We thank the referees for their judgements and comments. Our detailed
response is given below and the corresponding changes to the paper are
marked in bold face.

> ----------------------------------------------------------------------
> Report of Referee A -- LM17948/Bhat
> ----------------------------------------------------------------------
> This paper explores a number of simulations of small and large scale
> dynamo, both helical and nonhelical, with and without shear. Based on
> the behavior of the large-scale magnetic field in these, it proposes a
> new dynamo phase, the quasi-kinematic LSD, that occurs before after
> the SSD saturates, but while the large-scale field is "possibly still
> unaffected by the Lorentz force". They then make the claim that this
> alleviates the well-known "quenching" problem in LSD theory, which has
> beleaguered LSD theory for a couple of decades now.

There appears some confusion about "quenching" problem. There are two
issues: The first one (which we will refer to as problem A) is related
to magnetic helicity conservation, and what in literature is called
"catastrophic quenching" of the LSD. We do not claim here that this
"quenching" problem has been alleviated. It may still require resistive
dissipation of small scale field, magnetic helicity fluxes, or, as in
the recent work of Bhat 2021, effects of anisotropy.

The second "quenching" problem (B) is related to whether the faster
growth of the small-scale field to near equipartition levels due to
SSD action, compared to the growth of the large-scale field, leads to
a suppression of the LSD. We present evidence here for the operation
of a quasi-kinematic LSD, which shows that this "quenching" problem is
indeed alleviated. This is an important step in understanding large-scale
dynamos. Indeed, we have said in the concluding paragraph: "In conclusion,
we have demonstrated via direct numerical simulations the presence of a
novel second stage growth of large-scale field (QKLSD regime), one that
occurs between the kinematic stage driven by the SSD and the saturation
stage driven by magnetic helicity decay."

> These claims are strong, and if true, could mark a big advance in
> dynamo theory. However, I do not believe this paper presents
> sufficient evidence to justify such strong claims, and thus cannot
> recommend publication. Perhaps a different journal, with the results
> and figures extended into a longer format that properly explored the
> claims, would make a nice addition to the literature. My main concerns
> are

In the revised text we have now added a line explicitly using the word
quenching (see the last sentence of the 3rd paragraph on page 1) in order
to distinguish this from the problem discussed in the paper. The judgement
of the referee seems to have been based on this misunderstanding, but we
hope that the above clarification helps in justifying publication in PRL.

> - There is no explanation proffered for why previous arguments for
> quenching should not apply in this regime, aside from mention of the
> suppression principle (but then, the plots show cases with and without
> shear). Indeed standard MFT is used to estimate growth rates, while
> the proffered "toy model" isn't really a model, just an interpolation
> between growth rates. More analysis, of e.g. small-scale helicity or
> fluxes should be needed to make such strong claims. - In a similar
> vein, there is no comparison of why this hasn't been seen or
> interpreted before. Other simulations have been run at 512^3 (and
> higher), yet quenching has still been considered a big issue - what's
> the difference here?

Again the confusion seems to be which problem is being alleviated, as
already stated above. While we have not yet given a physical understanding
of how the QKLSD stage arises, we have presented evidence that indeed
it does, and thus problem B referred to above is solved.

Our toy model is used to illustrate that the presence of two growth
rates does not simply lead to a cross-over between two instantaneous
growth rates, and that there is really an extended interval where both
growth rates affect the measured instantaneous growth rate. Note also
that Eq.(1) is still a differential equation and cannot be used as an
interpolation formula. Thus, we kept this word on page 2 of the paper.

> - Quenching is a high-Rm effect, which, in order to apply results to
> real astrophysical systems, must be understood through the scaling of
> growth/saturation with Rm. This paper considers one run with lower Rm,
> which is only shown in the supplementary material, but in order to
> understand this properly, we need the scaling with Rm. Given the
> general noisiness of the QKLSD growth rates, this is insufficient to
> make such a strong claim. It also does not address the amplitude of
> the resulting fields, which is a key part of the issue.

As mentioned in our previous clarifications above, our claim is not
to have solved the catastrophic quenching problem, but a different one
involving competition between small-scale fields and large-scale fields.
This issue is also mentioned in the review by Tobias 2021, section 5.4

> - The saturation of the SSD is often discussed in terms of the
> hierarchical saturation of different scale eddies, literature that is
> not mentioned in the current paper (e.g. references in Rincon review
> 2019, section 3.5.2). Given the forcing scale that is not too far
> below the box in these simulations, along with the shear (effectively
> a very-large-scale eddy) it seems that the observed behavior could be
> something along these lines. This would still be an interesting
> result, as these models have had conflicting agreement with simulation
> results, but it does not fit with the claims and analysis made.

Thanks, we have added the reference mentioned by the referee. Regarding
scale separation, note that the run with Galloway-Proctor forcing (Run
GP600Pm1a, Figure 3), has a scale separation of 8 between the box and
forcing scale, and one still obtains clear evidence for the QKLSD stage.
This simulation with a completely different code, different type of
forcing, twice the scale separation with a similar Rm, reinforces the
conclusion that our results are indeed robust.

Moreover, the alpha-effect implicated in the helical LSD (Moffatt 1978)
is well known not to involve a "local" scale-by scale inverse cascade, but
rather a generation of large-scale fields directly from a smaller scale
helical forcing. We thus do not expect LSDs to have a saturation behavior
same as the SSDs. Further, even though one may wish to think of shear as
a "large-scale" eddy, the magnetic energy does not reach equipartition
with this kinetic energy, but rather with the turbulent energy; the
weak link in the LSD cycle being the strength of random helical motions
(or alpha effect) rather than the strength of the large-scale shear.

> ----------------------------------------------------------------------
> Report of Referee B -- LM17948/Bhat
> ----------------------------------------------------------------------
> 
> The paper provides a numerical study of the applicability of the
> kinematic large-scale dynamo in the presence of the small-scale dynamo
> in the saturated regime. This is an important astrophysical problem
> that is relevant, for instance, to the evolution of the magnetic field
> in spiral galaxies and accretion disks. The authors numerically apply
> helical turbulence with and without shear, and report what they call
> "novel quasi-kinematic large-scale dynamo", that operates while the
> turbulent dynamo gets into the saturation regime.
> As I explain below, the paper faces many problems and cannot be
> published in the PRL in the present form. I would like to give the
> authors a chance to address these issues and edit the paper
> accordingly.

Thanks for this opportunity, we clarify below various issues raised
by referee.

> The first surprising issue is why the authors believe that they deal
> with the “novel” regime of the large-scale dynamo. It has been
> discussed for decades that the magnetic field generated by the
> small-scale dynamo can serve as the seed field of the mean-field
> dynamo (Ruzmaikin et al. 1990). It is definitely not right to say that
> this possibility was overlooked in the earlier research.

There seems to be some confusion here. The idea that the small-scale
dynamo (SSD) generated field can serve as a seed for the large-scale
dynamo (LSD) has indeed been discussed earlier and we are aware of
this. Also in our own simulations, in the kinematic stage the large-scale
field arises as the tail of the SSD eigenfunction. We are not saying
that this aspect is new.

The new result of our work has to do with whether LSD can operate
in the presence of a strong SSD and the finding of a regime where a
'quasi-kinematic' LSD operates even though Lorentz forces have become
important.

As we explain, a potential concern for LSDs is that the SSD-generated
field can typically grow on a faster time-scale compared to that due to
the LSD. Then magnetic fluctuations can come into near equipartition
with the turbulence and, due to Lorentz forces, suppress the LSD much
before the large-scale field has grown sufficiently. We have shown here
explicitly that this problem does not arise.

Moreover, we have found evidence for operation of both dynamos in the
same system, one after another; first SSD operates and the large-scale
field grows together with small-scale fields as the "tail" of the same
eigenfunction. Then the small-scale field saturates and then large-scale
fields continue to grow with a slower growth rate (and oscillations when
shear is present), interestingly as expected from naive mean-field theory
(after incorporating an efficiency factor). These are indeed new findings.

Since the large-scale dynamo operates in a regime where the Lorentz
force due to the small-scale field is already strong enough to saturate
the SSD, we have called this new regime of large-scale dynamo action,
"quasi-kinematic".

> A much more serious problem of the paper is that the theory that the
> authors compare their results with is the mean-field dynamo (see
> Moffatt 2000). This theory is not applicable to highly conducting
> fluid as it violates the constrain provided by the helicity
> conservation (Vishniac & Cho 2001, Vishniac & Shapovalov 2014). It is
> surprising that the authors get the correspondence between the
> physically flawed theory and their simulations (see Fig. 5), but this
> cannot make the theory right. The magnetic helicity is known to be
> well conserved in high-conductivity fluids, e.g., astrophysical
> fluids. Thus, the correspondence, as one possibility, might suggest
> some issue with the conservation of the magnetic helicity by the codes
> that the authors employ. In fact, such sort calculations are only
> instructive if the authors could show that their codes conserve
> helicity properly.

Both the Pencil Code and Dedalus do conserve magnetic helicity. Regarding
the ability of numerical codes to conserve magnetic helicity, we have
added a comment in paragraph 4 on page 1. Further, we do not claim
here that the "quenching" problem, arising from helicity conservation,
has been alleviated. This problem may still exist in our simulations,
whose solution requires magnetic helicity fluxes, or in the recent work
of Bhat 2021, effects of anisotropy.

Note that magnetic helicity equation is used to constrain the nonlinear
saturation of the LSD (Gruzinov & Diamond 94, Blackman & Field 2002,
Blackman & Brandenburg 2002, Subramanian 2002). In this paper, on the
other hand, we are comparing a kinematic mean-field theory with results
from a 'quasi-kinematic' regime which is not yet nonlinear for the fields
at large-scale. Nonetheless we admit that we are applying a kinematic
theory to a system already affected by Lorentz forces (mentioned on page
4). We believe this caveat results in the efficiency factor and thus an
imperfect correspondence between the kinematic theory and simulations.

Regarding whether such a kinematic theory is applicable to large-Rm
regimes, we note that it has been shown by Dittrich et al 1984
(Astron. Nachr., 305, 119-125) using path integral methods that random
helical renovating flows can lead to the kinematic LSD equation with same
set of alpha and turbulent diffusion. Of course (as we ourselves have
emphasized extensively in earlier papers), magnetic helicity conservation
becomes crucially important to quench the LSD during nonlinear stages.

Moreover, even in the direct simulations of supernova-driven turbulence
where one follows large-scale field generation to the nonlinear regime,
turbulent transport coefficients like the alpha-effect have been measured,
and when used in the LSD equations, can reproduce the large-scale field
generation (Gressel et al (MNRAS, 429, 967-972); Bendre et al (2015, 2020)
(Astron. Nachr., 336, 991; MNRAS, 491, 3870-3883)). Thus, concepts from
mean-field theory do appear useful in all these contexts.

> The issue with kinetic helicity is another issue in the paper. It
> is known (Gruzinov & Diamond 1994, Vishniac & Cho 2001) that the
> injection of kinetic helicity cannot violate the magnetic helicity
> conservation and therefore its injection cannot result in sustainable
> magnetic field generation. Maybe it works, but if so, it has nothing
> to do with real astrophysical dynamos as the corresponding time scales
> are absurdly long for astrophysical systems. If the authors claim that
> they see such a transient magnetic field increase, they should compare
> their simulations with the corresponding theoretical expectations.

We agree that kinetic helicity injection does not violate magnetic
helicity conservation; our simulations do conserve magnetic helicity.
"Quenching" implied by such magnetic helicity conservation is expected to
show up when the field becomes strong enough, such that the Lorentz force
can resist the twisting due to the driven helical motions. This indeed
is the reason why the quasi-kinematic LSD begins to saturate. But before
this happens, we are presenting evidence that the SSD already saturates
and the system transits to a stage where large-scale fields continue to
grow due to a QKLSD. Note also that, since QKLSD does not yet require
magnetic helicity dissipation, the timescales involved in this growth are
dynamical timescales (a gamma_mean of 0.04 in Table 1, and correspond to
a growth time of about 10 eddy turn over timescale and are not the long
resistive time scale (which is of order Rm times the eddy turn over time).

In addition, as mentioned earlier, the simulation results for mean field
growth rate and oscillation frequencies during the QKLSD stage are in
reasonable agreement with expectations of the kinematic LSD equations.

> There are statements in the paper that are confusing: p. 1. The
> statement that the non-linear dynamo happens on the resistive
> time-scale that determines the decay of small-scale helicity is
> incorrect. The helicity tends to go to a larger scale and accumulate
> there. One requires Hubble time or more to resistively dissipate
> helicity on such scales (Vishniac & Cho 2001).

What we have said is: "Significant large-scale fields tend to arise,
but on long resistive time scales due to nonlinear growth governed by
the slow resistive decay of small-scale helicity"

What happens nonlinearly at late times after the QKLSD phase, is the
following. The small-scale magnetic helicity H+(t) actually goes to a
constant value at late times as its generation by the EMF is balanced
by the resistive term in the equation for the small-scale helicity.
The same EMF term, now resistively quenched, is also generating
large-scale magnetic helicity of the opposite sign. This leads to the
slow growth of the amplitude of the large-scale helicity H_(t) on
resistive time-scales. In turn implying a growth of the large-scale field
only on resistive time scales. These features are seen quite clearly in
the evolution curves of the small and large-scale helicities from these
simple helically forced dynamo simulations with periodic boundaries. In
fact such a plot is available in Fig. 5 in Supplementary material,
where we have also given a detailed discussion of both magnetic and
current helicity evolution.

Note that what determines the resistive evolution of the large-scale
helicity (and thus slow growth of large-scale field) is the fact that the
EMF term is balanced by the resistive term in the small-scale helicity
equation. The small-scale helicity could have been removed from the system
faster than resistive decay if say helicity fluxes are available (not so
in periodic domains). Thus, the resistive decay of small-scale helicity
is what limits and governs the resistive time-scale for large-scale
dynamo. We hope this clarifies the referee's confusion.

> P. 1. The concern that small-scale dynamo overwhelms the large-scale
> one is not motivated, as the mechanisms of the magnetic field
> generation are very different for the turbulent and large-scale dynamo
> (Xu & Lazarian 2016).

We have motivated the concern in the introduction section of the paper.
In the first paragraph, we have mentioned that the small-scale dynamo
grows fields much more rapidly. This implies that in a system where
both the dynamos operate, the LSD is expected to be overwhelmed by the
faster SSD. So while the mechanisms for growth of magnetic field are
very different between SSD and LSD, they operate simultaneously in such
helically forced turbulent systems.

As we detail the previous results obtained from studies of such systems,
we build up to the concern more explicitly in the second paragraph,
where the second last line says: "However, observations demand a
fast dynamo growing large-scale fields unhindered by rapidly growing
small-scale fields." This concern has been discussed extensively in our
earlier papers referred to in the introduction as references 4 and 5.
This issue is also mentioned in the review by Tobias 2021, section 5.4

As mentioned in our earlier answers above, the large-scale field grows
as the tail of the SSD in the beginning. But emerges explicitly after
SSD saturates in the quasi-kinematic regime. We have now modified the
introduction to make this concern clearer.  (Please see in bold font)

> The authors should explain how they are dealing with the helicity
> constrain within their studies. Only after that, the paper can be
> considered for publication.

As mentioned in an earlier answer, this paper is not related to the
study of the helicity issue of quenching. The helicity constraint becomes
important only in the fully nonlinear regime. Whereas our work is limited
till the 'quasi-kinematic' regime and its discovery. We are yet to assess
the saturation of quasi-kinematic LSD and how is that inter-related with
the constraints brought in by the helicity equation.

To this end, we have added a line in the introduction saying the paper
doesn't deal with helicity issue of quenching. Also, we will be looking
at the quenching issue in our future studies.

We hope we have clarified to the referee that we are addressing a regime
that is not fully nonlinear and thus does not involve a discussion of
the helicity issue. This was also why we did not refer to the extensive
literature on this explicitly.