Self-Similar Solution of Hot Accretion Flow with Anisotropic Pressure

Round  1

Reviewer 1 Report

Referee Report for manuscript 473308 "Self-similar solution of hot accretion flow with anisotropic pressure" by De-Fu Bu et al.

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The paper describes the steady-state dynamics of accretion disk and associated outflows considering an anisotropic pressure term. The application, as claimed in the paper, is the case of hot accretion disks  for which the mean-free path of ions is comparatively large.

 I find the results interesting and potentially important. Also, to my knowledge, a similar application of the theory has not been published before. Therefore I suggest the paper for publication.

However, I suggest a revision. No substantial revision required, but a major revision in the sense that many details have to be improved.

These are my comments:

1) There are quite many typos or grammar issues in the manuscript. Very often the article is missing "Magnetic field is..." instead of "The magnetic field is ...",  "in hot accretion flow" instead of "in hot accretion flows" or in a hot accretion flow".

2) A number of statements need to be further backup-ed or detailed.
For example:
- "extremely low accretion rate": what are the numbers?
- "mean-free path larger than gyro-radius": it would be interesting to have numbers
- "hot accretion flow": what are the temperatures considered, what are the densities, how do you constrain these densities?
- how to constrain the field strength, please give typical values
- "viscosity coefficient alpha ~0.1  ~B^2/p". A derivation must be provided for the latter relation, as it is an essential quantity for the paper.
- magnetic field is toroidally dominated. That is fine for me, however, we know solutions with a substantial poloidal field as well (see the MAD disk structure)

3) The introduction needs to be widened somewhat. I miss a more general discussion of disk accretion, not only about the hot flows. Simulations have been published for thin disk jet launching (e.g. Casse & Keppens; Zanni et al.; Sheikhnezami et al; and many more), or for relativistic disks (e.g. De Villier & Hawley; Gammie et al. 2003; Noble et al. 2011; Sadowski et al.).
A relevant paper is also Takahashi et al. (1990).
The authors need to put their work in a broader context. A nice review is Hawley et al. 2015.

4) The results must be presented in a clearer way:
- the figures are too small, in particular the labels
- the figure captions should exactly tell, what is plotted: Instead of "properties" in Fig.1 it must be told what the variable plotted on the y-axis actually represents. I suggest to tell in the caption again what xsi, alpha, c1 etc stands for.

- An example: text line 142 says that "in Fig.1 we see ... the radial infall velocity ... increases". I don't see an infall velocity mentioned in the caption of Fig. 1.  Also, where is that velocity measured? At which radius? Since self-similarity the variables are scale-free, but this should be mentioned again.
- Then: what does the increase of infall velocity imply? Is that a leading parameter? Does it become super Alfvenic for example? Would not the mass accretion rate be a better parameter (thus plotting \rho v_r)? The sound speed is mentioned as well, but no relation to the infall speed is made (sub/super-sonic?, does this change)

5) I strongly suggest the authors to discuss some astrophysical applications. They start off with mentioning AGN and the Galactic center, but Sect. 3 ends with discussion number values of xsis and alphas, and also in the summary not connections is made to astrophysics. What is the xsi for M87? What is the difference in impact for xsi = 1 or xsi = 1.2? How relevant is that in terms of astrophysical sources?

Author Response

Dear Referee,

Thank you very much for reviewing our paper. We think your comments are very constructive and helpful to improve this paper. We have modified the manuscript according to your reports. We hope you can be satisfied with our revision. The followings are our detailed responses to your comments. All changes of our manuscript are in boldface.

Best regards,

De-Fu Bu, Peiyao Xu, Bocheng Zhu

Your comment:

Point 1: There are quite many typos or grammar issues in the manuscript. Very often the article is missing "Magnetic field is..." instead of "The magnetic field is ...",  "in hot accretion flow" instead of "in hot accretion flows" or in a hot accretion flow".

Reply: We are sorry for the grammar mistakes. In the revised version, we have corrected the grammar mistakes.

Your comments:

Point 2:  A number of statements need to be further backup-ed or detailed. For example:
- "extremely low accretion rate": what are the numbers?

Reply: The accretion rate of accretion flow in Galactic center is 10²¹g/s. The accretion rate of accretion flow in M87 galaxy is 1.8*10²⁵g/s.

We have given the values of accretion rate for the accretion flows in Galactic center (Sgr A*) and M87 galaxy in Section 1, in paragraph 4.in line 42-46.

- "mean-free path larger than gyro-radius": it would be interesting to have numbers

Reply: The mean free path for accretion flow in Galactic center is approximately 10¹⁷cm. If we assume magnetic pressure is 0.1 times gas pressure, the gyro-radius is approximately 10⁶cm.

We have given the values of ions mean-free path and gyro-radius for the accretion flow in Galactic center in Section 1.in line 66-69.

- "hot accretion flow": what are the temperatures considered, what are the densities, how do you constrain these densities?

Reply: The temperature of gas in hot accretion flow almost equals to Virial temperature. In the inner most region (10 Schwarzschild radius), the gas can have temperature approximately 10¹¹ K (Yuan & Narayan 2014). The density is very low and depends on the mass accretion rate. In the accretion flow in Galactic center, in the inner most region, gas density is approximately 10^-18 g/cm³.

We have given the values in Section 1,in line 25-29.

- how to constrain the field strength, please give typical values

Reply: Currently, it is quite difficult to measure or even to constrain the strength of the magnetic field. Among the existing methods, Faraday rotation measure (RM) can provide the integration of electron density and magnetic field along the line of sight. RM combined with spectrum can roughly give magnetic field strength. For example, the RM of Sgr A*, together with the radio-up to-millimeter spectrum, constrains the magnetic field strength in the accretion flow to be 20 Gauss at 10 Schwarzschild radius. (Yuan et al. 2003).

We have included the information in Section 1.in line 78-83.  

- "viscosity coefficient alpha ~0.1  ~B^2/p". A derivation must be provided for the latter relation, as it is an essential quantity for the paper.

Reply: The magnetic stress for angular momentum transfer is induced by turbulence driven by magnetorotational instability (Balbus & Hawley 1991, 1998). The stress is Br*B_\phi/(4 \pi) ~ B². The viscous coefficient (alpha_1) is stress divided by gas pressure. Therefore, we have \alpha_1 ~ B^2/p.

We have included this information in Section 1.in line 84-88.

- magnetic field is toroidally dominated. That is fine for me, however, we know solutions with a substantial poloidal field as well (see the MAD disk structure).

Reply: We totally agree that a substantial poloidal field is present for the MAD disk structure. We have included two sentences to introduce this point in Section 2, in line 107-109.

Your comments:

Point 3: The introduction needs to be widened somewhat. I miss a more general discussion of disk accretion, not only about the hot flows. Simulations have been published for thin disk jet launching (e.g. Casse & Keppens; Zanni et al.; Sheikhnezami et al; and many more), or for relativistic disks (e.g. De Villier & Hawley; Gammie et al. 2003; Noble et al. 2011; Sadowski et al.).

A relevant paper is also Takahashi et al. (1990).
The authors need to put their work in a broader context. A nice review is Hawley et al. 2015.

Reply: We totally agree that we should also introduce works about thin disk. In the revised version, we have introduced this point and cite all the papers mentioned here in paragraph 1 of the “Introduction” section, in line 13-20.

Your comments:

Point 4: The results must be presented in a clearer way:

- the figures are too small, in particular the labels
- the figure captions should exactly tell, what is plotted: Instead of "properties" in Fig.1 it must be told what the variable plotted on the y-axis actually represents. I suggest to tell in the caption again what xsi, alpha, c1 etc stands for.

Reply: In the revised version, we make the figure larger. Also, in the caption, we introduced what are exactly plotted. We also include what xsi, alpha, and beta mean in the caption.

- An example: text line 142 says that "in Fig.1 we see ... the radial infall velocity ... increases". I don't see an infall velocity mentioned in the caption of Fig. 1.  Also, where is that velocity measured? At which radius? Since self-similarity the variables are scale-free, but this should be mentioned again.

Reply: In the revised version, in caption of Fig.1 we now introduce clearly what the panels plot (top-left panel is for radial infall velocity). For the self-similar solution, at any radii, the ratio of infall velocity to Keplerian velocity is a constant. In the figures, we plot the ratio of infall velocity to Keplerian velocity; therefore, the results (infall velocity plotted in top-left panle of Fig 1) can be applied at any radii.

We have included this point in Section 3,in line 173-180.

- Then: what does the increase of infall velocity imply? Is that a leading parameter? Does it become super Alfvenic for example? Would not the mass accretion rate be a better parameter (thus plotting \rho v_r)? The sound speed is mentioned as well, but no relation to the infall speed is made (sub/super-sonic?, does this change)

Reply: From Fig.1, we see with the increase of alpha_2, both the radial infall velocity and sound speed increase. Comparing the top-left and below panels, we see the infall velocity is always smaller than sound speed by one order of magnitude at any given value of alpha_2. Therefore, the infall velocity is sub-sonic. Because, we assume that magnetic pressure is 10 times smaller than gas pressure. Correspondingly, the Alfvenic velocity is smaller than gas sound speed by a factor of 3.3. Therefore, the gas infall velocity is sub-Alfvenic.

We have included this information in Section 3,in line 199-204.

Because, the mass density is a function of radius, the mass accretion rate (\rho*v_r) is a function of radius. Therefore, in order to plot mass accretion rate, we need to clearly state at which radius the figures are plotted. Because, the accretion flow is assumed to be self-similar, it seems that plotting velocity (in unit of Keplerian velocity) is more useful. This is because that velocity (in unit of Keplerian velocity) can be applied at any radii.

Your comment:

Point 5: I strongly suggest the authors to discuss some astrophysical applications. They start off with mentioning AGN and the Galactic center, but Sect. 3 ends with discussion number values of xsis and alphas, and also in the summary not connections is made to astrophysics. What is the xsi for M87? What is the difference in impact for xsi = 1 or xsi = 1.2? How relevant is that in terms of astrophysical sources?

Reply: Thank you for raising this important point.

For a hot accretion flow, the gas temperature is too high that gas is fully ionized. Observationally, it is very hard to detect outflow directly through absorption line. There are just some indirect evidences that outflow should be present for a hot accretion flow (Wang et al. 2013; Ma et al. 2019; Park et al. 2019). The properties (velocity, temperature) of outflows can not be given by observations. Numerical simulations give us the properties of outflows. The values of xsi are mainly from numerical simulations results (Yuan et al. 2012).

We have included this information in paragraph 1 of Section 3, in line 165-170. 

Author Response File: Author Response.pdf

Reviewer 2 Report

This manuscript deals with pressure differences in the directions parallel and normal to the magnetic field in low-luminosity AGNs on the dynamics of accretion flows, and the effects on the accretion flow and the generation of outflows.

I find that the paper is clear and is mostly well written.Thus Ionly report a few issues to take into account.

1. Bernoulli parameter. It is introduced in the abstract and then in the conclusions. I have not found any else reference in the main text about this topic, not even its definition, a mathematical expression, or a citation. A short explanation to introduce this parameter as the specific energy of the flow and a constant of motion is needed. Also a clear explanation in the results section is missing of why the Bernoulli parameter increases in the presence of anisotropic pressure (the increasing of the kinetic energy andenthalpyare already explained, thus the explanation is trivial). 

2. Section 1, lines 35 – 40:  

"However, particle-in-cell simulations show that the wave-particle interactions can effectively increase the collision rate of particles (Kunz, Schekochihin & Stone 2014; Riquelme, Quataert & Verscharen 2015; Sironi & Narayan 2015). Observations also support that the collision rate of particles can be relatively high even though the particles mean-free path is enormous. For example, galactic cosmic rays are quite isotropic despite their significantly long mean free path (e.g.Kulsrud2004)."  

The isotropic distribution of galactic cosmic rays is related to the tangled galactic magnetic field and the interaction with Alfven waves, rather than to collisions.Coulombiancollisions between cosmic rays and particles in the ISM are negligible. Are the authors are referring to wave-particle interactions rather than collisions between particles?

3. Section 1, line 61 – 62: Defines the *viscosity coefficient* $\alpha$.  Then in Section 3, line 160 defines $\alpha_2$ as the *strength of anisotropic pressure*, but it is first used much earlier, in equation (9). With respect to $\alpha_1$,  it is used first in the expression for $v_1$ in line 81, but.Please, unify definitions.

4. Equation (8), please clarify the meaning of **bb** and the operator ":" (also used in line 86).

5. Equation (10), and along the text, please check notation and explain components of $\Pi.$ (Not sure what $\Pi_{\OE\OE}$ means).

6. Section 2, lines 89 and 128 – 129: Repeat the same information.

7. Figures: Please print larger figures (particularly Y axis shows fuzzy).

Author Response

Dear Referee,

Thank you very much for reviewing our paper. We think your comments are very constructive and helpful to improve this paper. We have modified the manuscript according to your reports. We hope you can be satisfied with our revision. The followings are our detailed responses to your comments. All changes of our manuscript are in boldface.

Best regards,

De-Fu Bu, Peiyao Xu, Bocheng Zhu

Your comment:

1. Bernoulli parameter. It is introduced in the abstract and then in the conclusions. I have not found any else reference in the main text about this topic, not even its definition, a mathematical expression, or a citation. A short explanation to introduce this parameter as the specific energy of the flow and a constant of motion is needed. Also a clear explanation in the results section is missing of why the Bernoulli parameter increases in the presence of anisotropic pressure (the increasing of the kinetic energy andenthalpyare already explained, thus the explanation is trivial).

Reply: Thank you for raising this point. In the revised version, in the last paragraph of Section 3, in line 263-265. we defined the Bernoulli parameter and give some explanations about how Bernoulli parameter changes with anisotropic pressure.

Your comments:

2. Section 1, lines 35 – 40:  

"However, particle-in-cell simulations show that the wave-particle interactions can effectively increase the collision rate of particles (Kunz, Schekochihin & Stone 2014; Riquelme, Quataert & Verscharen 2015; Sironi & Narayan 2015). Observations also support that the collision rate of particles can be relatively high even though the particles mean-free path is enormous. For example, galactic cosmic rays are quite isotropic despite their significantly long mean free path (e.g.Kulsrud2004)."  

The isotropic distribution of galactic cosmic rays is related to the tangled galactic magnetic field and the interaction with Alfven waves, rather than to collisions. Coulombiancollisions between cosmic rays and particles in the ISM are negligible. Are the authors are referring to wave-particle interactions rather than collisions between particles?

Reply: We are sorry for this confusing point. The particle-in-cell simulations show that the collisions between particles can be effectively increased by wave-particle interactions. As you said, the Coulombiancollisions between cosmic rays and particles are negligible.

We have modified this point and deleted the sentence “Observations also support that the collision rate of particles can be relatively high even though the particles mean-free path is enormous. For example, galactic cosmic rays are quite isotropic despite their significantly long mean free path (e.g.Kulsrud2004). "

Your comment:

3. Section 1, line 61 – 62: Defines the *viscosity coefficient* $\alpha$.  Then in Section 3, line 160 defines $\alpha_2$ as the *strength of anisotropic pressure*, but it is first used much earlier, in equation (9). With respect to $\alpha_1$,  it is used first in the expression for $v_1$ in line 81, but.Please, unify definitions.

Reply: In the revised version, in Section 1, we define the *viscosity coefficient* as $\alpha_1$. It is same as that used in Equation (7). Below the original Equation 9 (equation 10 in current version), we give an explanation to $\alpha_2$.

Your comment:

4. Equation (8), please clarify the meaning of **bb** and the operator ":" (also used in line 86).

Reply: bb is a dyadic tensor. It has nine components. We give an exact expression for bb (see the new Equation (9)) . “:” is an operator of two rank-two tensors and we give an expression below Equation (9).

Your comment:

5. Equation (10), and along the text, please check notation and explain components of $\Pi.$ (Not sure what $\Pi_{\OE\OE}$ means).

Reply: $\Pi_{\OE\OE}$ should be $\Pi_{\phi\phi}$. We have corrected it.

Your comment:

6. Section 2, lines 89 and 128 – 129: Repeat the same information.

Reply: In the revised version, we have deleted the sentences in the original lines 89-91.

Your comment:

7. Figures: Please print larger figures (particularly Y axis shows fuzzy).

Reply: In the revised version, we make the figure larger. Now, it is much clear.

Author Response File: Author Response.pdf

Round  2

Reviewer 1 Report

I'm happy with the modifications that have been made.

However, I still recommend some little English language editing. The point I mentioned before concerning the articles is still present. I only gave two examples, but the paper has many more of these mistakes. Please have a look.

Author Response

Dear Referee:

Thank you for reviewing our paper again. We have read the paper very carefully and corrected the grammar mistakes.

Best wishes

De-Fu Bu, Pei-Yao Xu, Bo-Cheng Zhu

Author Response File: Author Response.pdf