}$ are encoded using two basic constituents, as previously
anticipated,

$\Gamma_{r}=\xi_{r}\int_{\mathcal{D}_{r}}d^{2}x\ \tilde{\Gamma}(\bm{x})\ .\ $
(2)

The variable $\xi_{r}$ is the coarse-grained squared-root dissipation rate,

$\xi_{r}=\frac{1}{\eta_{K}^{2}}\int_{\mathcal{D}_{r}}d^{2}x\sqrt{\frac{\varepsilon(\bm{x})}{\left\langle\varepsilon\right\rangle}}\
,\ $ (3)

modeled as a bounded GMC field, while $\varepsilon(\bm{x})$ is the local
energy dissipation rate and $\eta_{K}$ the usual Kolmogorov scale mori1 ;
mori4 . In addition, $\tilde{\Gamma}(\bm{x})$ is a Gaussian random field with
zero mean and pair correlation function

$\left\langle\tilde{\Gamma}(\bm{x})\tilde{\Gamma}(\bm{y})\right\rangle\sim|\bm{x}-\bm{y}|^{-\alpha}$
(4)

at inertial range scales (properly regularized for small scales
$|\bm{x}-\bm{y}|\ll\eta_{K}$). Using Eqs. (2-4), it is not hard to show that
$\left\langle|\Gamma_{r}|^{p}\right\rangle\sim r^{\lambda_{p}}$ with

$\lambda_{p}=\frac{(4-\alpha)}{2}p+\frac{\mu}{8}p(1-p)\ ,\ $ (5)

where $\mu\approx 0.17$ is the intermittency parameter related to the spatial
decay of the dissipation correlation function tang . The above equation is
only valid for low-order exponents $p\lesssim 6$. For higher-order statistics,
the linearization effect is accounted for by the bound imposed on the GMC
field mori1 ; mori4 .

A deeper understanding of the Eq. (3) was provided by mori4 , where the
authors showed that the number of vortices inside the domain $\mathcal{D}_{r}$
is statistically equivalent to the squared-root coarse-grained dissipation
$\xi_{r}$ for inertial range sized contours. In this sense, the statistical
repulsion among vortices at very small scales is phenomenologically related to
the breakdown of multifractality as pointed out in mori1 .

Note that the $4/5$-law for circulation, as conjectured through extensive
analysis of DNS data in iyer1 , which suggests
$\langle|\Gamma_{r}|^{3}\rangle\sim r^{4}$, does not concur with the naive
implementation of $\alpha=4/3$ in the VGM. This is due to a technical
difference between traditional cascade models and the GMC approach. In the
latter, the scaling exponents of the $p^{\text{th}}$-order moment do not
follow the same expected relation for simple cascade models:
$\langle(\sqrt{\varepsilon_{r}})^{p}\rangle\sim r^{\tau_{p/2}}$ when
$\langle\varepsilon_{r}^{p}\rangle\sim r^{\tau_{p}}$. In the GMC approach, the
evaluation of the coarse-grained dissipation rate $\varepsilon_{r}$ is
implicitly related to the intense local fluctuation of $\varepsilon(\bm{x})$.
Consequently, the only compatible way to use the GMC modeling for circulation
statistics is by setting $\alpha=4/3-\mu/2$, as already pointed out in mori4 .

The procedure of cluster summation consists of characterizing the contribution
of the circulation of each vortex and summing over neighboring vortices. It
was first introduced in the context of quantum turbulence muller2 , where the
velocity circulation is quantized, i.e., $\Gamma_{i}=\pm n\kappa$, with
$\kappa$ being the quantum of circulation barenghi . This procedure does not
fix the circulation contour, being independent of the particularities of the
local vortex distribution (or $\xi_{r}$ in classical turbulence), resulting in
the appearance of a self-similar behavior, ultimately understood as the degree
of polarization of chains of quantum vortices muller2 .

For classical turbulence, such a summation procedure is far more involved than
the quantum case. It can assume basically any value, and there may be not only
sign correlations but correlations among circulation magnitudes of the
vortices. Moreover, local vortex distribution is also supposed to be
correlated to the vortex circulation since, at small scales, there can be
decaying and/or reconnection processes occurring all over the flow. Therefore,
this kind of procedure is expected to show a scaling behavior for a
sufficiently large cluster.

Suppose one has a 2D slice of a 3D turbulent flow, with $N$ vortices at
positions $\bm{x}_{i}$ each of which carries a circulation $\Gamma_{i}$. We
define two types of cluster summations muller2 , labeled by a function $f$, as

$P^{(f)}_{i}(n)=\sum_{j=1}^{n}f\\!\left(\Gamma^{(i)}_{j}\right)\ ,\ $ (6)

where $n\leq N$ is the size of the cluster, spanning an area with a typical
size of $r\sim n^{1/2}$ due to the homogeneous behavior of the GMC scaling
exponents for $p=1$. To generate the cluster summation series, we choose a
specific vortex, for instance, the $i^{\text{th}}$ vortex. We label the
circulation of each vortex as $\Gamma^{(i)}_{j}$ and sort them based on their
relative distance $r_{ij}=|\bm{x}_{i}-\bm{x}_{j}|$ from the selected vortex.
Finally, we perform a summation over the increasing $j$ label. The ensemble
average is then defined by starting the series at different points
$\bm{x}_{i}$,

$\left\langle P_{n}\right\rangle=\frac{1}{N}\sum_{i=1}^{N}P^{(f)}_{i}(n)\ .\ $
(7)

As for the function $f:\mathbb{R}\rightarrow\mathbb{R}$, first we refer to a
“binarized” summation when only the sign of the vortices is considered, i.e.,
$f(x)=\text{sign}(x)$. This basically measures the degree of polarization of
the vortex cluster, exactly as in the case of quantum turbulence muller2 .
Lastly, we refer to the “continuum” cluster summation when $f(x)=x$. Note
that, however, the choice of this function instead of $|x|$ does not totally
exclude the polarization effects, and, the scaling exponents of these two
different series must be connected in some way. For both cases, we expect

$\left\langle\left|P_{n}\right|^{p}\right\rangle\propto n^{\beta p}\ .\ $ (8)

If $\beta=2/3$ for $f(x)=x$, K41 scaling is achieved (since $n\sim r^{2}$).
This result of disentangled scaling was observed in quantum turbulence to have
a K41-like property as shown by muller2 , but, as mentioned, the classical
case can be more involved, and a clear exploration of this scaling follows.

Let us assume the field $\tilde{\Gamma}(x)$ is the only responsible for the
scaling of the cluster summation in the VGM. Then, by the use of Eqs. (5) and
(8), one expects for the second order moment«

$4\beta=4-\alpha\rightarrow\beta=\frac{2}{3}+\frac{\mu}{8}\approx 0.688\ ,\ $
(9)

so that the correction introduced in the Gaussian field $\tilde{\Gamma}(x)$ to
reproduce the $4/5$-law directly influences cluster summation, i.e., the
partial polarization of the vortices muller2 . At this point, we note an
explicit phenomenological splitting between classical and quantum turbulence.
Although the correction to $\alpha$ makes the scaling exponents of Eq. (5)
numerically indistinguishable from those of circulation in the quantum case
iyer1 ; muller1 , it has completely different phenomenological bases. In the
case of quantum systems, the circulation can be thought of as the product of
partially polarized vortices (following Eq. (7) with $\beta=2/3$) by the local
distribution of vortices which is, to first order, related to
$\varepsilon_{r}$ as noted by muller2 . In the case of classical turbulence,
mori4  noticed that the vortex density is more accurately described by
$(\sqrt{\varepsilon})_{r}$, then to $\xi_{r}$. In this sense, an efficient
analysis of turbulent data should elucidate the structural differences between
these two systems.

##  III Data analysis

The open-access database used in this work is maintained by the Johns Hopkins
University (JHTDB) JHTD ; JHTD2 . We extracted 2D slices from a 3D, fully de-
aliased Navier-Stokes DNS of fully developed homogeneous and isotropic
turbulence, performed in a tri-periodic box of linear size $2\pi$ divided into
$N=4096$ collocation points. The Taylor-based Reynolds number is about
$R_{\lambda}\approx 610$ and the integral and Kolmogorov length scales are,
respectively, $L=1.3916$ and $\eta_{K}=1.3844\times 10^{-3}$.

The vortex identification method we choose is the standard swirling strength
criterion (SSC) which selects connected spots where $|\text{Im}(\lambda)|\geq
0.125\sigma_{\lambda}$, with $\lambda(\bm{x})$ being any of the complex
eigenvalues of the 2D velocity gradient tensor at position $\bm{x}$
($\lambda=0$ if they are all real), and $\sigma^{2}_{\lambda}$ its variance
zhou ; chakraborty .

The circulation of each vortex structure can be estimated as
$\Gamma_{i}=\omega_{i}^{max}A_{i}$, where $\omega^{max}$ is the maximum
vorticity inside the detected spot and $A_{i}$ its area. This procedure
usually overestimates circulation by discarding the effects of particular
spatial decay of vorticity inside the detected area. Nevertheless, it has been
shown to be a good approximation for the evaluation of circulation at large
contours, provided it is not too intense (see the supplemental material of
mori3 ).

###  III.1 Cluster Summation

Fig. 1 shows the typical realizations of the processes defined by Eq. (6) as
functions of the cluster size $n$. Despite showing similar large-scale
behavior, small-scale peculiarities are noticeably different between the
binarized and continuum cases. The effects of intensity correlations are
markedly seen by taking a closer look at, for instance, the shaded region in
Fig. 1: one notes that the binarized polarization oscillates around a mean
value, meanwhile, the continuum summation has a clear tendency to decrease.
This fact indicates, in this particular case, a strong correlation among
vortices with high negative intensity.

Figure 1: Typical cluster summation as functions of the cluster size for both
binarized (darker) and continuum (lighter) cases, for the same
$\Gamma^{(i)}_{j}$ realization. Each realization is normalized by its own
standard deviation to ease comparison. The shaded area highlights different
properties of the summation methods.

In Fig. 2, we show the statistical moments of cluster summation in both the
binarized and continuum cases, in the spirit of Eq. (8). The binarized
processes show a distinct scaling in the whole cluster size range studied
$(n\leq 10^{4})$, resulting in a general scaling exponent given by Eq. (8)
with $\beta=0.56\pm 0.01$. This means that the tangle of point vortices in
classical turbulence is less polarized than in quantum turbulence
($\beta=2/3$). In addition, not only the polarization of the vortices, but
their amplitude correlations are important for the K41 scaling, so that the
full scaling of the continuum process can be more fundamental than the sole
polarization of the vortices.

Figure 2: Statistical moments of cluster summation for the binarized process
(upper panel) and the continuum process (lower panel) as functions of the
cluster size $n$. Statistical moments are shown from $p=1$ (lighter colors) to
$p=8$ (darker colors).

In striking contrast to its binarized version, the continuum cluster summation
presents no clear scaling in the whole cluster size range. However, for very
large clusters one can identify a scaling regime depending on the moment
order, such that the higher the moment order, the shorter the scaling range.
The determination of the power law exponent in the large cluster scaling range
is difficult because of the substantial contamination due to the slow
crossover between small and large-scale behaviors. Moreover, a range of about
$10^{4}$ vortices has, in general, a spanning area comparable to integral
length scales, so the statistics can start to be affected by the energy
pumping mechanism, again, affecting the scaling region by further shortening
it, now from the right side of Fig. 2.

###  III.2 Small Cluster Regularization

One can try to understand the scaling range by noting that the behavior of the
cluster summation at very small contours is affected by the regularization of
$\tilde{\Gamma}(\bm{x}\rightarrow 0)$. Indeed, this was the leading mechanism
used in apol1  for the saturation of the kurtosis for $r\sim\eta_{K}$.

To further illustrate this point, we assume that, within the VGM framework,
the cluster summation is solely determined by the contribution of
$\tilde{\Gamma}(\bm{x})$, viz.,

$S_{n}=\sum_{i=1}^{n}\tilde{\Gamma}(\bm{x}_{i})\ .\ $ (10)

Writing the variance of $S_{n}$, one can split the contributions into

$\left\langle
S_{n}^{2}\right\rangle=\sum_{i=1}^{n}\left\langle\tilde{\Gamma}^{2}(\bm{x}_{i})\right\rangle+2\sum_{i=1}^{n}\sum_{j=i+1}^{n}\left\langle\tilde{\Gamma}(\bm{x}_{i})\tilde{\Gamma}(\bm{x}_{j})\right\rangle\
.\ $ (11)

Assuming now that the $\tilde{\Gamma}$ field is regularized, i.e.,
$\langle\tilde{\Gamma}(\bm{x}_{i})\tilde{\Gamma}(\bm{x}_{j})\rangle=\Gamma_{0}^{2}$
for $i=j$ and
$\langle\tilde{\Gamma}(\bm{x}_{i})\tilde{\Gamma}(\bm{x}_{j})\rangle=\Gamma_{0}^{2}|\bm{x}_{i}-\bm{x}_{j}|^{-\alpha}$
for $i\neq j$, the cluster summation’s variance reads

$\left\langle
S_{n}^{2}\right\rangle=\Gamma_{0}^{2}\left(n+\frac{4}{2-\alpha}(\pi\bar{\sigma})^{\alpha/2}n^{(4-\alpha)/2}\right)\
,\ $ (12)

where $\bar{\sigma}=N/L^{2}$ is the mean vortex density, whose value, obtained
numerically, is $\bar{\sigma}\approx 3\times 10^{-3}$ in lattice units. By the
usage of Eq. (9), the asymptotically dominant scaling exponent is
$(4-\alpha)/2=2\beta$ as expected. For our purposes, we need a sufficiently
large cluster, say, $n\gg n_{0}$ with

$n_{0}=\left(\frac{4}{2-\alpha}(\pi\bar{\sigma})^{\alpha/2}\right)^{2/(\alpha-2)}\approx
27\ ,\ $ (13)

for the self-similar exponent to dominate, since $(4-\alpha)/2>1$.
Unfortunately, the structures are expected to be less dense for higher
Reynolds numbers, which brings several difficulties for the scaling of cluster
summation. One may expect, for instance, that the cross-over point $n_{p}$
between small-scale and large-scale behavior of the cluster summation
statistical moments should grow with the moment order $p$. This can be grasped
by assuming a Gaussian behavior for $\tilde{\Gamma}(x)$, from where one may
apply Isserlis’ theorem isserl  to arrive at

Figure 3: ESS approach for the cluster summation scaling on the binarized
(dashed) and continuum (solid) cases. Colormaps are as in Fig. 2. Inset: ESS
exponents found on both cases compared to the Gaussian values $p/2$ (dashed).

$\left\langle
S_{n}^{2p}\right\rangle\sim\left(n+\frac{4}{2-\alpha}(\pi\bar{\sigma})^{\alpha/2}n^{(4-\alpha)/2}\right)^{p}\
,\ $ (14)

which reduces to

$\left\langle S_{n}^{2p}\right\rangle\propto n^{2\beta
p}\left[1+\left(\frac{n_{p}}{n}\right)^{(2\beta-1)}+\mathcal{O}\left(\left(\frac{1}{n^{2\beta-1}}\right)^{2}\right)\right]\
,\ $ (15)

with $n_{p}\equiv n_{0}p^{1/(2\beta-1)}$. Considering Eq. (9), one has
$n_{p}=n_{0}p^{\xi}$, with $\xi=12/(4+3\mu)\approx 2.66$. In short words, this
crossover would grow as a power law, due to the presence of the small-scale
regularization. However, the presence of highly correlated small-scale
clusters plays an important role in the crossover and the Gaussian behavior
over the whole range of $n$’s is known to be false. Indeed, the prediction for
the eighth-order moment $n_{8}\approx 6800$ is unrealistic, as seen in Fig. 2,
and hence this analysis is simply indicative.

###  III.3 Scaling Exponents and Asymptotic Gaussianity

An extended self-similarity (ESS) ESS  analysis was carried out for both the
binarized and continuum scalings and results are shown in Fig. 3. In the ESS
approach, one expects that although the scaling of the statistical moments as
a function of $n$ may not be clear-cut, the $p^{\text{th}}$ order moment has
the same small-scale behavior as the $q^{\text{th}}$ order moment, such that,
for the particular case $q=2$,
$\left\langle|P_{n}|^{q}\right\rangle\propto\left\langle|P_{n}|^{2}\right\rangle^{\zeta(p|2)}$.
This is a usual strategy of the ESS approach to extend the scaling region and
get better-resolved exponents. Nonetheless, as shown in Fig. 3, the continuum
process does not exhibit this property due to the collapsing moments for
$n\rightarrow 0$ as emphasized by Eq. (14).

A signature of Gaussian behavior in the binarized summation can be found on
the inset of Fig. 3, where the measured ESS exponents are depicted. Despite a
slight curvature for high-order moments, the Gaussian behavior for the
continuum cluster summation is restored at larger cluster sizes. One may note,
as suggested by Fig. 3, that the scaling of the continuum process tends
asymptotically to the binarized one, such that self-similarity is restored for
sufficiently big clusters.

Figure 4: Continuum cluster summation’s variance is shown in the upper frame,
while its subtracted version is shown in the bottom frame. Dashed lines are
power-law fits while shaded region represents a $2\sigma$-confidence interval
of the data. Figure 5: Standardized circulation PDFs of cluster summation for
various cluster sizes.

As a next step in fully characterizing the self-similar behavior of the
cluster summation, one needs to determine the scaling exponent of one of its
statistical moments, which we take, for simplicity, $p=2$, since $\zeta(p|2)$
was already computed. Furthermore, if one interprets the cluster summation as
Eq. (10) in the VGM, caution must be taken due to an unexpected numerical
coincidence between the power law $n^{5/4}$ and the curve provided by Eq. (12)
for the parameters $\bar{\sigma}$, $\alpha$ and $\Gamma_{0}$ in our data. This
exponent roughly fits the behavior of the variance as it may be inferred from
Fig. 4 (upper panel), where a similar scaling is depicted ($\sim n^{1.23}$).

The lower panel of Fig. 4 shows a properly subtracted scaling, following Eq.
(12). The scaling of the observables $\left(\left\langle
P^{2}_{n}\right\rangle-\left\langle P^{2}_{1}\right\rangle n\right)$ and
$\left\langle P^{2}_{n}\right\rangle$ are compatible with power laws
proportional to $n^{2\beta}$, with $\beta=0.686\pm 0.018$ and $\beta=0.615\pm
0.011$, respectively.

Further evidence supporting the Gaussian behavior of the large-scale continuum
summation is found in Fig. 5. Clusters achieve nearly Gaussian PDFs in the
range $n\approx$ 700–2000 where the exponents were fitted.

###  III.4 Spatial Correlation Function

To further validate the cluster summation methodology, we show in Fig. 6 the
vortex circulation two-point correlation function,
$\langle\tilde{\Gamma}(\bm{x}_{i})\tilde{\Gamma}(\bm{x}_{j})\rangle$, as a
function of the inter-vortex distance $r_{ij}=|\bm{x}_{i}-\bm{x}_{j}|$,
directly measured from DNS data.

Figure 6: Two-point correlation of the detected vortex circulation as a
function of the inter-vortex distance.

To achieve this, we measured individual vortices circulations, then took
pairwise products $\tilde{\Gamma}(\bm{x}_{i})\tilde{\Gamma}(\bm{x}_{j})$ on
each snapshot and computed averages over bins organized according to $r_{ij}$.
Results are considerably robust with regard to bin size, so we choose it as
twice the lattice parameter $dx$ of the DNS data, corresponding to $dx\approx
1.1\eta_{K}$. Error bars on the figure correspond to one standard error of the
mean.

While the high amount of fluctuations does not allow one to distinguish a
precise scaling, the fitted exponent $\alpha=1.31\pm 0.10$ turns out to be
compatible with the VGM prediction (shown as a solid line).

##  IV Discussion

Previous explorations of the building blocks of the VGM have revealed
important structural properties of the general statistical behavior of
circulation statistics in classical fully developed turbulence mori2 ; mori4 .
The so-far unexplored phenomenological field introduced in the VGM,
$\tilde{\Gamma}(\bf x)$, was considered in this work. The current findings
start to signal some fundamental differences in the building blocks of
circulation statistics in classical and quantum turbulence. By exploring the
degree of vortex polarization through the binarized cluster summation method,
we found clearly incompatible scaling exponents for these two systems, where
$\left\langle P^{2}_{n}\right\rangle\sim n^{1.120\pm 0.022}$ stands for the
classical case, while $\left\langle P^{2}_{n}\right\rangle\sim n^{4/3}$ to the
quantum one. Moreover, we found power law exponents to be compatible with the
correction predicted by the GMC approach, introduced in the VGM as
$\langle\tilde{\Gamma}(\bm{x})\tilde{\Gamma}(\bm{y})\rangle\sim n^{2\beta}$
with $\beta=2/3+\mu/8\approx 0.688$, to be in complete agreement with the
measured value $\beta=0.686\pm 0.018$ through the evaluation of the continuum
cluster summation.

The main structural similarities and differences between the modeling
constituents of quantum and classical turbulence can be summarized as: (i)
both systems can be modeled through the product of the partial polarization
(measured by the cluster summation) and the spatial distribution of vortices
(which brings the intermittency of circulation through the energy dissipation
ratio); (ii) vortex distribution in the classical system is related to the
field $\sigma\sim\xi_{r}\sim(\sqrt{\varepsilon})_{r}$, while, to first
approximation, in the quantum system it is related to
$\sigma_{r}\sim\varepsilon_{r}$; (iii) the partial polarization of the quantum
system displays K41 behavior, which is broken in the classical system due to
strong fluctuations in individual vortex intensities, while remaining self-
similar for big cluster sizes; (iv) circulation in both systems displays the
same overall scaling $\lambda_{p}^{classical}=\lambda_{p}^{quantum}$ at
inertial range scales. At this point, the rationale brought by the combination
of points (i), (iii), and (iv) is that the spatial distribution of vortices
must be different when comparing both systems. However, the connection of this
conclusion to point (ii) evokes a deeper understanding of the specific roles
of the underlying dynamics of Navier-Stokes and Gross-Pitaevskii equations on
the spatial distribution of vortices and how this is connected to the local
energy dissipation rate.

For the models based on the standard machinery of random cascades, such as the
mOK62 introduced by muller2 , the correction to the cluster summation is not
present since the scaling exponents of the coarse-grained dissipation exactly
vanish for $p=3$. On the other hand, the best bifractal fit achieved by iyer1
for low order moments $\lambda_{p}\approx(1.367\pm 0.009)p$ was referred to as
$1.4$ scaling, since the exponent $1.4$ collapses the circulation PDFs cores.
In the present work, we associate this value not only to the VGM’s prediction
but to the correction introduced by the GMC field, $\lambda_{p\rightarrow
0}\approx(4/3+3\mu/8)p=1.397p$.

Recently, experimental measures of the circulation scaling exponents in quasi-
two-dimensional turbulence were reported in zhu . Under the inverse energy
cascade regime, circulation shows very similar –not to say equal– scaling
behavior as standard three-dimensional homogeneous and isotropic turbulence.
These different systems, having similar overall statistical properties when
analyzed through the magnifying glass of the circulation variable, call
attention to the possibility of a broader unifying theory for circulation
statistics. The latter motivates the search for an extension of the VGM at
different phenomenological systems such as rotating turbulence or
magnetohydrodynamics, where the existence of a preferential direction breaks
isotropy. In this sense, the VGM setting and the role of a bounded GMC in the
vortex distribution can shed some light on the problems of clustering,
coalescence, and polarization of structures at different turbulent systems.

###### Acknowledgements.

V.J.V. thanks to G. Krstulovic and N. Müller for enlightening discussions
about the cluster summation method. This work was partially supported by CAPES
(grant no 88887.336246/2019-00) and CNPq.

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