Hot Accretion Flow in Two-Dimensional Spherical Coordinates: Considering Pressure Anisotropy and Magnetic Field

Round 1

Reviewer 1 Report

See report attached.

Comments for author File: Comments.pdf

Author Response

Dear referee,

Thank you for your valuable comments and concerns, which greatly help us improve our work. Now we reply to your concerns one by one.

Major concerns:

The current manuscript has significant overlap with the paper by De-Fu Bu et al. titled "Self-Similar Solution of Hot Accretion Flow with Anisotropic Pressure", which has also been published in Universe recently. However, the current manuscript does not refer to it at all, in spite of being a direct extension of that work (which does a 1D self-similar solution). In fact, large portions in the introductory sections are almost identical. There also seems to be significant overlap with the work by Wu et al, A&A 608, A114 (2017), which also the authors do not cite. The work by Wu et al is a 2D simulation having a very similar premise. This then begs the question of the motivation behind doing a self-similar solution. What is novel in the current work and where lies the difference with the previous work(s)?

Reply:

The differences between the present work and Bu et al (2019) lie in aspects of follows:  

We calculate a 2D solution, while Bu et al. (2019) obtained a 1D solution. The 2D solutions have significant advantages compared to 1D solutions. For example, in the 1D solution, outflows are assumed to be present. However, the detailed structures of outflow are unknown. In the 2D solutions, we have the detailed structure of outflows (e.g., the spatial distribution of outflow, velocity, temperature, density as the functions of spatial locations). The detailed structures of outflow(e.g., the opening angle) are important parameters in active galactic nuclei studies(Yuan F., Yoon D., Li Y., et al., 2018, ApJ, 857, 121). Therefore, it is quite important to obtain the 2D solutions of hot accretion flows with anisotropic pressure.

We add them on line 105-118.

As for the second paper Wu et al, A&A 608, A114 (2017). Now we introduced the differences between the present work and theirs.

In Wu et al(2017), the authors only studied accretion flow with extremely weak magnetic field. In their work, the magnetic pressure is at least four orders of magnitude smaller than the gas pressure. However, in reality, in hot accretion flows, the magnetic pressure is just smaller as a factor of 10(King et al. 2017). Thus, it is very necessary to study the hot accretion flow with anisotropic pressure in relatively stronger magnetic fields. In present work, we have studied the hot accretion flow with anisotropic pressure in a much more stronger magnetic field( beta=10,1000).

We added them on line 96-104.

Major concerns:

I could not find the reference Wu et al. 2017 on Pg. 3, line 86. Did the authors mean the reference I mentioned in point (1)?This should be corrected and a proper comparison should be made with this work in terms of the results obtained. Especially because this is one of the few references cited by the current work which tries to study the effect of anisotropic pressure on black hole accretion. It is imperative to establish what is already known and why is it necessary to do a further study.

Reply:

Sorry for our carelessness. It is the paper mentioned in point 1. We corrected it on the reference part on line 369.

As mentioned above, Wu et al. (2017) assumed an extremely weak magnetic field. Consequently, their mass flow of outflow is very weak; the mass inflow flow is 10 percent of the out mass flow. In our paper, when we assume a stronger magnetic field(the magnetic pressure is 1000 percent to the gas pressure), the outflow rate is quite large(the mass flux of outflow is 26 percent of inflow).

We add eq 48, 49 to calculate the ratio of mass inflow to outflow rate. The related discussions are on line 219-222, 224-248, 269-274.

Major concerns:

3. The current work considers a purely toroidal magnetic field, but it is known that a weak vertical field helps in the efficient transport of angular momentum through MRI (magnetorotational instability). How is angular momentum transport taking place in the current work?

Also, how do the authors think will the presence of a weak vertical field modify their results?

The magnetic field has several effects such as the formation of the wind/jet (Yuan et al. 2015), the convective stability of accretion flow (Yuan et al. 2012b and Narayan et al. 2012) and the transfer of angular momentum by MRI process in accretion flow (Balbus & Hawley 1998 )in the dynamical and observational appearance of the discs.

Reply:

Numerical simulations show that the magnetic field can be divided into a large-scale ordered

component and a small-scale turbulent component (e.g., Machida et al. 2000; Hirose et al. 2004;113 Bai & Stone 2013). In Equations (2), (4), and (5), B represents the large-scale component of the magnetic field. In Equation (2), ▽·T represents angular momentum transfer by the turbulent magnetic field.

We have illustrated these on line 125-128 on page 4.

In our work, in order to reduce complexity, we only assume that large scale magnetic field only has a toroidal component. In reality, the vertical component of the magnetic field should also be present. As you said, when the poloidal component of the magnetic field is present, wind/jet can be formed (Yuan et al. 2015). The convective stability of accretion flow (Yuan et al. 2012b and Narayan et al. 2012) can be suppressed and the transfer of angular momentum by MRI process in accretion flow (Balbus & Hawley 1998 ) in the dynamical and observational appearance of the discs.

We added them on line 277-284.

Major concerns:

\alpha_1 and \alpha_2 are not defined physically. Which of the alpha parameters can be

determined observationally? The authors set \alpha_1 = 0.1 in this work from observational

constraints I believe. Then, how do they justify their choices of \alpha_2? Is the final observed \alpha some kind of mean of the 2 \alpha coefficients defined here? If not, how are \alpha_1 and alpha_2 related? Since the whole work tries to basically study the dependence of all variables on \alpha_2, this should be very clearly explained/motivated/justified.

A related question is how large can be \alpha_2, or, physically, how much can anisotropic pressure contribute to an accretion flow realistically?

Reply:

The value of alpha_1=0.1 is from the observation. However, the value of alpha_2 cannot be constrained by observation. Theoretically, there is also no very constrained value of alpha_2. Therefore, we can change the value of alpha_2 to see how the properties of accretion flow change with the anisotropic pressure.

Major concerns:

No details are provided about the numerical method, kind of integrator, resolution etc. used in this work.

Reply:

We use the Runge–Kutta methods to integrate our equations. The resolution is 3* 10^-4.

We added it on line 157.

Major concerns:

The authors should briefly mention the kind of errors encountered when integrating all the way to \theta = 0. Is there some kind of singularity? More importantly, how small is \theta_a? No value is

given in the paper for this limiting angle. Is this "last reachable" angle dependent on resolution or any other parameter, so that one can understand it better?

Reply:

When we integrate our equations from 90 degree to 0 degree, we find at a critical point(theta_a), the results are quite unphysical. For example, at the region of theta <theta_a, the radial velocity can be several orders of magnitude higher than the Keplerian velocity. We can imagine when r is very small (r is several tens of Schwarzschild radius), the radial velocity can be much larger than the speed of light. The results in the region(theta<theta_a) is unphysical. Therefore, we stopped our integration at theta_a. The value of theta_a changes with the value of beta_phi0. For example, when beta_phi0=1000, theta_a~33, when beta_phi0=10, theta_a~50.

We have done tests and find that the value of theta_a does not change with resolution.

We added them on line 159-161.

Major concerns:

Figs 1 & 4: The magnetic field b_\phi should be plotted as a function of \theta, as this is the most interesting quantity in the current problem. Also, what equations are solved to obtain temperature as a function \theta?

Reply:

Thank you for your suggestion. We replot our figure and add b_phi and beta_phi. See figure 1,2,5,6.

The revisions relate to the new figure are on line 184-185, 190-194, 226-228, 233-235.

Sorry we forget to define the value. Temperature T=cs^2. We added it on line 199. The temperature can be solved by the energy equation eq 3. By using the self-similar assumption, the energy equation 3 can be rewritten as eq 32.

Major concerns:

What is "n" in Eqs. 21-23? It is not defined. Also, how is its value determined? Somewhere in the text it is written n=1.1, but no explanation as to why. Similarly, why is the resistivity coefficient set to be \eta_0 =0.1?

Reply:

-n is power-law index of density(eq 22). Numerical simulations of hot accretion flow show that the radial profile of density can be described as a power-law function of r as ρ is proportional to r^−n, with 0.5< n <1.5 (Stone, J. M., Pringle, J. E., & Begelman, M. C. 1999, MNRAS, 310, 1002); (Yuan, F., Bu, D., &Wu, M. 2012b, ApJ, 761, 130). In this paper, we choose n=1.1.

We added them on line 145-146, 176-178.

The value of the resistivity coefficient is not clear. In this paper, we set it to be 0.1.

Major concerns:

9. The plasma-\beta should be plotted as a function of \theta to see how it varies away from the midplane value \beta_0.

Reply:

Thank you for your suggestion. We replot our figure and add b_phi and beta_phi. See figure 1,2,5,6.

The revisions relate to the new figure are on line 184-185, 190-194, 226-228, 233-235.

Major concerns:

The authors just mention that the total outflow power is highest when \alpha_2 = 0.4 (e.g., Pg 11, lines 179-180; Pg 13, lines 197-198). However, no figure or further proof is provided. Yet, this is one of the major conclusions of this work.

Reply:

In figure 4 and figure 8, we plot the outflow power as a function of alpha_2. When alpha_2=0.4, the outflow power is the highest.

Major concerns:

Why is the overall energy flux for \beta_0 = 10 case (see Fig. 6, units 2x10^{-4}) lower than that for \beta_0 = 1000 (see Fig. 3, units 1.2x10^{-3})?

Reply:

In the case beta=10, the integration stopped at theta=50 degree, the outflow region is 50-60 degree. When beta=1000, the integration stopped at theta=33. The outflow region is 33-50. Due to the different size of outflow regions, the mass flux of outflow in the case beta=10 is smaller than that in the case beta=1000. From figure 4 and 8, we can see the thermal power dominates other powers. The thermal power equals to outflow mass flux times temperature, the temperature of outflow in the two cases (beta=10,1000) are quite similar. Due to the narrower regions we integrate, the outflow power in the case beta=10 is smaller.

Major concerns:

I think it is important to do a more rigorous investigation on the dependence of the variables on \beta_0. Only 2 values, 10 & 1000 are used currently, and the results are too similar to draw any strong conclusions (see e.g., Figs 1 & 4 which look almost identical). The authors should repeat their calculations with \beta_0 = 1 and 0.1 at least (if not a few more). A superthermal value of \beta_0=1 is justified by the simulations of Bai & Stone 2013 for e.g., and it would be interesting to see its effect in the light of recent interest in strongly magnetised accretion disks (e.g. Das & Begelman MNRAS 2018). A figure should also be provided for a fixed value of \alpha_2, which shows the dependence of different variables on \beta_0.

Reply:

Thank you for your important comments. In the revised version, we added figure 9. We have calculated the case beta_phi0=1, 10 & 1000 with a fixed alpha_2. For the hot accretion flows, when the flow is in the strongly magnetized MAD case, numerical simulations found that at the midplane the magnetic pressure is comparable to (but still slightly weaker than) the gas pressure (Narayan et al 2012), therefore, in our paper, we do not calculate the case beta=0.1.

We do not show a figure with beta_phi0=1 with different alpha_2. We can see from figure 9. The stopping point is theta=70 degree when alpha_2=0.1. When alpha_2 is higher, the region is narrower. Considering the narrowing region, we delete the figure with the case beta_phi0=1.

Minor issues:

Pg 3, lines 93-95: Reference(s) should be provided to justify this sentence "The effects of anisotropic pressure on accretion flow ....".

Reply:

We added a reference on line 97.

All variables should be defined on first appearance. For example, the anisotropic pressure \Pi and stress tensor {\bf T}, should be defined just after equations (1)-(5). Please check for similar occurrences.

Reply:

Thank you for reminding this. We define eta on line 132, the power-law index on line 145-146.  And we put the phrase (now on line 133-142) afterward to make sure all variables are defined on the first appearance.

Pg 8, Eqs. 37-40: These equations are suddenly in a different font and need to be changed to fit journal style.

Reply:

Thank you for reminding this.

We corrected the font. Now on eqs 38-41.

Pg 8, Eq. 41: Is p_g same as p? No need to introduce another variable to refer to the gas pressure.

Reply:

Thank you for reminding this.

Now it is corrected. Now on Eq. 42.

4.Pg 10, line 156: One should avoid using words like "always" in a scientific context, as all work remains to be tested.

Reply:

Thank you for reminding this.

Now “Always” is removed.

6.Pg 13: Last paragraph of the Conclusion should be rewritten. There are contradictory sentences saying once angular momentum can be transported by anisotropic viscosity and then the opposite; see lines 217-218 & line 220.

Reply:

Thank you for reminding this.

The sentence now is removed.

7.Pg 15, lines 306 & 307: Both the references are incorrect and should be rectified.

Reply:

Thank you for reminding this.

References now are corrected.

In addition, the title and many sentences are slightly revised to avoid similarities and better illustrate our work. They are all in boldface.

Title

Abstract: Line 18, 19, 20, 21, 22

Introduction

Line 32, 33, 40, 78, 79, 86.

We add eq.11,

2.3. Boundary equations: Line 151, 153-154, 155

Results: Line 157.

Best regards,

Huihong Deng, Defu Bu.

Reviewer 2 Report

Referee Report for manuscript 544436 "Two-dimensional self-similar solutions of hot accretion flow with anisotropic pressure" by Deng & Bu

This paper provides solutions of the self-similar steady-state dynamics of  a coupled system of an accretion disk with associated outflows.
The specific point here is to consider anisotropy of the pressure term. Hot disks of AGN could be the focus of this approach.

I have to state that although these results are potentially interesting, similar (very similar!) results have been recently published in Universe (2019, 5, 89) with one author being the co-authors of the present manuscript.

Further, many parts of the text, including title, abstract, introduction, results etc are in large part copied from the other paper it looks like.
The only difference to me seems to be the application of spherical coordinates (this manuscript) instead of cylindrical coordinates (the other Universe paper).
That Universe paper, is however not cited in the present manuscript.

Overall, this is not a scientifically proper way of publishing.

The conclusions mainly concentrate on the heating of the outflow caused by anisotropic pressure, the rest is statements on the equations and an obvious application to AGN. Heating was a topic already in the other paper.There are a number of language issues (mainly concerning the use of the arcticles "a", "the", or not using any of those), but that could be in  principle fixed.

Given the marginal scientific outcome of the paper, in particular  considering the previous Universe publication I recommend a rejection.
I do not believe that a major revision in particular discussion differences to the previous paper may help in order to provide an acceptable manuscript version.

Author Response

Dear referee,

Thank you for your comments. 

Huihong Deng, Defu Bu.

Round 2

Reviewer 1 Report

Report attached as file "report_v2.pdf"

Comments for author File: Comments.pdf

Author Response

Dear referee,

Thank you for your comments, which can significantly help us improve the quality of our paper. All the revisions are in boldface. Now we reply to your comments one by one.

Your comments:

1, In my previous report I had asked "how do the authors think will the presence of a weak vertical field modify their results?" By this I had meant, whether the effect of anisotropic pressure on the outflow power that the authors find in this work will change or not due to the presence of a vertical field. However, I did not receive any direct answer to this.

Reply:

We are sorry for not correctly answering your question.

In this paper, we find that the thermal energy flux of outflow is significantly higher than the kinetic energy flux of outflow. The anisotropic pressure plays a heating role. Therefore, with the increase of anisotropic pressure (alpha_2), the temperature of outflow increases (see the bottom right panel of Figure1,5). So, the (thermal) power of outflow increases with the increase of alpha_2. If a vertical magnetic field is present, the outflow will be stronger. The anisotropic pressure can still play a heating role in the presence of a vertical magnetic field. Therefore, with the increase of anisotropic pressure, the outflow temperature can still increase in the presence of the vertical magnetic field. We expect the (thermal) power of outflow can increase with the increase of anisotropic pressure even with the presence of the vertical magnetic field.

We have included this information on line 332-342. 

Your comments:

Lines 279-280: The authors state "The convective stability of accretion flow can be suppressed (Yuan et al. 2012b and Narayan et al. 2012)"

What exactly is meant by the above sentence, and if this is true what consequence will this have on the disk properties?

Reply:

We are sorry for not clearly expressing this sentence.

We mean that the convective motions can be suppressed by the magnetic field(Yuan et al. 2012b and Narayan et al. 2012). We explain more as follows. Hydrodynamical simulations of hot accretion flows(Stone et al. 1999) found that if the magnetic field is absent, the accretion flow is kinetically unstable, convective motions are present. Numerical simulations found that if the magnetic field is present, the convective motions disappears(Yuan et al. 2012b and Narayan et al. 2012), the turbulent motion of accretion flow is induced by MRI.

We have included this information on line 324-331. 

Your comments:

Lines 280-282: The authors state "the transfer of angular momentum by MRI process in accretion flow (Balbus & Hawley 1998) in the dynamical and observational appearance of the discs."

This sentence is incomplete and does not make any sense. What are the authors trying to convey here?

Reply:

We are sorry for this, now they have been deleted.

Your comments:

In response to one of my comments the authors replied that "The value of alpha_1=0.1 is from the observation." Corresponding references should be provided in the manuscript.

Reply:

We are sorry for this. The reference is King et al. 2007, MNRAS, 376, 1740.

Your comments:

In response to my question on alpha_2 the authors replied saying "Theoretically, there is also no very constrained value of alpha_2."

This is not acceptable. Even if there is no observational constraint on alpha_2, it must have some physical meaning and a theoretical constraint at least. The authors must explain this in order to motivate their work as this is the basis of their paper. Otherwise simply doing a parametric study by changing the value of alpha_2 is not enough.

Reply:

Thank you for your kind concerns.

In Chandra et al. 2015(ApJ,810,162), the authors did stability analysis to the accretion flow with anisotropic pressure. They found that for linear perturbations, the flow will be stable if alpha_2<9/8. Therefore, in our paper, the maximum value of alpha_2 is set to be 0.4, which is slightly smaller than 9/8.

We have added them on line 187-190.

Your comments: 

The authors should also comment on the connection between alpha_1 and alpha_2.

Reply:

Alpha_1 denotes the strength of viscous stress. As mentioned above, observations (King et al. 2007) show that the value of alpha_1~0.1. Apha_2 denote the strength of anisotropic pressure. As mentioned above, for a stable accretion flow, alpha_2 should be smaller than 9/8.

Your comments:

What order of Runge-Kutta method is used for the integration? Is the resolution 3x10^-4 same for all cases, irrespective of theta_a?

Reply:

We use two orders of Runge-Kutta method. The resolution is the same for all cases and irrespective of theta_a.

We have added this in line 163,164. 

Your comments:

The authors state in their previous response that "when beta_phi0=1000, theta_a~33, when beta_phi0=10, theta_a~50."

The range of integration thus seem to decrease drastically as the magnetic field becomes stronger --- as also seen from Figs. 1,2,5,6,9. This leads to a major concern --- can the low beta results be trusted? Especially as the authors are unable to pinpoint the exact reason for why the results become unphysical for theta < theta_a?

Reply:

In each model, the boundary conditions at the midplane are physical. Also, the solution in the region theta>theta_a satisfies the equations. Therefore, we think the solutions in the region theta>theta_a are trustable. The reason for the unphysical solution in the region theta

This information is on line 167-171,174-177. 

Your comments:

What are the units of pressure, density, temperature and magnetic field in Figs. 1,2,5,6, and 9?

Reply:

For the low accretion rate hot accretion flow, radiation can be neglected. In this case, from the hydrodynamical equations, we can see that the flow is density free. It means that the density (rho_0 in eq 22) can be any value. Consequently, the pressure can also be any value. The magnetic energy is scaled by gas pressure. Therefore, the magnetic field can be any value, too. However, the ratio of magnetic pressure to gas pressure is fixed. Now in the new figures1,5, we expressed temperature in the unit of virial temperature.

We have included these on line 177-182.

Your comments:

The anisotropic pressure seems to have no effect at all on the magnetic field of the disk (as clear from the bottom panels of Figs. 2 and 6) unlike the other parameters of the disk. Why is that? This feature needs to be explained properly.

Reply:

We have redrew the figure, see figure 2, 6. The magnetic field is replaced by the magnetic pressure (see the first panel). The magnetic pressure = p/beta. When the anisotropic pressure increases, both the gas pressure and beta increase. However, compared to other variables, beta is too large.Therefore, the change of the magnetic pressure and the magnetic field is not apparent.

We have included this on line 220-223.

Your comments:

In the new Eqs. 48 & 49 introduced by the authors, do the authors integrate theta from

0 to \pi or from 0 to theta_a? This needs to be clearly mentioned. Also, what are \rhomax and \rhomin?

Reply:

The integrate ranging is from theta_a to pi/2. It is not \rhomax and \rhomin, but rho*max(vr, 0) and rho*min(vr,0). Now we rewrite Eqs. 50&51(now) to avoid confusion.

Your comments:

Line 216: Figure number is missing in this sentence.

Reply:

Thank you for reminding this. Now the figure number is added.

Your comments:

In response to one of my comments the authors state that "For the hot accretion flows, when the flow is in the strongly magnetized MAD case, numerical simulations found that at the midplane the magnetic pressure is comparable to (but still slightly weaker than) the gas pressure (Narayan et al 2012), therefore, in our paper, we do not calculate the case beta=0.1."

The above is not true as demonstrated by the works of Bai & Stone 2013; Das & Begelman MNRAS 2018, which I had already mentioned in my previous report but the authors did not make a note of. Thus, there is enough motivation to do a case for beta = 0.1.

What the authors need to do is figure out the real reason why their integration region is becoming constricted with lower beta and only then they will be able to successfully simulate cases of stronger magnetic fields.

Reply:

Thank you for your valuable comment. Now we explain more as follows.

In both present works and Bai& Stone 2013, it is found that the value of beta decreases with the decrease of theta. Beta has the biggest value at the midplane. In Narayan et al. 2012( the simulations are about hot accretion flow), it is found that at midplane, beta is slightly larger than 1(at the midplane gas pressure dominates magnetic pressure). Note that at small theta region, Narayan et al.2012 also found that beta can be smaller than 1. In Bai& stone 2013 and Das et al. 2018 are about the cold thin disc, they also found that the value of beta is largest at the midplane. In Bai&Stone 2013, at the midplane, the lowest value of beta is 0.4(see figure 1 in their paper). The value of beta at midplane for hot accretion flow(Narayan 2012) is different from that at the midplane for cold thin discs(Bai 2013). The difference may be due to different physical conditions.

We have tried to calculate the case beta=0.1 at the midplane(we expect at the region theta<pi/2, the value of beta can be much smaller than 0.1). We cannot obtain a solution. The reason may be that for hot accretion flow, the value of beta at midplane is required to be larger than 1.

We have included these on line 278-290.

Your comments:

The entire manuscript needs to be thoroughly proof read as the authors revisions have introduced several grammatical errors.

We are sorry for this. Now we have corrected many grammar issues. They are in boldface.

Reviewer 2 Report

Referee Report for manuscript 544436
"Two-dimensional self-similar solutions of hot accretion flow with
anisotropic pressure"
by Deng & Bu

----------------------------------------------------

This report refers to the revised version of the above mentioned manuscript.

The new manuscript hast partly improved, in particular the relation to the Deng et al. 2019 paper is now clarifed.

Still, I do not see a major revision in order to improve the scientific quality of the paper.

What I am missing is an in-depth discussion of the physical results of the paper.

For example, at least the following points must be clarified:

1) What is the implication of only using a toroidal field? How "wrong" are the results by neglecting the poloidal field component? How applicable are the results to reality?

2) The numerical error mentioned in line 159 must be discussed in detail. Why does the error appear? It is not not sufficcient just to tell that others discovered the same error. Is that a problem of the code they use or is there a physical reason.
Why is the result found in this area "unphysical"?

3) Concerning there results, it would be essential to know whether the solution is super-sonic, super-fast magnetosonic.
The approach used is to solf the classical wind equation (modified by anisotropic pressure). So where are the critical points of the solution? Are the critical conditions considered in order to constrain the solution?

4) The solutions are given as parametrized with the viscous coefficients.
But what is the (astro)physical implication of that? Do we expect an outflow viscocity far from the disk?
The summary does not mention at all the impact of viscosity. It must be said that that the anisotropic pressure is considered by an effective viscosity.

5) I wondering whether there is no solution plotted with a radial profile? In the end we want to know the outflow accelerated from the disk, or also the radial profile of the disk structure.

6) When the impact of anisotropic pressure/ viscosity is mentioned, there authors only DESCRIBE the solution, e.g. "A stronger wind is drived".
However, they need to discuss the physical forces at work, that leads to the effects they claim to have discovered. Another example is the enhanced anergy flux (by a factor 20). No physical reason is given why the Bernoulli parameter is enlarged with alpha_2=0.4. For example, where does that enhanced power come from? Is the source of the outflow different?

6) The paper mentions several time black holes or AGN. However, there are
no relativistic effects mentioned or considered in the approach. So this is misleading.

7) There are still numerous language issues. It looks like that the authors did not care about the referees' comments.

8) Some last comments on the literature cited. There is essential work done in this field by Ferreira et al. starting in 1997. This work must be discussed and cited. Also recent papers on launching simulations are completely neglected (Murphy et al. 2010, Sheiknezami et al. 2012, Stepanovs et al. 2016). It will make more sense to cite and discuss these non-relativistic papers than to list GR-MHD simulations (lines, 38,39).
Also, the most fundamental paper on (self-similar) outflows is Blandford &
Payne. This needs to be discussed prominently.

Author Response

Dear referee,

Thank you for listing concerns which can significantly improve the scientific quality of our paper.

Now we have revised our work based on your concerns.

Your concerns:

1) What is the implication of only using a toroidal field? How "wrong" are the results by neglecting the poloidal field component? How applicable are the results to reality?

Reply:

Hirose et al. 2004(paper cited in our work) find in their simulations that in the main body region of the accretion flow, the magnetic field is mainly toroidal. A poloidal field is present in the region very close to the rotational axis. The presence of a poloidal field is crucial for jet formation. In our paper, we do not study jet. Therefore, we expect the neglection of a poloidal field cannot affect the properties of the main body of the accretion flow studied in this paper.

We have included this on line 139-144.

Your concerns:

2) The numerical error mentioned in line 159 must be discussed in detail. Why does the error appear? It is not not sufficcient just to tell that others discovered the same error. Is that a problem of the code they use or is there a physical reason.

Why is the result found in this area "unphysical"?

Reply:

We have deleted the ‘’unphysical’’ statement. Previous two-dimensional self-similar solutions (e.g., Jiao and Wu 2011 ApJ 733 112; Mosallanezhad et al. 2016 MNRAS 456 2877) also found an error when integrating. In these two works, the authors have also not obtained the solution from \pi/2 to 0. We think the reason for the numerical error should be that the self-similar assumption cannot be applied to the region close to the rotational axis.

We think the solution in the region of \theta_a<\theta<90 are still physical. There are two reasons. First, the solutions satisfy the equation and boundary conditions at  \theta = \pi/2. Second, the quantities derived at \theta_a are physical

We have included this information on line 167-171, 174-177.

Your concerns:

3) Concerning there results, it would be essential to know whether the solution is super-sonic, super-fast magnetosonic.

The approach used is to solf the classical wind equation (modified by anisotropic pressure). So where are the critical points of the solution? Are the critical conditions considered in order to constrain the solution?

Reply:

We plot the new figures (see figure 2, 6). Our solution is sub-sonic in both weak and strong magnetic field case. Our solution does not pass the fast magnetosonic point in the weak magnetic case but pass it in the strong magnetic case where theta~50. We didn’t consider the critical conditions before.

We have included this on line 225-229, 261-262.

Your concerns:

4) The solutions are given as parametrized with the viscous coefficients.

But what is the (astro)physical implication of that? Do we expect an outflow viscocity far from the disk?

The summary does not mention at all the impact of viscosity. It must be said that that the anisotropic pressure is considered by an effective viscosity.

Reply:

We found that with the increase of alpha_2, the specific power of outflow increases. The value of alpha_2 depends on the accretion rate. Lower accretion rate system has lower particle collisional rate and higher value of alpha_2. Therefore for hot accretion flow, the fraction of energy taken away from the accretion system by outflow increase with the decrease of accretion rate. Lower accretion rate system tends to drive stronger outflow/jet.

In the outflow region, the pressure is also anisotropic. Therefore, we still have a viscosity in an outflow far from the disk.

The anisotropic pressure is modeled by the anisotropic viscosity.

We have included this on line 303-307.

Your concerns:

5) I wondering whether there is no solution plotted with a radial profile? In the end we want to know the outflow accelerated from the disk, or also the radial profile of the disk structure

Reply:

For the self-similar solution, all of the physical quantities are described as a power-law function of radius. Therefore, it seems it is not so meaningful to plot the radial profile of physical quantities. For example, the Mach number(vr/cs) is independent of radius. If we plot the radial profile of the Mach number, we will get a straight line with the slope equal to zero. Therefore, in this paper, we have not plotted the radial profile of the disc structure.

Your concerns:

6) When the impact of anisotropic pressure/ viscosity is mentioned, there authors only DESCRIBE the solution, e.g. "A stronger wind is drived".

However, they need to discuss the physical forces at work, that leads to the effects they claim to have discovered. Another example is the enhanced anergy flux (by a factor 20). No physical reason is given why the Bernoulli parameter is enlarged with alpha_2=0.4. For example, where does that enhanced power come from? Is the source of the outflow different?

Reply:

The reason is that for hot accretion flow, gas pressure gradient force is mainly responsible for driving outflow (Yuan et al. 2012b). When the anisotropic pressure increases, the gas pressure gradient increases. Therefore, the wind/outflow turns stronger.

For the Bernoulli parameter, the enthalpy term dominates. The enthalpy is proportional to temperature. From Figure.1, we can see, with the increase of alpha_2, temperature increases. Therefore, the enthalpy increases with alpha_2. Bernoulli parameter increases with alpha_2.

We have included this on line 209-211, 235-237.

7) Your concerns:

the paper mentions several time black holes or AGN. However, there are

no relativistic effects mentioned or considered in the approach. So this is misleading.

Reply:

The relativistic effects are only important in the region very close to the black hole(r<20 Schwarzschild radius). In the region not so close to the black hole, the gravity of the black hole is approximately Newtonian. In our present paper, we look for a self-similar solution. The self-similar solution can also not be applied to the region very close to the black hole. Therefore, in this paper, we use Newtonian potential.

Your concerns:

8) There are still numerous language issues. It looks like that the authors did not care about the referees' comments.

Reply:

We are sorry about this. Now we corrected many mistakes. They are in boldface.

Your concerns:

9) Some last comments on the literature cited. There is essential work done in this field by Ferreira et al. starting in 1997. This work must be discussed and cited. Also recent papers on launching simulations are completely neglected (Murphy et al. 2010, Sheiknezami et al. 2012, Stepanovs et al. 2016). It will make more sense to cite and discuss these non-relativistic papers than to list GR-MHD simulations (lines, 38,39).

Also, the most fundamental paper on (self-similar) outflows is Blandford &

Payne. This needs to be discussed prominently.

Reply:

Outflow/jet generated from a cold thin disk has also been studied by both analytical (Ferreira et al. 1997; Stepanovs and Fendt 2016; Blandford & Payne) and simulation (Murphy et al. 2010, Sheikhnezami et al. 2012; Stepanovs et al. 2014) works.

Now we cited and discussed the important works you listed. See line 34-38.

Round 3

Reviewer 1 Report

Report attached as file report_v3.txt

Comments for author File: Comments.pdf

Author Response

Please see the attachment.

Author Response File: Author Response.pdf

Reviewer 2 Report

I'm happy with the revised version and can suggest publication.

I only have two comments:

1) on comment 7) in the authors' reply. What I meant is that the authors need to clearly state that is is a non-relativistic apporach. It is confusing to mention black holes and AGN and cite relativistic papers more than non-relativistic papers, but then applying a non-relativistic approach. As mentioned in their answer, they need to clearly state that the self-similar approach ich not feasible close to the black hole.

2) on comment 5) I still don't understand, why the Mach number should be constant in radius. For example the classical Blandford & Payne solution clearly show the increase of teh Alfven Mach number. Please check your results, and please consider to plot the Mach number and velocity profiles as a function of radius. Of course a discussion of these issues will be needed as well.

Author Response

Please see the attachment.

Author Response File: Author Response.pdf