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\shorttitle{Scaling of the Hosking integral}
\shortauthor{H. Zhou, R. Sharma, and A. Brandenburg}

\title{Response to the referees' reports on the manuscript entitled ``Scaling of the Hosking integral in decaying magnetically-dominated turbulence''}

\author{
Hongzhe Zhou\aff{1}\corresp{\email{hongzhe.zhou@su.se}},
Ramkishor Sharma\aff{1,2}
\and
Axel Brandenburg\aff{1,2,3,4}}

\affiliation{
\aff{1}Nordita, KTH Royal Institute of Technology and Stockholm University,
Hannes Alfv\'ens v\"ag 12, SE-10691 Stockholm, Sweden
\aff{2}The Oskar Klein Centre, Department of Astronomy,
Stockholm University, AlbaNova, SE-10691 Stockholm, Sweden
\aff{3}McWilliams Center for Cosmology \& Department of Physics,
Carnegie Mellon University, Pittsburgh, PA 15213, USA
\aff{4}School of Natural Sciences and Medicine, Ilia State University,
3-5 Cholokashvili Avenue, 0194 Tbilisi, Georgia
}

\begin{document}

\maketitle

We thank the referees for their detailed and helpful comments and suggestions.
Based on the reports, we have carefully revised our manuscript, and the edits have been highlighted in the newly submitted version.
Before we address the referees' specific comments,
we wish to explain a few other changes we have made to the manuscript that became necessary.

\begin{enumerate}\itemsep+3mm
\item
Following conversations with Keith Moffatt and Alex Schekochihin, we
have changed the naming of the conserved quantity and have also changed
the title.
To explain this to the reader, we have now added the following comment
in the footnote on page~3:
``We are grateful to Keith Moffatt for alerting us
to the fact that the formerly used term `Saffman helicity invariant'
may be misleading, because the term helicity invariant is reserved
for integrals which are chiral in character.
Moreover, Saffman never considered helicity in his papers.
The term `magnetic helicity density correlation integral' may be more
appropriate but rather clumsy.
Following Schekochihin (2020), we now refer to it as the Hosking
integral.
We use the term integral instead of invariant as long as we are not in
the ideal limit.''

\item
We have corrected the values in the third column
in table~1, which previously were the initial time rather than magnetic energy density.

\item

We have added some discussions in section~4.3 regarding
the spectral slope in the inertial range,
and two new reference lines in Figure 3(a).
The slope becomes slightly closer to $k^{-2}$ at late times, in agreement with previous works.
This also slightly changes the normalization of $I_H$
at late times, and we have added equations~(4.10) and (4.13).
The latter is now also plotted in Figure 3(d), in the dashed curve.
In addition, we have corrected a missing factor of $2\pi$ in the solid curve in Figure 3(d).

\item
In the first paragraph of section~4.4, we have rephrased
to say ``an asymptotic value close to unity'', which is more appropriate.

\item
In figures~4(a) and 7,
we have distinguished the low-
and high-resolutions runs using
dashed and solid curves, respectively.

\item
In the previous version of figure~4(a), 
there were only two data points for run K200D3t
at
small $t$: $t\simeq0.003$ and $t=1$,
which made the curve look rather steep.
We have now added a few data points at $t<1$
to make the curve more continuous.

\item
We have added some discussions regarding the
Hosking cross helicity integral in section~4.4.
Figure~4(c) now plots $p_{AB}$ and $p_C$;
previously they were $p_{AB}$ and $p_H$.
Since $p_H$ has already been shown in figure~4(b),
there is no loss of information.


\end{enumerate}

We now present our detailed response to each of the three referees.
Their comments are reproduced in blue and are followed by our remarks.

\section{Response to the first referee's report}

\noindent
1. \blue{The paper appears to have an internal inconsistency in its definition of $I_H$.}

$\to$ We have now used $\intIH$ for the integral definition [e.g., equation~(2.1)],
and $\cIH$ for its $R\to\infty$ limit in open domains,
or its $\xiM\ll R\ll L_\text{box}$ limit in finite domains.
We have also added a subscript $R$ for $w(\bmk)$ in, e.g., equation~(2.6),
to emphasize that it depends on $R$.

\vspace{3mm}
\noindent
2. \blue{I think it would be beneficial for the authors to add a few
sentences to the Introduction about magnetic reconnection, in order
to motivate the comparison between reconnection-controlled and ideal
decay timescales later in the paper.
In particular, I think it would be useful to provide a brief explanation
of why one might in principle expect the reconnection timescale to
set the decay timescale (i.e., because topological constraints on
the magnetic field prevent ideal relaxation).
It should be noted that the role of magnetic reconnection in
setting the timescale of magnetic decay was first suggested by Zhou
et. al. [Phys. Rev. Res. 1, 012004 (2019) and J. Plasma Phys. 86,
535860401 (2020)] and Bhat et. al. [Mon. Not. R. Astron. Soc. 501, 3074
(2021)] --- these papers should be cited in this discussion.}

$\to$ We have now added a corresponding discussion at the bottom of page~2.

\vspace{3mm}
\noindent
3. \blue{I think it plausible that it results from an initial non-self-similar
evolution during which the $t=0$ magnetic field relaxes locally to
a state that minimizes the Lorentz force as much as possible while
respecting topological constraints.
If the resulting field structure is approximately force free in local
patches, and thus close to maximally helical (locally!), then one should
expect such an increase in $I_H / B^4 \xi_M^5$ from its initial value.
Do the authors consider that their measurement of the kurtosis rules
out this interpretation?}

$\to$ We cannot rule out this possibility, and we have now added a
paragraph in section~4.5 on page 13 to address this point.

\vspace{3mm}
\noindent
\blue{The authors might consider evaluating (4.4)
numerically for the spectra that they measure rather than for the
piecewise spectrum (4.6), and comparing the measured value of $I_H$
to the result of this calculation in order to unambiguously assess the
role of non-Gaussianity.}

$\to$ We did such a calculation and found that non-Gaussianity may
actually be the dominant contribution of deviations in figures~3(c) and (d).
We have added figure~5 and discussions below equation~(4.10) to address this.

\vspace{3mm}
\noindent
4. \blue{The phrasing of the last sentence of the abstract (particularly the 'however') suggests that a reconnection-controlled timescale is contrary to the prediction of HS21 --- a slight rephrasing might be beneficial to resolve this.}

$\to$ We have rephrased this.

\vspace{3mm}
\noindent
5. \blue{In the introduction: ``This may be of relevance for explaining
the inverse cascading in non-helical magnetically dominated decaying
turbulence...".
HS21 show that the inverse transfer {\em must} occur if both (i) $I_H$
conservation and (ii) self-similarity hold.
The present paper confirms both of these for non-helical decay, and
therefore provides direct evidence that $I_H$ conservation is the cause
of the inverse transfer. It might be worth clarifying this point.}

$\to$ We have revised the text to be clearer on this point.

\vspace{3mm}
\noindent
6. \blue{To avoid possible confusion, it may be worth the authors commenting explicitly in the text of Section 4.1 that the run in figure 1 is at low resolution, being designed only to compare the different schemes for computing $I_H$, and thus should not be expected to give good conservation of $I_H$.}

$\to$ We have added some comments in the middle of page~2 on this point.

\vspace{3mm}
\noindent
7. \blue{The parameter $\beta$ is mentioned just below (3.4), but not defined until (3.7).}

$\to$ We have now rephrased the text to avoid introducing $\beta$ at this early point.


\section{Response to the second referee's report}

\noindent
1. \blue{A plot of the helicity correlation $\langle h(x)h(x+r)\rangle$
for the isotropic case would help a lot in the definitions that follow.}

$\to$ We have added figure 2(b) and mentioned it in section~4.2.

\vspace{3mm}
\noindent
2. \blue{In equation~(2.2) the authors use the integral length scale as the typical
correlation length scale of helicity.
This is not  exact as figures 3a and 3b suggest.
A more appropriate definition would be using the helicity spectrum rather
than  the energy spectrum in 2.3.}

$\to$ The helicity correlations are now shown in figure~2(b).
From the inset it is shown that $4\pi R^2C_h(R)$ peaks at approximately $2\xiM$. 
Thus using $\xiM$ as computed from energy spectra is justified to be appropriate.
We have also computed the ratio $\xiM/\xi_h$, the latter being defined as
\beq
\xi_h\equiv \frac{\int k^{-1}\Sp{h}\ \text{d}k}
{\int \Sp{h}\ \text{d}k}.
\eeq
This is now plotted in figure~7 of appendix~A.
The ratio is of order unity and scales roughly as $\sim\ln t$,
which is much slower than the power-law decay of the magnetic energy.

\vspace{3mm}
\noindent
3. \blue{Then the authors proceed to give X different definitions of $I_H$
some of which depend on $R$ and some do not a,d as figure 1 suggests
the different definitions give results that differ by an order of
magnitude! What value is there in $I_H$ if not all definitions converge
to one value for the same plot?}

$\to$ $\cIH$ from different methods have different prefactors because
they have different weight functions, $w_R(\bm k)$.
Those methods do not fix the constant.
What matters is the constancy of the quantity as the turbulence decays.
Indeed, all methods have the same decay exponent, as demonstrated in figure~1(b), justifying that they are equally valid.

\vspace{3mm}
\noindent
\blue{A plot that demonstrates the measurements of $I_H(R)$ based on its
different  definitions is required for a typical run.}

$\to$ We have now made such a plot in figure~1(a) for run~K60D1c.

\vspace{3mm}
\noindent
\blue{It should also be explained which value of $R$ is chosen.
Figure 2 suggest that $R=0.2L \sim 0.4L$ might be an acceptable choice, but how
do they pick?}

$\to$ For figure~1 we have now explained our choice $R=0.115L$ in the caption.
We have also added it in figure~3.
The choice is made such that for all $t$, the chosen point is within
the asymptotic regime of $\intIH$ which is roughly independent of $R$.

\vspace{3mm}
\noindent
\blue{If the different definitions do not give the same value then something
must be wrong in the Hosking and Schekochihin theory there was only
one $I_H$.}

$\to$ Consider the behavior of $\intIH$ when $R$ increases from $0$ to $R\gg\xiM$.
It grows like $\propto R^3$ when $R\ll\xiM$, because the integrated
regime roughly sees a constant magnetic helicity density $h$.
Once $R\gg \xiM$, the boundary of $V_R$ sweeps through both regions with
positive and negative $h$, and hence $\intIH$ stops growing and saturates.
Thus the saturation value of $\intIH$ depends very much on its value
when $R\ll \xiM$, which is sensitive to the shape of $V_R$ and therefore
$w_R(\bmk)$.
As an example, consider a spherical region with radius $R_0$, filled with constant $h$.
Then consider two cases of $V_R$, both centered at the center of the sphere,
but one is spherical and one is cubic.
The magnetic helicity contained in $V_R$ will be different when
the boundary of $V_R$ touches the boundary of the sphere.

Nevertheless, the crucial points are (i) $\intIH$ saturates when $R\gg\xiM$,
(ii) $\cIH$ is conserved in the ideal MHD limit, and (iii) it scales with
the magnetic field strength as $\cIH\propto B^4\xiM^5$.
The absolute value of $\intIH$ is less important in determining the
dynamics of decaying turbulence.
We have now made some corresponding changes in Sect.~4.1.

\vspace{3mm}
\noindent
4. \blue{Section 3.2,3.3. Introducing the basic assumptions here like d $E_M/dt = - E_M^{3/2}/ \xi_M$ (for the Alfven time decay) and so on for the other cases would help a lot the discussion about the exponents here.}

$\to$ We have added some explanations of the derivations in section~3.3.
The relation of the referee has been added in a footnote on page~7.

\vspace{3mm}
\noindent
5. \blue{Section 4.4. Results in this section need to be presented better. For example in fig 4b and 4c the data are so sparse that convey no message and should be removed and 4a should be discussed more.}

$\to$ We think figures 4(b) and (c) are in fact among the most important
results of our work, and we explain our reasons here.

Figure 4(b) plots in circle symbols the time evolution
(increasing marker size) of the decay exponent of $\cIH$,
and compares them with the decay exponent of the magnetic energy.
It shows a clear distinction between the quantities,
namely that $\cIH$ becomes better conserved in the high $\Lu_n$ limit,
whereas the energy doesn't.
Hence figure 4(b) conveys one of the most crucial messages of the current work,
that there is numerical evidence to believe $\cIH$ is indeed an ideal
invariant of MHD.

Figure 4(c) is similar to 4(b), but now it plots 
the decay exponents of the magnetic helicity and
the Hosking cross helicity integral (replacing
$p_H$ in the previous version of the manuscript), the former being
a well-known ideal invariant.
That $p_H$ and $p_{AB}$ share a similar scaling law with $\Lu_n$ further
strengthens the belief that $\cIH$ is a true invariant.

\vspace{3mm}
\noindent
\blue{K60D1bt seems to be constant here while from fig 3d the authors seem to draw a different conclusion in the end of section 4.5.}

$\to$ In figure 3(d), we plot the normalized invariant $\tilde I_H$,
whose time evolution is further subject to our choice of the
time-dependent normalization factor.
That $\tilde I_H$ increases is now concluded to be a result of
the non-Gaussianity, as suggested by the first referee and one of
the new plots, figure~5.
Hence figures~3(d) and 4(a) are not in disagreement.

\vspace{3mm}
\noindent
6. \blue{Section 4.6. I find it hard to conclude something from fig 5 (and I believe so do the authors). 
I believe it would be better if the authors focused on the highest resolution runs. Show plots of 
$E_M$,  $\xi_M$,  $I_H$ as a function of time; then if a power-law range appears to be present one can talk about the exponents, $p,q$.}

$\to$ The power-law exponents $p$ and $q$ here are defined as the
instantaneous ones, equation~(3.6), and their evolution is shown in
figure~6 (it was figure 5 in the previous version)
with increasing marker sizes.
Therefore an overlapping area in the $pq$ diagram would imply a
straight line of the evolution curve on a log-log scale plot.

In figures~6(c) and (d), it is quite clear that the data points
saturate along the line $\beta=3/2$, which is in agreement with
the theoretical expectation that $\cIH$ governs the decay.

\section{Response to the third referee's report}

\noindent
1. \blue{Presumably the simulations are performed without initial
  helicity. This should be stated in the description.}

$\to$ In the last paragraph of the introduction
starting with ``The goal of the present paper is ...,'' we have now
emphasized ``due to magnetic helicity fluctuations, but in the absence
of net magnetic helicity,'' where ``fluctuations'' and ``net'' are written in italics.

\vspace{3mm}
\noindent
2. \blue{On the same issue: It would be instructive to compare the
  non-helical runs with runs with maximal and fractional
  helicity. }

$\to$ To address this point, we have now added at the end of section 4.6 the
paragraph ``To compare with the case of a fully helical turbulent magnetic...''
We have also added a footnote in section~4.1 on this point, saying
``For this reason, partially or fully helical magnetic fields would not result in
a meaningful definition of $\cIH$. In those cases,
the conservation of magnetic helicity determines $\beta$.''

\end{document}
