Issue 
A&A
Volume 615, July 2018



Article Number  A169  
Number of page(s)  10  
Section  Astrophysical processes  
DOI  https://doi.org/10.1051/00046361/201731710  
Published online  07 August 2018 
Nonresonant Alfvénic instability activated by high temperature of ion beams in compensatedcurrent astrophysical plasmas
^{1}
Main Astronomical Observatory, NASU, Kyiv, Ukraine
email: malovichp@i.ua
^{2}
SolarTerrestrial Centre of Excellence, Royal Belgian Institute for Space Aeronomy, Ringlaan 3, 1180 Brussels, Belgium
email: voitenko@oma.be
Received:
3
August
2017
Accepted:
23
April
2018
Context. Compensatedcurrent systems are established in response to hot ion beams in terrestrial foreshock regions, around supernova remnants, and in other space and astrophysical plasmas.
Aims. We study a nonresonant reactive instability of Alfvén waves propagating quasiparallel to the background magnetic field B_{0} in such systems.
Methods. The instability is investigated analytically in the framework of kinetic theory applied to the hydrogen plasmas penetrated by hot proton beams.
Results. The instability arises at parallel wavenumbers k_{z} that are sufficiently large to demagnetize the beam ions, k_{z}V_{Tb}/ω_{Bi} ≳ 1 (here V_{Tb} is the beam thermal speed along B_{0} and ω_{Bi} is the ioncyclotron frequency). The Alfvén mode is then made unstable by the imbalance of perturbed currents carried by the magnetized background electrons and partially demagnetized beam ions. The destabilizing effects of the beam temperature and the temperature dependence of the instability threshold and growth rate are demonstrated for the first time. The beam temperature, density, and bulk speed are all destabilizing and can be combined in a single destabilizing factor α_{b} triggering the instability at α_{b} > α_{b}^{thr}, where the threshold value varies in a narrow range 2.43 ≤ α_{b}^{thr} ≤ 4.87. New analytical expressions for the instability growth rate and its boundary in the parameter space are obtained and can be directly compared with observations. Two applications to terrestrial foreshocks and foreshocks around supernova remnants are briefly discussed. In particular, our results suggest that the ions reflected by the shocks around supernova remnants can drive stronger instability than the cosmic rays.
Key words: plasmas / waves / instabilities / solar wind / ISM: supernova remnants
© ESO 2018
1. Introduction
Diluted ion beams propagating along the background magnetic field B_{0} are widespread in space and astrophysical plasmas, including solar wind (Marsch 2006, and references therein), terrestrial foreshocks (Paschmann et al. 1981, and references therein), supernova remnants (Bell 2005, and references therein), and many other astrophysical environments (Zweibel & Everett 2010, and references therein). As the plasmas are typically quasineutral, the background electrons tend to follow the beam ions compensating their current. Depending on particular settings, the compensating currents can also be provided by other plasma components, like costreaming electron beams injected simultaneously with the ion beams. Plasma instabilities developing in such compensatedcurrent systems not only regulate the plasma and beam parameters keeping them close to the marginally unstable states, but can also be important sources for the background plasma heating, energetic particle acceleration, and amplification of the background magnetic field.
Plasma waves in the compensatedcurrent systems can be driven unstable by resonant (Duijveman et al. 1981; Gary 1985; Vojtenko et al. 1990) and nonresonant (Winske & Leroy 1984; Bell 2004; Achterberg 2013) wave–particle interactions. Resonant kinetic instabilities of various wave modes, driven by the beam ions, have been studied extensively in the past. Parallelpropagating Alfv én and fast waves have been found to be strongly unstable for beam velocities higher than a few Alfvén velocities (e.g., Gary 2005; Marsch 2006, and references therein). Concurrent instabilities of oblique (kinetic) Alfvén waves come into play at lower (but still superAlfvénic) beam velocities (Voitenko 1998).
The abovementioned instabilities can be driven directly by the beam ions (Sentman et al. 1981; Winske & Leroy 1984; Gary 1985) or by the electron return currents (Winske & Leroy 1984; Bell 2004; Chen & Wu 2012, and references therein). The noncompensated electron currents flowing along B_{0}, may also drive both the resonant (Voitenko 1995) and nonresonant (Malovichko & Iukhimuk 1992; Malovichko 2007) instabilities of Alfvén waves. The simplest case of purely parallel propagating Alfvén waves has been considered in application to the currentcarrying coronal loops (Malovichko & Iukhimuk 1992), where these waves appeared to always be unstable. Later on, the analysis was extended by accounting for the oblique propagation (Malovichko 2007) and the currents curried by lowdensity beams (Malovichko 2010), and applied to the terrestrial magnetosphere and coronal loops.
Selfconsistent modifications of the background magnetic field by the electric currents, neglected by Malovichko & Iukhimuk (1992), Malovichko (2007); and Voitenko (1995), may reduce or even stabilize current instabilities. This issue does not concern instabilities developing in compensatedcurrent systems. Such systems, formed around supernova remnants by highenergy streaming cosmic rays (CRs), were studied by Bell (2004), who found a new nonresonant Alfvénic instability (hereafter Bell instability). Since then, the Bell instability and its modifications have attracted a lot of interest (see, e.g., Amato & Blasi 2009; Bret 2009; Zweibel & Everett 2010; Schure et al. 2012; Achterberg 2013; Kobzar et al. 2017, and references therein). Following Bell (2004), the primary focus was on the unstable modes with finite k_{z}V̄_{bz}/ω_{Bi} propagating along B_{0} (where V̄_{bz} is a characteristic velocity of the beam ions along the mean magnetic field B_{0}  z; k_{z} is the parallel wavenumber; and ω_{Bi} is the ioncyclotron frequency). Compensated currents can also drive an oblique Alfvén instability (Malovichko et al. 2014) for which the perpendicular wave dispersion due to finite k_{⊥}V_{Tb⊥}/ω_{Bi} is essential (k_{⊥} and V_{Tb⊥} are the perpendicular wavenumber and beam thermal velocity in the plane ⊥ B_{0}).
Other electrostatic and electromagnetic instabilities may develop in compensatedcurrent systems (see, e.g., Gary 2005; Bret 2009; Brown et al. 2013; Marcowith et al. 2016, and references therein). Which wave modes grow fastest critically depends on the beam and plasma parameters. In the case of cold diluted proton beams propagating along B_{0}, the electrostatic twostream and Buneman instabilities are much faster than the electromagnetic Alfvénic instabilities (see, e.g., Fig. 44 in Bret et al. 2010). Nevertheless, as is noted by Bret et al. (2010), these electrostatic instabilities are quickly saturated, and then electromagnetic Alfvénic/Bell instabilities come into play. In the hot beam/plasma systems, where the twostream/Buneman instabilities cannot develop, the electromagnetic Alfvénic/Bell instabilities dominate.
The Bell instability has a maximum growth rate γ_{Bell} ≃ 0.5j̄_{b}ω_{Bi}, where j̄_{b} = n_{b}V_{b}/ (n_{0}V_{A}) is the beam current normalized by the Alfvén current. This maximum is attained at the parallel wavenumber k_{zm} V_{A}/ω_{Bi} = 0.5j̄_{b} and the perpendicular wavenumber k_{⊥} = 0. These expressions are exactly the same as for the instability studied earlier by Winske & Leroy (1984) in application to the terrestrial foreshock. The difference is that the role of V̄_{bz} in the setting considered by Winske & Leroy (1984) is played by the bulk velocity of the beam V_{b} rather than the large velocity spread of CRs. Both the Winske–Leroy and Bell instabilities grow fastest when the wave vector k is parallel to B_{0}; they are physically the same instability that can be named the compensatedcurrent parallel instability (CCPI).
The physical mechanism of the CCPI is related to the fact that for sufficiently small parallel wavelengths and sufficiently high V̄_{bz}, the beam protons become partially demagnetized (unfrozen off the perturbed magnetic field). The demagnetization reduces the beam contribution to the fluctuating currents δj ⊥ B_{0} flowing along the twisted perturbed magnetic field lines, whereas the electron currents remain magnetized, thus providing the noncompensated fluctuating transversal currents. These currents amplify the initial perturbations via the positive feedback loop giving rise to CCPI. This kind of instability is sometimes called reactive.
Surprisingly, despite its importance in astrophysical applications, the CCPI theory is still poorly developed. Many important properties of the instability (the wavenumber dependence of the instability growth rate, behavior of the maximum growth rate in the parameter space, instability boundaries in the parameter spaces, etc.) have not been fully investigated. In the present paper, we study CCPI of Alfvén waves in more detail in the framework of kinetic theory. We consider a simple model of the compensatedcurrent system where the hydrogen plasma is hydrogen plasma is penetrated by the lowdensity proton beam and the beam current and charge are compensated by the background electrons. Despite its simplicity, this model is applicable to the reactive CCPI driven by compensated currents in many space and astrophysical environments.
2. Plasma model and dispersion equation for Alfvén waves
We consider a threecomponent plasma consisting of the background steady ion component (i), the lowdensity ion beam (b) propagating with velocity V_{b} along z  B_{0}, and the electron component (e) providing the neutralizing current and charge:(1) (2)
We assume here that the beam ions (b) and the background ions (i) are protons. All plasma components are modeled by the shifted Maxwellian velocity distributions(3)
where n_{s}, V_{s}, , T_{s}, and m_{s} are the mean number density, parallel bulk velocity, thermal velocity, temperature, and particle mass of the plasma species s, and v = (v_{x}, v_{y}, v_{z}) – velocityspace coordinates. The subscripts z and ⊥ indicate directions parallel and perpendicular to B_{0}. The plasma model defined by Eqs. ((1)–(3)) has been extensively used in the past (see, e.g., Gary 2005, and references therein). The neutralizing current can also be provided by the copropagating electron beam (see, e.g., Zweibel & Everett 2010 and references therein); however it does not alter the reactive CCPI for lowdensity ion beams n_{b} ≪ n_{0} (Amato & Blasi 2009).
The nontrivial solutions to the Maxwell–Vlasov set of equations exist if the wave frequency ω and the wave vector k = (k_{x}, k_{y}, k_{z}) satisfy the following dispersion equation (see, e.g., Alexandrov et al. 1984)(4)
where ε_{ij} is the dielectric tensor and δ_{ij} is the Kronecker delta. For the parallelpropagating modes with k_{x} = k_{y} = 0, the components of the dielectric tensor given by Alexandrov et al. (1984) reduce to(5)
where ξ_{s,n} = (ω − k_{z}V_{s} + nω_{Bs}) / (k_{z}V_{Ts}), ω_{Ps} (ω_{Bs}) is the plasma (cyclotron) frequency. Instead of the plasma dispersion function W (x), we use the function(6)
introduced by Alexandrov et al. (1984). It has the following asymptotic expansions:(7)
where η = 0 for Im x > 0, η = 1 for Im x = 0, and η = 2 for Im x < 0.
In the case of parallel propagation, the dispersion Eq. (4) splits into two independent equations,(9)
describing lefthand (sign −) and righthand (sign +) polarized electromagnetic waves. In what follows we consider the lefthand polarized Alfvén branch undergoing the compensatedcurrent instability. Taking into account quasineutrality (2) and current compensation (1), Eq. (9) for Alfvén waves can be written as(10)
In the following sections we consider important limits of (10) typical for the reactive CCPI instability.
3. Dispersion relation for parallelpropagating waves
As we are going to analyze the reactive nonresonant instability, we neglect the contribution of the imaginary part of J_{+} (ξ_{b,−1}). Furthermore, we consider a lowfrequency instability with ω/ω_{Bi} smaller than other terms in ξ_{b,−1}, which allows us to neglect the ωdependent part in the argument of function J_{+}. In this case (10) reduces to the following quadratic equation with respect to ω/ω_{Bi}:(11)
ζ_{b} = −V_{b}/V_{Tb} − 1/ (k_{z}ρ_{Tb}), and ρ_{Tb} = V_{Tb}/ω_{Bi}. To avoid misunderstanding, we note that although ρ_{Tb} looks like the ion beam gyroradius, it is defined by the parallel beam temperature rather than the perpendicular one, and here it has a different physical meaning.
Equation (11) is the secondorder eigenmode equation for Alfvén waves modified by the ion beam and return electron current (second and fourth terms, respectively). Its solution is straightforward:(13)
From (13) it is obvious that the instability can be driven by the last term under the square root when k_{z}V_{b} > 0. In what follows we assume V_{b} > 0 considering potentially unstable waves with k_{z} > 0 (in the case of V_{b} < 0, the identical instability develops for k_{z} < 0 ). In the absence of the beam, Eq. (13) reduces to the Alfvén wave dispersion, ω = k_{z}V_{A} at n_{b} = 0.
The wave with dispersion (13) becomes unstable when the last term under the square root dominates. This term represents effects due to the electron current. The growth rate γ = Im[ω] of the corresponding instability is(14)
Here we introduce the cumulative destabilizing parameter(15)
that includes all beam parameters. One can think of it as of product of the normalized current j̄_{b} = n_{b}V_{b}/ (n_{0}V_{A}) and velocity spread V̄_{Tb} = V_{Tb}/V_{A} of the beam ions. The growth rate (14) is analyzed below analytically and numerically, and its scalings are found in some important limits. It is interesting to note that the righthand polarized magnetosonic instability can be obtained from the above equation by changing the sign of the first term under the square root (the magnetosonic instability hence requires k_{z}V_{b} < 0).
4. Compensatedcurrent instability driven by hot ion beams
By hot beams we mean the beams whose thermal velocity is significantly larger than the bulk velocity, V_{Tb} ≫ V_{b}. For such beams, the growth rate (14) can be simplified by neglecting the small term V_{b}/V_{Tb} in ξ_{b,−1}. The argument of J_{+} is then simplified to ξ_{b,−1} ≈ −1/ (k_{z}ρ_{Tb}) ≡ ζ_{b}. In this case γ_{k} depends on the normalized parallel wavenumber k_{z}ρ_{Tb} and two dimensionless bulk parameters, V_{A}/V_{b} and α_{b}. Then the (maximum) instability growth rate γ_{m} = max_{k}γ appears to be a function of α_{b} and V_{A}/V_{b} only, whereas the dependence on the general multiplier V_{A}/V_{Tb} is trivial and can be excluded by the renormalization of γ_{m}. We note that the hotbeam condition V_{Tb} > V_{b} restricts the applicability range of the analytical results obtained below, but in general it does not restrict the instability range (see also Sect. 6).
4.1. Instability areas in the parameter space
Here we find the instability threshold and the instability area in the parameter space (α_{b}, V_{A}/V_{b}). To this end, we present the growth rate (14) in the following useful form:(16)
From (16), the instability condition is obtained as(17)
Since the righthand side of (17) is positive, it is obvious that only superAlfvén beams, V_{b} > V_{A}, can trigger instability. Therefore, the absolute threshold for the beam velocity is and the system is stable with respect to reactive CCPI for all V_{b} < V_{A}.
Using (17), it is also possible to find the threshold for _{b} analytically. First, solving (17) with respect to the k_{k}dependent term A_{k}/ (k_{z}ρ_{Tb}), we find that the unstable wavenumbers k_{z} should satisfy(18)
When the velocity threshold is exceeded, V_{b} > V_{A}, the right boundary of (18) is always larger than the left boundary making the interval between them nonempty. As the function A_{k}/ (k_{z}ρ_{Tb}) is limited by the maximum value max_{k} [A_{k}/ (k_{z}ρ_{Tb})] ≈ 0.411 achieved at , the condition (18) can only be satisfied for sufficiently large α_{b}. From the lefthand inequality, it immediately follows the instability condition for α_{b} and the corresponding threshold:(19)
The instability condition is satisfied above the threshold curve defined by (19), which is shown in Fig. 1 by the solid line. The unstable area above this curve in the parameter space (α_{b}, V_{A}/V_{b}) is shaded. The dependence of the threshold on V_{A}/V_{b} is rather weak; it grows from the minimum value at V_{A}/V_{b} → 0 to the maximum value at V_{A}/V_{b} → 1. The absolute threshold for α_{b} below which the system is stable is . The meaning of the right boundary in (18), shown in Fig. 1 by the dashed line, is clarified in the following subsection.
Fig. 1.
Instability threshold in the parameter space (α_{b}, V_{A}/V_{b}) (solid line); the CCPI develops at all . The dashed line shows the split threshold above which there are two separate ranges of unstable wavenumbers k_{z} 

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In terms of the normalized beam current j̄_{b} = n_{b}V_{b}/ (n_{0}V_{A}) and thermal velocity V̄_{Tb} = V_{Tb}/V_{A}, (19) can be written as . Then the instability condition for the beam thermal velocity reads as(20)
This thresholdlike condition is an important new result quantitatively demonstrating the destabilizing effect of the beam temperature. It shows the threshold above which the velocity spread of the beam ions triggers the instability even for weak beams.
Similarly, the threshold condition for the beam current can be written as(21)
which quantifies the range of unstable beam currents. Again, it is seen that even a weak ion beam can activate CCPI if the beam thermal velocity is sufficiently high. In particular, the beam current required for the instability can be many orders of magnitude smaller than the Alfvén current.
We note that varies slowly for fast superAlfv énic beams and can be approximated as at V_{b}/V_{A} > 3. For rough estimations, in all velocity ranges can be replaced by its average value 3.5.
4.2. Unstable wavenumber ranges
Properties of CCPI are illustrated further by Figs. 2 and 3 showing all three terms of the condition (18): the left and right boundaries, and the function (k_{z}ρ_{Tb})^{−1}A_{k}. The unstable ranges where (18) is satisfied are shaded. A regular singlepeak behavior of the function (k_{z}ρ_{Tb})^{−1}A_{k}, as seen in Figs. 2 and 3, allows us to investigate how the unstable wavenumber range evolves with α_{b}.
Fig. 2.
Illustration of the condition (18) for V_{A}/V_{b} = 0.9 and α_{b} = 6. In this case and there is only one unstable wavenumber range (shaded area). 

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Fig. 3.
Illustration of the condition (18) for V_{A}/V_{b} = 0:9 and α_{b} = 10. In this case and there two unstable wavenumber ranges presented by two shaded areas. 

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When α_{b} increases but is still smaller than , the left boundary of (18) decreases remaining above the maximum of (k_{z}ρ_{Tb})^{−1}A_{k}. In this case there are no unstable wavenumbers and the system is stable. Once α_{b} rises above , the decreasing left boundary of (18 ) drops below the maximum of (k_{z}ρ_{Tb})^{−1}A_{k} and the unstable wavenumber range k_{z1} < k_{z} < k_{z2} appears, where k_{z1} and k_{z2} are the lower and upper roots of the equation(22)
As long as α_{b} is not far from the threshold , there is a single unstable wavenumber interval surrounding . This situation is illustrated in Fig. 2, where V_{A}/V_{b} = 0.9, , and . However, when α_{b} increases further, the right boundary of (18) also drops below the maximum of (k_{z}ρ_{Tb})^{−1}A_{k}, which happens at(23)
In this case, shown in Fig. 3 for α_{b} = 9, the righthand side inequality of (18) is not satisfied in the range , where and are the lower and upper roots of the equation(24)
Instead of unstable, we have now a prohibited wavenumber range around . As a result, the unstable wavenumber range splits into two: the first unstable range is and the second .
The split threshold (23) is shown in Fig. 1 by the dashed line. For parameter values above this line, the instability develops in two wavenumber ranges, as mentioned above. These unstable ranges are shown in Fig. 3 by the shaded areas.
Furthermore, Fig. 4 shows the α_{b} dependence of the unstable wavenumber ranges, where the outer and inner boundaries are respectively defined by the lefthand and righthand margins of (18). It is seen that below there is no instability, at there is a single unstable range of k _{z}, and above there are two unstable ranges.
Fig. 4.
Unstable wavenumber ranges in the (α_{b}, k_{z}) plane for V_{A}/V_{b} = 0.9. The outer boundary is defined by the lefthand side and the inner boundary by the righthand side of condition (18). It is seen that below there is no instability, at there is a single unstable range of k_{z}, and above there are two unstable ranges. 

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From Fig. 3 it is obvious that k_{z1}ρ_{Tb} and are located between k_{z}ρ_{Tb} ≈ 0.77, where (k_{z}ρ_{Tb})^{−1}A_{k} is zero, and , where (k_{z}ρ_{Tb})^{−1}A_{k} is maximum. This wavenumber range corresponds to −1.3 < ζ_{b} < −0.65, where ReJ_{+} (ζ_{b}) can be approximated by the liner numerical fit(25)
Using this in (22) and (24), we find k_{z1} and as(26) (27)
From these expressions we see that with increasing α_{b} the difference between k_{z1} and decreases, , and the first unstable range becomes very narrow.
On the other hand, the roots k_{z2}ρ_{Tb} and bounding the second unstable range, are located above , where ζ_{b} > −0.65. Then, using the small argument expansion (7) for ReJ_{+} (ζ_{b}), we find(28) (29)
At large values of α_{b}, both the width of the second unstable range and the gap between the unstable ranges grow linearly with α_{b}.
In summary, the most important analytical result obtained here is the instability boundary in the parameter space (V_{A}/V_{b}; α_{b}), which can be used directly to analyze observational data. The compensatedcurrent systems with V_{A}/V_{b} < 1 and are unstable. The unstable area in the parameter space (V_{A}/V_{b}; α_{b}) is divided further by into two unstable subareas: with one unstable wavenumber range, and with two unstable wavenumber ranges.
4.3. Instability growth rate
Once α_{b} rises above , an unstable range between k_{z1} and k_{z2} appears. The instability growth rate (14) as function of k_{z} is shown in Fig. 5. The plasma parameters α_{b} and V_{A}/V_{b} in this figure are chosen in such a way as to illustrate the behavior of CCPI in the unstable wavenumber ranges found above. So, the case α_{b} = 6 with one unstable wavenumber range is shown by the dashed line and the case α_{b} = 10 with two unstable wavenumber ranges is shown by the solid lines. The dotted curve in Fig. 5 is for the case α_{b} = 8, which is close to the splitting threshold. It is seen that when the right instability boundary in (18) approaches the maximum of function (k_{z}ρ_{Tb})^{−1}A_{k}, the valley and the second peak in γ_{k} appear. This happens at , where(30)
Fig. 5.
Wavenumber dependence of the instability growth rate driven by superAlfvénic ion beams with V_{A}/V_{b} = 0.9 for three values of α_{b}: α_{b} = 6, 8, and 10. For larger α_{b}, the unstable area and the maximum growth rate extend to larger k_{z}ρ_{Tb}. 

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is the value of α_{b} at which a local “plateau” in γ_{k} occurs at the wavenumber where ∂γ_{k}/∂k_{z} = 0 and . For all , the secondary peak of γ_{k} exists at . Since , the secondary peak arises before the interval of prohibited wavenumbers appears.
It is seen that CCPI is stronger and the most unstable wavenumbers are larger for larger α_{b}. The secondary peak that appears at is lower than the main peak. These trends are confirmed below analytically.
The most unstable wavenumber and the corresponding maximum growth rate γ_{max} can be found by maximizing (14) with respect to k_{z}, γ_{max} = max_{k} (γ_{k}), which we call the CCPI growth rate. The normalized CCPI growth rate γ_{max}/ω_{Bi} as a function of n_{b}/n_{0} and V_{b}/V_{A} is shown in Fig. 6 for hot beam with V_{Tb}/V_{A} = 10^{2}. It is seen that γ_{max} increases rapidly, roughly proportional to both n_{b}/n_{0} and V_{b}/V_{A}, which means it is proportional to the current n_{b}V_{b}. This behavior agrees with the current nature of CCPI confirmed below analytically by (34) and (35).
Fig. 6.
Normalized growth rate γ_{max}/ω_{Bi} as a function of n_{b}/n_{0} and V_{b}/V_{A} for hot beam with V_{Tb}/V_{A} = 10^{2}. γ_{max} increases regularly with both n_{b}/n_{0} and V_{b}/V_{A} once the threshold is exceeded. 

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The threshold for n_{b}/n_{0} (V_{b}/V_{A}) is lower for smaller V_{b}/V_{A} (n_{b}/n_{0}), in agreement with (19). In particular, the velocity threshold decreases with n_{b}/n_{0} and reaches the minimum value when n_{b}V_{Tb}/ (n_{0}V_{A}) → 1.
Dependence of γ_{max} on the thermal velocity V_{Tb} is somewhat different (see Fig. 7). First, near the threshold, γ_{max} increases very rapidly with V_{Tb}. But this increase quickly slows down as V_{Tb} departs further from the threshold. Already at , γ_{max} becomes virtually independent of V_{Tb}.
Fig. 7.
Normalized growth rate γ_{max}/ω_{Bi} as a function of V_{Tb}/V_{A} for n_{b}V_{b}/ (n_{0}V_{A}) = 0.05 (solid curve), 0.1 (dotted curve), and 0.15 (dashed curve). Starting from zero, the growth rate γ_{max} increases rapidly with V_{Tb}, but this increase is quickly saturated. Larger currents n_{b}V_{b}/ (n_{0}V_{A}) result in larger γ_{max} for all V_{Tb}. 

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To understand this behavior, we proceed with the analytical analysis. Here we take into account that in the wavenumber range , where the growth rate attains its maximum, the lowζ_{b} approximation ReJ_{+} (ζ_{b}) ≈ −(k_{z}ρ_{Tb})^{−2} is valid. Thus, using A_{k} = 1 + ReJ_{+} (ζ_{b}) ≈ 1 − (k_{z}ρ_{Tb})^{−2} in (16), we find the following approximation for the maximum of γ_{k}:(31)
The last term in the square parentheses in (31) is adjusted by replacing the approximate numerical value by to make it compatible with the exact α_{b} threshold (19). We verified numerically that the approximation (31) is good for arbitrary α_{b}, both near the threshold and far from it. In general, with the larger beam velocity and/or temperature, the smaller beam density is needed for instability.
The explicit dependence of the instability growth rate on the beam thermal velocity V̄_{Tb} follows from (31):(33)
It is seen that γ_{max} increases quickly with V̄_{Tb} once the threshold is overcome, . The fast increase of γ_{max} reflects the instability response to the progressive demagnetization of the beam ions as their velocity spread increases above the threshold.
However, when V̄_{Tb} becomes large enough, , the term containing it becomes negligibly small and γ_{max} becomes virtually independent of V̄_{Tb}. In this hightemperature regime the beam ions are fully demagnetized and the further increase of V̄_{Tb} no longer affects the instability. This regime corresponds to the asymptotic overthreshold limit , where γ_{max} simplifies to(34)
The familiar threshold velocity of the beam, , is still present in (34), but the temperature dependence is already missed, as can be observed in Fig. 7 at large V_{Tb}.
The maximum growth rate (34) simplifies further for the fast beams with V_{b}/V_{A} > 3,(35)
with the most unstable parallel wavenumber . The asymptotic scaling (35) recovers the scaling obtained by Bell (2004). As is seen from Fig. 7, expressions (34) and (35) provide good estimations for γ _{max} at , which also quantifies the meaning of “asymptotic regime” in terms of V_{Tb}. It appears that the expressions found by Bell are only valid in this asymptotic regime.
For , the secondary peak arises at k_{z}ρ_{Tb} < 1.54, where we can use approximation (25). Then for this peak we obtain the local maximum(36)
where and we took into account that . The ratio of this peak to the main peak is(38)
Taking into account that (at V_{A}/V_{b} → 1), we see that the peak γ_{m2} is always significantly smaller than the main peak γ_{max}. The maximum ratio γ_{m2}/ γ_{max} ≈ 0.45 is achieved at and V_{A}/V_{b} ≲ 1.
We note that the unstable fluctuations also have a small oscillatory part Re[ω] = 0.5 (n_{b}/n_{0}) ω_{Bi}. For most unstable wavenumber , the real frequency is smaller than the frequency of the normal Alfvén mode . Since , the instability is aperiodic.
5. Parallel Alfvén instability in particular compensatedcurrent systems
Let us consider two feasible applications of CCPI. First we apply our results to the solar wind upstream of the quasiparallel terrestrial shock where the plasma conditions are relatively well documented. Then we extend the analysis to the interstellar medium around supernova remnants, assuming the similar scalings of the beam parameters as in the terrestrial foreshock.
5.1. Quasiparallel terrestrial foreshock
Hot ion beams with V_{Tb} > V_{b} > V_{A} are regularly observed in the solar wind upstream of the terrestrial bow shock where the shock normal is quasiparallel to the interplanetary magnetic field B_{0} (Paschmann et al. 1981; Tsurutani & Rodriguez 1981). This ordering of characteristic velocities suggests that CCPI driven by hot ion beams can develop in the quasiparallel foreshocks.
More specifically, we use the following scalings for characteristic beam velocities, V_{b} ≲ V_{shock} and V_{Tb} ~ 3V_{shock}, where the shock velocity is equal to the solar wind speed, V_{shock} = V_{SW}. These scalings are compatible with observations reported by Paschmann et al. (1981) and Tsurutani & Rodriguez (1981). Yet another beam parameter, number density n_{b}, does not vary much around n_{b} = 0.1 cm^{−3} (Paschmann et al. 1981). In terms of the background solarwind density n_{0} ~ 5–10 cm^{−3}, this gives n_{b}/n_{0} ~ 0.01–0.02. Taking the typical value of Alfvén velocity, V_{A} ≈ 0.1V_{SW}, we obtain the cumulative destabilizing parameter α_{b} ~ 2.5–5, which is slightly overthreshold depending on the particular value of V_{b}. Such proximity of the system to the CCPI threshold can be a signature of CCPI operating in the foreshock and relaxing the beam parameters towards the threshold.
On the other hand, as is seen from Fig. 6, even slight deviations of α_{b} from the threshold can make CCPI strong. So, for V_{A}/V_{b} ~ 0.1 and α_{b} = 6 the maximum growth rate is already high, γ_{max} ≈ 0.07ω_{Bi}, with the most unstable wavenumbers k_{zm}ρ_{Tb} ≳ 2. Narita et al. (2006) and Hobara et al. (2007) analyzed properties of electromagnetic fluctuations observed around terrestrial bow shock. Most straightforwardly, our results can be compared with the wavenumber distribution of the fluctuations in the quasiparallel foreshocks shown in Fig. 9 by Narita et al. (2006), where the measured wavenumbers are normalized by the ion gyroradius. In terms of the background ion gyroradius ρ_{Ti}, with the typical temperature of the diffuse ions T_{b}/T_{i} = 4 × 10^{2}, our most unstable wavenumbers k_{zm}ρ_{Ti} ~ k_{zm}ρ_{Tb}/20 ~ 0.1 map upon the major peak observed at k_{z}ρ_{Ti} = 0.1 (see upper panel of Fig. 9 in Narita et al. 2006).
In the quasiparallel foreshock region, Narita et al. observed another, subdominant peak at k_{z}ρ_{Ti} = 0.6. To explain this peak by CCPI we need a significantly lower beam temperature, T_{b}/T_{i} ~ 10, which is more typical for quasiperpendicular foreshocks. We can speculate that CCPI can also generate this second peak. First, the CCP instability develops in the quasiperpendicular foreshock region where the beams have required temperatures T_{b}/T_{i} ~ 10, which is supported by the observed enhancement at k_{z}ρ_{Ti} ≈ 0.4. Then the unstable fluctuations are convected in the quasiparallel foreshock region where their observed wavenumbers are k_{z}ρ_{Ti} ≈ 0.6.
The above estimations suggest that CCPI can contribute to electromagnetic fluctuations observed in the quasiparallel terrestrial foreshock and can impose limitations on the parameters of the beams formed by reflected ions. Further direct confrontations of observed values of α_{b} with the stability diagram in Fig. 1 are needed to clarify the role of CCPI in the regulation of ionbeam parameters in the foreshock.
5.2. Foreshock regions around supernova remnants
Supernova remnants expanding in the interstellar medium develop bow shocks at their boundaries. These shocks propagate with high velocities V_{shock} ~ 2 × 10^{9} cms^{−1} providing a feasible source of energy for the cosmic ray acceleration, and also for the magnetic field amplification. By analogy with the terrestrial bow shock, we assume that the reflected ions also occur in the supernova foreshocks setting up a compensatedcurrent system. CCPI can develop in supernova foreshocks if parameters of reflected ions (subscript b) satisfy , defined by (19).
For reasonable background density n_{0} = 10^{−2}− 1 cm^{−3} and magnetic field B_{0} ~ 10^{−7} − 10^{−5} G (Zweibel & Everett 2010), the Alfvén velocity varies in the range V_{A} = 2 × 10^{4}−2 × 10^{7}cm/s. Then the resulting Alfvén Mach number in supernova remnants M_{A} = V_{shock}/V_{A} = 10^{2}−10^{5} is much larger than in Earth’s bow shock. For the similar scalings as in the terrestrial foreshocks, n_{b}/n_{0} ~ 0.01, V_{b} ~ 0.5V_{shock}, and V_{Tb} ~ 2V_{shock}, even with the most unfavorable V_{shock}/V_{A} = 10^{2} the destabilizing parameter α_{b} ~ 10^{2} is much larger than the threshold . In this far overthreshold range, the CCPI operates in the asymptotic regime (35) with very high growth rate γ_{max}/ω_{Bi} ~ 0.5. We note that this value is already at the edge of applicability of our lowfrequency approximation. Such a high growth rate suggests that the instability modifies the beam parameters strongly, in particular reducing the beam velocity towards the local Alfvén velocity, V_{b} ≳ V_{A}.
Let us compare the instability driven by the reflected ions with the similar instability driven by cosmic rays around supernova remnants (Bell 2004; Zweibel & Everett 2010). Taking the background magnetic field B_{0} ≳ 10^{−6} G and the cosmicrays flux n_{CR}V_{b} ~ 10^{4} cm^{−2} s^{−1} (Zweibel & Everett 2010), we estimate the normalized current and the corresponding growth rate around supernova remnants. With ω_{Bi} ≃ 0.03 s^{−1}, we get in absolute numbers.
The above estimations show that the CCPI instability driven by reflected ions is much stronger than the instability driven by cosmic rays. Therefore, the former instability can be a more efficient amplifier for magnetic fields around supernova remnants. On the other hand, a fraction of the beam ions can be scattered back to the shock by electromagnetic fluctuations generated by CCPI, thus providing a seed population for the further Fermi acceleration to high cosmicray energies.
6. Discussion
A number of competing electrostatic and electromagnetic instabilities may arise when different plasma species move with respect to each other (see Gary 2005; Bret 2009, and references therein). The hierarchical structure of these instabilities depends on many parameters and remains an open question (see further discussions in Bret et al. 2010; Brown et al. 2013; Marcowith et al. 2016).
In our setting with hot ion beams, the fast twostream/ Buneman instabilities are quenched by the large thermal velocities, which are larger than the streaming velocities. Inspection of Fig. 3.20 by Gary (2005) shows that the thresholds of electrostatic ionacoustic and ioncyclotron instabilities are significantly higher than the Alfvénic threshold for V_{Ti}/V_{A} ~ T_{e}/T_{i} ~ 1 typical in the terrestrial foreshock. Among them, the electron/ion cyclotron instability has the lowest threshold velocity, which is still very high, for n_{b} < 0.1n_{e}. The ion/ion acoustic instability is suppressed further by large beam temperatures, as is seen from Fig. 3.15 by Gary (2005). Therefore, these highfrequency electrostatic instabilities cannot compete with CCPI in the wide range of beam velocities 1 < V_{b}/V_{A} < 10^{2}. At higher beam velocities, V_{b}/V_{A} > 10^{2}, the ionacoustic and ioncyclotron harmonic waves can be generated by the electronion relative motion. However, even in this velocity range CCPI can develop independently as long as the mean parameters reside in the unstable area (Fig. 1), whereas the kinetic instabilities are quickly saturated by the local quasilinear plateaus.
Parallelpropagating lefthand and righthand polarized instabilities have been studied by Gary et al. (1984) and Gary (1985). Using numerical solutions of the dispersion equation, it has been observed that the lefthand polarized Alfvénic instability becomes competitive or even dominant when the beam ions are sufficiently hot (see Fig. 8 by Gary et al. 1984). The condition ξ_{b,−1} < 1 was used by Gary et al. to categorize this instabilas ionbeam resonant, i.e., resulting from the direct resonant coupling of the unstable mode with the beam ions. However, kinetic and reactive effects have not been distinguished for this mode, which did not allow us to realize that above threshold (19) the instability transforms from kinetic resonant to reactive nonresonant (see Fig. 8 and related discussions below). In the reactive regime, the meaning of the condition ξ_{b,−1} ≈ k_{z}ρ_{Tb}^{−1} < 1 is reversed: here it indicates that the unstable perturbations are on small enough scales to decouple from the beam ions by the demagnetization effect. The resulting Alfvén instability is then driven not by the resonant interactions with the beam ions, but by the bulk return current of the magnetized electrons. The current nature of this instability is similar to the nature of related current instability (Malovichko & Iukhimuk 1992) that can develop in the absence of any beams.
Fig. 8.
Contribution of the reactive CCPI growth rate (dashed curve) to the total growth rate (dotted curve) for V_{Ti}/V_{A} = T_{e}/T_{i} = 1, V_{b}/V_{A} = 10, V_{Tb}/V_{A} = 25, and n_{b}/n_{0} = 0.02. It is seen that the reactive destabilizing effects dominate the instability growth rate for this set of parameters. The wave frequency is shown by the solid line. 

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Interplay of the reactive and resonant lefthand Alfvénic instabilities also needs further investigations. Our preliminary estimations indicate that the relative importance of the reactive versus kinetic destabilizing effects increases quickly once α_{b} rises above the threshold . In Fig. 8. we show the contribution of the reactive CCPI to the total growth rate for reference plasma parameters that may occur in foreshocks: V_{Ti}/V_{A} = T_{e}/T_{i} = 1, V_{b}/V_{A} = 10, V_{Tb}/V_{A} = 25, and n_{b}/n_{0} = 0.02. The corresponding total growth rate in Fig. 8 is given by Eq. (14) with the imaginary part of J_{+} taken into account. It therefore includes both the reactive effects due to the bulk currents and the resonant waveparticle interactions. It is seen that the destabilizing reactive response becomes stronger than the resonant wave response when α_{b} is still not far from the threshold ( in Fig. 8). The instability is thus driven mainly by the reactive effects and can be analyzed ignoring kinetic resonant effects, as we did in the present study. The same approach can also be applied in the immediate vicinity of the reactive threshold if the quasilinear plateaus or other local deformations of the velocity distributions weaken destabilizing kinetic effects. Analytical treatment becomes more tangled when reactive and kinetic effects are of comparable efficiency and have to be accounted for simultaneously, in which case the evolution of the system becomes more complex (cf. Yoon & Sarfraz 2017).
There are also left and righthand polarized instabilities driven by cold ion beams in the ioncyclotron frequency range (Mecheri & Marsch 2007). These instabilities are strong when the velocity spread of the beam ions is so small that all the beam ions (and hence the beam as a whole) are resonant. In our settings with hot ion beams these instabilities are quenched similarly to the twostream/Buneman instabilities.
In the considered case of hot ion beams, V_{b}/V_{Tb} < 1, the analytical treatment of wavenumbers k_{z}ρ_{Tb} < V_{Tb}/V_{b} is simplified by neglecting the term ~V_{b}/V_{Tb} in ξ_{b,−1}. As the most unstable wavenumber scales as k_{z}ρ_{Tb} ≈ α_{b}/2 (32), this restriction is not stringent:(39)
This condition is the opposite of the firehose instability condition (see Eq. (14) by Malovichko et al. 2014), which means that the CCPI can operate in a wide range of parameters below the firehose threshold. For cooler beams, where the condition V_{b}/V_{Tb} < 1 is violated (e.g., in the quasiperpendicular foreshock regions), the analysis should be extended by accounting for corresponding terms.
The compensatedcurrent parallel instability can also affect other processes in space. For example, it can limit the fieldaligned currents generated by Alfvénwave fluxes in the inner magnetosphere and plasma sheet boundary layer (Artemyev et al. 2016). In the solar wind, CCPI can contribute to the regulation of relative motion of different plasma species. It was found that many states of beaming structures in the solar wind are close to the thresholds of magnetosonic and Alfvén instabilities (Marsch & Livi 1987; Gary et al. 2000) and the firehose instability (Chen et al. 2016). Since CCPI can operate close to these thresholds (and sometimes below them), a refined analysis is needed to decide its role in the solar wind, as compared to the magnetosonic and firehose instabilities. These are subjects for future studies.
7. Conclusions
We investigated reactive nonresonant compensatedcurrent parallel instability (CCPI) of lefthand polarized Alfvén waves in compensatedcurrent systems established by hot diluted ion beams. Ionbeam demagnetization due to finite k_{z}ρ_{Tb} is crucial for CCPI (ρ_{Tb} is based on the parallel beam temperature, and thus does not represent the beam ion gyroradius). New analytical expressions for the instability growth rate (31) and threshold (19) are found and analyzed.
The most important new properties of CCPI can be summarized as follows:

Reactive nonresonant CCPI depends on all bulk parameters of the beam: beam density n_{b}, bulk velocity V_{b}, and thermal velocity V_{Tb}. These parameters all increase the instability growth rate and can be combined in the single destabilizing parameter α_{b} = (n_{b}/n_{0}) (V_{b}/V_{A}) (V_{Tb}/V_{A}). The instability develops at , where the instability threshold (19) varies from at V_{A}/V_{b} → 0 to at V_{A}/V_{b} → 1. The analytical threshold (19) can be directly compared with satellite data to analyze the stability of beamplasma systems in space.

CCPI is strongly affected by the velocity spread of the beam ions V_{Tb}. It defines the range of unstable beam currents, , with the current threshold varying in the range .

The instability growth rate γ_{max} (33) increases sharply with V_{Tb} once the threshold (20) is overcome (Fig. 7). This fast increase is caused by the fast demagnetization of the beam ions, in which case they cannot compensate for the perturbed currents of fully magnetized electrons. In a more distant overthreshold range the temperature dependence weakens because of the nearly saturated demagnetization.

From the growth rate γ_{max} (31) it follows that the instability can be strong, γ_{max} ≳ 0.1ω_{Bi}, even for modest not far from the threshold. The most unstable wavenumber k_{z}ρ_{Tb} ≳ 1.54 near the threshold , but increases with α_{b} quickly approaching the asymptotic scaling k_{z}ρ_{Tb} ~ _{b}/2. In this asymptotic regime, our growth rate reduces to (35), the same as was obtained by Bell (2004).

Two particular applications to the terrestrial foreshocks and supernova remnants show that the reactive CCPI can operate there. An analysis of Sect. 5.2 suggests that the ions reflected from the shocks around supernova remnants can drive stronger instability than the cosmic rays. In the terrestrial foreshock, CCPI can regulate beam parameters generating electromagnetic fluctuations observed at k_{z}ρ_{Ti} ≈ 0.1.
Our results complement and extend previous studies on electromagnetic instabilities and their role in space and astrophysical plasmas. CCPI can develop around supernova remnants expanding into interstellar medium, participating in the braking process, and heating and redistributing energy in the supernova shocks. The same concerns the solarwind regions upstream of the terrestrial bow shock, and other heliospheric shocks, where CCPI can bound the beam parameters and contribute to the lowfrequency electromagnetic turbulence. Similarly, CCPI can affect other space and astrophysical environments containing superAlfvénic ion beams and return currents.
Acknowledgments
This research was supported by the Belgian Science Policy Office (through Prodex/Cluster PEA 90316 and IAP Programme project P7/08 CHARM).
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All Figures
Fig. 1.
Instability threshold in the parameter space (α_{b}, V_{A}/V_{b}) (solid line); the CCPI develops at all . The dashed line shows the split threshold above which there are two separate ranges of unstable wavenumbers k_{z} 

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In the text 
Fig. 2.
Illustration of the condition (18) for V_{A}/V_{b} = 0.9 and α_{b} = 6. In this case and there is only one unstable wavenumber range (shaded area). 

Open with DEXTER  
In the text 
Fig. 3.
Illustration of the condition (18) for V_{A}/V_{b} = 0:9 and α_{b} = 10. In this case and there two unstable wavenumber ranges presented by two shaded areas. 

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In the text 
Fig. 4.
Unstable wavenumber ranges in the (α_{b}, k_{z}) plane for V_{A}/V_{b} = 0.9. The outer boundary is defined by the lefthand side and the inner boundary by the righthand side of condition (18). It is seen that below there is no instability, at there is a single unstable range of k_{z}, and above there are two unstable ranges. 

Open with DEXTER  
In the text 
Fig. 5.
Wavenumber dependence of the instability growth rate driven by superAlfvénic ion beams with V_{A}/V_{b} = 0.9 for three values of α_{b}: α_{b} = 6, 8, and 10. For larger α_{b}, the unstable area and the maximum growth rate extend to larger k_{z}ρ_{Tb}. 

Open with DEXTER  
In the text 
Fig. 6.
Normalized growth rate γ_{max}/ω_{Bi} as a function of n_{b}/n_{0} and V_{b}/V_{A} for hot beam with V_{Tb}/V_{A} = 10^{2}. γ_{max} increases regularly with both n_{b}/n_{0} and V_{b}/V_{A} once the threshold is exceeded. 

Open with DEXTER  
In the text 
Fig. 7.
Normalized growth rate γ_{max}/ω_{Bi} as a function of V_{Tb}/V_{A} for n_{b}V_{b}/ (n_{0}V_{A}) = 0.05 (solid curve), 0.1 (dotted curve), and 0.15 (dashed curve). Starting from zero, the growth rate γ_{max} increases rapidly with V_{Tb}, but this increase is quickly saturated. Larger currents n_{b}V_{b}/ (n_{0}V_{A}) result in larger γ_{max} for all V_{Tb}. 

Open with DEXTER  
In the text 
Fig. 8.
Contribution of the reactive CCPI growth rate (dashed curve) to the total growth rate (dotted curve) for V_{Ti}/V_{A} = T_{e}/T_{i} = 1, V_{b}/V_{A} = 10, V_{Tb}/V_{A} = 25, and n_{b}/n_{0} = 0.02. It is seen that the reactive destabilizing effects dominate the instability growth rate for this set of parameters. The wave frequency is shown by the solid line. 

Open with DEXTER  
In the text 
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