# Hosking integral in nonhelical Hall cascade

Axel Brandenburg  Isaac Newton Institute for Mathematical Sciences, 20
Clarkson Road, Cambridge CB3 0EH, UK; and Nordita, KTH Royal Institute of
Technology and Stockholm University, Hannes Alfvéns väg 12, SE-10691
Stockholm, Sweden

###### Abstract

The Hosking integral, which characterizes magnetic helicity fluctuations in
subvolumes, is known to govern the decay of magnetically dominated turbulence.
Here we show that, when the evolution of the magnetic field is controlled by
the motion of electrons only, as in neutron star crusts, the decay of the
magnetic field is still controlled by the Hosking integral, but now it has
effectively different dimensions than in ordinary magnetohydrodynamic (MHD)
turbulence. This causes the correlation length to increase with time $t$ like
$t^{4/13}$ instead of $t^{4/9}$ in MHD. The magnetic energy density decreases
like $t^{-10/13}$, which is slower than in MHD, where it decays like
$t^{-10/9}$. These new analytic results agree with earlier numerical
simulations for the nonhelical Hall cascade.

##  1 Introduction

The x-ray emission from neutron stars during the first hundreds of years is
believed to be powered by magnetic dissipation within their outer crusts
(Gourgouliatos et al., 2018). Since the ions are immobile in neutron star
crusts, electric currents are transported by electrons alone (Cho & Lazarian,
2009). Their velocity is $\bm{u}=-\bm{J}/en_{e}$, where
$\bm{J}={\bm{\nabla}}\times\bm{B}/\mu_{0}$ is the current density, $\bm{B}$ is
the magnetic field, $e$ is the elementary charge, $n_{e}$ is the electron
density, and $\mu_{0}$ is the permeability. The evolution of $\bm{B}$ is then
governed by the induction equation where the electromotive force
$\bm{u}\times\bm{B}$ is given by $-\bm{J}\times\bm{B}/en_{e}$. The induction
equation therefore takes the form (Goldreich & Reisenegger, 1992)

$\frac{\partial\bm{B}}{\partial
t}={\bm{\nabla}}\times\left(-\frac{1}{en_{e}}\bm{J}\times\bm{B}-\eta\mu_{0}\bm{J}\right),$
(1)

where $\eta$ is the magnetic diffusivity. The $\bm{J}\times\bm{B}$
nonlinearity in this equation leads to a cascade toward smaller scales—similar
to the turbulent cascade in hydrodynamics turbulence (Goldreich & Reisenegger,
1992). This model is therefore referred to what is called the Hall cascade.
There has been extensive work trying to quantify the amount of dissipation
that occurs (Gourgouliatos et al., 2016, 2020; Gourgouliatos & Hollerbach,
2018; Igoshev et al., 2021; Anzuini et al., 2022). Idealized simulations in
Cartesian geometry resulted in power law scaling for the resistive Joule
dissipation (Brandenburg, 2020, hereafter B20). It depends on the typical
length scale of the turbulence, the electron density, the magnetic field
strength, and the presence or absence of magnetic helicity. Denoting volume
averages by angle brackets, the decay of the magnetic energy density ${\cal
E}=\langle\bm{B}^{2}\rangle/2\mu_{0}$ with time $t$ tends to follow power law
behavior, ${\cal E}\propto t^{-p}$, where the exponent $p$ is smaller than in
magnetohydrodynamic (MHD) turbulence. Here, $\langle...\rangle$ denotes volume
averaging over the spatial coordinates $\bm{x}$. In the helical case, it was
found that $p=2/5$, while for the nonhelical case, B20 reported $p\approx
0.9$. The correlation length of the turbulence, $\xi$, increases with time
like $\xi\propto t^{q}$, where $q=2/5$ in the helical case, i.e., $q=p$, and
$q\approx 0.3$ in the nonhelical case. In the helical case, the exponent $2/5$
was possible to explain on dimensional grounds by noting that the magnetic
field does not correspond to a speed (the Alfvén speed, as in MHD) with
dimensions $\,{\rm m}\,{\rm s}^{-1}$ in SI units, but to a diffusivity with
dimensions $\,{\rm m}^{2}\,{\rm s}^{-1}$.

The decay properties of the nonhelical Hall cascade were not yet theoretically
understood at the time. In the last one to two years, however, significant
progress has been made in describing the decay of magnetically dominated
turbulence, where a new conserved quantity has been identified, which is now
called the Hosking integral (see Zhou et al., 2022, for details regarding the
naming). The purpose of the present paper is to propose the scaling of the
Hall cascade under the assumption that it is governed by the constancy of the
Hosking integral, which now has different dimensions than in MHD.

##  2 Hosking integral and scaling for the Hall cascade

The Hosking integral $I_{\rm H}$ is defined as the asymptotic limit of the
magnetic helicity density correlation integral ${\cal I}_{\rm H}(R)$ for
scales $R$ large compared to the correlation length of the turbulence, $\xi$,
but small compared to the system size $L$. The original work on this integral
is that by Hosking & Schekochihin (2021), who subsequently applied it to the
magnetic field decay in the early universe (Hosking & Schekochihin, 2022); see
also Brandenburg et al. (2015) and Brandenburg & Kahniashvili (2017) for
earlier work were inverse cascading of magnetically dominated nonhelical
turbulence was first reported. The function ${\cal I}_{\rm H}(R)$ is given by

${\cal I}_{\rm H}(R)=\int_{V_{R}}{\rm d}{}^{3}r\,\langle
h(\bm{x})h(\bm{x}+\bm{r})\rangle,$ (2)

where $V_{R}$ is the volume of a ball of radius $R$ and $h=\bm{A}\cdot\bm{B}$
is the magnetic helicity density with $\bm{A}$ being the magnetic vector
potential, so $\bm{B}={\bm{\nabla}}\times\bm{A}$. We recall that
$\langle...\rangle$ denotes averaging over $\bm{x}$. The function ${\cal
I}_{\rm H}(R)$ is thus the integral over the volume $V_{R}=4\pi R^{3}/3$. For
small $R$, it increases proportional to $R^{3}$, but for large $R$, it levels
off. This is the value of $R$, which determines the Hosking integral $I_{\rm
H}\equiv{\cal I}_{\rm H}(R)$. In practice, it is chosen empirically and must
still be small compared with the size of the domain; see Hosking &
Schekochihin (2021) for various examples and Zhou et al. (2022) for a
comparison of different computational techniques for obtaining ${\cal I}_{\rm
H}(R)$.

What matters for the Hall cascade is the fact that the dimensions of $h$ are
$[h]=[B]^{2}[x]$, and therefore the dimensions of ${\cal I}_{\rm H}$ and
$I_{\rm H}$ are

$[I_{\rm H}]=[B]^{4}[x]^{5}.$ (3)

However, as already noted in B20, using $e=1.6\times 10^{-19}\,{\rm A}\,{\rm
s}$, $\mu_{0}=4\pi\times 10^{-7}\,{\rm T}\,{\rm m}\,{\rm A}^{-1}$, and
$n_{e}\approx 2.5\times 10^{40}\,{\rm m}^{-3}$ for neutron star crusts, we
have $en_{e}\mu_{0}\approx 5\times 10^{15}\,{\rm T}\,{\rm s}\,{\rm m}^{-2}$,
and therefore

$\frac{B}{en_{e}\mu_{0}}=\frac{B}{5\times 10^{15}\,{\rm T}}\,\frac{\,{\rm
m}^{2}}{\,{\rm s}},$ (4)

which is why we say $B$ has dimensions of $\,{\rm m}^{2}\,{\rm s}^{-1}$.111 In
MHD, by comparison, the ion density $\rho$ is a relevant quantity. Using
$\rho=10^{3}\,{\rm kg}\,{\rm m}^{-3}$ for solar surface plasmas, and the
identity $1\,{\rm T}=1\,{\rm kg}\,{\rm s}^{-2}\,{\rm A}^{-1}$, we have
$\mu_{0}=4\pi\times 10^{-7}\,{\rm T}^{2}\,{\rm s}^{2}\,{\rm m}\,{\rm
kg}^{-1}$, and therefore $\rho\mu_{0}\approx 3.5\times 10^{-2}\,{\rm T}\,{\rm
s}\,{\rm m}^{-1}$, or $B/\sqrt{\rho_{0}\mu_{0}}=(B/3.5\times 10^{-2}\,{\rm
T})\,\,{\rm m}\,{\rm s}^{-1}$, which is why we say that in MHD, $B$ has
dimensions of $\,{\rm m}\,{\rm s}^{-1}$. Therefore, the dimensions of $I_{\rm
H}$ are

$[I_{\rm H}]=[x]^{13}[\,{\rm s}]^{-4}.$ (5)

Thus, given that $I_{\rm H}={\rm const}{}$ in the limit of small magnetic
diffusivity, a self-similar evolution must imply that all relevant length
scales, and in particular the magnetic correlation length $\xi(t)$, must
increase with time like $\xi\sim t^{4/13}$. Since $4/13\approx 0.31$, this is
indeed close to the behavior $\xi\sim t^{q}$ with $q\approx 0.3$ found in Sec.
3.2 of B20 (their Run B), as already highlighted in the introduction of the
present paper.

To demonstrate that the energy spectra $E(k,t$ at different times are indeed
self-similar, we collapse them on top of each other by plotting them versus
$k\xi(t)$. Here, $\xi(t)={\cal E}^{-1}\int k^{-1}E(k,t)\,{\rm d}{}k$ is the
weighted integral of $k^{-1}$, where the spectra are normalized such that
$\int E(k,t)\,{\rm d}{}k={\cal E}(t)$ is the magnetic energy density. This
ensures that the maxima of $E(k\xi)$ are always approximately near
$k\xi(t)=1$. In addition, we must also compensate for the decay in amplitude
by multiplying the spectra by a time-dependent function, e.g.,
$\xi(t)^{\beta}$, where $\beta$ is a suitable exponent, so that the
compensated spectra all have the same height. In this way, we find a universal
spectral function by plotting

$\left[\xi(t)\right]^{\beta}E\big{(}k\xi(t),t\big{)}\equiv\phi\big{(}k\xi(t)\big{)}.$
(6)

As an example, we show in Figure 1 the compensated spectra for Run B of B20,
which we discuss in more detail below. At this point, we just note that these
were solutions to Eq. (1), where the initial condition consists of a
nonhelical magnetic field with a spectrum $E(k)\propto k^{4}$ for $k\ll
k_{0}$, with $k_{0}$ being the initial peak wavenumber. For $k\gg k_{0}$, we
assume a decaying spectrum, here with $E(k)\propto k^{-5/3}$, although this
particular choice of the exponent was not important. After some time, the
spectral slopes of both subranges change: At small $k$, the spectrum steepens
from $k^{4}$ to $k^{5}$. Beyond the peak, it falls off with a $k^{-7/3}$
inertial range, as was already found by Biskamp et al. (1996).

Figure 1:  Compensated spectra for Run B of B20, which corresponds to Run B1
in the present paper. Here, $\beta=1.7$ has been used as the best empirical
fit parameter.

To determine the theoretically expected value of $\beta$, we invoke the
condition that the compensated spectra be invariant under rescaling, $t\to\tau
t^{\prime}$, $\bm{x}\to\tau^{q}\bm{x}^{\prime}$, where $\tau$ is an arbitrary
scaling factor. We recall that the dimensions of $E(k,t)$ are
$[x]^{5}[t]^{-2}$, so rescaling yields a factor $\tau^{5q-2}$. In addition,
expressing $E^{\prime}$ in terms of its universal spectral function
$\phi(k\xi)$, the factor $\xi^{\beta}$ on the left-hand side of the Eq. (6)
produces a factor $\xi^{-\beta}$ on the right-hand side of Eq. (6), and
therefore, after rescaling, a $\tau^{-\beta q}$ factor, i.e.,

$E(k)\to\tau^{5q-2}\,E^{\prime}(k)\propto\xi(t)^{-\beta}\tau^{-\beta
q}\phi(k\xi).$ (7)

Therefore, $5q-2=-\beta q$ must be satisfied in order that the compensated
spectra remain invariant under rescaling. Thus, $\beta=2/q-5$, as already
found in B20. Inserting now $q=4/13$ yields $\beta=3/2$. The total magnetic
energy density is therefore

${\cal E}=\int\xi^{-\beta}\phi(k\xi)\,{\rm
d}{}k=\xi^{-(\beta+1)}\int\phi(k\xi)\,{\rm d}{}(k\xi)\propto
t^{-(\beta+1)\,q},$ (8)

and since ${\cal E}\propto t^{-p}$, we have222An equivalent, but conceptually
simpler route to the $t^{-10/13}$ decay law, suggested by David Hosking
(private communication), is to use Eq. (1) to argue that the timescale $t_{\rm
dec}$ for magnetic decay is proportional to $\xi^{2}/B$. Since $t_{\rm dec}$
is proportional to $t$ for selfsimilar decay, we get, using
$\xi^{5}B^{4}=[B^{2}(\xi^{2}/B)^{10/13}]^{13/4}=(B^{2}t^{10/13})^{13/4}={\rm
const}{}$, the scaling $B^{2}\propto t^{-10/13}$. The difference between MHD
and Hall cascade is that, in the former, $t_{\rm dec}$ is proportional to
$\xi/B$, but $\propto\xi^{2}/B$ in the latter. $p=(\beta+1)\,q$. Using
$\beta=2/q-5$, we have $p=2\,(1-2q)$; see Eq. (28) of B20. For $q=4/13\approx
0.31$, we have $p=2\,(1-8/13)=10/13\approx 0.78$, which is not quite as close
to the value reported in B20 as that of $q$, but this could be ascribed to the
lack of scale separation and also the magnetic field no longer being strong
enough so that the Lundquist number,333Note that, unlike the case of MHD, in
the present case of Hall cascade, no wavenumber factor enters in the
definition of the Lundquist number.

$\mbox{\rm Lu}=B_{\rm rms}/en_{e}\mu_{0}\eta,$ (9)

is no longer in the asymptotic regime. This also resulted in the empirical
value of $\beta$ being slightly larger than the theoretical one, as we will
see next.

##  3 Comparison with simulations for different diffusivities

In B20, various simulations of the Hall cascade have been presented, including
forced and decaying simulations, helical and nonhelical ones, with constant
and time-varying magnetic diffusivities, with and without stratification, etc.
The main purpose of that work was to understand the dissipative losses that
would lead to resistive heating in the crust of a neutron star. One of those
simulations is particularly relevant for the present paper: his Run B, which
had a relatively strong initial magnetic field, no helicity, large scale-
separation, and a magnetic diffusivity that decreased with time in a power law
fashion, allowing the simulation to retain a higher Lundquist number as the
magnetic field decreases.

In the present paper, we analyze his Run B, which is here called Run B1. It is
actually a new run, because we now have calculated the Hosking integral during
run time. We also compare with another run (Run B2), where we decreased the
magnetic diffusivity by a factor of two. As in B20, $\eta$ is assumed to
decrease with time proportional to $t^{-3/7}$. We kept, however, the same
resolution of $1024^{3}$ mesh points for both runs, but we must keep in mind
that this can lead to artifacts resulting from a poorly resolved diffusive
subrange for Run B2.

Run | Lu | $t_{1}/[t]$ | $t_{2}/[t]$ | $\tilde{\eta}$ | $\tilde{B}_{\rm rms}$ | $\tilde{\epsilon}$ | $p$ | $q$ | $\beta$ | $p_{\rm H}$
---|---|---|---|---|---|---|---|---|---|---
B1 | 650 | 0.2 | 500 | 0.024 | 600 | $3\times 10^{6}$ | $0.8\pm 0.1$ | $0.3\pm 0.1$ | $1.7\pm 0.1$ | 0.16
B2 | 1300 | 3 | 200 | 0.011 | 700 | $6\times 10^{6}$ | $0.78\pm 0.05$ | $0.31\pm 0.05$ | $1.6\pm 0.05$ | 0.11
Table 1:  Summary of runs discussed in this paper.

In Table 1, we compare several characteristic parameters: the start and end
times, $t_{1}$ and $t_{2}$, respectively, of the interval for which averaged
data have been accumulated, nondimensional measures of the magnetic
diffusivity, the magnetic field strength, and the dissipation, $\tilde{\eta}$,
$\tilde{B}_{\rm rms}$, and $\tilde{\epsilon}$, respectively, and the
instantaneous scaling exponents $p$, $q$, and $\beta$. For $\tilde{\eta}$,
$\tilde{B}_{\rm rms}$, and $\tilde{\epsilon}$, we compute the following time-
averaged ratios:

$\tilde{\eta}\equiv\langle t\eta/\xi^{2}\rangle_{t},\quad\tilde{B}_{\rm
rms}\equiv\langle B_{\rm
rms}/(en_{e}\mu_{0}\eta)\rangle_{t},\quad\mbox{and}\quad\tilde{\epsilon}\equiv\langle\epsilon/(e^{2}n_{e}^{2}\mu_{0}\eta^{3}/\xi^{2})\rangle_{t},$
(10)

where $\xi(t)={\cal E}^{-1}\int k^{-1}E(k,t)\,{\rm d}{}k$ is the correlation
length and $\epsilon=\eta\mu_{0}\langle\bm{J}^{2}\rangle$ is the magnetic
dissipation with $\eta=\eta(t)$, as noted above. These were also computed in
B20. Time is given in diffusive units, $[t]=(\eta k_{0}^{2})^{-1}$. In the
runs of series B of B20, the value of $k_{0}$ is 180 times larger than the
lowest wavenumber $k_{1}\equiv 2\pi/L$ of our cubic domain of size $L^{3}$.

Figure 2:  Compensated spectra for Run B2.  Figure 3:  $pq$ diagrams for Runs
B1 (open red symbols) and B2 (closed blue symbols). Larger symbols denote
later times.  Figure 4:  Evolution of $I_{\rm H}(t)$, showing first a slight
increase and then a decline proportional to $t^{-0.16}$ for Run B1 and
proportional to $t^{-0.11}$ for Run B2. The inset shows the evolution of
${\cal I}_{\rm H}(R;t)$ as a function of $R$ for increasing values of $t$
(indicated by the arrow) for Run B2. The abscissae of the main plot and the
inset are normalized by $\eta k_{0}^{2}$ and $k_{0}$, respectively.

It turns out that a lower resistivity is important for obtaining the expected
scaling. We therefore now consider Run B2, where $\mbox{\rm Lu}\approx 1300$.
The result is shown in Figure 2, where we used $\beta=1.6$ as the best fit,
which is still slightly larger than the expected value of 3/2, but it goes in
the right direction. Therefore, we show in Figure 2 the resulting compensated
spectra for Run B2, where the magnetic diffusivity is half that of Run B1 and
Lu is now twice as large a before; see Table 1. We use the Pencil Code (Pencil
Code Collaboration et al., 2021) and, in both cases, we use a resolution of
$1024^{3}$ mesh points.

Another comparison between Runs B1 and B2 is shown in Figure 3, where we
compare their evolution in the $pq$–diagram. While Run B1 clearly evolves
along the $\beta\approx 1.7$ line, Run B2 tends to be closer to the
$\beta\approx 3/2$ line. Note also that both runs settle near the $p=2(1-2q)$
self-similarity line (B20), although we begin to see departures near the end
of the run, which is due to the finite size of the domain.

Finally, we show in Figure 4 the scalings of $I_{\rm H}(t)$, where we see that
the decay exponent $p_{\rm H}\equiv-{\rm d}{}\ln I_{\rm H}/{\rm d}{}\ln t$ is
about $p_{\rm H}\approx 0.16$ for Run B1 and about $0.11$ for Run B2. Earlier
work by Zhou et al. (2022) showed that $p_{\rm H}$ decreases as the Lundquist
number increases, and is, in MHD, around 0.2 for $\mbox{\rm Lu}\approx
10^{3}$, and decreases to $p_{\rm H}\approx 0.01$ for $\mbox{\rm Lu}\approx
4\times 10^{7}$. Such large values can currently only be obtained with
magnetic hyper-diffusivity (Hosking & Schekochihin, 2021; Zhou et al., 2022),
but this has not been attempted in the present work.

MHD | $[B]=[x]/[t]$ | $[\langle h\rangle]=[x]^{3}/[t]^{2}$ |  $q=2/3$ (helical)
---|---|---|---
|  | $[I_{\rm H}]=[x]^{9}/[t]^{4}$ |  $q=4/9$ (nonhel)
| $[E(k,t)]=[x^{3}][t]^{-2}$ | $\beta=2/q-3$ | $\beta=2\cdot 9/4-3=3/2$
|  | $p=2\,(1-q)$ | $p=2\,(1-4/9)=10/9$
Hall | $[B]=[x]^{2}/[t]$ | $[\langle h\rangle]=[x]^{5}/[t]^{2}$ |  $q=2/5$ (helical)
|  | $[I_{\rm H}]=[x]^{13}/[t]^{4}$ |  $q=4/13$ (nonhel)
| $[E(k,t)]=[x^{5}][t]^{-2}$ | $\beta=2/q-5$ | $\beta=2\cdot 13/4-5=3/2$
|  | $p=2\,(1-2q)$ | $p=2\,(1-2\cdot 4/13)=10/13$
Table 2:  Comparison of the scalings for MHD and the Hall cascade.

As already noted by Zhou et al. (2022), there is an initial increase in
$I_{\rm H}(t)$. This is explained by the fact that the magnetic field obeys
Gaussian statistics initially, but not during the later evolution. The inset
shows the $R$ dependence of ${\cal I}_{\rm H}(R;t)$ for different $t$. The
relevant value of $R$ is deemed to be at the location where the local slope of
${\cal I}_{\rm H}(R)$ is minimum at late times.

##  4 Conclusion

The present work has highlighted the power of dimensional arguments, which
were here applied to the case of the Hall cascade without helicity, where the
magnetic field is naturally represented as a quantity with units of a magnetic
diffusivity. The magnetic helicity density has units of $\,{\rm m}^{5}\,{\rm
s}^{-2}$ and the Hosking integral has units of $\,{\rm m}^{13}\,{\rm s}^{-4}$,
which yields $q=4/13$, $\beta=3/2$, and $p=10/13$. Comparing with standard
MHD, where the magnetic field has units of $\,{\rm m}\,{\rm s}^{-1}$, our
exponents $p$ and $q$ are now smaller, but $\beta$ is still the same in both
cases; see Table 2 for a comparison between Hall cascade and MHD. The
empirically determined value of $\beta$ is somewhat larger, but this can be
explained by finite scale separation and small Lundquist numbers.

The decay properties of the Hall cascade are important in understanding
resistive heating in neutron stars while producing at the same time larger
scale magnetic fields at a certain speed through inverse cascading (B20). Such
simulations have already been done in spherical geometry (Gourgouliatos et
al., 2020), but the magnetic field in those simulations did not yet exhibit
clear forward or inverse cascading. This is presumably due to their initial
magnetic field being strongly localized at intermediate length scales. Using
an initial broken power law, as done here, would help producing the expected
forward or inverse cascadings, but this may also require much larger
resolution than what is currently possible. Similarly, of course, the values
of $n_{e}$ and $\eta$ are depth dependent in real neutron stars, but the work
of B20 showed that this did not affect the scaling behavior of the magnetic
decay. Therefore, the importance of the Hosking integral may well carry over
to real neutron stars.

The possible role of reconnection in the Hall cascade remains still an open
question. In the case of MHD, reconnection has been discussed extensively by
Hosking & Schekochihin (2021), making reference to earlier work by Zhou et al.
(2019, 2020) and Bhat et al. (2021). Also in the Hall cascade there is the
possibility that the decay of ${\cal E}$ could be slowed down in an
intermediate range of values of the Lundquist number. As shown in Zhou et al.
(2022), this could lead to a termination line in the $pq$ diagrams that is
different from the $p=2(1-2q)$ self-similarity line discussed here. At the
moment, however, there is no compulsory evidence for deviations of the
solutions from the self-similarity line.

## Acknowledgements

I would like to thank the Isaac Newton Institute for Mathematical Sciences,
Cambridge, for support and hospitality during the programme “Frontiers in
dynamo theory: from the Earth to the stars” where the work on this paper was
undertaken. I am also grateful to the two referees for their comments, which
have led to improvements in the presentation.

## Funding

This work was supported by EPSRC grant no EP/R014604/1 and the Swedish
Research Council (Vetenskapsrådet, 2019-04234). Nordita is sponsored by
Nordforsk. We acknowledge the allocation of computing resources provided by
the Swedish National Allocations Committee at the Center for Parallel
Computers at the Royal Institute of Technology in Stockholm and Linköping.

## Declaration of Interests

The authors report no conflict of interest.

## Data availability statement

The data that support the findings of this study are openly available on
Zenodo at doi:10.5281/zenodo.7357799 (v2022.11.24). All calculations have been
performed with the Pencil Code (Pencil Code Collaboration et al., 2021);
DOI:10.5281/zenodo.3961647.

## Author ORCID

A. Brandenburg, https://orcid.org/0000-0002-7304-021X

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