the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
| 14 Jul 2025
Collisional Transport Coefficients In Kappa-Distributed Plasmas
Abstract. In this paper, we present a set of transport equations (continuity, momentum, and energy) using the Kappa velocity distribution as our zeroth-order function within the framework of the five-moment approximation. Then, we derive the corresponding transport momentum and energy coefficients using the Boltzmann collision integral. The results are expressed in terms of hypergeometric functions. These calculations have been done for three types of collisions: Coulomb collisions, hard-sphere interactions, and Maxwell molecules collisions. Furthermore, we explore the transport coefficients in two special cases: (1) the limiting case when kappa index approaches infinity, where the results converge to those of the Maxwellian distribution, and (2) the case of a non-drifting Kappa distribution. Finally, we discuss the behaviour of the transport coefficients in the case of Coulomb collisions for the Kappa distribution and compare it with the result of Maxwellian distribution.
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RC1: 'Comment on egusphere-2025-2874', Horst Fichtner, 25 Jul 2025
- AC2: 'Reply on RC1', Mahmood Jwailes, 26 Aug 2025
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RC2: 'Comment on egusphere-2025-2874', Marina Stepanova, 25 Jul 2025
GENERAL COMMENTS:
The manuscript by Jwailes et al., "Collisional transport coefficients in kappa-distributed plasmas," presents a closed system of transport equations obtained using a five-moment approximation for drifting kappa distributions. The authors evaluate transport coefficients for density, momentum, and energy for three types of collisions: Coulomb collisions, hard-sphere interactions, and Maxwell molecule collisions, considering that both types of particles are described by a drifting Kappa distribution. The resulting transport coefficients are also expressed in terms of the hypergeometric function. The authors further analyze two special cases: (1) the limiting case where the kappa index approaches infinity, resulting in convergence to the Maxwellian distribution, and (2) the case of a non-drifting Kappa distribution. Finally, the authors analyze in detail how the transport coefficients vary as a function of number density, temperature, and the difference between drift velocities, for different values of kappa, assuming that both species have the same kappa value.
The manuscript is well-written, presents interesting results, and I recommend its publication after the following comments have been addressed:
SPECIFIC COMMENTS:
- The introduction is too brief and contains a mixture of basic principles of kinetic plasma theory, an introduction to the Kappa distribution function, and a short list of observations of Kappa distributions in various space plasmas. From my perspective, it is necessary to better articulate the significance of “it’s significantly important to see how the transport equations and the transport coefficients will be affected when we expand the velocity distribution function around the Kappa distribution”. Specifically, what prior work has been done in this specific topic, what gap does this study aim to address?
- Sections 2 and 3 provide a summary of equations and definitions commonly found in advanced plasma physics textbooks. While this summary is useful for providing the reader with a smooth introduction to the topic, I suggest shortening these sections by moving a significant portion of the equations to the Appendix. This would allow the authors to better highlight the novel results presented in the subsequent sections.
- I would also suggest expanding the conclusions section to emphasize what new insights we have gained regarding collisional transport and its relevance to various applications in space physics.
TECHNICAL CORRECTIONS:
- In Figures 1-3, please add labels near the colorbars. The information regarding temperature and kappa can be embedded within the color plots themselves.
- As expected, in Figure 4, the lines generated for Kappa distributions with kappa values of 100 and 1000 are indistinguishable from the Maxwellian distribution. Please explore the possibility of using dotted and dash-dotted lines to determine if it is possible to differentiate this overlap visually.
Citation: https://doi.org/10.5194/egusphere-2025-2874-RC2 -
AC1: 'Reply on RC2', Mahmood Jwailes, 26 Aug 2025
The authors would like to thank Prof. Marina Stepanova for their feedback and for providing insightful recommendations on improving the motivation of the study and the clarity of the text. We have responded to the reviewer's comments in the pdf attached.
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- 1
The authors present an evaluation of transport coefficients in suprathermal,
so-called kappa-distributed plasmas. While I (Horst Fichtner, hf@tp4.rub.de)
think that the manuscript is well prepared and well written, some improvements
should be made before publication. I have the following remarks and suggestions:
Major:
(1) The authors have not related their study to closely related literature on
the same topic. There is a study by Du (2013, Phys. Plasmas, 20, 092901),
who studied transport coefficients in Lorentz plasmas with the (standard)
kappa-distribution. These results should be compared to those obtained in
the present study.
(2) Note, however, that the results derived by Du (2013) were based on a
variant of the kappa distribution that assumes kappa-independent temperature
rather than kappa-independent thermal velocity and that, thus, it may not
be suitable for various space plasmas, as discussed in Lazar et al. (2016,
Astron. Astrophys. 589, id.A39). Therefore, Husidic et al. (2021, Astron.
Astrophys. 654, A99) have revisited the task employing the alternative
(standard) kappa distribution. The results obtained in the present study
should also be related to those obtained in Husidic et al. (2021).
(3) Independent of which representation for the standard kappa distribution is
chosen, it is well-known that it has various serious limitations (kappa
greater than 3/2, diverging velocity moments, non-extensive entropy), see,
e.g., the discussion in Scherer et al. (2019, Astrophys. J. 881:93). To
overcome these problems, Scherer et al. (2017, Europhys. Lett., 120, 50002)
have introduced the regularised kappa distribution (RKD). The latter has
been employed in Husidic et al. (2022, Astrophys. J. 927:159) for deriving
more realistic transport coefficients in suprathermal space plasmas. The
limitations of the use of standard - as opposed to regularized - kappa
distributions should be mentioned.
Minor:
(a) The reference Burgers (1969) appears to be incomplete: missing publisher.
(b) Some citations should be in parantheses in the text.
(c) Note that even the Burgers' results for the Maxwellian case are not exact
but an approximation, see the discussion in Fichtner et al. (1996, J.
Plasma Phys. 55, 95).
(d) Given the brief outlook provided in section 8 of the present manuscript,
it should be noted that also the case of a magnetized space plasma has
already been studied in the context of transport coefficients (Guo & Du
2019, Physica A: Stat. Mech. Appl. 523, 156). That work has its limitations
but, nonetheless, may be of interest for later comparison purposes.
In summary, while I consider the results to be publishable, the study would
be significantly improved by putting it into the context of the existing
literature and to emphasise the differences to the latter and/or the new
aspects added.