Alpha/proton Instability in the Presence of Proton and Alpha Temperature Anisotropy and its Application to the Deceleration of Alpha Particles in the Solar Wind

Observations from Helious, Wind and PSP spacecraft have revealed that alpha particles are omnipresent in the solar wind (Bame et al. 1977; Pizzo et al. 1983; Marsch & Richter 1984; Li & Li 2006). The alpha particles comprise about 5% of ions and they stream faster than the ambient protons (Steinberg et al. 1996; Neugebauer et al. 1996; Feldman et al. 1996). The drift velocity of the alpha beams with respect to the ambient solar wind is usually equal to or less than the local Alfvén velocity, and these beams propagate in directions nearly parallel to the ambient magnetic field (Marsch & Livi 1987; Verniero et al. 2020). In the weakly collisional plasma environment of the solar wind, the alpha beams are typically at a higher temperature than the background protons by a factor of around four or even larger (Ebert et al. 2009; Borovsky 2016). The local Alfvén velocity of solar wind will decrease with the heliocentric distance when propagating from the Sun into the interplanetary space, and the limitation of the alpha beam drift velocity implies a continuous deceleration of the alpha beams throughout the whole interplanetary space (Kasper et al. 2008; Bourouaine et al. 2013; Ďurovcová et al. 2019). Moreover, these alpha particles play a particularly important role in the energization processes of particles in the solar wind (Chandran et al. 2013; Verscharen et al. 2015; Ofman 2019; Markovskii et al. 2019). However, the formation and evolution mechanism of the alpha particles in the solar wind are still unclear.

The drift velocity of the alpha beams, if it is large enough, can excite two different types of electromagnetic alpha/proton modes: the magnetosonic/whistler wave and the Alfvén/ion-cyclotron wave (Revathy 1978; Verscharen et al. 2013a, 2013b). The former is prior to generate in the plasma with β > 1, and it propagates along the ambient magnetic field B ₀ , where β is the ratio of plasma thermal to magnetic pressures (Gary et al. 2000a, 2000b). The latter is inclined to arise in the plasma with β < 1, and it is oblique to the B ₀ field. These alpha beam instabilities are believed to be an efficient mechanism in constraining the beam velocity (Lu et al. 2009; Rehman et al. 2020). As soon as the threshold condition is satisfied, the corresponding wave is effectively induced, which leads to an exchange of energy between energetic beams and waves via the wave-particle interactions, and the beam drift velocity decreases (Gary et al. 2001; Xiang et al. 2018). In this way, these instabilities can put an upper bound on the alpha-beam drift velocity. However, the observed alpha-beam drift velocity is far less than the theoretical prediction (Reisenfeld et al. 2001; Bourouaine et al. 2013).

On the other hand, ions usually present a bi-Maxwellian velocity distribution in the solar wind, which can be represented as different temperature components T_⊥ ≠ T_∥ with respect to B ₀ , such as the proton temperature anisotropy T_i⊥/T_i∥ (Kasper et al. 2003; Bale et al. 2009; Maruca et al. 2012; Seough et al. 2013; Huang et al. 2020; Hellinger et al. 2006) and the alpha temperature anisotropy T_α⊥/T_α∥ (Marsch et al. 1982b; Gary et al. 2016). Due to the propagating-wind expansion, the perpendicular and parallel temperature components can be governed by the CGL mechanism (Chew et al. 1956). In some plasma environments with T_i⊥ ≠ T_i∥, the value of T_i⊥/T_i∥ is often located in the range of 1 to 2 in the fast wind, while it is less than 0.5 in the slow solar wind (Marsch et al. 1982a; Marsch 2012; Jian et al. 2016). In addition, in some plasma environments with T_α⊥ ≠ T_α∥, the value of T_α⊥/T_α∥ is smaller than 1 in the slow wind, and sometimes it is larger than 1 in the fast solar wind (Gary et al. 2016). Therefore, the temperature anisotropies of both proton and alpha can be smaller or larger than 1 in different solar wind plasma environments.

In the previous studies of Gary et al. (2000a, 2000b), the linear theory and hybrid simulations were used to study the growing modes of the alpha beam instability and the role of the wave-particle scattering on the ion heating and acceleration. However, their studies examined the properties of the alpha beam instability in simple Maxwellian or dirft-Maxwellian plasmas. In this study, we investigate the alpha beam instability in the presence of anisotropic proton and alpha and analyze the effects of proton and alpha temperature anisotropies on these instabilities at arbitrary propagation angles relative to B ₀ . Our present results can provide a physical mechanism to explain the deceleration of alpha beams in the solar wind. The rest of this paper is organized as follows. In Section 2, the basic model and the Vlasov equation are presented. In Section 3, we investigate the effects of the proton temperature anisotropy on the alpha beam instability. In Section 4, we also study the effects of the alpha temperature anisotropy on the alpha beam instability. Finally, in Section 5, the discussion and conclusion are given.

In the alpha-beam return-current system magnetized by the background magnetic field B ₀ = B₀ e _z , let us consider a collisionless plasma with three particle species: the static proton (i) with anisotropic temperatures, the alpha beam (α) with anisotropic temperatures and field-aligned drift velocity v_α , and the background electron (e) with isotropic temperatures and field-aligned drift velocity v_e . Taking into account both the beam and temperature anisotropy, the equilibrium distribution function, which is assumed in the drift bi-Maxwellian distribution, can be given as follows (Chen & Wu 2012):

$$\begin{eqnarray}&&{f}_{s0}=\displaystyle \frac{{n}_{s}}{{(2\pi )}^{3/2}{v}_{{T}_{s\perp }}^{2}{v}_{{T}_{s\parallel }}}{e}^{-\tfrac{{v}_{\perp }^{2}}{2{v}_{{T}_{s\perp }}^{2}}-\displaystyle \frac{{({v}_{\parallel }-{v}_{s})}^{2}}{2{v}_{{T}_{s\parallel }}^{2}}},\end{eqnarray} \tag{ 1 }$$

where n_s and v_s represent the number density and the drift velocity for species s, respectively. The subscripts s = i, α and e denote the ambient proton, alpha beam and ambient electron, respectively, and the subscripts ⊥ and ∥ correspond to the perpendicular and parallel directions with respect to B ₀ . Here ${v}_{\perp }=\sqrt{{v}_{x}^{2}+{v}_{y}^{2}}$ and v_∥ = v_z represent the perpendicular and parallel velocities, respectively. While ${v}_{{T}_{s\perp (\parallel )}}=\sqrt{\tfrac{{T}_{s\perp (\parallel )}}{{m}_{s}}}$ is the perpendicular (parallel) thermal velocity, and m_s and T_s⊥(∥) represent the mass and the perpendicular (parallel) temperature, respectively.

In collisionless space and astrophysical plasmas, waves and instabilities can be derived from the following equations (Wu 2012)

$$\begin{eqnarray}&&\begin{array}{l}\left[\displaystyle \frac{\partial }{\partial t}+{\boldsymbol{v}}\cdot {\rm{\nabla }}+\displaystyle \frac{{q}_{s}}{{m}_{s}}\left({\boldsymbol{v}}\times {{\boldsymbol{B}}}_{0}\right)\cdot {{\rm{\nabla }}}_{{\boldsymbol{v}}}\right]{f}_{s1}\\ \ =\,-\displaystyle \frac{{q}_{s}}{{m}_{s}}\left({{\boldsymbol{E}}}_{1}+{\boldsymbol{v}}\times {{\boldsymbol{B}}}_{1}\right)\cdot {{\rm{\nabla }}}_{{\boldsymbol{v}}}{f}_{s0},\\ \quad {\rm{\nabla }}\times {{\boldsymbol{B}}}_{1}={\mu }_{0}{\boldsymbol{j}}+\displaystyle \frac{1}{{\varepsilon }_{0}{\mu }_{0}}\displaystyle \frac{\partial {{\boldsymbol{E}}}_{1}}{\partial t},\\ \qquad {\rm{\nabla }}\times {{\boldsymbol{E}}}_{1}=-\displaystyle \frac{\partial {{\boldsymbol{B}}}_{1}}{\partial t},\end{array}\end{eqnarray} \tag{ 2 }$$

where q_s , f_s0, and f_s1 represent the charge, unperturbed distribution function, and perturbed distribution function. ${\vec{E}}_{1}$ and ${\vec{B}}_{1}$ are the electric field fluctuation and magnetic field fluctuation, respectively.

By combining Equations (1) and (2), the general dispersion relation in a homogeneous plasma can be obtained as follows (Stix 1992)

$$\begin{eqnarray}\left|\begin{array}{lcc}{\varepsilon }_{{xx}}-{c}^{2}{k}_{z}^{2}/{\omega }^{2} & {\varepsilon }_{{xy}} & {\varepsilon }_{{xz}}+{c}^{2}{k}_{x}{k}_{z}/{\omega }^{2}\\ {\varepsilon }_{{yx}} & {\varepsilon }_{{xx}}-{c}^{2}{k}^{2}/{\omega }^{2} & {\varepsilon }_{{yz}}\\ {\varepsilon }_{{zx}}+{c}^{2}{k}_{x}{k}_{z}/{\omega }^{2} & {\varepsilon }_{{zy}} & {\varepsilon }_{{zz}}-{c}^{2}{k}_{x}^{2}/{\omega }^{2}\end{array}\right|=0,\end{eqnarray} \tag{ 3 }$$

where

$$\begin{eqnarray}\begin{array}{rcl}\varepsilon & = & I+\displaystyle \sum _{s}\displaystyle \frac{{\omega }_{{ps}}^{2}}{{\omega }^{2}}\displaystyle \sum _{n}\left[{\zeta }_{s0}Z({\zeta }_{{sn}})-\left(1-\displaystyle \frac{{T}_{s\perp }}{{T}_{s\parallel }}\right)\right)\\ & & \left.(,1,+,{\zeta }_{{sn}},Z,(,{\zeta }_{{sn}},),)\right]{Y}_{{sn}}+\,\displaystyle \sum _{s}\displaystyle \frac{2{\omega }_{{ps}}^{2}}{{\omega }^{2}}{\zeta }_{s0}^{2}\displaystyle \frac{{T}_{s\parallel }}{{T}_{s\perp }}{e}_{z}{e}_{z},\end{array}\end{eqnarray} \tag{ 4 }$$

and

$$\begin{eqnarray}{Y}_{{sn}}=\left[\begin{array}{ccc}{n}^{2}{{\rm{\Gamma }}}_{{sn}}/{\lambda }_{s} & {in}{{\rm{\Gamma }}}_{{sn}}^{{\prime} } & -n{\left(\displaystyle \frac{{T}_{s\parallel }}{{T}_{s\perp }}\right)}^{1/2}\displaystyle \frac{\omega +n{\omega }_{{cs}}}{{k}_{z}{v}_{{T}_{s\parallel }}}{{\rm{\Gamma }}}_{{sn}}/\sqrt{{\lambda }_{s}}\\ -{in}{{\rm{\Gamma }}}_{{sn}}^{{\prime} } & {n}^{2}{{\rm{\Gamma }}}_{{sn}}/{\lambda }_{s}-2{\lambda }_{s}^{2}{{\rm{\Gamma }}}_{{sn}}^{{\prime} } & i{\left(\displaystyle \frac{{T}_{s\parallel }}{{T}_{s\perp }}\right)}^{1/2}\displaystyle \frac{\omega +n{\omega }_{{cs}}}{{k}_{z}{v}_{{T}_{s\parallel }}}\sqrt{{\lambda }_{s}}{{\rm{\Gamma }}}_{{sn}}^{{\prime} }\\ -n{\left(\displaystyle \frac{{T}_{s\parallel }}{{T}_{s\perp }}\right)}^{1/2}\displaystyle \frac{\omega +n{\omega }_{{cs}}}{{k}_{z}{v}_{{T}_{s\parallel }}}{{\rm{\Gamma }}}_{{sn}}/\sqrt{{\lambda }_{s}}-i{\left(\displaystyle \frac{{T}_{s\parallel }}{{T}_{s\perp }}\right)}^{1/2}\displaystyle \frac{\omega +n{\omega }_{{cs}}}{{k}_{z}{v}_{{T}_{s\parallel }}}\sqrt{{\lambda }_{s}}{{\rm{\Gamma }}}_{{sn}}^{{\prime} } & \displaystyle \frac{\omega +n{\omega }_{{cs}}}{{k}_{z}{v}_{{T}_{s\parallel }}}\sqrt{{\lambda }_{s}}{{\rm{\Gamma }}}_{{sn}}^{{\prime} } & (\displaystyle \frac{{T}_{s\parallel }}{{T}_{s\perp }})\displaystyle \frac{{\left(\omega +n{\omega }_{{cs}}\right)}^{2}}{{k}_{z}^{2}{v}_{{T}_{s\parallel }}^{2}}{{\rm{\Gamma }}}_{{sn}}\end{array}\right].\end{eqnarray} \tag{ 5 }$$

In the above expressions, ${\lambda }_{s}=\tfrac{{k}_{x}^{2}{v}_{{T}_{s\perp }}^{2}}{{\omega }_{{cs}}^{2}}$, ${\zeta }_{s0}=\tfrac{\omega -{k}_{z}{v}_{s}}{\sqrt{2}{k}_{z}{v}_{{T}_{s\parallel }}}$, ${\zeta }_{{sn}}\,=\tfrac{\omega -{k}_{z}{v}_{s}-n{\omega }_{{cs}}}{\sqrt{2}{k}_{z}{v}_{{T}_{s\parallel }}}$, ${\omega }_{{cs}}=\tfrac{{q}_{s}{B}_{0}}{{m}_{s}}$, ${\omega }_{{ps}}=\sqrt{\tfrac{{n}_{s0}{q}_{s}^{2}}{{\varepsilon }_{0}{m}_{s}}}$, and ${{\rm{\Gamma }}}_{{sn}}={I}_{n}({\lambda }_{s}){e}^{-{\lambda }_{s}}$. Here I_n (λ_s ) is the modified Bessel function of the first kind of nth-order, and $Z({\zeta }_{{sn}})$ is the plasma dispersion function. Under the plane wave assumption, the wavevector is k = k_x e_x + k_z e_z . From nontrivial solutions of Equation (3), all plasma modes in arbitrary directions relative to B ₀ can be derived in the kinetic Valsov system.

Several numerical methods were developed to solve the kinetic dispersion relation of Equation (3), such as WHAMP (Roennmark 1982), NHDS (Verscharen et al. 2013b) and KUPDAP (Sugiyama et al. 2015). In this study, we adopt a new numerical solver, PDRK (Xie & Xiao 2016; Xie 2019), to find the dispersion relation of the plasma modes in Equation (3). Unlike other solvers by using the Newton iterative method, the PDRK solver is based on the standard matrix eigenvalue method to solve an equivalent linear equation system that was derived from the linear transformation of Equation (3). Therefore, all physical solutions can be obtained by running the PDRK solver without requiring any initial guess value, which provides an excellent opportunity to examine the plasma instability. Recently, using the PDRK solver, several plasma instabilities have been investigated in various active phenomena in space and solar plasmas, including the proton and electron temperature anisotropy (Sun et al. 2019, 2020) and the proton beam instability (Xiang et al. 2020, 2021) in the solar wind, and the electron beam instability in flare loops (Chen et al. 2020).

In the numerical calculations, we assume both the charge-neutral and zero-current conditions in the alpha beam-return current system, i.e., ∑_s q_s n_s = 0 and ∑_s q_s n_s v_s = 0. We also perform calculations in the center of the background-proton frame. The plasma parameters n_e = 5 × 10⁶ m⁻³, B₀ = 5 ×10⁻⁹ T, and 4T_i∥ = 4T_e∥ = T_α∥ have been adopted. Here both the background proton and the alpha beam with temperature anisotropies are considered, T_i⊥ ≠ T_i∥ and T_α⊥ ≠ T_α∥, while the background electron is assumed to be isotropic, T_e⊥ = T_e∥.

In this section, we investigate the wave dispersion and polarization properties of the alpha beam instability accompanied with the proton temperature anisotropy and systemically analyze the influences of the proton temperature anisotropy on the wave frequency, growth rate and threshold condition.

Figure 1 displays the distribution of the dispersion relation and polarization properties in the plane of the propagation angle θ and normalized wavenumber k λ_i for the oblique Alfvén/ion-cyclotron (OA/IC) wave in the plasma with low-beta β_e∥ = 0.1 (${\beta }_{e\parallel }=2{\mu }_{0}{n}_{e}{T}_{i\parallel }/{B}_{0}^{2}$ being the parallel electron beta), where panels (a), (b) and (c) correspond to different proton temperature anisotropies T_i⊥/T_i∥ = 0.25, 1 and 2, respectively. The plasma parameters are n_α /n_e = 0.05, v_α /v_A = 1.5, T_α∥ = 4T_e∥ = 4T_i∥ and T_α⊥ = 4T_e⊥ = 4T_i∥. As can bee seen from Figure 1, the proton beam can effectively drive the OA/IC wave in the low-beta case, i.e., β_e∥ = 0.1, and its propagation angle is located in the region of 30° ≤ θ ≤ 70°. Moreover, as the proton temperature anisotropy T_i⊥/T_i∥ increases from 0.25 to 2, the OA/IC wave properties of ω_r /ω_ci , γ/ω_ci and E_y /iE_x do not vary obviously.

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Figure 2 shows the distribution of the dispersion relations and polarization properties in the plane of the propagation angle θ and normalized wavenumber k λ_i for the parallel magnetosonic/whistler (PM/W), backward magnetosonic/whistler (BM/W), parallel Alfvén/ion-cyclotron (PA/IC), OA/IC and mirror waves in the plasma with high-beta β_e∥ = 1.5. Here planes (a), (b) and (c) correspond to different proton temperature anisotropies T_i⊥/T_i∥ = 0.25, 1 and 2, respectively. Other plasma parameters are the same as Figure 1. From Figure 2, it can be found that in the case of β_e∥ = 1.5 and T_i⊥/T_i∥ = 1, the alpha beams can drive both the PM/W and OA/IC waves, which correspond to different propagation angles θ ≤ 10° and 40° ≤ θ ≤ 65°, respectively. As T_i⊥/T_i∥ = 0.25, besides the PM/W and OA/IC waves, the proton temperature anisotropy provides an additional driving source to generate the BM/W wave in the direction of anti-parallel propagation. Compared to that of T_i⊥/T_i∥ = 1, both the OA/IC and PM/W waves at T_i⊥/T_i∥ = 0.25 have wider propagation angle ranges for 20° ≤ θ ≤ 70° and θ ≤ 20°, respectively. As T_i⊥/T_i∥ = 2, three kinds of modes exist, i.e., the PA/IC, OA/IC and mirror waves. The PA/IC, OA/IC and mirror waves dominate in different regimes of θ ≤ 20°, 25° ≤ θ ≤ 55° and 55° ≤ θ < 90°, respectively.

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In the following, the influences of the proton temperature anisotropy on the threshold conditions are also investigated. Figure 3 shows the distributions of the dispersion relations (ω_r /ω_ci and ${\gamma }_{\max }/{\omega }_{{ci}}$) in the plane of the parallel electron beta β_e∥ and the alpha-beam drift velocity v_α /v_A for different values of the proton temperature anisotropy T_i⊥/T_i∥ = 0.25 ((a), (d)), 1 ((b), (e)) and 2 ((c), (f)). Here panels (a)–(c) represent the PM/W, BM/W and PA/IC waves at the parallel propagation case of θ = 0, and panels (d)–(f) stand for the OA/IC and mirror waves at the oblique propagation case of θ = 55°. In the maximum growth rate distributions, the black solid lines correspond to the contour values of ${\gamma }_{\max }/{\omega }_{{ci}}={10}^{-3}$, 10⁻² and 10⁻¹, and the curved line with ${\gamma }_{\max }/{\omega }_{{ci}}={10}^{-3}$ is defined as the instability threshold. Figure 3 clearly displays the dispersion relation and threshold distributions of the PM/W, BM/W, PA/IC, OA/IC and mirror waves, and their dependence on β_e∥ and v_α /v_A , as well as T_i⊥/T_i∥.

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For parallel propagation cases (θ = 0) shown in Figures 3(a)–(c), the velocity threshold and the real frequency of the PM/W wave decrease as β_e∥ increases in the absence of the proton temperature anisotropy, i.e., T_i⊥/T_i∥ = 1. As T_i⊥/T_i∥ = 0.25, the PM/W and BM/W waves can be generated, and the unstable region is enlarged obviously compared to the case of T_i⊥/T_i∥ = 1. In particular, the BM/W wave dominates in the unstable region at a smaller drift velocity of v_α /v_A , whereas the PM/W wave with a larger growth rate becomes very strong at a larger v_α /v_A . As T_i⊥/T_i∥ = 2, both the PM/W and PA/IC waves are unstable, but they have different unstable regions. The threshold velocity v_α /v_A of the PA/IC wave is much smaller than that of the PM/W wave. For oblique propagation cases (θ = 55°) shown in Figure 3(d)–(f), only the OA/IC wave appears at T_i⊥/T_i∥ = 1. As T_i⊥/T_i∥ = 0.25, the threshold velocity of the OA/IC wave extends to a lower value (i.e., v_α /v_A < 1), especially in the β_e∥ > 2 regime. As T_i⊥/T_i∥ = 2, both the OA/IC and mirror waves are induced in different v_α /v_A and β_e∥ regimes. The OA/IC wave with a larger growth rate tends to dominate in the unstable region at a larger v_α /v_A , while the mirror wave with a lower threshold velocity is easily excited at a smaller v_α /v_A and a larger β_e∥.

The variations of instability thresholds with the alpha-beam drift velocity v_α /v_A and parallel electron beta β_e∥ are shown in Figure 4, where the top and bottom panels correspond to two different situations of (a) T_i⊥/T_i∥ ≤ 1 and (b) T_i⊥/T_i∥≥1 , respectively. In Figure 4(a) the solid, dotted and dashed lines correspond to the PM/W, BM/W and OA/IC waves, and in Figure 4(b) the solid, dotted, dashed and dotted–dashed lines represent the PM/W, PA/IC, OA/IC and mirror waves, respectively. For the cases of T_i⊥/T_i∥ ≤ 1 shown in Figure 4(a), there exist three kinds of waves: PM/W, OA/IC and BM/W waves. At T_i⊥/T_i∥ = 1, both PM/W and OA/IC waves can be induced by the alpha beams, but they have different unstable regimes. The OA/IC wave is more likely to excite at β_e∥ < 0.5, while the PM/W wave is more inclined to arise at β_e∥ > 0.5, which is in agreement with previous results in Gary et al. (2000a, 2000b). As T_i⊥/T_i∥ decreases from 1 to 0.25, the threshold velocity of both OA/IC and PM/W waves obviously decreases, especially for the OA/IC wave at β_e∥ > 2. Also, the BM/W wave appears at T_i⊥/T_i∥ = 0.25 and 0.5, implying that the proton temperature anisotropy with T_i⊥/T_i∥ < 1 can provide additional free energy contributing to the generation and growth of BM/W waves. The threshold condition of the BM/W wave is reduced to a lower v_α /v_A and a lower β_e∥ at T_i⊥/T_i∥ = 0.25. Besides, for the cases of T_i⊥/T_i∥ ≥ 1 shown in Figure 4(b), four types of waves exist: PM/W, OA/IC, PA/IC and mirror waves. As T_i⊥/T_i∥ increases from 1 to 2, the threshold v_α /v_A of PM/W waves increases, while that of OA/IC waves first increases slightly then decreases sharply. Moreover, the presence of T_i⊥/T_i∥ > 1 leads to lower thresholds v_α /v_A for the PA/IC and mirror waves, which can be reduced to near or below v_A .

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We investigate the alpha-beam instability associated with the alpha temperature anisotropy and perform the effects of alpha temperature anisotropy on these instabilities at arbitrary propagation angles with respect to B ₀ . In this section, we assume the alpha beam with an anisotropic temperature, while the background proton and electron are taken as with an isotropic temperature.

Figure 5 shows the distribution of the dispersion relations and polarization properties in the plane of the propagation angle θ and normalized wavenumber k λ_i for the OA/IC wave in the plasma with low-beta β_e∥ = 0.1, where panels (a), (b) and (c) correspond to different alpha temperature anisotropies T_α⊥/T_α∥ = 0.25, 1 and 2, respectively. The plasma parameters are n_α /n_e = 0.05, v_α /v_A = 1.5, 4T_i∥ = 4T_e∥ = T_α∥ and 4T_i⊥ = 4T_e⊥ = T_α∥. From Figure 5, it can be seen that only the OA/IC wave survives at β_e∥ = 0.1. Moreover, the real frequency, growth rate and polarization of the OA/IC wave are insensitive to the alpha temperature anisotropy T_α⊥/T_α∥, implying that the influence of T_α⊥/T_α∥ on wave properties of OA/IC waves is minor at β_e∥ = 0.1.

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Figure 6 shows the distribution of the dispersion relations and polarization properties in the plane of the propagation angle θ and normalized wavenumber k λ_i for the PM/W, OA/IC, PA/IC and mirror waves in the plasma with high-beta β_e∥ = 1.5. Here planes (a), (b) and (c) correspond to different alpha temperature anisotropies T_α⊥/T_α∥ = 0.25, 1 and 2, respectively. From Figure 6, one finds that the PM/W wave dominates in the unstable region at θ ≤ 10°, while the OA/IC wave distributes at 40° ≤ θ ≤ 60° when T_α⊥/T_α∥ = 1, which is consistent with the results of Gary et al. (2000a, 2000b). Moreover, the unstable ranges of θ and k λ_i of PM/W and OA/IC waves strongly depend on the alpha temperature anisotropy. For example, as T_α⊥/T_α∥ = 1 (0.25), the propagation angle is θ ≤ 10° (θ ≤ 20°) for the PM/W wave and 40° ≤ θ ≤ 60° (30° ≤ θ ≤ 70°) for the OA/IC wave. As T_α⊥/T_α∥ = 2, the PA/IC and mirror waves are unstable, while the PM/W and OA/IC waves disappear. This indicates that the presence of the anisotropic alpha beams with T_α⊥/T_α∥ > 1 can lead to an enhancement of the growth of PA/IC and mirror waves, but an inhibition of the growth of the PM/W and OA/IC waves.

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To understand more general wave properties, we present in Figure 7 the distributions of the dispersion relations (ω_r /ω_ci and ${\gamma }_{\max }/{\omega }_{{ci}}$) in the plane of the parallel electron beta β_e∥ and the alpha-beam drift velocity v_α /v_A for different values of alpha temperature anisotropy T_α⊥/T_α∥ = 0.25 ((a), (d)), 1 ((b), (e)) and 2 ((c), (f)). Here panels (a)–(c) represent PM/W and PA/IC waves at the parallel propagation case of θ = 0, and panels (d)–(f) stand for the OA/IC, oblique magnetosonic/whistler (OM/W) and mirror waves at the oblique propagation case of θ = 55°. In the maximum growth rate distributions, the black solid lines correspond to the contour values of ${\gamma }_{\max }/{\omega }_{{ci}}={10}^{-3}$, 10⁻² and 10⁻¹, respectively. For parallel propagation cases shown in Figures 7(a)–(c), there are two kinds of waves, i.e., the PM/W and PA/IC waves. For oblique propagation cases shown in Figures 7(d)–(f), three kinds of waves exist, that is, the OA/IC, OM/W, and mirror waves. These waves distribute in different plasma parameter regimes.

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As shown in Figures 7(a)–(c), the PM/W wave has an unstable region wider at T_α⊥/T_α∥ = 0.25 than at T_α⊥/T_α∥ = 1. As T_α⊥/T_α∥ = 2, the PA/IC wave arises, which extends to lower threshold conditions (i.e., v_α /v_A < 1). In addition, the oblique waves are further presented in Figures 7(d)–(f). Compared with the T_α⊥/T_α∥ = 1 case, the threshold v_α /v_A of the OA/IC wave can extend to a lower value in the case of T_α⊥/T_α∥ = 0.25 (i.e., v_α /v_A ≤ 1 for β_e∥ ≥ 1). Moreover, besides the OA/IC wave, both mirror and OM/W waves can be excited at T_α⊥/T_α∥ = 2. The mirror wave dominates in the higher velocity regime of 0.8 < v_α /v_A < 2, while the OM/W wave dominates in the lower velocity regime of v_α /v_A < 0.8.

In addition, we discuss the possible relevance of the alpha temperature anisotropy to the threshold conditions. Figure 8 displays the instability thresholds of the normalized alpha-beam drift velocity v_α /v_A versus the parallel electron beta β_e∥, where the top and bottom panels correspond to two different situations of (a) T_α⊥/T_α∥ ≤ 1 and (b) ≥ 1, respectively. In Figure 8(a) the solid and dashed lines correspond to the PM/W and OA/IC waves, and in Figure 8(b) five types of lines represent the PM/W, OA/IC, PA/IC, OM/W and mirror waves, respectively. For an excess of parallel temperature T_α⊥/T_α∥ ≤ 1 shown in Figure 8(a), it can be found that the threshold velocities of both PM/W and OA/IC waves are shifted to lower values as T_α⊥/T_α∥ decreases. This tide is more prominent for the PM/W wave at β_e∥ > 0.5. The threshold velocity is v_α /v_A ≤ 1 (1.2) for the PM/W wave (the OA/IC wave) when T_α⊥/T_α∥ ≤ 0.5 and β_e∥ ≥ 1. On the other hand, for an excess of perpendicular temperature T_α⊥/T_α∥ ≥ 1 presented in Figure 8(b), the threshold velocities of both PM/W and OA/IC waves increase as T_α⊥/T_α∥ increases. As T_α⊥/T_α∥ = 2, the PA/IC, OM/W, and mirror waves arise, and their instability thresholds extend to lower values (i.e., v_α /v_A < 1). This implies that the presence of the anisotropic alpha beams with T_α⊥/T_α∥ > 1 can lead to a broader unstable region at a lower β_e∥ and/or lower v_α /v_A .

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In-situ observations by space satellites have revealed that the alpha beams often stream faster than the background protons, and the drift velocity between two components is typically less than the local Alfvén velocity (Marsch et al. 1982a; Marsch & Livi 1987; Steinberg et al. 1996; Neugebauer et al. 1996; Feldman et al. 1996; Verniero et al. 2020). Because the local Afvén velocity decreases with the increase of the heliocentric distance (Marsch et al. 1982b), the observed limitation of the alpha beams with v_α /v_A ≤ 1 implies that the drift velocity of the alpha beams v_α is continuously reduced when the solar wind flows from the Sun into the interplanetary space (Chandran et al. 2013; Alterman et al. 2018). The alpha beam instability has been proposed to be responsible for decelerating the alpha-beam drift velocity in the solar wind (Revathy 1978; Gary et al. 2000a, 2000b). However, there is still no consensus on the details of deceleration processes for the solar wind alpha beams. Also, the effects of the ion temperature anisotropy on the alpha beam instability have not been examined. In this study, we investigate the properties of the alpha beam instability in the presence of both proton and alpha temperature anisotropies in the solar wind and consider the effects of T_i⊥/T_i∥ and T_α⊥/T_α∥ on these instabilities at arbitrary propagation angles.

It is found that the wave dispersion relations and polarization properties of ω_r /ω_ci , γ/ω_ci and E_y /iE_x are strongly dependent on T_i⊥/T_i∥, T_α⊥/T_α∥ and β_e∥. Moreover, the effects of T_i⊥/T_i∥ and T_α⊥/T_α∥ on these waves are minor in the region with low-beta β_e∥ < 1, but obvious in the region with high-beta β_e∥ ≥ 1. For the condition of both proton temperature anisotropy and alpha beam, there exist different types of unstable modes, i.e., the PM/W, BM/W and OA/IC waves at T_i⊥/T_i∥ < 1 and the PM/W, PA/IC, OA/IC and mirror waves at T_i⊥/T_i∥ > 1. For T_i⊥/T_i∥ < 1, as T_i⊥/T_i∥ decreases, the growth rates of the BM/W and OA/IC waves are enhanced by T_i⊥/T_i∥, and their instability thresholds of both two waves are extended to a lower v_α /v_A and a lower β_e∥ at the propagation angle of θ = 0 and 55°, respectively. For T_i⊥/T_i∥ > 1, as T_i⊥/T_i∥ increases, the instability thresholds of the PA/IC, OA/IC and mirror waves are diverted to a lower v_α /v_A and a lower β_e∥.

For the condition of both alpha temperature anisotropy and alpha beam, there are different kinds of unstable modes: the PM/W and OA/IC waves at T_α⊥/T_α∥ < 1 and the PM/W, OA/IC, PA/IC, OM/W and mirror waves at T_α⊥/T_α∥ > 1. For T_α⊥/T_α∥ < 1, with the decrease of T_α⊥/T_α∥, the growth rates of the PM/W and OA/IC waves increase, and the threshold conditions of two waves are extended to a smaller v_α /v_A , especially for the PM/W wave at β_e∥ > 0.5. For T_α⊥/T_α∥ > 1, with the increase of T_α⊥/T_α∥, the growth rates of the PM/W and the OA/IC waves decrease, while those of the PA/IC, OM/W and mirror waves increase. Moreover, the unstable regimes of the PA/IC, OM/W and mirror wave are enlarged to a lower v_α /v_A and a lower β_e∥.

In addition, we compare the present results with those of the previous studies in Gary et al. (2000a, 2000b). The existence of the temperature anisotropy of the proton and alpha particles can lead to not only the enhanced alpha beam instabilities, but also the excitation of extra wave modes. The proton and/or alpha temperature anisotropy exerts a strong influence on the threshold of the alpha beam instabilities, resulting in a larger growth rate and a lower threshold velocity. Therefore, the alpha beam instability accompanied with the proton and/or alpha temperature anisotropy may reduce the alpha-beam drift velocity to near or less than the local Alfveń velocity, which shows a good fit to the observations in the solar wind.

In summary, our physical model could give a rational explanation for the alpha-beam deceleration in the solar wind. When the solar wind plasma flows into the interplanetary space, the alpha-beam drift velocity v_α does not vary obviously, but the local Afvén velocity v_A certainly decreases, which leads to an increase of the normalized drift velocity v_α /v_A . As v_α /v_A exceeds the threshold of the alpha beam instability, the beam energy is transferred to the wave energy via the OA/IC and PM/W wave excitation. However, when v_α /v_A < 1.5, the alpha beam instability accompanied with proton and alpha temperature anisotropies could work in concert to deceleration alpha beams because its threshold condition undergoes major changes in the presence of the anisotropic proton and alpha. In general, both the proton and alpha temperature anisotropies are common intrinsic phenomena in the solar wind (Yoon et al. 2019). In plasma environments with T_i⊥/T_i∥ ≠ 1, the deceleration of the alpha beams is governed by the BM/W and OA/IC waves at T_i⊥/T_i∥ < 1 and by the PA/IC, OA/IC and mirror waves at T_i⊥/T_i∥ > 1. While in plasma environments with T_α⊥/T_α∥ ≠ 1, the dissipation of the alpha beams is regulated by the PM/W wave at T_α⊥/T_α∥ < 1 and by the PA/IC, OM/W and mirror waves at T_α⊥/T_α∥ > 1. The presence of both the proton and alpha temperature anisotropies further limits the alpha-beam drift velocity to less than the local Alfvén velocity, which shows a good agreement with the solar wind observations. Our results are potentially important for comprehending the deceleration and evolution mechanism of alpha particles in the solar wind.

This work was supported by the National Natural Science Foundation of China under grants 12103018, 11690034, and 12075084. L. Xiang also was supported by the Science and Technology Innovation Program of Hunan Province (2020RC2049). We give special thanks to Dr. J. F. Tang at XAO for the valuable suggestions.