arxiv: 2605.13056 · v1 · submitted 2026-05-13 · 🌌 astro-ph.SR · physics.plasm-ph

Recognition: 2 theorem links

· Lean Theorem

Resonant shear-flow instability in anisotropic supersonic plasmas with heat flux

Namig S. Dzhalilov

Authors on Pith no claims yet

Pith reviewed 2026-05-14 18:46 UTC · model grok-4.3

classification 🌌 astro-ph.SR physics.plasm-ph

keywords resonant instabilityshear flowtemperature anisotropyheat fluxcollisionless plasmasolar wind16-moment equationssupersonic flow

0 comments

The pith

A resonant instability develops in supersonic shear flows of anisotropic collisionless plasmas and disappears in the vortex sheet limit.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper analyzes the linear stability of oblique waves in a magnetized plasma with a smooth hyperbolic shear flow profile aligned to the magnetic field, using 16-moment fluid equations that include temperature anisotropy and heat flux. It finds an infinite set of complex eigenfrequencies corresponding to resonant modes that grow fastest when the wave phase velocity equals the mean flow velocity. The growth rate weakens at higher Mach numbers and for higher mode numbers, while heat flux effects are minor in supersonic conditions. This resonant mechanism vanishes for discontinuous flows, setting it apart from the Kelvin-Helmholtz instability, and it may account for observed transitions between isotropic and anisotropic proton temperatures in the solar wind.

Core claim

Within the 16-moment transport model for collisionless plasmas, the wave equation for disturbances in a homogeneous-density, uniform-field supersonic shear flow with a hyperbolic velocity profile reduces to an exactly solvable equation whose solutions form a discrete spectrum of complex frequencies. The resulting instability is resonant, attaining maximum growth when the phase velocity matches the flow speed, and the growth rate falls both with rising Mach number toward an asymptote and with increasing mode order; heat flux contributes negligibly. The unstable modes cease to exist in the vortex-sheet limit, unlike classical Kelvin-Helmholtz instability.

What carries the argument

Reduction of the governing wave equation to a solvable form with special functions for the hyperbolic shear velocity profile within the 16-moment closure.

If this is right

The growth rate of the instability approaches a constant asymptotic value as the Mach number increases.
Parallel heat flux exerts negligible influence on the instability under supersonic flow conditions.
The instability is absent in the discontinuous vortex sheet limit, distinguishing it from the Kelvin-Helmholtz mechanism.
This resonant instability can explain boundaries between isotropic and anisotropic proton temperature regions in low-beta solar wind plasma.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

This mechanism could be tested in other astrophysical contexts involving smooth supersonic shears, such as in stellar winds or jet boundaries.
Nonlinear saturation of these modes might lead to local isotropization of the proton temperature distribution.
Extensions to time-dependent or inhomogeneous backgrounds could reveal how the discrete spectrum evolves in realistic solar wind conditions.

Load-bearing premise

The unperturbed state must have spatially homogeneous density and magnetic field together with a smooth hyperbolic velocity profile so that the wave equation reduces to an exactly solvable form.

What would settle it

Observation of growing waves in solar wind data where phase velocity matches the local flow velocity at boundaries between isotropic and anisotropic proton temperature regions would support the resonant instability; the absence of such waves in detailed observations would challenge the proposed explanation.

Figures

Figures reproduced from arXiv: 2605.13056 by Namig S. Dzhalilov.

**Figure 1.** Figure 1: Overview of the solar wind corotating interaction regions (CIR) structure. The superposed averages of OMNI2 measurements for the 27 CIR events flow velocities of the solar wind are plotted as a function of time. The zero epoch is the CIR stream interface as determined by the maximum in the plasma vorticity. The plots extend from 2 days prior to the passage of the stream interface (in the slow wind) to 3 da… view at source ↗

**Figure 2.** Figure 2: Asymptotic behaviour of the complex dispersion function F(p, η) with respect to the real part of the spectral parameter η (ηR = Re(η)) for the case of large p (here p = 100) and ηi = Im(η) = 0.02. The simultaneous condition Re(F) = 0 and Im(F) = 0 corresponds to the existence of an infinite sequence of eigenvalues. It can be easily demonstrated that the equation F(p, η) = 0 possesses an infinite number of … view at source ↗

**Figure 3.** Figure 3: Exact numerical solutions of the dispersion equation F(p, η) = 0 as a function of the parameter p. The real parts of the eigenvalues ηR = Re(η) are plotted versus p. The curves ηR(p) correspond to individual modes n = 0, 1, 2, 3, . . . (from bottom to top, respectively). For higher modes (n ≥ 10), the dependencies ηR(p) become linear (red lines) and coincide exactly with the asymptotic analytical expressio… view at source ↗

**Figure 4.** Figure 4: Exact numerical solutions of the dispersion equation F(p, η) = 0 as a function of the parameter p. The imaginary parts of the eigenvalues ηi = Im(η) are plotted versus p. The curves from left to right correspond to the first ten modes n = 0, 1, 2, . . . , 10. For higher modes, instability (ηi > 0) appears at larger values of the parameter p. For p > 5, the growth rate rapidly approaches the asymptotic valu… view at source ↗

**Figure 5.** Figure 5: The wave-flow resonance condition, ηR(p, n) = 0 is represented on the plot. The curve illustrates the values of the parameter p for the first 20 modes, n = 0, 1, · · · , 20. The locations of the points based on the exact solution (44) and the asymptotic formula (47) are practically identical. by the Wind spacecraft indicate that the actual temperature anisotropy at 1 AU remains confined within certain limi… view at source ↗

**Figure 6.** Figure 6: Growth rates Γ(p) of the first ten modes (n = 0, 1, . . . , 10). The maxima correspond to the resonance conditions of the modes with the mean flow. region, concentrated around βp = 1. The dependence of proton temperature anisotropy on β∥p, is presented in figure 7, which is borrowed from Hellinger et al. (2006). As can be seen in the figure, the rhomboidal region (α, β∥p) of anisotropy is bounded within 10… view at source ↗

**Figure 7.** Figure 7: A color scale plot of the relative frequency of (β∥p, T⊥p/T∥p) in the WIND/SWE data (1995–2001) for the solar wind velocity 600 km/s. The (logarithmic) color scale is show on the right. The over plotted curves show the contours of the maximum normed growth rate in the corresponding bi-Maxwellian plasma (left) for the proton cyclotron instability (solid curves) and the parallel fire hose (dashed curves) and… view at source ↗

**Figure 8.** Figure 8: The curves representing the left boundaries, extracted from the figure 7, indicate the transition between isotropic and anisotropic regions for low plasma beta conditions, βp < 1. At the marked points along these boundaries, the maximum growth rates, Γmax, were determined [PITH_FULL_IMAGE:figures/full_fig_p017_8.png] view at source ↗

**Figure 9.** Figure 9: All potential values of kz/σ, representing the ratio of the characteristic scale of flow velocity shear to the wavelength along the flow direction, are determined at the anisotropy boundaries shown in the previous figure 8 for the case of maximum growth rate of the fundamental mode with n = 0. The two plotted curves, corresponding to α < 1 (red curve) and α > 1 (blue curve), exhibit almost complete overla… view at source ↗

**Figure 10.** Figure 10: Possible values of the ratio kz/σ (numbers indicated on the curves) within the observed anisotropy range (βp < 100, α < 5), for which the growth rate of the fundamental mode (n = 0) reaches its maximum value. Outside the interval 0.01 < kz/σ < 3, the shear-flow instability does not occur. mechanisms governing collisionless dissipation and the subsequent heating of the coronal and solar wind plasmas remain… view at source ↗

read the original abstract

This work is devoted to the study of the influence of temperature anisotropy and parallel heat flux on the stability of supersonic shear flow in collisionless plasmas. Within a fluid-based framework, we employ the 16-moment transport equations -- derived from the Vlasov-Maxwell system -- to describe the plasma dynamics. By performing a modal analysis we investigate the oblique propagation of linear disturbances within a magnetized plasma characterized by a shear flow of arbitrary profile aligned with the ambient magnetic field. In the unperturbed state, both the plasma density and the magnetic field are assumed to be homogeneous. For a smooth, hyperbolic velocity profile representing supersonic shear, the governing wave equation is reduced to a form amenable to an exact analytical solution. Analytical solutions are expressed in terms of special functions that yield an infinite discrete spectrum of complex eigenfrequencies ($n = 0, 1, 2, \dots$). The instability is identified as resonant, peaking when the wave phase velocity matches the mean flow velocity, with the growth rate decreasing for higher-order modes. The results indicate that, while heat flux exerts a negligible influence under conditions of supersonic flow, the growth rate decreases and approaches an asymptotic value as the Mach number increases. Notably, the instability vanishes in the vortex sheet limit, distinguishing it from the classical Kelvin-Helmholtz mechanism. These findings suggest that this specific instability holds significant potential for explaining the problem of observed boundaries between isotropic and anisotropic proton temperature regions in a low-beta solar wind plasma.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper reduces the 16-moment wave equation for a specific hyperbolic supersonic shear profile to special-function solutions that give a discrete resonant spectrum, but the solar-wind application rests on background assumptions whose validity is not shown.

read the letter

The main result is an exact analytical treatment of oblique linear modes in a magnetized supersonic shear flow that includes temperature anisotropy and parallel heat flux. For the tanh velocity profile with uniform density and B, the governing equation reduces to a form solved by special functions, producing an infinite discrete set of complex eigenfrequencies. The growth rate peaks when the phase speed matches the local flow speed, falls for higher-order modes, and becomes negligible once the Mach number is large; heat flux itself has little influence. The mode disappears in the vortex-sheet limit, which separates it from classical Kelvin-Helmholtz instability. That analytical spectrum for this combination of effects is new relative to the cited literature. The setup is straightforward and the distinction from KH is cleanly stated. The soft spots are the assumptions required for the reduction: spatially constant density and magnetic field, a smooth hyperbolic profile, and continued validity of the 16-moment closure during growth. The abstract gives no explicit derivation steps or numerical checks against the original system, so it is not possible to judge how sensitive the spectrum is to small changes in the background or closure. The proposed link to observed isotropic-anisotropic boundaries in the low-beta solar wind therefore remains suggestive rather than demonstrated. This is for readers who work with fluid closures for collisionless plasmas and want analytical spectra for shear flows. It is worth sending to peer review because the exact solution is concrete enough to be examined and extended, even if the physical range needs tighter justification.

Referee Report

2 major / 2 minor

Summary. The manuscript investigates the linear stability of supersonic shear flows in anisotropic collisionless plasmas using the 16-moment transport equations derived from the Vlasov-Maxwell system. For a hyperbolic velocity profile U(x) = U0 tanh(x/L) with uniform background density and magnetic field, the governing wave equation for oblique perturbations is reduced to a form solvable exactly in terms of special functions, producing an infinite discrete spectrum of complex eigenfrequencies. The resulting mode is identified as resonant, with maximum growth when the phase velocity equals the local flow velocity; growth rates decrease with mode order, become negligible with increasing Mach number, and vanish in the vortex-sheet limit, distinguishing the mechanism from classical Kelvin-Helmholtz instability. Heat-flux effects are reported as negligible under supersonic conditions, and the instability is proposed as a candidate explanation for observed boundaries between isotropic and anisotropic proton-temperature regions in low-beta solar wind.

Significance. If the analytical reduction is verified to satisfy the original 16-moment system and the closure remains valid, the work supplies an exactly solvable resonant instability with a discrete spectrum and clear parametric dependence on Mach number and anisotropy. This offers falsifiable predictions for growth rates and mode structure that could be tested against solar-wind observations, providing a potential fluid-based mechanism for temperature-anisotropy transitions without invoking kinetic effects.

major comments (2)

[Derivation of the wave equation and eigenfrequency spectrum] The central claim rests on reducing the linearized 16-moment system to a wave equation solvable by special functions for the specific hyperbolic profile. The manuscript must supply the explicit intermediate steps of this reduction and a direct substitution verification showing that the obtained eigenfrequencies satisfy the original 16-moment equations (including the heat-flux and anisotropy evolution equations) rather than an approximated or truncated form.
[Analysis of the vortex-sheet limit] The assertion that the instability vanishes in the vortex-sheet limit and is thereby distinct from Kelvin-Helmholtz relies on the exact solvability for the smooth profile. The limiting procedure (L → 0) must be shown explicitly, confirming that the growth rate approaches zero while preserving the resonance condition and that no singular KH-like branch emerges within the 16-moment closure.

minor comments (2)

[Analytical solution] Clarify the precise definition and normalization of the special functions employed in the solution; include their recurrence relations or asymptotic forms used to extract the complex eigenfrequencies.
[Parametric study] The statement that heat flux exerts negligible influence should be quantified by showing the scaling of the growth rate with the heat-flux parameter across the reported Mach-number range.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of the manuscript and the constructive major comments. We address each point below and agree that expanding the derivation details and the limiting analysis will improve clarity and rigor. The revised manuscript will incorporate these additions.

read point-by-point responses

Referee: [Derivation of the wave equation and eigenfrequency spectrum] The central claim rests on reducing the linearized 16-moment system to a wave equation solvable by special functions for the specific hyperbolic profile. The manuscript must supply the explicit intermediate steps of this reduction and a direct substitution verification showing that the obtained eigenfrequencies satisfy the original 16-moment equations (including the heat-flux and anisotropy evolution equations) rather than an approximated or truncated form.

Authors: We acknowledge that the current manuscript presents the final reduced wave equation and its special-function solutions without all intermediate algebraic steps. In the revised version we will add a dedicated appendix that starts from the linearized 16-moment equations, eliminates the perturbation variables step by step (including the heat-flux and anisotropy evolution equations), and arrives at the wave equation. We will also include an explicit substitution verification for the fundamental mode (n=0) that confirms the eigenfrequency satisfies the full original system; the structure of the solution ensures the same holds for higher modes. This addition directly addresses the request for verification beyond any possible truncation. revision: yes
Referee: [Analysis of the vortex-sheet limit] The assertion that the instability vanishes in the vortex-sheet limit and is thereby distinct from Kelvin-Helmholtz relies on the exact solvability for the smooth profile. The limiting procedure (L → 0) must be shown explicitly, confirming that the growth rate approaches zero while preserving the resonance condition and that no singular KH-like branch emerges within the 16-moment closure.

Authors: We agree that an explicit demonstration of the L → 0 limit is required. In the revised manuscript we will add a subsection that takes the analytic dispersion relation obtained from the special-function solution and shows that the imaginary part of the frequency tends to zero as L decreases while the resonance condition (phase velocity equal to local flow velocity) remains satisfied. We will further demonstrate that, within the 16-moment closure, the boundary conditions remain regular and no additional singular branch analogous to classical Kelvin-Helmholtz appears. This explicit limiting analysis will confirm the distinction from KH instability. revision: yes

Circularity Check

0 steps flagged

No significant circularity; analytical reduction is self-contained

full rationale

The paper starts from the standard 16-moment transport equations (derived from Vlasov-Maxwell), states the assumptions of homogeneous density and magnetic field in the unperturbed state, prescribes the hyperbolic velocity profile U(x) = U0 tanh(x/L), and reduces the resulting wave equation to an exactly solvable form in special functions. The discrete spectrum, resonant character (phase velocity matching mean flow), decreasing growth rates with mode number and Mach number, negligible heat-flux effect, and vanishing growth in the vortex-sheet limit all follow directly from this eigenvalue solution. None of these results are fitted to data, redefined by construction, or justified only via self-citation chains. The 16-moment closure and homogeneous-background assumptions are declared upfront rather than derived from the eigenfrequencies themselves. This is a standard analytical derivation with no load-bearing circular steps.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the 16-moment closure being an adequate description of collisionless plasma dynamics and on the choice of a smooth hyperbolic velocity profile that permits exact solution.

axioms (1)

domain assumption 16-moment transport equations derived from the Vlasov-Maxwell system constitute a valid fluid closure for collisionless anisotropic plasmas
Invoked at the outset to replace the full kinetic description.

pith-pipeline@v0.9.0 · 5562 in / 1237 out tokens · 35131 ms · 2026-05-14T18:46:41.721740+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

the instability vanishes in the vortex sheet limit, distinguishing it from the classical Kelvin-Helmholtz mechanism

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

120 extracted references · 120 canonical work pages

[1]

G. K. Batchelor , title =. J. Fluid Mech. , year =

work page
[2]

and Drazin, P

Blumen, W. and Drazin, P. G. and Billings, D. F. , title=. J. Fluid Mech. , volume =. 1975 , doi=

work page 1975
[3]

and Zank, G

Hunana, P. and Zank, G. P. , title =. Astrophys. J. , volume =. 2017 , doi =

work page 2017
[4]

and Tenerani, A

Hunana, P. and Tenerani, A. and Zank, G. P. and Khomenko, E. and Goldstein, M. L. and Webb, G. M. and Cally, P. S. and Collados, M. and Velli, M. and Adhikari, L. , title =. J. Plasma Phys. , volume =. 2019 , doi =

work page 2019
[5]

and Settino, A

Malara, F. and Settino, A. and Perrone, D. and Pezzi, O. and Guzzi, G. and Valentini, F. , title =. Astrophys. J. , volume =. 2022 , doi =

work page 2022
[6]

and Settino, A

Maiorano, T. and Settino, A. and Malara, F. and Pezzi, O. and Pucci, F. and Valentini, F. , title =. J. Plasma Phys. , volume =. 2020 , doi =

work page 2020
[7]

Drazin, P. G. and Reid, W. H. , title =

work page
[8]

and Cerri, S

Henri, P. and Cerri, S. S. and Califano, F. and others , title =. Phys. Plasmas , volume =. 2013 , doi =

work page 2013
[9]

Ismayilli, R. F. and Dzhalilov, N. S. and Shergelashvili, B. M. and Poedts, S. and Pirguliyev, M. S. , title =. Phys. Plasmas , volume =. 2018 , doi =

work page 2018
[10]

Belyaev, M. A. and Rafikov, R. R. , title =. Astrophys. J. , volume =. 2012 , doi =

work page 2012
[11]

Ray, T. P. , title =. Mon. Not. R. Astron. Soc. , volume =

work page
[12]

, title =

Glatzel, W. , title =. Mon. Not. R. Astron. Soc. , volume =

work page
[13]

Galtier, S and Nazarenko, SV and Newell, AC and Pouquet, A , Title =

work page
[14]

C. J. Brownell and L. K. Su , title =. AIAA Paper 2004-2335 , year =

work page 2004
[15]

C. J. Brownell and L. K. Su , title =. AIAA Paper 2007-1314 , year =

work page 2007
[16]

S. C. R. Dennis , title =. Ninth Intl Conf. on Numerical Methods in Fluid Dynamics , publisher =. 1985 , editor =

work page 1985
[17]

Borovsky, J. E. and Denton, M. H. , title =. J. Geophys. Res. , volume =. 2010 , doi =

work page 2010
[18]

Hwang and E

L.-S. Hwang and E. O. Tuck , title =. J. Fluid Mech. , year =

work page
[19]

M. G. Worster , title =. In Interactive dynamics of convection and solidification , publisher =. 1992 , editor =

work page 1992
[20]

Koch , title =

W. Koch , title =. J. Sound Vib. , year =

work page
[21]

Lee , title =

J.-J. Lee , title =. J. Fluid Mech. , year =

work page
[22]

C. M. Linton and D. V. Evans , title =. Phil.\ Trans.\ R. Soc.\ Lond. , year =

work page
[23]

P. A. Martin , title =. 1980 , journal =

work page 1980
[24]

R. S. Rogallo , title =

work page
[25]

Ursell , title =

F. Ursell , title =. 1950 , journal =

work page 1950
[26]

On the oscillations near and at resonance in open pipes , year =

L. On the oscillations near and at resonance in open pipes , year =. J. Engng Maths , volume =

work page
[27]

P. L. Miller , title =. 1991 , address =

work page 1991
[28]

and Pritchett, P

Miura, A. and Pritchett, P. L. , title =. J. Geophys. Res. , volume =. 1982 , doi =

work page 1982
[29]

and Lovelace, R

Roy Choudhury, S. and Lovelace, R. V. E. , title =. Astrophys. J. , volume =. 1986 , doi =

work page 1986
[30]

, title =

Camporeale, E. , title =. Space Sci. Rev. , volume =. 2012 , doi =

work page 2012
[31]

Kaghashvili, E. K. , journal =

work page
[32]

Kaghashvili, E. K. and Esser, R. , title =. Astrophys. J. , volume =. 2000 , doi =

work page 2000
[33]

, title =

Rogava, A. , title =. Astrophys. Space Sci. , volume =. 2004 , doi =

work page 2004
[34]

Li, J. W. and Chen, Y. and Li, Z. Y. , title =. Phys. Plasmas , volume =. 2006 , doi =

work page 2006
[35]

Shergelashvili, B. M. and Poedts, S. and Pataraya, A. D. , title =. Astrophys. J. , volume =. 2006 , doi =

work page 2006
[36]

and Burgess, D

Camporeale, E. and Burgess, D. and Passot, T. , title =. Phys. Plasmas , volume =. 2009 , doi =

work page 2009
[37]

Osman, K. T. and Matthaeus, W. H. and Hnat, B. and Chapman, S. C. , title =. Phys. Rev. Lett. , volume =. 2012 , doi =

work page 2012
[38]

Talwar, S. P. , title =. Phys. Fluids , volume =. 1965 , doi =

work page 1965
[39]

and Patel, V

Roy Choudhury, S. and Patel, V. L. , journal =

work page
[40]

and El-Rabii, H

Peyrichon, G. and El-Rabii, H. and Ripoll, J.-F. and Casner, A. and Michael, A. and Merkin, V. G. and Hurricane, O. A. , title =. Phys. Plasmas , volume =. 2025 , doi =

work page 2025
[41]

and Travnicek, P

Stverak, S. and Travnicek, P. and Maksimovic, M. , title =. J. Geophys. Res. , volume =. 2008 , doi =

work page 2008
[42]

Balbus, S. A. and Hawley, J. F. , title =. Astrophys. J. , volume =. 1991 , doi =

work page 1991
[43]

Maruca, B. A. and Kasper, J. C. and Bale, S. D. , title =. Phys. Rev. Lett. , volume =. 2011 , doi =

work page 2011
[44]

and Shaaban, S

Lazar, M. and Shaaban, S. M. and Poedts, S. and Stverak, S. , title=. Mon. Not. R. Astron. Soc. , volume =. 2017 , doi=

work page 2017
[45]

and Lazar, M

Vafin, S. and Lazar, M. and Fichtner, H. and Schlickeiser, R. and Drillisch, M. , title =. Astron. Astrophys. , volume =. 2018 , doi =

work page 2018
[46]

and Osman, K

Servidio, S. and Osman, K. T. and Valentini, F. , title=. Astrophys. J. , volume =. 2014 , doi=

work page 2014
[47]

and Schlickeiser, R

Ibscher, D. and Schlickeiser, R. , title=. Phys. Plasmas , volume =. 2014 , doi =

work page 2014
[48]

and Skoda, T

Schlickeiser, R. and Skoda, T. , title =. Astrophys. J. , volume =. 2010 , doi =

work page 2010
[49]

and Riazantseva, M

Vafin, S. and Riazantseva, M. and Pohl, M. , title =. Astrophys. J. Lett. , volume =. 2019 , doi =

work page 2019
[50]

Yoon, P. H. and Lazar, M. and Salem, C. and Seough, J. and Martinovi. Boundary of the Distribution of Solar Wind Proton Beta versus Temperature Anisotropy , journal =. 2024 , doi =

work page 2024
[51]

Zhao, G. Q. and Li, H. and Feng, H. Q. and Wu, D. J. and Li, H. B. and Zhao, A. , title =. Astrophys. J. , volume =. 2019 , doi =

work page 2019
[52]

and Zhao, J

Sun, H. and Zhao, J. and Liu, W. and Xie, H. and Wu, D. , journal =

work page
[53]

Martinovic, M. M. and Klein, K. G. and Durovcova, T. and Alterman, B. L. , title =. Astrophys. J. , volume =. 2021 , doi =

work page 2021
[54]

and Hellinger, P

Matteini, L. and Hellinger, P. and Landi, S. and Travn. Ion Kinetics in the Solar Wind: Coupling Global Expansion to Local Microphysics , journal =. 2012 , doi =

work page 2012
[55]

Cranmer, S. R. , title =. Astrophys. J. Suppl. , volume =. 2014 , doi =

work page 2014
[56]

and Ofman, L

Ozak, N. and Ofman, L. and Vi. Ion Heating in Inhomogeneous Expanding Solar Wind Plasma: The Role of Parallel and Oblique Ion-Cyclotron Waves , journal =. 2015 , doi =

work page 2015
[57]

Chen, C. H. K. and Matteini, L. and Schekochihin, A. A. and Stevens, M. L. and Salem, C. S. and Maruca, B. A. and Kunz, M. W. and Bale, S. D. , title =. Astrophys. J. Lett. , volume =. 2016 , doi =

work page 2016
[58]

and Klein, K

Verscharen, D. and Klein, K. G. and Maruca, B. A. , title =. Living Rev. Sol. Phys. , volume =. 2019 , doi =

work page 2019
[59]

Kasper, J. C. and Lazarus, A. J. and Gary, S. P. , title =. AIP Conf. Proc. , pages =. 2003 , doi=

work page 2003
[60]

Bale, S. D. and Kasper, J. C. and Howes, G. G. , title =. Phys. Rev. Lett. , volume =. 2009 , doi =

work page 2009
[61]

Maruca, B. A. and Kasper, J. C. and Gary, S. P. , title =. Astrophys. J. , volume =. 2012 , doi =

work page 2012
[62]

and Yoon, P

Seough, J. and Yoon, P. H. and Kim, K. H. , title =. Phys. Rev. Lett. , volume =. 2013 , doi =

work page 2013
[63]

and Kasper, J

Huang, J. and Kasper, J. C. and Vech, D. , title =. Astrophys. J. Suppl. , volume =. 2020 , doi =

work page 2020
[64]

Hellinger, P. and Tr. Solar wind proton temperature anisotropy: Linear theory and WIND/SWE observations , journal =. 2006 , doi =

work page 2006
[65]

and Travnicek, P

Hellinger, P. and Travnicek, P. M. , title =. Astrophys. J. Lett. , volume =. 2014 , doi =

work page 2014
[66]

and Arregui, I

Hillier, A. and Arregui, I. and Matsumoto, T. , title =. Astrophys. J. , volume =. 2024 , doi =

work page 2024
[67]

Phillips, J. L. and Gosling, J. T. , title =. J. Geophys. Res. , volume =. 1990 , doi =

work page 1990
[68]

and Belmont, G

Chus, T. and Belmont, G. , title =. Phys. Plasmas , volume =. 2006 , doi =

work page 2006
[69]

Coleman, P. J. , title =. Astrophys. J. , volume =. 1968 , doi =

work page 1968
[70]

Belcher, J. W. and Davis, L. , title =. J. Geophys. Res. , volume =. 1971 , doi =

work page 1971
[71]

Terry, P. W. , title =. Rev. Mod. Phys. , volume =. 2000 , doi =

work page 2000
[72]

Viall, N. M. and DeForest, C. E. and Kepko, L. , title =. Front. Astron. Space Sci. , volume =. 2021 , doi =

work page 2021
[73]

Birch, M. J. and Hargreaves, J. K. , title =. Adv. Space Res. , volume =. 2021 , doi =

work page 2021
[74]

Dzhalilov, N. S. and Kuznetsov, V. D. and Staude, J. , title =. Contrib. Plasma Phys. , volume =. 2011 , doi =

work page 2011
[75]

Cerri, S. S. and Pegoraro, F. and Califano, F. and Del Sarto, D. and Jenko, F. , title =. Phys. Plasmas , volume =. 2014 , doi =

work page 2014
[76]

Kuznetsov, V. D. and Dzhalilov, N. S. , title =. Plasma Phys. Rep. , volume =. 2009 , doi =

work page 2009
[77]

Dzhalilov, N. S. and Kuznetsov, V. D. , title =. Plasma Phys. Rep. , volume =

work page
[78]

Fejer, J. A. , title =. Phys. Fluids , volume =. 1964 , doi =

work page 1964
[79]

Gerwin, R. A. , title =. Rev. Mod. Phys. , volume =

work page
[80]

and Hasegawa, A

Chen, L. and Hasegawa, A. , title =. J. Geophys. Res. , volume =. 1974 , doi =

work page 1974

Showing first 80 references.