Thermodynamics and collisionality in firehose-susceptible high-$\beta$ plasmas

A. F. A. Bott; E. Quataert; J. Squire; L Arzamasskiy; M. W. Kunz

REVIEW 3 major objections 4 minor 42 references

Thermodynamics and collisionality in firehose-susceptible high-$\beta$ plasmas

T0 review · 3 major / 4 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read The paper claims that firehose-susceptible high-β plasmas saturate into three distinct states, including a newly identified Alfvén-enabling state in which linear Alfvén waves survive.

desk verdict Despite a calibrated prefactor and an untested seed dependence in N_fold, the three-regime picture of firehose saturation is credible and worth refereeing seriously. read the letter →

arxiv 2501.13663 v2 pith:JQYSXA3F submitted 2025-01-23 astro-ph.HE physics.plasm-phphysics.space-ph

A. F. A. Bott , M. W. Kunz , E. Quataert , J. Squire , L Arzamasskiy This is my paper · ORCID

classification astro-ph.HEphysics.plasm-phphysics.space-ph

keywords firehoseinstabilitypressureanisotropyAlfvén-enablingstateAlfvén-inhibitingeffectivecollisionalityhigh-betaplasmahybridparticle-in-cellsimulationsAlfvénwaves

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper seeks to establish that the firehose instability, driven by excess parallel pressure in a high-$\beta$ collisionless plasma, has more than one saturation outcome. Depending on the plasma $\beta$ $\beta_i$ and the product $\tau\Omega_i$ of the macroscopic evolution time with the ion Larmor frequency, the saturated state is ultra-high-$\beta$, Alfvén-inhibiting, or the newly identified Alfvén-enabling state. In the Alfvén-enabling state, reached when $\tau$ exceeds $\tau_{\mathrm{cr}}\sim \beta_i^{1.6}\Omega_i^{-1}$, the pressure anisotropy is held near $-1.6/\beta_i$, so the effective Alfvén speed retains roughly 20% of its isotropic value. If correct, this matters because slowly evolving high-$\beta$ astrophysical plasmas are precisely the ones that remain able to support Alfvén waves and Alfvénic turbulence, with their thermodynamics set by an anomalous scattering rate $\nu_{\mathrm{eff}}\sim\beta_i/\tau$.

What carries the argument

The load-bearing object is the wavenumber-dependent firehose threshold together with an e-folding criterion built on it. Kinetic oblique modes at $k_\parallel\rho_i\approx0.45$ and $k_\perp\rho_i\approx0.35$ are destabilized at $\Delta\approx -1.35/\beta_i$, which is less negative than the long-wavelength fluid threshold $\Delta=-2/\beta_i$ at which the Alfvén restoring force is cancelled. Because the oblique growth rate near threshold is $\gamma\approx0.3\Omega_i\beta_i^{-0.6}$, the condition $\gamma\Delta t\approx N_{\mathrm{fold}}$ with $N_{\mathrm{fold}}\approx5$ converts into $\tau_{\mathrm{cr}}\approx27\beta_i^{1.6}\Omega_i^{-1}$. This threshold separates the Alfvén-inhibiting state, where the broad spectrum of long-wavelength modes is destabilized before regulation, from the Alfvén-enabling state, where ion-Larmor-scale modes regulate the pressure anisotropy first. In the Alfvén-enabling state the effective collision operator is a quasilinear pitch-angle scattering operator whose rate depends on parallel velocity.

What would settle it

A decisive test is to compare the first-minimum anisotropy at fixed $\beta_i$ across shearing and expanding simulations; the paper's model predicts that $(\Delta_i)_{\min}\beta_i$ should follow one master curve, approximately $-1.35 - 5.1\,\beta_i/(\tau\Omega_i)^{0.625}$, with the Alfvén-enabling transition at $\tau\Omega_i\approx27\beta_i^{1.6}$. If shearing and expansion data at matched parameters do not fall on the same curve, the e-folding calibration is not universal.

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Extended reading notes

Core claim

The paper's central claim is that the saturation of the ion firehose instability in a high-$\beta$ collisionless plasma produces three qualitatively distinct thermodynamic states. The previously known ultra-high-$\beta$ state occurs at $\tau\lesssim\beta_i/\Omega_i$; the Alfvén-inhibiting state occurs for $\beta_i/\Omega_i \ll \tau \lesssim \tau_{\mathrm{cr}}(\beta_i)$, with $\Delta_{\mathrm{sat}}\approx -2/\beta_i$ so the effective Alfvén speed vanishes; and the newly identified Alfvén-enabling state occurs for $\tau\gtrsim\tau_{\mathrm{cr}}$, where $\Delta_{\mathrm{sat}}\approx -1.6/\beta_i$ and $v_{\mathrm{A,eff}}^2/v_A^2\approx0.2$. The critical timescale is $\tau_{\mathrm{cr}}\approx27\beta_i^{1.6}\Omega_i^{-1}$ for $1\ll\beta_i\ll10^5$, derived by requiring that oblique ion-Larmor-scale firehose modes, unstable already at $\Delta\approx -1.35/\beta_i$, grow for about $N_{\mathrm{fold}}\approx5$ e-foldings before the anisotropy reaches the fluid threshold $-2/\beta_i$. In the Alfvén-enabling state the magnetic fluctuations live at ion-Larmor scales, split between oblique firehose modes and secondary quasi-parallel modes, and their backreaction is approximated well by a quasilinear pitch-angle scattering operator with box-averaged rate $\nu_{\mathrm{eff}}\approx0.4\beta_i/\tau$ but with much weaker scattering of suprathermal ions. The paper shows that the solar wind, the intracluster medium, and black-hole accretion flows are all in the Alfvén-enabling regime.

Load-bearing premise

The phase boundary between the Alfvén-inhibiting and Alfvén-enabling states rests on the assumption that oblique ion-Larmor-scale firehose modes need about five e-folding times to begin regulating the pressure anisotropy, a number calibrated on the same simulations used to test the boundary.

Editorial extensions

If this is right

For $1\ll\beta_i\ll10^5$, the transition to the Alfvén-enabling state occurs at $\tau_{\mathrm{cr}}\approx27\beta_i^{1.6}\Omega_i^{-1}$; beyond it, $\Delta_{\mathrm{sat}}\approx -1.6/\beta_i$ and $v_{\mathrm{A,eff}}^2/v_A^2\approx0.2$.
In the Alfvén-enabling state the box-averaged effective collisionality is $\nu_{\mathrm{eff}}\approx0.4\beta_i/\tau$, corresponding to a Braginskii viscosity $\mu_B\approx0.8\tau B^2/4\pi$.
Firehose fluctuations in that state are concentrated at ion-Larmor scales and split into oblique modes and secondary quasi-parallel modes, with $\delta B^2/B_0^2\sim\beta_i^{1/4}(\tau\Omega_i)^{-1/2}$.
Applied to the solar wind, the intracluster medium, and black-hole accretion flows, the conclusion is that all three sit in the Alfvén-enabling regime and can support Alfvén waves and Alfvénic turbulence.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

An open extension is whether the $\tau_{\mathrm{cr}}\sim\beta_i^{1.6}\Omega_i^{-1}$ boundary survives under shearing rather than uniformly expanding background flows; the model suggests it should, but the paper does not run that comparison.
If the Alfvén-enabling state is generic, MHD-scale Alfvénic turbulence models for cluster and accretion-flow plasmas remain viable, but the secondary quasi-parallel firehose modes identified here could act as a velocity-selective sink of turbulent energy at ion-Larmor scales.
The velocity-dependent scattering operator implies suprathermal ions stay more anisotropic than thermal ions in saturation, a signature that might be observable in velocity-resolved spectra of the intracluster medium or accretion flows.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Colleague —

This is the paper that puts the Alfvén-inhibiting vs Alfvén-enabling firehose saturation states on a phase map, and it mostly holds up. The new material is real: a quantitative transition timescale τ_cr ≈ 27 β_i^1.6 Ω_i^{-1}, an FLR-based derivation of the kinetic threshold Δ_cr ≈ −1.35/β_i, the identification of secondary parallel firehose modes, and a quasilinear pitch-angle-scattering operator for the Alfvén-enabling state. The simulation campaign is large — 24 HEB runs spanning β_i = 6–200, τΩ_i ~ 2e2–5e4 — and the collapse of the measured (Δ_i)_min onto a single function of τΩ_i/β_i^1.6 (Fig. 5) is real evidence that this parameter controls the transition.

What I think is solid: the FLR threshold story is a genuine analytic step tested against the linear solver; the Alfvén-enabling state is credible for slowly evolving high-β plasmas; and the conclusion that the ICM, accretion flows, and solar wind sit in that regime follows from their enormous τΩ_i, so the astrophysical punchline is robust to order-unity shifts in the prefactor.

Soft spots, in proportion. First, the boundary is partly calibration, not derivation: N_fold ≈ 5 is set from the same simulations used to validate the curve (eqs. 3.5–3.7), and the β^1.6 exponent inherits the empirical growth-rate fit (eq. 2.7), whose error bars are not reported. The stress-test concern about seed amplitude is legitimate: in PIC runs the seed is numerical noise, and if the seed level varies with β, the inferred power law picks up a logarithmic correction. This should be tested with N_ppc/noise variation and, ideally, 3D runs. It does not change the qualitative regime assignment for astrophysical plasmas. Second, the δB^2 scaling in the Alfvén-inhibiting state is explicitly unresolved — Melville et al. get (β/τΩ)^{1/2} while these runs get β/τΩ — so Table 1's entry carries a question mark. To their credit, the paper says so plainly and offers plausible causes, but the simulation is 2.5D and this is where a 3D check matters most. Third, they disclose their own limitations: d_i is unresolved in the β_0 = 200 runs (acceptable for ρ_i-scale firehose modes), and numerical collisionality has a quantitative effect on some results (App. B). Sections 5–8 were truncated in my copy, so I could not fully check the collision-operator details; nothing I saw suggests a load-bearing flaw.

Who it is for: plasma astrophysicists working on high-β microphysics, turbulence, ICM, and accretion theory. It deserves a serious referee — the claim is important, the evidence base is large, and the soft spots are fixable rather than fatal. Send it to review, and tell the referee to push on the N_fold derivation and seed sensitivity.

Referee Report

3 major / 4 minor

Summary. The paper studies the nonlinear saturation of the ion firehose instability in high-beta collisionless plasmas using linear kinetic theory, a suite of hybrid expanding-box (HEB) particle-in-cell simulations spanning beta_i = 6-200 and tau Omega_i ~ 2e2-5e4, and a quasilinear scattering model. The central claim is that the post-saturation state of a firehose-unstable plasma falls into three qualitatively distinct regimes - ultra-high-beta, Alfvén-inhibiting, and a newly identified Alfvén-enabling regime - depending on beta_i and tau Omega_i. The Alfvén-enabling regime is reached when tau exceeds a critical value tau_cr ~ beta_i^{1.6} Omega_i^{-1}, in which case the saturated anisotropy is Delta_sat ~ -1.6/beta_i, so that v_A,eff^2/v_A^2 ~ 0.2 and linear Alfvén waves and Alfvénic turbulence can still propagate. The paper also characterizes the magnetic fluctuation spectra, the effective collision operator, and the velocity-space structure of the ion distribution, and it concludes that plasmas such as the solar wind, the intracluster medium, and black-hole accretion flows lie in the Alfvén-enabling regime.

Significance. If correct, the identification and characterization of the Alfvén-enabling state is an important step forward in understanding firehose saturation in high-beta, low-collisionality astrophysical plasmas: it changes the expected effective Alfvén speed, wave propagation, and turbulent transport properties of these systems. The paper's strengths include a broad simulation campaign with a clean scaling collapse in Fig. 5, a careful characterization of the magnetic-energy spectra and of the non-Maxwellian distribution functions, the identification of secondary parallel firehose modes, and a transparent discussion of the empirical ingredients in the theory (e.g., N_fold, growth-rate fits). The paper is honest about the empirical nature of some inputs, which makes its claims more testable rather than less.

major comments (3)

[§3.2, Eq. (3.7) and §4.3.1, Eq. (4.11)] The transition timescale tau_cr ≈ 5 N_fold beta_i^{1.6} Omega_i^{-1} is not determined ab initio: the number of e-foldings N_fold ≈ 5 is stated as found 'in our simulations', and the fit (4.11) subsequently uses N_fold ≈ 5.4 to reproduce the simulated values of (Delta_i)_min. Since the same HEB simulations are used to set N_fold and to validate the phase boundary (Fig. 5), the scaling collapse is partly a test of internal consistency rather than an independent confirmation of the functional form. Please report the sensitivity of the predicted boundary to N_fold, for example by varying the initial fluctuation seed amplitude or N_ppc, or by presenting the expected shift in tau_cr if N_fold is not universal across beta_i and tau Omega_i.
[§2.3, Eqs. (2.7)-(2.8)] The exponent 1.6 in tau_cr inherits the empirical power-law fits gamma_peak ≈ 0.3 Omega_i beta_i^{-0.6} and gamma_peak ≈ 0.4|Delta_i - Delta_cr|^{0.6} Omega_i, but no error bars or residuals are quoted for these fits. Because the beta^{1.6} scaling in Eq. (3.7) and the beta-independent scaling of (Delta_i)_min - Delta_cr with (tau Omega_i)^{-0.625} in Eq. (3.8) are the two central quantitative predictions of the paper, please quantify the uncertainty in the fitted exponents and constants, and propagate these uncertainties through to tau_cr and to the prediction (4.11).
[§3.1.3 and Table 1] The saturation amplitude at the Alfvén-enabling boundary scales as delta B^2/B^2 ~ beta_i^{1/4}(tau Omega_i)^{-1/2} ~ beta_i^{-0.55} when tau Omega_i ~ beta_i^{1.6}. If the initial fluctuation seed amplitude is roughly independent of beta_i (which is typical for PIC thermal noise at fixed N_ppc), then the effective number of e-foldings obeys N_fold ≈ const - 0.5 ln(delta B_sat^2/delta B_0^2) ≈ const - 0.275 ln beta_i, introducing a logarithmic correction to the argument of Eq. (3.7). The paper does not discuss this correction. Please estimate its magnitude over the simulated range and assess whether it is absorbed by the empirical fits or would contaminate the apparent pure beta^{1.6} power law; if it is negligible, state why.

minor comments (4)

[Introduction (p. 3, near Eq. 1.1)] The sentence 'If instead the feedback of the firehose instability regulates the pressure anisotropy such that Delta_i ≈ -2/beta_i, an Alfvén-enabling state would result' contradicts the immediately preceding definition of the Alfvén-inhibiting state (Delta_i = -2/beta_i). It should read Delta_i ≈ -1.6/beta_i or, more generally, Delta_i > -2/beta_i.
[Figure 3 caption] The caption refers to blue and red points but does not state what the colors denote; please make the caption self-contained by explaining that blue (red) marks runs that remain Alfvén-enabling (reach Delta_i < -2/beta_i).
[§4.2.2] The paper usefully corrects a previously misreported Hall term in Hellinger & Trávníček (2005) and Bott et al. (2021). It would be helpful to state explicitly whether any quantitative results in Bott et al. (2021) are affected by that typo, or whether the implementation used there was already correct.
[Eq. (4.11) and Fig. 5] The empirical fit (4.11) is quoted with no uncertainties. Adding confidence intervals or stating that the constants are point estimates from visual inspection would make the degree of support clearer, especially because Fig. 5(b) shows visible scatter about the dashed line.

Circularity Check

1 steps flagged · score 5.0 of 10

Partial circularity in the Alfvén-enabling boundary: Nfold is set by the same simulations used to validate τcr, and the factor 27 is empirical.

fitted input called prediction [Section 3.2, Eqs. (3.5)-(3.7); Section 4.3.1, Eq. (4.11), Figs. 3/5; Table 2 caption]
"(in our simulations, we find Nfold ≈ 5). This implies that τcr(βi) ≈ 1.5Nfoldβiγ−1⊥f . (3.5) ... τcr(β∥i) ≈ 5Nfoldβ1.6i Ω−1i (3.7). ... The empirical factor of 27 is introduced so that runs with ˜τeff ≳ 1 are at all times in an Alfvén-enabling state (see section 4.3.1)."

The transition timescale τcr is presented as a prediction, but its only free parameter Nfold is fixed by the simulations that are then used to validate the boundary: the paper states Nfold ≈ 5 from "our simulations," and Eq. (4.11) is fitted to the same runs with Nfold ≃ 5.4. Table 2 hard-codes the same threshold by introducing "the empirical factor of 27" so that τ˜eff ≳ 1 marks Alfvén-enabling runs, and Fig. 5(a) then uses this 27β^{1.6} normalization to claim that Eq. (3.7) is "an excellent fit" and that Eq. (3.8) is "consistent" with Eq. (4.11). The prefactor of τcr and the detailed Δmin(τ) relation are therefore calibrated to the data they are said to test, not independently predicted.

full rationale

The paper is not generally circular. The linear-theory sections compute thresholds and growth rates from the hot-plasma dispersion relation, and the saturated values Δsat, νeff, δB^2 spectra, and distribution functions are measured in new HEB simulations rather than imported from the conclusion. The three-regime classification is tied to v_A,eff by definition, which is legitimate. The one substantive circular element is the quantitative Alfvén-enabling boundary: Nfold is calibrated from the simulations, and the same simulations are then presented as confirming Eq. (3.7) and Eq. (4.11), with the factor 27 explicitly empirical. This makes the boundary's prefactor and its detailed Δmin scaling partly self-confirmatory. However, the β^{1.6} power law and the kinetic threshold Δcr≈−1.35/β are independent linear-theory results, and the qualitative conclusion that very slowly evolving high-β plasmas are Alfvén-enabling is robust to order-unity changes in Nfold. The circularity is therefore partial and does not invalidate the overall regime picture.

Assumptions & free parameters 3 free parameters · 6 assumptions · 0 invented entities

The central phase-boundary prediction depends on one genuinely fitted parameter (N_fold) and on empirical power-law fits to linear growth rates. The linear dispersion relation, CGL double-adiabatic evolution, and 2.5D hybrid simulation methodology are prior-knowledge assumptions. No new physical entities are postulated.

free parameters (3)

N_fold = approximately 5.4 (set to match simulation threshold)
Number of e-foldings of oblique firehose growth required for backreaction; introduced in eq. (3.5) to define tau_cr and adjusted so that tau_cr Omega_i = 27 beta_i^1.6 matches the simulation phase boundary in Fig. 3. This is a fit to the data the paper predicts.
growth-rate power-law constants = gamma_peak approximately 0.3 Omega_i beta_i^{-0.6}; gamma_peak approximately 0.4 |Delta_i-Delta_cr|^0.6 Omega_i
Empirical fits to numerical solutions of the hot-plasma dispersion relation (eqs. 2.7 and 2.8), used to derive tau_cr proportional to beta_i^1.6. They are not derived from first principles.
prefactor in (Delta_i)_min fit = 5.1 in eq. (4.11)
Best-fit coefficient for (Delta_i)_min beta_i approximately -1.35 - 5.1 beta_i/(tau Omega_i)^0.625, used to locate the boundary; the theory (3.8) gives 1.8, so the prefactor is empirical.

assumptions (6)

standard math Hot-plasma dispersion relation and linear Vlasov theory are valid for collisionless bi-Maxwellian plasma
Used throughout section 2 to compute growth rates and thresholds; standard plasma physics.
domain assumption Double-adiabatic (CGL) evolution before instability onset
Eqs. (4.1) and (4.3) assume no heat fluxes or collisionality prior to firehose growth; appropriate for the expanding-box setup.
domain assumption 2.5D hybrid-kinetic simulations capture the relevant firehose physics
Spatial gradients restricted to (x,z) plane; particles and fields are 3D; stated in section 4.2.2. May miss 3D mode couplings.
domain assumption Numerical collisionality does not qualitatively alter the saturated state
Authors state numerical collisionality has a quantitative but not qualitative effect on some results (section 4.2.2 and Appendix B). Not independently verified.
ad hoc to paper Quasilinear pitch-angle scattering approximates the effective collision operator in the Alfvén-enabling state
Section 6 (abstract: 'well approximated'). The scattering operator is a modeling assumption for the firehose fluctuations, not derived from first principles.
domain assumption Macroscopic evolution type does not affect saturation when timescales are separated
Stated in section 4.2.1 and revisited in section 7; expansion versus shear can alter deltaB^2/B0^2 as acknowledged in section 3.1.2.

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Cite this review

Pith. "Pith review of Thermodynamics and collisionality in firehose-susceptible high-$\beta$ plasmas." pith.science (2026). https://pith.science/paper/JQYSXA3F

@misc{pith2026250113663,
  author       = {Pith},
  title        = {Pith review of: Thermodynamics and collisionality in firehose-susceptible high-$\beta$ plasmas},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/JQYSXA3F}},
  note         = {Machine review of arXiv:2501.13663}
}

abstract

We study the evolution of collisionless plasmas that, due to their macroscopic evolution, are susceptible to the firehose instability, using both analytic theory and hybrid-kinetic particle-in-cell simulations. We establish that, depending on the relative magnitude of the plasma $\beta$, the characteristic timescale of macroscopic evolution, and the ion-Larmor frequency, the saturation of the firehose instability in high-$\beta$ plasmas can result in three qualitatively distinct thermodynamic (and electromagnetic) states. By contrast with the previously identified `ultra-high-beta' and `Alfv\'en-inhibiting' states, the newly identified `Alfv\'en-enabling' state, which is realised when the macroscopic evolution time $\tau$ exceeds the ion-Larmor frequency by a $\beta$-dependent parameter, can support linear Alfv\'en waves and Alfv\'enic turbulence because the magnetic tension associated with the plasma's macroscopic magnetic field is never completely negated by anisotropic pressure forces. We characterise these states in detail, including their saturated magnetic-energy spectra. The effective collision operator associated with the firehose fluctuations is also described; we find it to be well approximated in the Alfv\'en-enabling state by a simple quasilinear pitch-angle scattering operator. The box-averaged collision frequency is $\nu_{\rm eff} \sim \beta/\tau$, in agreement with previous results, but certain sub-populations of particles scatter at a much larger (or smaller) rate depending on their velocity in the direction parallel to the magnetic field. Our findings are essential for understanding low-collisionality astrophysical plasmas including the solar wind, the intracluster medium of galaxy clusters and black-hole accretion flows. We show that all three of these plasmas are in the Alfv\'en-enabling regime of firehose saturation and discuss the implications of this result.

Figures

Figures reproduced from arXiv: 2501.13663 by the authors.

**Figure 1.** Phase-space map of high-βi firehose-susceptible plasmas in βi and τΩi. In this paper, we put forward a comprehensive theory for how the firehose instability grows, saturates and then affects the thermodynamics and collisionality of high-β plasma. We claim that, depending on the relative magnitude of βi and τΩi , there are three qualitatively distinct regimes: ultra-high-β, Alfvén-inhibiting and Alfvén-enabling. For … view at source ↗

**Figure 2.** (a) Linear growth rate γ of firehose-unstable modes as a function of parallel and perpendicular wavenumber for a range of different ∆i at βi = 200, mi/me = 1836, Te = T∥i , and vthe/c = 0.05. The growth rates are calculated on a 4002 grid in (k∥ρi, k⊥ρi), with equal logarithmic spacing in both directions. (b) Critical value of ∆i below which firehose instability onsets, ∆cr, as a function of parallel and perpendicul… view at source ↗

**Figure 3.** Phase-space maps of various simulations of high-βi firehose-susceptible plasmas, which indicate whether an Alfvén-enabling state (∆i > −2/β∥i) is maintained at all times (blue points) or not (red points). In the left map, we include the HEB simulations completed for this paper (denoted by ‘×’), as well as the shearing-box hybrid-kinetic simulations reported in Kunz et al. (2014a) (‘+’) and Melville et al. (2016) (‘ … view at source ↗

Figures from the paper (21 more)

**Figure 4.** Figure 4: Time evolution of (a) the firehose-instability parameter −∆iβ∥i and (b) the squared and normalised effective Alfvén speed v 2 A,eff /v2 A for all of the D runs (βi0 = 50). On panel (a), the dotted black line denotes the threshold ∆iβ∥i = −1.35 of the oblique firehose i…

**Figure 5.** Figure 5: Panel (a): values of the firehose-instability parameter −∆iβ∥i at the time tmin at which the pressure anisotropy attains its first minimum, (∆i)min, for all runs, as a function of τexp,effΩi/β∥i(tmin) 1.6 . The dotted (dashed) black line denotes the threshold ∆iβ∥i = −…

**Figure 6.** Figure 6: Panel (a): values of the firehose-instability parameter −∆iβ∥i at the time tsat at which the square of the perturbed magnetic-field strength δB2 f /B2 0 associated with the firehose fluctuations attains its maximum value, (δB2 f /B2 0 )max, for all runs as a function o…

**Figure 7.** Figure 7: 2D visualisations of the out-of-plane component of the perturbed magnetic field in two simulations at βi0 = 50 representing (a) an Alfvén-enabling state (run DVI) and (b) an Alfvén-inhibiting state (run DIII). The perturbed field appears at three different times: in th…

**Figure 8.** Figure 8: (a) Time evolution of the square of the perturbed magnetic-field strength δB2 f /B2 0 associated with the firehose fluctuations for all of the D runs (βi0 = 50). (b) Maximum value of δB2 f /B2 0 , (δB2 f /B2 0 )max, as a function of τexp,effΩi/β∥i(tsat) 1.6 , for all r…

**Figure 9.** Figure 9: Two-dimensional magnetic-energy spectra EB(k∥, k⊥) of the firehose fluctuations at a selection of different times during the firehose instability’s evolution: linear phase (far left), nonlinear phase (near left), and two times during the saturated state (near and far r…

**Figure 10.** Figure 10: Time evolution of the square of the perturbed magnetic-field strength δB2 f /B2 0 (solid black line) associated with the firehose fluctuations, along with the analogous quantity δB2 f,pl/B2 0 for quasi-parallel fluctuations (solid red line) and δB2 f,ob/B2 0 for obliq…

**Figure 11.** Figure 11: figure 11. We find that [PITH_FULL_IMAGE:figures/full_fig_p027_11.png]

**Figure 11.** Figure 11: (a) Numerically determined (half)-period τosc of oscillation of the perturbed magnetic energy δB2 f /B2 0 associated with all firehoses modes (black), of the magnetic energy δB2 f,pl/B2 0 associated with parallel modes (red), and the magnetic energy δB2 f,ob/B2 0 asso…

**Figure 12.** Figure 12: Domain-averaged ion-distribution function f(v∥, v⊥) in a simulation representative of an Alfvén-enabling state (run DVI) during the (a) linear (t = 0.042τexp), (c) nonlinear (t = 0.07τexp) and (e) saturated (t = 0.375τexp) stages of the firehose instability. The right…

**Figure 13.** Figure 13: Domain-averaged ion-distribution function f(v∥, v⊥) in a simulation representative of an Alfvén-inhibiting state (run DIII) during the (a) linear (t = 0.075τexp), (c) nonlinear (t = 0.2τexp) and (e) saturated (t = τexp) stages of the firehose instability. As in figure…

**Figure 14.** Figure 14: Pitch-angle gradient of the ion distribution function f divided by 2˜vfM (solid red line), where v˜ = v/vthi, and fM is a Maxwellian distribution with the same temperature as f, averaged over v⊥. The solid blue line is the analogous quantity, but calculated using fbiM…

**Figure 15.** Figure 15: (a) Values of the effective collisionality νeff measured directly in the simulations (solid lines) with Alfvén-enabling states (runs BIV, CIV, and DVI). The expansion time in these simulations is τexpΩi0 = 2 × 104 . The effective collisionalities predicted by the simp…

**Figure 16.** Figure 16: (a) Values of the effective collisionality ⟨νeff ⟩sat measured directly in all simulations, averaged over the time interval between the time at which the firehose fluctuations attain their peaks strength and the time at which the next local minimum is obtained. The da…

**Figure 17.** Figure 17: (a) Two-dimensional magnetic-energy spectra of the firehose fluctuations in run CV at a selection of different times around the emergence of the parallel secondary firehose instability. (b) Pitch-angle gradient of the ion distribution function f divided by 2˜vfM (soli…

**Figure 18.** Figure 18: (a) 1D (parallel) magnetic-energy spectrum EB(k∥) of all firehose fluctuations in the saturated, Alfvén-enabling state of run CV (solid red line). Also plotted are the magnetic-energy spectra of non-quasi-parallel fluctuations (blue solid line) and quasi-parallel ones…

**Figure 19.** Figure 19: (a) Best-fit estimates for wavenumber parameters introduced in (6.6) for all of our Alfvén-enabling simulations as a function of the expansion time. (b) Same as in panel (a), but as a function of βi. (c) Best-fit estimates for spectral amplitude parameters introduced …

**Figure 20.** Figure 20: Panel (a): slice plots at fixed k⊥ of the frequency-dependent magnetic-energy spectrum EB(k∥, k⊥, ϖ) of the firehose fluctuations in run CV, averaged over the saturated state. Panel (b): fluctuation-energy-weighted average value of real frequency ϖsim of the firehose …

**Figure 21.** Figure 21: Panel (a): slice plot of the pitch-angle gradient of the ‘saturated’ ion distribution function fi,sat divided by 2˜vfM in an Alfvén-enabling state (run CV). Here, fi,sat is the domain-averaged ion-distribution function f(v∥, v⊥) time-averaged over the saturated period…

**Figure 22.** Figure 22: Fokker–Planck coefficients A(v, ξ) (top row) and B(v, ξ) (bottom row) obtained two different ways: using tracked-particle data from run CV to calculate the jump moments (6.22) assuming either ∆t = 4πΩ−1 i (left column) or ∆t = 8πΩ−1 i (middle column); and comparing ou…

**Figure 23.** Figure 23: (a) Values of the effective collisionality νeff measured directly in all simulations at the time tc at which the oblique firehose threshold is reached. The dashed line indicates the effective (time-averaged) value νeff = β∥i/6τexp,eff of the collisionality predicted i…

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Pith tools

Reviewed August 10, 2026 · model on record in the stance chip above.

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