Exploring the Impact of Self-Excited Alfvén Waves on Transonic Winds: Applications in Galactic Outflows

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

I would like more discussion in the introduction  of wave mixing of non-WKB Alfven waves in the introduction, including the incompressible wave mixing equations of Zhou and Matthaeus (1990), and of Heinemann and Olbert (1980) (see also my attached report). The model, as far as I can see it does not describe the interaction of the Alfv\'en waves with the magneto-acoustic modes.

Comments for author File: Comments.pdf

Author Response

Reviewer 1

1. Much of the paper is based on a discussion of cosmic ray two fluid equations given by Ko (1992): Ko, C.M. 1992, A note on the hydrodynamical description of cosmic ray propagation, Astron. Astrophys., 259, pp 377-381.

Our response:

 

We thank to the respected referee for this observation. However, there is slight correction that the hydrodynamic approach used in this study is based on the framework developed by Ko(1992, 2001) as the four fluid model. The four fluid model is the magnetohydrodynamic description of the cosmic-ray driven plasma system. The model consist of four components namely thermal plasma, cosmic-rays, self-excited propagating forward and backward Alfvén waves by which all of them are treated as fluids. The description of the components and their coupled interactions are mainly expressed as the energy density or pressure. In this paper the four fluid hydrodynamic approach is applied to study the role of the Alfvén waves in effecting dynamics of the galactic wind outflows with taking account the effects of the gravitational field from the center, non-linear Landau damping and bremsstrahlung cooling as well.

 

2. It is useful to mention, that WKBJ, unidirectional Alfv´en wave driven winds were developed by Belcher (1971), Belcher and Davis (1971), Alazraki and Couturier (1971), and later by Hollweg (1978,1996) and McKenzie (1994) as models for the Solar Wind. A non-WKBJ bi-directional Alfv´en wave driven wind model was described by Heinemann and Olbert (1980). Later Zhou and Matthaeus and and Marsch and Tu (1989) developed incompressible turbulence equations for the Els¨asser variables Z µρ, corresponding to outward (−) and inward (+) propagating Alfv´enic disturbances (here we use the convention of Velli (1991) for the Els¨asser variables Z ± hich is opposite to that adopted by Zhou and Matthaeus (1990)). The Zhou and Matthaeus (1990a,b) equations contain the Heinemann and Olbert (1980) model Alfv´en wave driven wind model as a special case where the magnetic field and fluid velocity are purely poloidal. The interaction of the Alfv´en modes with the magneto-acoustic modes in this model are suppressed as being negligible, but this is a questionable assumption, unless one invokes Landau damping of the magneto-acoustic modes. In the incompressible turbulence equations of Zhou and Matthaeus (1990a,b), the backward and forward waves interact with each other via the wave mixing tensor, which depends on the gradients of the background flow. The Heinemann and Olbert (1980) equations and the Zhou-Matthaeus (1990a,b) equations are also discussed by Webb et al. (2024) (Webb, G.M., Anco, S.C., Meleshko, S.V. and Kaptsov, E.I. 2024, Reviews of Modern Plasma Physics, 2024 8:33 see also references therein). This is not meant to suggest the above paper should necessarily be referenced, but it does contain a good reference list.

Our response:

 

We thank to the respected referee for the recommendation to include the discussion of the Alfvén wave-driven models in the Introduction section and providing us the comprehensive review of the Alfvén wave-driven models as in the form of the suggestions. In our revised manuscript, we have included the paragraph about the Alfvén wave-driven models in the Introduction section where we comprehensively discussed about the nature of the Alfvén waves propagation and the developments of the model in the form of mini-literature review. As mentioned in the paragraph the propagation of the Alfvén waves can be modeled in two categories namely (i) WKBJ (ii) Non-WKBJ. In WKBJ approximation, the Alfvén waves model are primarily described by the small wavelength (in comparisons with the characteristics length of the plasma) and uni-directional flow that were mainly  developed by by Belcher (1971), Belcher & Davis (1971), and Alazraki & Couturier (1971), and later extended by Hollweg (1978, 1996) and McKenzie (1994). But the prediction of the Alfvén waves propagation in the WKBJ fails or deviates in highly erratic inhomogeneous plasma background especially in the turbulent environment and so one must resort use the non-WKBJ approach to capture the dynamics of the Alfvén waves. The non-WKBJ Alfvén waves models features large wavelength (in comparisons with the characteristics length scale of the plasma system), bi-directional, mixing and coupling interactions between forward and backward  self-propagating Alfvén waves that were developed by Heinemann & Olbert (1980), Matthaeus (1990a,b) and Marsch & Tu (1989). We have extensively discussed these developments to place our present four-fluid, steady-state treatment in the broader context of Alfvén wave-driven wind theories. But we also emphasized in our manuscript that in the four-fluid framework usually neglects turbulent conditions and so the model is put into the treatise in the simple WKBJ model. We have cited the foundation papers as per suggestion and in addition we have also included from the other relevant papers as well.

(The detailed paragraph about the Alfvén wave-driven models is found at the page 2  the in the Introduction section in the second paragraph)

 

3. A further classic reference to linearized MHD model equations used in discussions of MHD stability that might be useful are the Frieman and Rotenberg equations (Friemann, E.A. and Rotenberg (1960), On hydromagnetic stability of stationary equilibria, Rev. Mod. Phys., 32 (4), 898-902.

Our response:

 

We thank to the respected referee for the invaluable suggestion. In our revised manuscript we acknowledged and cited the work Friemann and Rotenberg (1960) and briefly mentioned about the formulation of the generalized linearized MHD model that examine the behavior of hydromagnetic stability  at stationary equilibria. However, we also discussed about the formulation and framework of the four fluid hydrodynamic model forwarded by Ko. (1992, 2001) is usually based on the steady state condition that ignores time dependent variables. Although the model has the criterion to check the validity and admitting the physical solutions based on the morphological profile reaching asymptotically to uniform states but it thereof lacks capability to check the stability of the morphological profile at the uniform or equilibrium state. (The detailed discussion is given at page 5 in the last paragraph highlighted in blue)

 

 

References:

1. Ko C. M. A note on the hydrodynamical description of cosmic ray propagation. Astronomy and Astrophysics, 1992; Vol. 259, pp. 377–381

2.. Ko C. M. Continuous solutions of the hydrodynamic approach to cosmic-ray propagation. Journal of Plasma Physics, 2001; Vol. 65, pp. 305–317

3. Frieman, E. Rotenburg, M. On Hydromagnetic Stability of Stationary Equilibria. Rev. Mod. Phys., 2001; Vol. 32 pp. 898
4. Ko C. M. A note on the hydrodynamical description of cosmic ray propagation. Astronomy and Astrophysics, 1992; Vol. 259, pp. 377–381
5. Ko C. M. Continuous solutions of the hydrodynamic approach to cosmic-ray propagation. Journal of Plasma Physics, 2001; Vol. 65, pp. 305–317

 

 

 

4. It should be noted that the MHD equations possess conservation laws. The perturbed form of the conservation laws should also be satisfied in some sense for the perturbed MHD equations.

Our response:

We thank to the respected referee for highlighting the validity and consistency of the conservation laws. Given the basic framework of the magnetohydrodynamic equations whether it is the four fluid model is embedded in the mass conservation, energy conservation, continuity equation, momentum equation and magnetic flux it must always satisfy conservation principles. When considering the perturbations that involves changes of these basic equations with respect to time are also expected to obey conservation laws. But in the present work, our analysis is solely restricted to the steady state four fluid hydrodynamic model that is used to explore the impact of the Alfven waves in driving cosmic-driven galactic outflows and so we are mainly focused to analyze morphological structure of the outflows in terms of pressure and velocity variations along  only spatial scales. Therefore the model offers no treatise the perturbed form of conservation laws as temporal scales or time dependent variables are considered to be ignored.  

 (for the conservation equations in terms of mass flux, momentum equation, energy density and flux equations please refer to eqs 1 – 9 at page 4)

Author Response File: Author Response.pdf

Reviewer 2 Report

Comments and Suggestions for Authors

Studies on cosmic rays contribute to understanding the energetic plasma outflows ejected from galactic centers against strong gravitational potential wells. Cosmic rays interact with thermal plasma. Based on the assumption that thermal plasma, cosmic rays, and self-excited Alfvén waves are treated as fluids, the authors used a multi-fluid model to investigate steady-state transonic solutions under specific boundary conditions at the base of the potential well. The results show that galactic outflows have three stages: coupling, decoupling, and re-coupling. It is found that due to cooling and wave damping effects, the four-fluid model exhibits characteristics such as non-monotonic velocity distribution different from the three-fluid model, which is significant for understanding galactic evolution.

If the authors consider potential revisions, this article can be published.

 

Main issues:

1. The outflow is divided into coupling, decoupling, and re-coupling stages, with the coupling stage bounded by (10 < \xi < 15\) . What is the physical unit of \(\xi\) (e.g., kpc, pc)?
2. The manuscript contrasts the four-fluid model with Ko et al. [26]’s three-fluid model (e.g., non-monotonic velocity profiles in ). To isolate the role of backward waves, could comparisons with control cases (e.g., "four-fluid without cooling" or "three-fluid with cooling") demonstrate that backward waves, not just enhanced cooling/damping, drive these differences?
3. The model assumes steady-state transonic solutions, but real galactic outflows may be time-dependent (e.g., due to intermittent supernovae). What is the range of validity for the steady-state assumption? Would time-dependent effects alter the three-phase evolution?

 

Minor issues:

1. The abstract should include a description of self-excited Alfvén waves.
2. The concluding part of the abstract should supplement the research conclusions of the reference model.
3. the numbering order of the references is chaotic.
4. There is no legend in Figure 2 of this paper.

Author Response

Reviewer 2 (Major)

1. The outflow is divided into coupling, decoupling, and re-coupling stages, with the coupling stage bounded by (10 < \xi < 15\) . What is the physical unit of \(\xi\) (e.g., kpc, pc)?

Our response:

Thank you. \Xi is the perpendicular distance from the galactic disk. We are using the normalized units in our study.

2. The manuscript contrasts the four-fluid model with Ko et al. [26]’s three-fluid model (e.g., non-monotonic velocity profiles in). To isolate the role of backward waves, could comparisons with control cases (e.g., "four-fluid without cooling" or "three-fluid with cooling") demonstrate that backward waves, not just enhanced cooling/damping, drive these differences?

Our response:

Thank you for asking this specific question. We have presented the cases with and without cooling and wave damping in the previous published article Ko et al. (2021, A&A 654, A63). Four fluid model is technically different then three fluid model as it is fact that with four-fluid model we have extra stochastic acceleration term attached (so we may not confuse three fluid model with four fluid is the current essence of the study). As we can see that cooling and wave damping terms acting oppositely in to the model sometime support acceleration sometime deceleration happens. So, the role of backward propagating waves is playing a crucial role in shifting regions (coupling/decoupling/recoupling far/near to the base of the gravitational potential well).

Our response:

3. The model assumes steady-state transonic solutions, but real galactic outflows may be time-dependent (e.g., due to intermittent supernovae). What is the range of validity for the steady-state assumption? Would time-dependent effects alter the three-phase evolution?

Our response:

Cosmic rays couple to plasma via magnetic fluctuations. The minimum fluctuation needed for the coupling is very small. Thus, it is easy for cosmic rays to couple with the plasma. On the other hand, the strength of the fluctuation cannot be much larger than the mean strength otherwise the quasilinear theory behind the model may breakdown. One of the consequences of the coupling is cosmic ray streaming instability, which may excite or damp hydromagnetic fluctuations or waves (depending on the anisotropy or pressure gradient of cosmic ray). We suppose the waves stand for turbulence. To keep the model to a manageable level we do not consider other wave or turbulent generating mechanisms (we consider wave damping mechanism, such as nonlinear Landau damping though). This approach has been adopted by, e.g., Breitschwerdt et al. (1991, A&A 245, 79), Dorfi & Breitschwerdt (2012, A&A 540, A77). The fluctuation level they used is about 0.1 to 0.5 which is similar to ours.

The solutions we studied are steady state solutions. Supposedly as all transients are gone, there is no causality in time. All solutions have to satisfy the boundary conditions. The boundary conditions could be interpreted as the sole driver of the system. It is possible that the boundary conditions include both forward and backward propagating waves, thus we might regard the boundary conditions as the driver of the waves. It is somewhat similar to the standing wave of a rope. Note that the backward propagating wave goes to zero at very large distances making stochastic acceleration ineffective.

 

Thank you for asking very crucial question regarding time dependency of our current model. At the galactic scales the steady state model is good approximation to study the galactic winds and we do think time dependence may influence the evolution of the galactic disk but more feasible approach to study time dependency is simulations-based approach for supernovae type of cases.

Minors

 

1.The abstract should include a description of self-excited Alfvén waves.

Our response:

Thank you, the introduction section has been updated. (p.1, Abstract section)

2. The concluding part of the abstract should supplement the research conclusions of the reference model.

Our response:

Thank you, the abstract section has been updated. (p.1, end of the Abstract section)

 

3. the numbering order of the references is chaotic.

Our response:

Thank you, the reference section has been updated with proper reference style. (p.14-16, Reference section)

4. There is no legend in Figure 2 of this paper.
   Our response:

Thank you, the Figure Legend has been updated. (p.13, Figure 2; )

Author Response File: Author Response.pdf

Round 2

Reviewer 2 Report

Comments and Suggestions for Authors

Work is suitable  for publication in universe.