1.1. Velocity distribution (VD) functions

Quasi-thermal particle populations in magnetised plasmas are most often described by anisotropic Maxwellian distributions. The non-drifting bi-Maxwellian distribution function $f_M$ is given by

(1.1) \begin{align} f_M(v_{\|}, v_{\perp })= N_{\!M} \exp\! {\left (- \frac {v_{\|}^2}{\theta _{M \|}^2}- \frac {v_{\perp }^2}{\theta _{M,\perp }^2}\right )}, \end{align}

where $v_{\|}$ and $v_{\perp }$ represent velocity components parallel and perpendicular to the background magnetic field. The thermal speeds are $\theta _{M \|, \perp } = \sqrt {2k_B T^M_{\|, \perp }/m_{\!j}}$ with the Maxwellian temperature $ T^M_{\|, \perp }$ , the Boltzmann constant $k_B$ , the particle rest mass $m_{\!j}$ of species $j$ and the normalisation constant reads $N_{\!M}=1/(\pi ^{3/2} \theta _{M,\|} \theta _{M, \perp }^2 )$ .

In situ observations in space plasmas also reveal suprathermal particle populations which manifest as high-energy tails of the observed distributions, that cannot be captured by Maxwellian models alone. These suprathermal tails are often described using $\kappa$ -distributions, originally introduced by Olbert (Reference Olbert1968) and widely applied in studying kinetic instabilities (Meneses, Gaelzer & Ziebell Reference Meneses, Gaelzer and Ziebell2018; Shaaban et al. Reference Shaaban, Lazar, López, Yoon and Poedts2021; López et al. Reference López, Shaaban and Lazar2021a ). The standard bi- $\kappa$ -distribution (SKD) can be written as (Lazar, Fichtner & Yoon Reference Lazar, Fichtner and Yoon2016)

(1.2) \begin{align} f_{\mathrm{SKD}}\left (v_{\|}, v_{\perp }, \kappa \right )=N_{\mathrm{SKD}}\left (1+\frac {v_{\|}^2}{\kappa \theta _{\|}^2}+\frac {v_{\perp }^2}{\kappa \theta _{\perp }^2}\right )^{-\kappa -1}, \end{align}

with normalisation

(1.3) \begin{align} N_{\mathrm{SKD}}=\frac {1}{\pi ^{3 / 2} \theta _{\|} \theta _{\perp }^2} \frac {\varGamma (\kappa +1)}{\kappa ^{3 / 2} \varGamma (\kappa -1 / 2)} , \end{align}

where $\varGamma$ denotes the gamma function. The parameter $\kappa$ controls the strength of the suprathermal tail and the distribution approaches a Maxwellian in the limit $\kappa \rightarrow \infty$ . The thermal speeds are $\theta _{ \|, \perp } = \sqrt {2k_B T^{\kappa }_{\|, \perp }/m}$ , with $T^{\kappa }_{\|, \perp }= ({\kappa }/({\kappa -3/2})) T^{M}_{\|, \perp } \gt T^{M}_{\|, \perp }$ and $\kappa \gt 3/2$ . A well-known limitation of the SKD is that higher-order velocity moments (including essential macro-parameters such as temperature, heat flux and transport coefficients) diverge (Scherer, Fichtner & Lazar Reference Scherer, Fichtner and Lazar2017), whereas the observed distributions may exhibit stronger tails corresponding to lower values of $\kappa$ (Štverák et al. Reference Štverák, Trávníček, Maksimovic, Marsch, Fazakerley and Scime2008; Gloeckler et al. Reference Gloeckler, Fisk, Mason, Roelof and Stone2012; Wilson et al. Reference Wilson III2019a , Reference Wilson IIIb ).

To overcome these shortcomings, the regularised $\kappa$ -distribution (RKD) has been introduced (Scherer et al. Reference Scherer, Fichtner and Lazar2017, see also Lazar & Fichtner (Reference Lazar and Fichtner2021)), modifying the SKD by introducing an exponential cutoff that guarantees the convergence of all velocity moments.

(1.4) \begin{align} \begin{aligned} f_{\mathrm{RKD}}\left (v_{\|}, v_{\perp }, \kappa , \alpha \right )= N_{\mathrm{RKD}}\!\left (1+\frac {v_{\|}^2}{\kappa \varTheta _{\|}^2}+\frac {v_{\perp }^2}{\kappa \varTheta _{\perp }^2}\right )^{-\kappa -1} \exp \!\left (-\frac {\alpha ^2 v_{\|}^2}{\varTheta _{\|}^2}-\frac {\alpha ^2 v_{\perp }^2}{\varTheta _{\perp }^2}\right ). \end{aligned} \end{align}

In this case the normalisation constant is (Scherer et al. Reference Scherer, Lazar, Husidic and Fichtner2019)

(1.5) \begin{align} N_{\mathrm{RKD}}=\frac {1}{\pi ^{3 / 2} \varTheta _{\|} \varTheta _{\perp }^2 W}, \end{align}

with

(1.6) \begin{align} W = U\!\left (\frac {3}{2}, \frac {3-2\kappa }{2},\alpha ^2 \kappa \right ), \end{align}

where $U$ denotes the Tricomi function (Oldham, Myland & Spanier Reference Oldham, Myland and Spanier2009). The parameter $\alpha \gt 0$ controls the exponential cutoff and is independent of $\kappa$ . The ‘thermal’ speeds are defined as $\varTheta _{\|, \perp }$ and differ from those defined in the SKD case. The RKD retains the desirable suprathermal behaviour of $\kappa$ distributions while ensuring that all velocity moments remain finite. Consequently, macroscopic plasma quantities are always consistently defined through velocity moments. The SKD and Maxwellian distributions are recovered as limiting cases for $\alpha \rightarrow 0$ and $\kappa \rightarrow \infty$ , respectively, see Scherer et al. (Reference Scherer, Fichtner and Lazar2017); Schröder et al. (Reference Schröder, Fichtner, Lazar, Verscharen and Klein2025a ). Applications of the RKD include the macroscopic description of suprathermal populations, modelling of anisotropic distributions measured in situ in the solar wind and the evaluation of transport properties in non-equilibrium plasmas (Lazar et al. Reference Lazar, Scherer, Fichtner and Pierrard2020; Scherer et al. Reference Scherer, Husidic, Lazar and Fichtner2021; Husidic et al. Reference Husidic, Scherer, Lazar, Fichtner and Poedts2022; Hau, Chang & Lazar Reference Hau, Chang and Lazar2023). A more detailed discussion of the RKD and its properties and applications can be found in Schröder et al. (Reference Schröder, Fichtner, Lazar, Verscharen and Klein2025a , Reference Schröder, Lazar, López and Fichtnerb ).

In the present study we extend the investigation of RKD plasmas within linear wave theory by considering arbitrary propagation angles with respect to the magnetic field. Earlier studies were largely restricted to parallel electromagnetic modes (Husidic et al. Reference Husidic, Lazar, Fichtner, Scherer and Astfalk2020; Schröder et al. Reference Schröder, Fichtner, Lazar, Verscharen and Klein2025a , Reference Schröder, Lazar, López and Fichtnerb ). Here, we include oblique propagation and focus on the electron and proton firehose instabilities (EFHI and PFHI, respectively).

Parallel firehose modes are oscillatory or periodic with finite real frequency ( $\omega _r\neq 0$ ), whereas oblique propagation also allows for purely non-propagating, aperiodic modes ( $\omega _r=0$ ). Linear theory predicts that the fastest-growing electron firehose modes occur at oblique angles and are typically aperiodic (Yoon, Wu & de Assis Reference Yoon, Wu and de Assis1993; Li & Habbal Reference Li and Habbal2000; Shaaban et al. Reference Shaaban, Lazar, López, Fichtner and Poedts2019). Hybrid simulations support these predictions (Gary & Nishimura Reference Gary and Nishimura2003; López et al. Reference López, Lazar, Shaaban, Poedts, Yoon, Viñas and Moya2019). For protons, aperiodic firehose modes can likewise dominate near instability thresholds when mixing with anisotropic electrons (López et al. Reference López, Micera, Lazar, Poedts, Lapenta, Zhukov, Boella and Shaaban2022), which is consistent with solar wind observations (Kasper et al. Reference Kasper, Lazarus and Gary2002; Bale et al. Reference Bale, Kasper, Howes, Quataert, Salem and Sundkvist2009). With isotropic electrons, periodic proton firehose modes are expected to dominate (Hellinger & Matsumoto Reference Hellinger and Matsumoto2000).

While dispersion relations for SKD plasmas can be solved analytically or numerically, e.g. via the DIS-K solver (López et al. Reference López, Shaaban and Lazar2021; López et al. Reference López, Shaaban and Lazar2021b ), a full (semi-)analytical solvable dispersion tensor and plasma dispersion functions for anisotropic RKD plasmas for oblique propagation have not yet been derived. However, such systems can be investigated numerically using the Arbitrary Linear Plasma Solver (ALPS) (Verscharen et al. Reference Verscharen, Klein, Chandran, Stevens, Salem and Bale2018). This solver determines, for arbitrary propagation angles, the linear Vlasov–Maxwell dispersion relation directly from tabulated background distribution functions and does not require explicit further analytical expressions.

1.2. Linear dispersion theory

First, we assume that the plasma fluctuations, specifically those of the electric and magnetic fields ( $\boldsymbol{E}$ and $\boldsymbol{B}$ , respectively), are small enough to justify the application of linear theory. In order to investigate plasma instabilities, we solve the kinetic dispersion relation to obtain the complex frequency

(1.7) \begin{align} \omega (\boldsymbol{k}) = \omega _r(\boldsymbol{k}) + i\gamma (\boldsymbol{k}), \end{align}

where $\boldsymbol{k}$ is the wave vector corresponding to plane-wave-like fluctuations, and $\omega _r$ denotes the real and $\gamma$ the imaginary part of the frequency.

To obtain this solution, we use the linearised Vlasov equation and the linearised set of Maxwell’s equations, providing an expression for the plasma susceptibilities $\boldsymbol{\chi}_{\!j}$ for the $j$ th species (see Appendix A). These susceptibilities are then related to the plasma’s dielectric tensor $\boldsymbol{\epsilon }$ through

(1.8) \begin{align} \boldsymbol{\epsilon } = \unicode {x1D7D9} + \sum _j \boldsymbol{\chi}_{\!j}. \end{align}

From that, we can obtain the wave equation

(1.9) \begin{align} (\boldsymbol{k}c/\omega ) \times [(\boldsymbol{k}c/\omega ) \times \boldsymbol{E}] + \boldsymbol{\varepsilon } \boldsymbol{\cdot }\boldsymbol{E} \equiv \mathcal{D} \boldsymbol{\cdot }\boldsymbol{E} = 0 ,\end{align}

where $c$ is the speed of light. Solving $\text{det}\ \mathcal{D} = 0$ with $\boldsymbol{E}\neq 0$ provides non-trivial solutions. To obtain the latter, the ALPS code (Verscharen et al. Reference Verscharen, Klein, Chandran, Stevens, Salem and Bale2018) will be used, which offers a robust numerical approach, providing solutions for arbitrary linear plasma set-ups. ALPS has proven (and further proven with this study) to be a valuable tool for studying a wide range of plasma instabilities. For further details on its implementation and capabilities, refer to Verscharen et al. (Reference Verscharen, Klein, Chandran, Stevens, Salem and Bale2018) and Klein & Verscharen (Reference Klein and Verscharen2025).

The structure of the paper is as follows. We begin with a brief description of our set-up in § 2. In the first part of this section, the ALPS implementation is validated against a series of previous results, in particular, those obtained for oblique EFHI modes in Maxwellian plasmas (Li & Habbal Reference Li and Habbal2000; Maneva et al. Reference Maneva, Lazar, Viñas and Poedts2016), SKD plasmas (Shaaban et al. Reference Shaaban, Lazar, López, Fichtner and Poedts2019) and for oblique PFHI in Maxwellian plasmas (Hellinger & Matsumoto Reference Hellinger and Matsumoto2000; Maneva et al. Reference Maneva, Lazar, Viñas and Poedts2016). The new RKD results are presented and discussed in the third section. All findings are summarised in the concluding § 4.

2.1. Validation. The OEFHI with Maxwellian and SKD

To begin with, the ALPS set-up is validated against previous results for EFHI driven by bi-Maxwellian electrons (Li & Habbal Reference Li and Habbal2000), which here are reproduced using DIS-K, imposing the same plasma parameters. For the electrons, $\beta _{e}=5$ and $T_{e,\perp }/T_{e,\parallel }=0.5$ , while the protons are described by an isotropic Maxwellian with $\beta _{p}=1$ . Figure 1 presents the (O)EFHI dispersion curves for different propagation angles ( $0^\circ$ , $17^\circ$ , $20^\circ$ , $22^\circ$ and $24^\circ$ , from top to bottom), obtained with ALPS (solid lines) and DIS-K (dots). Both the wave frequency (left) and growth rate (right) are shown as functions of the wavenumber $k=\sqrt {k^2_{\|} + k^2_{\perp }}$ . Overall, the agreement between the two solvers is very good across all propagation angles (less than $0.2\,\%$ deviation for the maximum growth rate $\gamma _m$ and the corresponding wavenumber $k_m$ ). Note that these results were also replicated by NHDS (Verscharen & Chandran (Reference Verscharen and Chandran2018)) and PLUME (Klein, Howes & Brown (Reference Klein, Howes and Brown2025)), agreeing with DIS-K and ALPS.

For parallel propagation, the agreement between figure 1(a) in Li & Habbal (Reference Li and Habbal2000) and our results is excellent. However, for oblique propagation, noticeable differences arise, as evidenced by a comparison of the unstable solutions in our figure 1 and figure 1 in Li & Habbal (Reference Li and Habbal2000). The present results can be regarded as an updated version, and a detailed comparison between the results reported by Li & Habbal (Reference Li and Habbal2000) and the present ALPS/DIS-K solutions is summarised in table 1. For the periodic branch, the largest deviations are found at finite propagation angles, where the present solutions exhibit systematically higher growth rates and larger ranges of unstable wavenumbers. Differences in the aperiodic branch are generally smaller, although with some noticeable deviations in the transition region between periodic and aperiodic modes, see the results for $\theta = 22^\circ$ and $\theta = 24^\circ$ . At $\theta = 22^\circ$ , in the present ALPS/DIS-K solutions the periodic peak remains slightly dominant with $\gamma _m$ approximately $1.25$ times larger than the aperiodic peak, in contrast to the dominant aperiodic branch (with $\gamma _m$ approximately $1.5$ times larger than the periodic one) reported by Li & Habbal (Reference Li and Habbal2000).

More significant differences are found at $\theta = 24^\circ$ . While Li & Habbal (Reference Li and Habbal2000) show a strongly dominant aperiodic branch with $\gamma _m$ approximately $6.7$ times larger than the periodic branch with a non-developed peak, the present solutions instead exhibit two distinguishable peaks, with only a moderate dominance of the aperiodic one (approximately $1.27$ times larger than the periodic peak). In addition, the periodic branch obtained here remains unstable over a wider wavenumber interval with a markedly higher and broader peak.

Figure 2.

Normalised growth rates as functions of the normalised wavenumber for aperiodic OEFHI at different propagation angles ( $30^\circ$ , $40^\circ$ , $50^\circ$ , $60^\circ$ , $70^\circ$ , $75^\circ$ and $80^\circ$ ), obtained with ALPS (solid lines) and DIS-K (dots) for the Maxwellian case.

Figure 2 shows the growth rates of aperiodic modes at higher propagation angles, with ALPS results indicated by solid lines and DIS-K with dots. The agreement between both solvers remains very good in all cases (less than $0.1\,\%$ deviation in terms of $\gamma _m$ and $k_m$ ). The overall solutions are in good qualitative agreement with the results in Li & Habbal (Reference Li and Habbal2000) (see their figure 3), including the shift of the growth rates toward higher wavenumbers and an increase of $\gamma _m$ up to $\theta =70^\circ$ , followed by a decrease at larger angles. However, we find notable quantitative differences, which are summarised in table 1.

The present solutions systematically predict broader unstable wavenumber intervals and, in most cases, enhanced maximum growth rates compared with Li & Habbal (Reference Li and Habbal2000). Despite the qualitative agreement, differences are quantitatively noticeable for the periodic branch at finite angles, and for the extent of the unstable range and the maximum growth rates of high-angle aperiodic solutions.

Figure 3.

Normalised growth rates as functions of the normalised wavenumber, for the oblique (aperiodic) EFHI at propagation angles $60^\circ$ , $70^\circ$ , $75^\circ$ and $80^\circ$ for SKD with $\kappa =3.0$ , obtained with ALPS (solid lines) and DIS-K (dots).

Now we validate ALPS with OEFHI solutions obtained for SKD electrons with DIS-K; see also previous results in Shaaban et al. (Reference Shaaban, Lazar, López, Fichtner and Poedts2019). Thus, figure 3 shows the comparison of the normalised growth rate between ALPS and DIS-K for the aperiodic OEFHI for different angles ( $60^\circ$ , $70^\circ$ , $75^\circ$ and $80^\circ$ ) for SKD electrons with $\kappa = 3.0$ . The agreement between both solvers is very good for all cases (less than $0.4\,\%$ deviation in terms of $\gamma _m$ and $k_m$ ). For bi-Maxwellian electrons, Li & Habbal (Reference Li and Habbal2000) imposed an electron–proton temperature ratio of $T_{e,\|}/T_{p, \|} = 5$ . To minimise the effect of the cyclotron resonant protons for the $\kappa$ -distributed plasma cases, we use $\beta _e = \beta _p = 4.0$ for the further study of the OEFHI.

In figure 4, contour plots, derived with DIS-K, of the OEFHI normalised frequency (left panel) and growth rate (right panel), in the $k$ – $\theta$ -plane are shown for the Maxwellian case (top) and for the SKD with $\kappa =3.0$ (bottom). While the Maxwellian case has unstable periodic solutions up to $\theta \approx 18^\circ$ and $kc/\omega _{pp} \approx 8.1$ , the SKD case has unstable periodic solutions up to an increased $\theta \approx 31^\circ$ and decreased $kc/\omega _{pp} \approx 4.1$ . The aperiodic mode is unstable for the Maxwellian case up to $\theta \approx 73^\circ$ and $kc/\omega _{pp} \approx 37$ . The maximum growth rate is reached at $\theta _m \approx 65.1^\circ$ with $\gamma _m \approx 122.5$ at $k_m c/\omega _{pp}\approx 24.1$ . For the SKD case, the aperiodic mode is unstable until $\theta \approx 89^\circ$ and $kc/\omega _{pp} \approx 36.5$ . The maximum growth rate is reached at $\theta _m \approx 72.9^\circ$ with $\gamma _m \approx 195.6$ at $k_m c/\omega _{pp}\approx 23.2$ .

The validation for the OPFHI, inspired by Hellinger & Matsumoto (Reference Hellinger and Matsumoto2000), can be found in Appendix C, showing again very good agreement. These applications to the OEFHI and OPFHI validate both the ALPS and the DIS-K code as reliable tools, because the same results were obtained with rather different numerical algorithms. We note that ALPS is computationally more demanding than DIS-K (or general (semi-)analytic solvers), since it evaluates the dispersion relation through more complex numerical integration schemes over tabulated distribution functions, whereas DIS-K relies on analytical expressions for the SKD plasma dispersion functions. In addition, the findings for the modes in an SKD plasma can now serve as a reference for the RKD ones discussed in the following.

Figure 4.

Contour plot of the OEFHI normalised frequency (left panel) and growth rate (right panel) in the $k$ – $\theta$ -plane for the Maxwellian case (top) and SKD case with $\kappa =3.0$ (bottom), obtained with DIS-K.

Figure 4. Long description

The image contains four contour plots. Panel A and Panel B show the Maxwellian case, while Panel C and Panel D show the SKD case with kappa equal to 3.0. Each panel is divided into two subplots: the left subplot displays the normalized frequency, and the right subplot displays the growth rate. The x-axis in all subplots is labeled ‘kc/ωpp’ and ranges from 0 to 35. The y-axis is labeled ‘θ’ and ranges from 0 to 80 degrees. The color scale at the top of each subplot indicates the values of the normalized frequency and growth rate, with blue representing lower values and red representing higher values. In the Maxwellian case, the left subplot shows a non-propagating region, while the right subplot distinguishes between periodic and aperiodic regions. Similarly, the SKD case with kappa equal to 3.0 shows non-propagating regions in the left subplot and periodic and aperiodic regions in the right subplot.

3.1. Oblique electron firehose instability

Figure 5 shows for the (O)EFHI normalised frequencies in the left panels and normalised growth rates in the right panels for different propagation angles ( $0^\circ$ , $10^\circ$ , $16^\circ$ and $30^\circ$ , from top to bottom) and for several VD functions: Maxwellian (blue), SKD with $\kappa =3.0$ (black), RKD with $\kappa =3.0$ and $\alpha =0.15$ (brown), RKD with $\kappa =2.0$ and $\alpha =0.10$ (red), RKD with $\kappa =1.5$ and $\alpha =0.1$ (green) and RKD with $\kappa =1.0$ and $\alpha =0.10$ (magenta). Figure 6 shows the same results for $16^\circ$ and $30^\circ$ , but with more compact plotting scales for better visibility of the SKD and RKD solutions. Because all RKDs have different $\kappa$ values, they will be distinguished only by their $\kappa$ values throughout this analysis, for convenience.

The parallel EFHI case, where only periodic modes are present, has been discussed by Schröder et al. (Reference Schröder, Fichtner, Lazar, Verscharen and Klein2025a ). The present results are consistent with those findings and can be summarised as follows. Compared with the Maxwellian case, the use of an SKD with $\kappa =3.0$ leads to an increase in the maximum growth rate and a shift toward lower wavenumbers, accompanied by a reduction of the unstable wavenumber range. The real frequency exhibits a similar shift toward lower wavenumbers, with enhanced values. Using an RKD with the same $\kappa =3.0$ value and $\alpha =0.15$ , thus reducing the suprathermal tails of the VD, shifts both the real and imaginary solutions towards the Maxwellian solution. The maximum growth rate is slightly enhanced, since the moderate cutoff increases the effective anisotropy. Using RKDs with lower $\kappa$ values and a moderate cutoff ( $\alpha =0.1$ ) maintains the pattern observed for the SKD. As $\kappa$ decreases (while keeping $\alpha$ constant), corresponding to an enhanced population of suprathermal particles, the maximum growth rate increases further and shifts to lower wavenumbers, resulting in a more pronounced and narrower peak. For example, while the Maxwellian case yields $\gamma _m/\varOmega _p = 0.55$ at $k_m c/\omega _{pp} = 6.36$ , the RKD case with $\kappa =1.0$ gives $\gamma _m/\varOmega _p = 0.86$ at $k_m c/\omega _{pp} = 0.61$ , corresponding to an increase by a factor of $1.56$ and a reduction in $k_m$ by approximately one order of magnitude. Further values regarding the maximum growth rates for the different (O)EFHI cases can be found in table 2.

At $\theta = 10^\circ$ , the Maxwellian case exhibits a distinct aperiodic peak with $\gamma _m/\varOmega _p = 0.516$ at $k_m c/\omega _{pp} = 4.024$ , which competes with a periodic branch that remains dominant (we refer to a mode as dominant if its maximum growth rate is higher than the maximum growth rates of all other solutions to the linear dispersion relation), with approximately $11.6\,\%$ higher $\gamma _m/\varOmega _p = 0.576$ at $k_m c/\omega _{pp} = 6.68$ . Compared with the parallel case, the periodic peak shows a very similar maximum growth rate (increase of less than $5\,\%$ ), with a slight shift toward higher $k_m$ . For the SKD case, a marginal aperiodic peak appears at low wavenumbers, with $\gamma _m/\varOmega _p = 0.239$ at $k_m c/\omega _{pp} = 0.363$ . The periodic branch clearly dominates, reaching a significantly higher $\gamma _m$ , approximately $182\,\%$ larger than the aperiodic peak. Relative to the parallel case, the propagating mode remains nearly unchanged in $\gamma _m$ , with only a minor shift toward higher $k_m$ . The same behaviour is seen for the RKD with $\kappa = 3.0$ . The aperiodic peak is slightly more pronounced, but is lower in terms of $\gamma _m$ ( $\gamma _m/\varOmega _p = 0.203$ at $k_m c/\omega _{pp} = 0.408$ ) compared with the SKD cases. All three RKD cases with lower $\kappa$ -values exhibit only a periodic peak. In comparison with the parallel case, this peak is essentially unchanged in terms of $\gamma _m$ (increase of below $0.5\,\%$ ), with only a slight shift toward higher $k_m$ ( ${\lesssim}4\,\%$ ). Notably, the periodic peaks of all SKD and RKD cases exceed both the periodic and aperiodic peaks of the Maxwellian case. At certain values of $k$ , where $\gamma \lt 0$ , the solutions corresponding to the unstable branch jump to a different branch for the RKDs with $\kappa = 2.0$ and $\kappa = 1.5$ . This behaviour is not shown here (nor at other angles), as it is not the focus of the present analysis.

At $\theta = 16^\circ$ , the Maxwellian case is clearly dominated by the aperiodic branch, with $\gamma _m/\varOmega _p = 2.971$ , approximately $382\,\%$ higher than the periodic peak ( $\gamma _m/\varOmega _p = 0.617$ ). The latter shows only a slight increase compared with the parallel case and $\theta = 10^\circ$ . However, as the aperiodic branch extends to higher wavenumbers, the periodic branch becomes effectively truncated. For the SKD case, the aperiodic mode becomes more pronounced and slightly enhanced compared with $\theta = 10^\circ$ , reaching $\gamma _m/\varOmega _p = 0.251$ at $k_m c/\omega _{pp} = 0.408$ . Nevertheless, it remains significantly weaker than the periodic mode, which attains $\gamma _m/\varOmega _p = 0.669$ at $k_m c/\omega _{pp} = 1.844$ . As in the previous cases, the periodic branch shows only a modest increase in $\gamma _m$ and a slight shift toward higher $k_m$ . The RKD with $\kappa =3.0$ shows the same development with respect to the periodic peak ( $\gamma _m/\varOmega _p = 0.780$ at $k_m c/\omega _{pp} = 2.594$ ) and aperiodic peak ( $\gamma _m/\varOmega _p = 0.214$ at $k_m c/\omega _{pp} = 0.467$ .) The RKD with $\kappa =2.0$ develops a weak aperiodic mode at $k_m c/\omega _{pp} = 0.307$ with $\gamma _m/\varOmega _p = 0.300$ (approximately $19.5\,\%$ higher than in the SKD case), together with a dominant periodic mode reaching $\gamma _m/\varOmega _p = 0.781$ , approximately $160\,\%$ larger. The periodic branch remains very similar to the corresponding peaks at lower angles. The RKDs with $\kappa =1.5$ and $\kappa =1.0$ exhibit only a periodic mode, with values closely matching those obtained at smaller propagation angles. Although the periodic modes in the SKD and RKD cases exceed the corresponding periodic peak of the Maxwellian (by a similar factor as for lower angles), the Maxwellian aperiodic mode has the largest growth rate overall. In particular, it exceeds the periodic mode of the RKD with $\kappa =1.0$ by approximately $245\,\%$ , and is roughly an order of magnitude larger than the aperiodic mode of the RKD with $\kappa =2.0$ .

At $\theta = 30^\circ$ , the Maxwellian case exhibits only a non-propagating mode, with a strongly enhanced growth rate reaching $\gamma _m/\varOmega _p = 21.15$ at $k_m c/\omega _{pp} = 9.599$ , together with an extended range of unstable wavenumbers. In contrast to lower angles, the SKD case shows a dominant aperiodic mode, with $\gamma _m/\varOmega _p = 2.705$ at $k_m c/\omega _{pp} = 0.717$ , exceeding the periodic mode ( $\gamma _m/\varOmega _p = 1.167$ at $k_m c/\omega _{pp} = 1.984$ ) by approximately $63\,\%$ . Using an RKD with $\kappa =3.0$ , the periodic peak is barley developed, due to the competition with the aperiodic mode, which gets significantly enhanced (more than an order of a magnitude compared with $\theta =16^\circ$ ), with $\gamma _m=3.564$ at $k_m c/\omega _{pp} = 3.520$ exceeding those of the other SKD/RKD cases (by roughly $670\,\%$ ) and showing a clear trend towards the Maxwellian solutions due to the cutoff of the suprathermal tails. For the RKD with $\kappa =2.0$ , the non-propagating peak is enhanced, reaching $\gamma _m/\varOmega _p = 0.463$ , which is approximately $54\,\%$ higher than at $\theta = 16^\circ$ . Nevertheless, the propagating mode remains dominant, with $\gamma _m/\varOmega _p = 0.893$ , approximately $93\,\%$ larger. The RKD cases with lower $\kappa$ values ( $\kappa =1.5$ and $\kappa =1.0$ ) develop aperiodic modes at this angle, in contrast to the behaviour at lower $\theta$ . These aperiodic contributions become less pronounced as $\kappa$ decreases. The maximum growth rates, however, remain similar across the RKD cases, with $\gamma _m/\varOmega _p = 0.458$ for $\kappa =1.5$ and $\gamma _m/\varOmega _p = 0.430$ for $\kappa =1.0$ and approximately two times smaller than those of the corresponding periodic ones. The periodic peaks are slightly enhanced and shifted toward higher wavenumbers. For example, relative to the parallel case, the RKD $\kappa =1.0$ case shows an increase of approximately $16\,\%$ in $\gamma _m$ and $72\,\%$ in $k_m$ . Overall, the maximum growth rate in the Maxwellian case exceeds those of the SKD and RKD cases by more than an order of magnitude.

Figure 7.

Normalised growth rates as functions of the normalised wavenumber, for the (aperiodic) OEFHI at propagation angles $60^\circ$ , $70^\circ$ , $75^\circ$ and $80^\circ$ for Maxwellian and different RKDs.

Figure 7 presents the normalised growth rates for higher propagation angles ( $60^\circ$ , $70^\circ$ , $75^\circ$ and $80^\circ$ ) for the Maxwellian case and for RKDs, obtained with ALPS. At these angles, only non-propagating modes are found, which are in general more than two orders of a magnitude higher then the periodic growth rates at parallel propagation. At $\theta = 60^\circ$ , the RKD cases exhibit higher growth rates than the Maxwellian case, in contrast to the behaviour at intermediate angles. As $\kappa$ decreases, the maximum growth rate increases and shifts toward lower wavenumbers. This trend is consistently observed for all angles considered here. The RKD with $\kappa =3.0$ shows always lower maximum growth rates (approximately $2\,\%$ – $18\,\%$ ) and a shift towards the Maxwellian solution compared with the SKD case. At $\theta = 70^\circ$ , the Maxwellian case has already passed its maximum growth rate (see figure 4, where the maximum occurs at $\theta \approx 65^\circ$ ) and begins to decrease. In contrast, all RKD cases show a significant amplification of the growth rate, accompanied by a shift toward higher wavenumbers. For example, the RKD case with $\kappa = 1.0$ reaches $\gamma _m/\varOmega _p = 282.1$ , corresponding to an increase of $103\,\%$ relative to $\theta = 60^\circ$ ( $\gamma _m/\varOmega _p = 138.7$ ), and an increase of $170\,\%$ compared with the Maxwellian value $\gamma _m/\varOmega _p = 104.6$ at $\theta = 70^\circ$ . The Maxwellian case, however, spans a broader range of unstable wavenumbers. The RKD with $\kappa =3.0$ reaches its maximum growth rate at $\theta = 70^\circ$ with $\gamma _m =178.0$ At $\theta = 75^\circ$ , both $\gamma _m$ and $k_m$ increase further for the RKD cases with lower $\kappa$ -values, leading to broad and pronounced peaks. The cases with $\kappa = 2.0$ and $\kappa = 1.5$ reach their maximum growth rates near this angle, with $\gamma _m/\varOmega _p = 228.2$ and $\gamma _m/\varOmega _p = 266.9$ , respectively. The Maxwellian case exhibits no unstable solutions at this angle (see figure 4, where instability is limited to $\theta \approx 73^\circ$ ). At $\theta = 80^\circ$ , the maximum growth rates for $\kappa = 3.0$ and $\kappa = 1.5$ decrease and shift toward higher wavenumbers. In contrast, the case with $\kappa = 1.0$ attains its maximum near this angle, producing a broad peak that extends over a wide wavenumber interval (up to $kc/\omega _{pp} \approx 38$ ), with $\gamma _m/\varOmega _p = 339.2$ at $k_m c/\omega _{pp} = 23.6$ , which is an increase of the growth rate of a factor of roughly 400 compared with the corresponding parallel propagation. These results would not be possible to achieve using SKDs for which the condition $\kappa \gt 1.5$ must be fulfilled.

Figure 8.

Normalised wave frequencies (left) and normalised growth rates (right) as functions of the normalised wavenumber for OPFHI at different propagation angles ( $0^\circ$ , $10^\circ$ , $35^\circ$ , from top to bottom), obtained with ALPS for Maxwellian, SKD and different RKDs.

Figure 8. Long description

Panel A: The top left graph is a line graph showing normalized wave frequencies as a function of normalized wavenumber for a propagation angle of 0 degrees. The x-axis is labeled ‘kc/ωp,p’ and the y-axis is labeled ‘ωr/Ωp’. Multiple lines represent different distributions: Maxwellian, SKD, and various RKDs. Panel B: The top right graph is a line graph showing normalized growth rates as a function of normalized wavenumber for a propagation angle of 0 degrees. The x-axis is labeled ‘kc/ωp,p’ and the y-axis is labeled ‘γ/Ωp’. Multiple lines represent different distributions: Maxwellian, SKD, and various RKDs. Panel C: The middle left graph is a line graph showing normalized wave frequencies as a function of normalized wavenumber for a propagation angle of 10 degrees. The x-axis is labeled ‘kc/ωp,p’ and the y-axis is labeled ‘ωr/Ωp’. Multiple lines represent different distributions: Maxwellian, SKD, and various RKDs. Panel D: The middle right graph is a line graph showing normalized growth rates as a function of normalized wavenumber for a propagation angle of 10 degrees. The x-axis is labeled ‘kc/ωp,p’ and the y-axis is labeled ‘γ/Ωp’. Multiple lines represent different distributions: Maxwellian, SKD, and various RKDs. Panel E: The bottom left graph is a line graph showing normalized wave frequencies as a function of normalized wavenumber for a propagation angle of 35 degrees. The x-axis is labeled ‘kc/ωp,p’ and the y-axis is labeled ‘ωr/Ωp’. Multiple lines represent different distributions: Maxwellian, SKD, and various RKDs. Panel F: The bottom right graph is a line graph showing normalized growth rates as a function of normalized wavenumber for a propagation angle of 35 degrees. The x-axis is labeled ‘kc/ωp,p’ and the y-axis is labeled ‘γ/Ωp’. Multiple lines represent different distributions: Maxwellian, SKD, and various RKDs.

3.2. Oblique proton firehose instability

Figure 8 shows the (O)PFHI dispersion relations (normalised frequencies in the left panels and normalised growth rates in the right panels) for different propagation angles ( $0^\circ$ , $10^\circ$ and $35^\circ$ , from top to bottom) and for several VD functions: Maxwellian (blue), SKD with $\kappa =3.0$ (black), RKD with $\kappa =3.0$ and $\alpha =0.15$ (brown), RKD with $\kappa =2.0$ and $\alpha =0.1$ (red), RKD with $\kappa =1.5$ and $\alpha =0.1$ (green) and RKD with $\kappa =1.0$ and $\alpha =0.1$ (magenta).

For the parallel case, where only periodic modes are present, the results are in qualitative agreement with those reported by Husidic et al. (Reference Husidic, Lazar, Fichtner, Scherer and Astfalk2020). They can be summarised as follows. Using an SKD with low $\kappa$ enhances the growth rate and shifts $\gamma _m$ toward lower wavenumbers compared with the Maxwellian case. The real frequency is also enhanced at low to intermediate wavenumbers. This trend persists when employing RKDs with lower $\kappa$ (at fixed, moderate $\alpha$ ), corresponding to an increased population of suprathermal particles in the tails. For example, the Maxwellian case yields $\gamma _m/\varOmega _p = 0.123$ at $k_m c/\omega _{pp} = 0.279$ , whereas the RKD with $\kappa = 1.0$ gives $\gamma _m/\varOmega _p = 0.169$ at $k_m c/\omega _{pp} = 0.131$ , corresponding to an increase of approximately $37\,\%$ in $\gamma _m$ and a reduction of approximately $113\,\%$ in $k_m$ . Using an RKD with $\kappa =3.0$ , the overall growth rate gets slightly reduced (approximately $2\,\%$ in terms of $\gamma _m$ ) and shifted towards the Maxwellian case. The range of unstable wavenumbers remains approximately the same for all VDs.

At $\theta = 10^\circ$ , only propagating modes are unstable, and only minor differences relative to the parallel case are observed. The maximum growth rate decreases slightly across all cases (by approximately $3$ %– $6\,\%$ ), while $k_m$ is reduced by ${\lesssim}2\,\%$ . Consequently, the unstable wavenumber range is also slightly reduced.

At $\theta = 35^\circ$ , more pronounced differences emerge. In the Maxwellian case, only an unstable aperiodic mode is present, with $\gamma _m/\varOmega _p = 0.0809$ at $k_m c/\omega _{pp} = 0.200$ , corresponding to reductions of approximately $34\,\%$ and $28\,\%$ relative to the parallel (periodic) case. The unstable wavenumber range is reduced by more than a factor of two. In the SKD case, a dominant non-propagating branch appears at low wavenumbers, with $\gamma _m/\varOmega _p = 0.0774$ , which is approximately $14\,\%$ higher than for the propagating mode ( $\gamma _m/\varOmega _p = 0.0679$ ). The latter is reduced by approximately $50\,\%$ compared with the parallel case. Both periodic and aperiodic peaks remain below the Maxwellian $\gamma _m$ , although the SKD spans a broader unstable wavenumber range. The RKD case with $\kappa =3.0$ exhibits an $5\,\%$ increased aperiodic mode ( $\gamma _m/\varOmega _p = 0.0816$ at higher $k_m c/\omega _{pp} = 0.1088$ ) and approximately $10\,\%$ decreased periodic mode ( $\gamma _m/\varOmega _p = 0.0614$ at $k_m c/\omega _{pp} = 0.1884$ ) compared with the SKD case, shifting the behaviour towards the Maxwellian solution where only the aperiodic mode is unstable. For all three RKD scenarios with lower $\kappa$ values, unstable aperiodic branches are present. As $\kappa$ decreases, these peaks become less pronounced and shift toward lower wavenumbers. In all cases, however, the periodic branch remains dominant (e.g. by approximately $15\,\%$ for $\kappa =2.0$ and $47\,\%$ for $\kappa =1.0$ ). The aperiodic $\gamma _m$ values for the RKDs are smaller than those of the SKD (by approximately $4$ %– $9\,\%$ ) and the Maxwellian (by approximately $8$ %– $13\,\%$ ). In contrast, the periodic peaks in all RKD cases exceed those of the SKD (both periodic and aperiodic) and the Maxwellian (aperiodic), although they are reduced and shifted toward lower wavenumbers compared with the corresponding parallel RKD cases. For instance, for $\kappa = 1.0$ , the periodic peak with $\gamma _m/\varOmega _p = 0.1094$ is approximately $61\,\%$ higher than the periodic SKD peak, but approximately $35\,\%$ lower than in the parallel RKD $\kappa = 1.0$ case. The unstable wavenumber range in the RKD cases is reduced by approximately a factor of $1.7$ compared with the parallel propagation.

Figure 9 shows the normalised growth rates at higher propagation angles ( $60^\circ$ , $70^\circ$ , $75^\circ$ and $80^\circ$ ) for the Maxwellian and SKD cases and several RKDs, computed with ALPS. In this angular range, only unstable non-propagating modes are observed. The maximum growth rates in the RKD cases exceed those of the Maxwellian case at all angles. As $\kappa$ decreases, $\gamma _m$ increases and shifts toward lower $k_m$ . For example, at $\theta = 70^\circ$ , the RKD case with $\kappa = 1.0$ yields $\gamma _m/\varOmega _p = 0.1233$ at $k_m c/\omega _{pp} = 0.2629$ , which is approximately $38\,\%$ higher and corresponds to a $36\,\%$ reduction in $k_m$ compared with the Maxwellian case ( $\gamma _m/\varOmega _p = 0.0897$ at $k_m c/\omega _{pp} = 0.4093$ ). The RKD with $\kappa =3.0$ shows slightly lower maximum growth rates (up to $6\,\%$ ) for all cases compared with the SKD, with differences becoming more pronounced at the highest angles, shifting the behaviour towards the solution of the Maxwellian VD.

As $\theta$ increases, $\gamma _m$ increases up to a maximum angle, while $k_m$ also increases and the peaks become significantly broader (particularly at $\theta = 80^\circ$ ), extending over a wider interval of unstable wavenumbers. For all VDs, the maximum aperiodic growth rates remain below the corresponding maximum periodic growth rates in the parallel case. For instance, the Maxwellian case reaches its maximum aperiodic growth rate around $\theta = 55^\circ$ with $\gamma _m/\varOmega _p = 0.1088$ , which is approximately $12\,\%$ lower than the parallel value $\gamma _m/\varOmega _p = 0.123$ .

In all RKD cases, the maximum aperiodic growth rate is attained at slightly larger angles, around $\theta = 60^\circ$ . For example, for $\kappa = 1.0$ , $\gamma _m/\varOmega _p = 0.1298$ , which is approximately $23\,\%$ lower than the corresponding parallel value $\gamma _m/\varOmega _p = 0.169$ at $\theta = 0^\circ$ . At $\theta = 80^\circ$ , the RKD with $\kappa = 1.0$ exhibits a maximum growth rate that is approximately $229\,\%$ higher than in the Maxwellian case, along with an increase of about $50\,\%$ in the range of unstable wavenumbers.

Figure 9.

Normalised growth rates as functions of the normalised wavenumber, for the (aperiodic) OPFHI at propagation angles $55^\circ$ , $60^\circ$ , $70^\circ$ and $80^\circ$ for Maxwellian and different RKDs.