Dear Maarit, 

Thanks for the update and pointing us to the revised draft on arXiv, which I’ve now had a chance to properly look through. 

Indeed, there are plenty of similarities in the solutions, despite of 
us being at the different ends of the parameter regime compressibility-wise. 
The main difference seems to be that we do not recover equally coherent 
patterns as you in the butterfly diagrams, when magnetic forcing is 
included in addition to the kinetic one. Whether this then means 
that the dynamos operating in our systems are of different nature 
will be difficult to assess (at least with the PC, as incompressible 
simulations are difficult to perform). As per the referee's request,
we also ended up adding magnetically forced full MHD runs to the paper, 
and the butterfly diagrams from those are very similar to our SMHD cases, that is,
with much less coherent large-scale structures than you get.
You may check the revised version of the manuscript in the arXiv now.

I’m actually pretty surprised by these differences you see here, so I wonder if we could do a bit more investigation. In particular, it seems to me that we ran very similar numerical experiments – except, as you say, the compressibility, perhaps a slightly different Rm, and a different Pm – so it’s quite surprising that the results look quite different. You convincingly measure eta_yx<0 in the kinetic one, where we definitely measured eta_yx>0 using the kinematic TFM with NL sims (points in figure 5 from the ApJ paper), so I’m a bit confused here. One possibility is that yours have somewhat higher Re (or that Pm has an effect), since eta_yx does seem to want to turn over at the higher Re in our plot.

In the interests of narrowing this down – both the MSC and the kinematically forced runs – I would be keen to run a few simulations at identical parameters with snoopy (as long as these aren’t too expensive). While I can get most of the parameters from the paper, there were a couple I wasn’t sure about (is the mean density is one?) – so would you mind providing your simulation parameters (just to make sure I don’t stuff it up)? E.g., in the code units: box size, urms, etc.? Then I’ll try and run identical snoopy simulations, to see whether it is an incompressibility thing (which would be surprising), or a difference with parameters. I won’t be able to do any of the TFM stuff, but we can at least easily look at the structures.

Does that sound okay to you?


> As for the magnetic runs, although we of course never tried either 
> the NL TFM or Burger's-like MHD, the pressure force was discussed quite 
> a bit in the JPP article as being a key component of the magnetic 
> shear-current mechanisms (and also the well-known lack of magnetic 
> quenching in unsheared turbulence). The diagrams in figures 2 and 3 
> of our JPP are a bit confusing (it all turned out to be quite complicated...), 
> but a quite general conclusion is that the pressure-induced response 
> is effectively the cause of the MSC effect (correlation between an 
> initial small-scale b perturbation and the pressure response in u causes 
> the EMF to enhance the large-scale field). Perhaps you could add something 
> to this effect to the conclusion (or elsewhere) of your paper? I think 
> it is probably quite an important point for these types of 
> magnetically-induced transport coefficients. If it would be helpful, we 
> could have a chat on zoom soon to discuss some of this? 

We have also looked at this analytically, even though not as thoroughly as you.
Even without the pressure term, there is a contribution to eta_{yx}, 
through the magnetic part of the background turbulence, at least in the 
ideal limit. Have/Could you easily repeat your analysis by suppressing 
the pressure term? It would be interesting to compare our analytical results 
as well. In any case, we have now referred to your analytical work, and
mentioned your conclusion about the pressure term being the main
 
driver of the MSC.

I took a quick look, and it might be a little non-trivial, but shouldn’t be too bad I think. We never did the full calculation, it just came from examining the origin of different terms (as in the toy model from the JPP paper). I’ll get back to you soon on this.

> Were you referring to the discussion of the appendix in the ApJ paper? 
> Because we certainly didn't mean to argue that the shear-stochastic-alpha  
> dynamo (fluctuating alpha + shear) can't provide amplification (e.g., field 
> growth figures 1 and 2 of our ApJ paper is attributed to fluctuations in 
> alpha + shear). The appendix was specifically referring to the Kraichnan-Moffat 
> mechanism, and various variants that have appeared in the literature. We 
> think that this KM mechanism is unlikely to be important in general situations, 
> but its quite different as shown by Axel & Mitra's 2012 paper.

We remain quite confused about your appendix in the ApJ paper, and further
discussion about it might be in order. Since we are confused, it is indeed
better not to refer to your conclusion in this paper, and perhaps come back to it
later. Hence, we merely now highlight that in SMHD, the diagonal components can, indeed, be
larger than the off-diagonal ones, which is different from what was reported,
e.g., by Brandenburg et al., 2008. 

Sure, we’d be happy to explain this, and apologies for any confusion! The basic point was just that there are two completely different mechanisms that people call “stochastic alpha” (or various variants of this). The first is the standard shear mechanism as in your equation 25 – this is strong sometimes and can grow large scale fields, as you find. The second is the Kraichnan-Moffat mechanism, which can nominally work even without shear. This one requires not just larger off-diagonal components, but actually that the correlation of the diagonal components (xx with yy), exceed both the dissipation and the off-diagonal components. In your measurements, the xx is bigger than the yy (figure 3), so they clearly can’t be that well correlated, so this seems unlikely. We have no disagreement that the mechanism of equation 25 could work though. 

Cheers,
Jono



... They [B08] also reported that the diagonal and off--diagonal components of the $\alpha$ tensor were
nearly equal. In the SMHD cases studied here, this is no longer the case, as is shown
in \Fig{fig:histo}, where the diagonal $xx$ component dominates.

> For the discussion in this paragraph of the introduction it would also be 
> nice to mention that the analytic results (from the PRE) agreed with the 
> reasonably strong magnetic SC contribution (unless, of course, you 
> completely disagree with the general approach of magnetic SOCA calculations).

Nishant is still working on the fully compressible analytics, and we 
might come back to you when he is done. Only then we can fully state
whether we agree or not, although our ideal pressureless case seems
to be consistent, at least when comparing with your equations (32) and (35).
As said, we now mention your work and the prediction of relatively strong
magnetic SC contribution with q values found in accretion disks and
galaxies. In the regime what we study here, that is omega=0, your
work seems to suggest only a weak contribution.

We will keep you posted on our progress; we are now testing the fully compressible 
TFM, and looking more into the significance of the pressure gradient term, 
for example.

BW, Maarit et al.