ENVIRONMENTS FOR MAGNETIC FIELD AMPLIFICATION BY COSMIC RAYS

Recently, a powerful instability which couples a high flux of cosmic rays to their host medium has been discovered (Bell 2004; Blasi & Amato 2008); we refer to this as the "Bell instability." The instability amplifies low frequency, right circularly polarized electromagnetic fluctuations with wavenumber parallel to the ambient magnetic field. In contrast to the classical cyclotron resonant streaming instability (Wentzel 1968; Kulsrud & Pearce 1969), in which cosmic rays with Lorentz factor γ amplify Alfvén waves with wavelength of the order of the cosmic ray gyroradius $r_{\rm {cr}}\sim \gamma c/\omega _{\rm {ci}}$ (where $\omega _{\rm {ci}}$ is the non-relativistic ion-cyclotron frequency), the characteristic wavelength of the Bell instability is much less than $r_{\rm {cr}}$. The Bell instability is thought to be an important ingredient of diffusive shock acceleration in supernova remnants. It might be responsible for amplifying the magnetic field by up to ∼2 orders of magnitude above its interstellar value and increasing the maximum energy to which cosmic rays can be accelerated, possibly up to the "knee" at ∼10¹⁵ eV (Drury 2005; Reville et al. 2008b). This is an important result, because it has been known since the work of Lagage & Cesarsky (1983) that standard models of diffusive shock acceleration in the interstellar medium fail to reach the energy of the knee.

Although the Bell instability has been applied primarily to supernova remnants, there are many other environments in which the cosmic ray flux may be large enough to excite it: galactic wind termination shocks, intergalactic shocks, and shocks in disks and jets. Cosmic rays streaming away from local sources or from their host galaxies may also constitute a sufficient flux. If the Bell instability exists in any of these environments it can amplify the magnetic field and transfer cosmic ray energy and momentum to the background plasma as well as increasing the efficiency of cosmic ray acceleration. In view of our current uncertainty as to how galactic and intergalactic magnetic fields originated and are maintained (Widrow 2002; Kulsrud & Zweibel 2008), any mechanism for amplifying them is of interest. Because fluctuations generated by the Bell instability have a definite helicity relative to the background magnetic field, they could be significant in amplifying the field at large scales (Pouquet et al. 1976). This paper assesses the conditions under which the Bell instability, or more generally any rapidly growing, nonresonant electromagnetic streaming instability, can exist.

In order to excite the Bell instability, the cosmic ray particle flux $n_{\rm {cr}}v_D$, thermal ion density n_i, and Alfvén speed v_A must satisfy the inequality (as we show in Section 2.2.1)

$$\begin{equation} n_{\rm {cr}}v_D > n_i\frac{v_A^2}{\langle \gamma \rangle c}, \end{equation} \tag{ 1 }$$

where 〈γ〉 is of the order of the mean cosmic ray Lorentz factor. This can be written in terms of the cosmic ray and magnetic energy densities $U_{\rm {cr}}$, U_B,

$$\begin{equation} \frac{U_{\rm {cr}}}{U_B}>\frac{c}{v_D}. \end{equation} \tag{ 2 }$$

Equations (1) and (2) express the requirement that the characteristic wavenumber of the Bell instability be much greater than the reciprocal of the mean cosmic ray gyroradius $c\langle \gamma \rangle /\omega _{\rm {ci}}$.

The Bell instability was originally derived for a cold plasma, which is valid when the thermal ion and electron gyroradii r_i, r_e, are much less than the characteristic wavelength k_Bell of the instability; k_Bellr_i,e → 0. Reville et al. (2008a) considered kr_i small but nonzero. Their condition that the instability is significantly modified by thermal effects can be written in terms of the cosmic ray flux and ion thermal velocity $v_i\equiv \sqrt{2k_BT/m_i}$ is (as we derive in Section 2.2.2)

$$\begin{equation} n_{\rm {cr}}v_D > n_i\frac{v_A^3}{v_i^2}. \end{equation} \tag{ 3 }$$

When the inequality (3) is satisfied, the wavenumber of maximum instability, k_wice (warm ions, cold electrons) decreases relative to k_Bell. And while the maximum growth rate of the Bell instability and the growth rate of the resonant streaming instability are independent of B as long as v_D much exceeds the Alfvén speed v_A, the growth rate of the thermally modified instability increases linearly with B.

In this paper, we extend Reville et al.'s (2008a) analysis to cases where kr_i is not small, include cyclotron damping, and investigate the properties of the instability by solving the plasma dispersion relation. The results are given schematically in Figure 1 and precisely for two representative environments, for 〈γ〉 = 1, in Section 3.1. Equations (1) and (3) define curves on the ($n_{\rm {cr}}v_D,B$) plane. We show that these two curves, together with the requirements $k_{\rm wice}r_{\rm {cr}}> 1$, k_wicer_i < 1, and k_Bellr_i < 1, divide the $(n_{\rm {cr}}v_D, B)$ plane into the domains delineated in the figure. First, we consider the case of $k_{\rm wice}r_{\rm {cr}}> 1$: if the cosmic ray flux is too low, resonant streaming instabilities can be excited, but nonresonant instabilities of the Bell or thermally modified Bell type are precluded. The threshold flux depends only on v_i and the ion density n_i, and is independent of B. Above this threshold, but below a second threshold, defined by k_wicer_i < 1, the plane is divided into three regions. If B is large enough that Equation (1) is violated, nonresonant instability is again precluded. Intermediate values of B, such that Equation (3) is violated but Equation (1) is not, define the range of the standard Bell instability. If B is small enough that Equation (3) holds, there is nonresonant instability of the thermally modified type. The condition k_wicer_i ⩽ 1 gives an upper limit to the flux at which nonresonant instability can be excited. Above this flux, the ions are unmagnetized and any instability which exists is driven solely by the electrons. This limit too, which is discussed further in Section 2 and in the Appendix, is independent of B and depends only on n_i and v_i. Evaluating these limits numerically, we find that there is a broad range of cosmic ray flux and magnetic field strength for which the instability operates efficiently, should mediate the transfer of cosmic ray momentum and energy to the background, and should be a powerful source of circularly polarized magnetic field fluctuations.

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In Section 2, we write down the general dispersion relation and solve it in various limits. In Section 3, we give quantitative versions of Figure 1 and apply the results to several different astrophysical environments. Section 4 is a summary and discussion. Gaussian cgs units are used throughout.

This paper is restricted to linear theory. The quasilinear and nonlinear evolution of the instability has been studied by a number of authors using both magnetohydrodynamic (Bell 2004, 2005; Pelletier et al. 2006; Zirakashvili et al. 2008) and kinetic models (Niemiec et al. 2008; Riquelme & Spitkovsky 2009; Ohira et al. 2009; Luo & Melrose 2009; Vladimirov et al. 2009). A host of saturation mechanisms has been investigated, including increase of the fluctuation scale to $r_{\rm {cr}}^{-1}$, acceleration of the thermal plasma, coupling to stable or damped modes, and relaxation of the exciting beam. Although understanding saturation is essential for predicting the outcomes of the instability, delineating the regimes in parameter space where the instability exists is the first step.

We analyze the situation considered by Bell; a singly ionized plasma with ion number density n_i and temperature T. There is a uniform magnetic field $\mathbf {B}=\hat{z} B$ and a population of proton cosmic rays with number density $n_{\rm {cr}}$ streaming along B with speed v_D relative to the thermal ions. We assume from now on that the mean Lorentz factor 〈γ〉 of the cosmic rays is of order unity. There are applications, such as a "layered" shock precursor in which the most energetic particles have penetrated furthest upstream, where locally 〈γ〉 ≫ 1, and our results can be scaled readily to this situation; that application is important for determining the maximum energy to which particles can be accelerated. However, the instability will be excited by the bulk cosmic ray population closer to the shock, which is important for field amplification.

There are two populations of electrons, one with density n_i which has no bulk velocity in the frame of the protons, and the other with density $n_{\rm {cr}}$ which drifts with the cosmic rays at speed v_D. Thus, the system is charge neutral and current free. This is the model used in Zweibel (2003) and Bell (2004). Amato & Blasi (2009) have considered the Bell instability when all the electrons drift at speed $(n_{\rm {cr}}/n_i)v_D$ and found results similar to those obtained for the two electron populations assumed here. But although the Bell instability in its original form is insensitive to the precise form of the thermal electron distribution function f_e(v), it does turn out to depend on f_e in a hot plasma.

We briefly consider the constraints on f_e in the Appendix. Based on an assessment of the electrostatic Langmuir instability, we argue that for very large drifts and high cosmic ray densities, such as are expected in young supernova remnants expanding into diffuse interstellar gas, a separate electron beam with density $n_{\rm {cr}}$ and drift velocity v_D is unstable, and would tend to relax. For more moderate shocks, or in outflows where the drift speed is less than the electron thermal velocity, such a beam is stable. In general, stability considerations alone do not determine f_e.

The fastest growing nonresonant cosmic ray streaming instabilities are sensitive to the precise form of f_e only in a plasma so hot and/or weakly magnetized that the thermal ions do not respond. Any instability present is then an instability of the electrons alone. Its physical significance is unclear, since it depends on the form of f_e.

2.1. Full Dispersion Relation

We are interested in right circularly polarized electromagnetic fluctuations which propagate parallel to B and depend on z and t as exp i(kz − ωt); thus, instability corresponds to Im(ω) ≡ ω_i>0. The dispersion relation for the fluctuations can be written as

$$\begin{eqnarray} \frac{c^2k^2}{\omega ^2}-\omega _{\rm {ci}}\frac{n_{\rm {cr}}}{n_i}\zeta _r\frac{\left(\omega -kv_D\right)}{\omega ^2}\frac{c^2}{v_A^2}&=&\frac{\omega _{pi}^2}{\omega kv_i} Z\left(\frac{\omega _{\rm {ci}}+\omega }{kv_i}\right) + \nonumber \\ & & \frac{\omega _{pe}^2}{\omega kv_e} Z\left(\frac{\omega _{ce}+\omega }{kv_e}\right), \end{eqnarray} \tag{ 4 }$$

where we have dropped the displacement current, ω_pe,i ≡ (4πn_ie²/m_e,i)^1/2 are the thermal electron and ion plasma frequencies, ω_ce,i ≡ ∓eB/m_e,ic are the electron and ion cyclotron frequencies, Z is the plasma dispersion function (Fried & Conte 1961)

$$\begin{equation} Z(z)\equiv \frac{1}{\sqrt{\pi }}\int _{-\infty }^{\infty }\frac{e^{-s^2}}{s-z}ds, \end{equation} \tag{ 5 }$$

and the quantity ζ_r is defined in Equation (A10) of Zweibel (2003) and is plotted in Figure 2. Although the exact behavior of ζ_r depends on the cosmic ray distribution function, it has the general property that when $kr_{\rm {cr}}\gg 1$, ζ_r → −1, with a small imaginary part of order $(kr_{\rm {cr}})^{-1}$. The physics underlying this behavior is that cosmic rays barely respond to disturbances with wavelengths much less than their gyroradius, but the electrons do respond, resulting in a large perturbed current. A fraction $(kr_{\rm {cr}})^{-1}$ of cosmic rays have large enough pitch angles that they can resonate, resulting in a small imaginary part. As to the background plasma terms, the first term on the right-hand side of Equation (4) represents the response of the thermal ions, and the second term represents the cold electrons.

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We have solved Equation (4) for a variety of ambient medium parameters and cosmic ray distribution functions, and have reproduced the plots of growth rate versus wavenumber in Bell (2004) and Blasi & Amato (2008). An example is shown in Figure 3, which plots growth rate versus wavenumber at fixed cosmic ray flux, ion density, and magnetic field strength for three different temperatures: T = 10⁴ K, T = 10⁶ K, and T = 10⁷ K. The other parameters, B = 3 μG, n_i = 1 cm⁻³, $n_{\rm {cr}}v_D=10^4$ cm⁻² s⁻¹, are similar to those chosen by these authors as representative of cosmic ray acceleration in a young supernova remnant.

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The lowest temperature is essentially the cold plasma result.⁴ The growth rate peaks at k_Bell and then plunges, but remains positive as k increases. The T = 10⁶ curve is similar to the T = 10⁴ K curve, except that the peak growth rate is reduced and occurs at a slightly smaller k, and the positive tail is damped. The T = 10⁷ K curve is markedly different. The wavelength of the fastest growing mode is longer, and the peak growth rate is lower, than at T = 10⁶ K. The instability cuts off abruptly at a longer wavelength than in the T = 10⁴ K and T = 10⁶ K cases, then re-emerges in a short interval of k before disappearing again. We explain these features with an analytical treatment in the following two subsections.

The effect of magnetic field strength on the fastest growing mode is shown in Figures 4(a) and 4(b). Figure 4(a) plots the maximum growth rate ω_i,fgm versus B at T = 10⁴ K, with other parameters as in Figure 3. For B ⩽ 1 μG, ω_i,fgm increases linearly with B. At larger B, ω_i,fgm is independent of B. Different behavior is seen for k_fgm: the wavenumber of the fastest growing mode, plotted in Figure 4(b), is independent of B at low field strength (here, for B ⩽ 1 μG) and decreases as B⁻¹ at larger B. These features, too, are derived in the following two subsections.

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2.2. Analytical Results

2.2.1. Standard Case

We recover the essential form of the Bell instability from Equation (4) by setting ζ_r = −1 and taking the zero temperature limit of the right-hand side. Using Z(z) → −1/z; |z| ≫ 1, Equation (4) becomes

$$\begin{equation} \frac{c^2k^2}{\omega ^2} + \omega _{\rm {ci}}\frac{n_{\rm {cr}}}{n_i}\frac{\left(\omega -kv_D\right)}{\omega ^2}\frac{c^2}{v_A^2}= -\frac{\omega _{pi}^2}{\omega (\omega _{\rm {ci}}+\omega)}- \frac{\omega _{pe}^2}{\omega (\omega _{ce}+\omega)}. \end{equation} \tag{ 6 }$$

We now assume $\omega \ll \omega _{\rm {ci}}$, approximate $(\omega _{\rm {ci}}+\omega)^{-1}$ by $\omega _{\rm {ci}}^{-1}(1-\omega /\omega _{\rm {ci}})$, and use $\omega _{pe}^2/\omega _{ce}=-\omega _{pi}^2/\omega _{\rm {ci}}$. Multiplying the resulting dispersion relation by ω²v²_A/c² yields

$$\begin{equation} k^2v_A^2 +\omega _{\rm {ci}}\frac{n_{\rm {cr}}}{n_i}\left(\omega -kv_D\right)=\omega ^2. \end{equation} \tag{ 7 }$$

The solution to Equation (7) is

$$\begin{equation} \omega =\frac{\omega _{\rm {ci}}}{2}\frac{n_{\rm {cr}}}{n_i}\pm \left[\left(\frac{\omega _{\rm {ci}}}{2}\frac{n_{\rm {cr}}}{n_i}\right)^2-\omega _{\rm {ci}}\frac{n_{\rm {cr}}}{n_i}kv_D+k^2v_A^2\right]^{1/2}. \end{equation} \tag{ 8 }$$

The fastest growing mode occurs at the Bell wavenumber k_Bell

$$\begin{equation} k_{\rm Bell}\equiv \frac{\omega _{\rm {ci}}}{2}\frac{n_{\rm {cr}}}{n_i}\frac{v_D}{v_A^2} \end{equation} \tag{ 9 }$$

and has

$$\begin{eqnarray} \omega (k_{\rm Bell})\equiv \omega _{\rm Bell}& = & \frac{\omega _{\rm {ci}}}{2}\frac{n_{\rm {cr}}}{n_i}\left[1+\frac{v_D}{v_A}\left(\frac{v_A^2}{v_D^2}-1\right)^{1/2}\right] \nonumber \\ && \rightarrow i \frac{\omega _{\rm {ci}}}{2}\frac{n_{\rm {cr}}}{n_i}\frac{v_D}{v_A}, \end{eqnarray} \tag{ 10 }$$

where the limiting expression following the arrow holds for v_D/v_A ≫ 1.

Equation (10) shows that the instability requires v_D/v_A>1. This is also the threshold for the classical resonant streaming instability. When the cosmic ray flux is too low to satisfy Equation (1), the dominant instabilities are resonant instabilities of Alfvén waves. The peak growth rate occurs at wavenumbers which resonate with cosmic rays near the mean energy, $k\sim r_{\rm {cr}}^{-1}$, and is of order $\omega _{\rm {ci}}(n_{\rm {cr}}/n_i)(v_D/v_A-1)$ (Kulsrud & Cesarsky 1971). For v_D/v_A ≫ 1, this expression agrees to within a factor of order unity with ω_Bell, but the two cannot be used simultaneously because they apply for opposite cases of the Equation (1).

At fluxes which satisfy Equation (1), the assumptions made in deriving the classical resonant growth rate—that the underlying waves are Alfvén waves and that the growth time is much longer than the wave period—are incorrect (Zweibel 1979, 2003; Achterberg 1983). At wavenumbers $k < r_{\rm {cr}}^{-1}$, the growth rate is of order $\omega _{\rm {ci}}(n_{\rm {cr}}/n_i)(v_D/v_A-1)^{1/2}$ for v_D slightly greater than v_A and peaks at $\omega _{\rm {ci}}(n_{\rm {cr}}v_D/n_ic)^{1/2}$ for v_D ≫ v_A (Zweibel 2003). Comparing this expression to ω_Bell, we see that $\omega _{\rm res}/\omega _{\rm Bell}\sim (n_i v_A^2/n_{\rm {cr}}c v_D)^{1/2}$. From Equation (1), we see that whenever the Bell instability operates, its growth rate exceeds the growth rate of the resonant instability, and the growth rate of the resonant instability is lower than predicted by the classical theory.

At k>2k_Bell, the nonresonant instability is stabilized by magnetic tension. Comparison with Figure 3 shows that although the growth rate plunges to low levels at this value of k, the instability does not completely disappear, as is predicted by Equation (7). The ω_i ∝ k⁻¹ tail seen in Figure 3 can be recovered by using the full ζ_r; for $kr_{\rm {cr}}\gg 1$, $\zeta _r\sim -1 +i/kr_{\rm {cr}}$.

The Bell instability is important in shock acceleration if the growth time is shorter than the time it takes the shock to travel through the layer within which the cosmic rays are confined. For acceleration at the maximum rate, the cosmic ray diffuse according to the Bohm formula with diffusion coefficient $D\sim cr_{\rm {cr}}$, which sets the convection time across the layer as $(\omega _{\rm {ci}}v_D^2/c^2)^{-1}$. The Bell instability will be able to grow if $(n_{\rm {cr}}v_D/n_iv_A)>v_D^2/c^2$.

2.2.2. Warm Ions, Cold Electrons

We now imagine decreasing B or increasing T such that $kv_i/\vert \omega _{\rm {ci}}+\omega \vert$ is less than unity but not infinitesimal. In this case, $Z(z_i)\sim -1/z- 1/2z^3+i\sqrt{\pi }e^{-z^2}$. The imaginary part represents ions for which the Doppler shifted wave frequency $\omega - kv=-\omega _{\rm {ci}}$; i.e. ions in cyclotron resonance. The −1/2z³ term represents the finite gyroradius of the ions, which partially decouples them from the field. Using this approximation, Equation (4) becomes

$$\begin{eqnarray} \omega ^2-&&\omega \left(\frac{k^2v_i^2}{2\omega _{\rm {ci}}}+\omega _{\rm {ci}}\frac{n_{\rm {cr}}}{n_i}-i\sqrt{\pi }\frac{\omega _{\rm {ci}}^2}{kv_i}e^{-\frac{\omega _{\rm {ci}}^2}{k^2v_i^2}} \right)\nonumber\\ &&-k^2v_A^2+\omega _{\rm {ci}}\frac{n_{\rm {cr}}}{n_i}kv_D=0. \end{eqnarray} \tag{ 11 }$$

Comparison of Equations (7) and (11) shows that the thermal ions have two effects: cyclotron resonance, which is represented by the imaginary term, and a pressure-like effect due to the finite ion gyroradius, which increases the wave speed and is known as ion gyroviscosity. In the absence of cosmic rays, the dispersion relation agrees with Foote & Kulsrud (1979), and in the limit $kr_{\rm {cr}}\gg 1$, with Reville et al. (2008a),⁵ except that ion cyclotron damping is neglected in both papers.

Equation (11) represents the behavior of the growth rate quite well. This is shown in Figure 5, which compares ω_i calculated from Equations (4) and (11).

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In the left panel, T = 10³ K, thermal effects are unimportant and the differences between the two curves are entirely due to the value adopted for ζ_r; taking ζ_r = −1 omits the effect of resonant particles at high k and underestimates the cosmic ray response at low k, thereby overestimating ω_i. In the right panel, T = 10⁷ K, thermal effects are important enough to modify the instability (see Figure 3 and below) but the analytical dispersion relation successfully reproduces the main features. Again, setting ζ_r = −1 overestimates the growth rate at low k.

It can be shown from Equation (11) that when

$$\begin{equation} \frac{v_A}{v_i} < \left(\frac{n_{\rm {cr}}v_D}{n_iv_i}\right)^{1/3}, \end{equation} \tag{ 12 }$$

the fastest growing mode is determined by competition between the drift term $\omega _{\rm {ci}}kv_Dn_{\rm {cr}}/n_i$ and the finite gyroradius term rather than the drift term and the magnetic tension term. This condition is equivalent to Equation (3). The wavenumber k_wice of the fastest growing mode in this regime is

$$\begin{equation} k_{\rm wice}\sim \frac{\omega _{\rm {ci}}}{v_i}\left(\frac{n_{\rm {cr}}}{n_i}\frac{v_D}{v_i}\right)^{1/3}. \end{equation} \tag{ 13 }$$

Equation (13) neglects cyclotron damping, and thus is valid only for $k_{\rm wice}r_i\sim (n_{\rm {cr}}v_D/n_iv_i)^{1/3} < 1$. As k_wicer_i → 1 cyclotron damping becomes large, and the instability shuts off. A further constraint is that the instability be nonresonant: $k_{\rm wice}r_{\rm {cr}}\sim (c/v_i) k_{\rm wice}r_i > 1$. Both requirements can be written as temperature and density dependent limits on $n_{\rm {cr}}v_D$; we return to them in Section 3.1 (Equations (17), (20)).

The growth rate ω_wice corresponding to k_wice is

$$\begin{equation} \omega _{\rm wice}\sim \omega _{\rm {ci}}\left(\frac{n_{\rm {cr}}}{n_i}\frac{v_D}{v_i}\right)^{2/3}, \end{equation} \tag{ 14 }$$

in agreement with Reville et al. (2008a).

It can be shown from Equations (10), (12), and (14) that ω_wice/ω_Bell < 1. At the limits of validity of the warm ion approximation, which is $k_{\rm wice}v_i/\omega _{\rm {ci}}=1$, ω_wice/ω_Bell = v_A/v_i. The suppression of the growth rate is due not to thermal ion cyclotron damping, which is weak for $\omega _{\rm {ci}}/kv_i\gg 1$, but due to the restoring force exerted by the warm ions. Cyclotron damping does, however, come into play at shorter wavelength, obliterating the resonant tail of the instability. This happens roughly where $(kv_i/\omega _{\rm {ci}})e^{-(\omega _{\rm {ci}}/kv_i)^2} > n_{\rm {cr}}v_D/n_i c$. For the cosmic ray flux and ion density assumed in Figures 3–5, this occurs at $kv_i/\omega _{\rm {ci}}\sim 0.27$, or $kc/\omega _{\rm {ci}}\sim 9\times 10^5 T^{-1/2}$. This is consistent with the behavior shown in Figure 3.

At T = 10⁷ K, something more complicated is going on: the instability growth rate decreases sharply after peaking near $kc/\omega _{\rm {ci}}\sim 10^2$, as predicted by Equation (13), but then has a brief resurgence. This is because the resonant ion cyclotron term overwhelms the stabilizing gyroviscous term for $\omega _{\rm {ci}}/kv_i\sim 3/2$, removing, in a small band of k space, gyroviscous stabilization.

As we discussed in Section 2.2.1, the instability can only efficiently amplify magnetic fields at a shock if its growth time is faster than the convection time $(\omega _{\rm {ci}}v_D^2/c^2)^{-1}$. From Equation (14) we see that this requires

$$\begin{equation} \omega _{\rm {ci}}\left(\frac{n_{\rm {cr}}}{n_i}\frac{v_D}{v_i}\right)^{2/3} > \omega _{\rm {ci}}\frac{v_D^2}{c^2}. \end{equation} \tag{ 15 }$$

The criterion (15) is independent of magnetic field strength and depends only on the cosmic ray flux, drift speed, and the density and temperature of the ambient medium.

2.2.3. Hot Ions

When kr_i ⩾ 1, Equation (11) becomes invalid. In this limit, the argument of the plasma dispersion function in Equation (4) becomes large, and $Z(z)\approx -z^{-1} + i\sqrt{\pi }$. Physically, this means the ions are responding very little to the perturbation. Ion cyclotron damping is also weak, because the slope of the distribution function is small at the resonant velocities.

Under these conditions, the instability, if it exists at all, is due entirely to properties of the electron distribution function. In deriving the cosmic ray response function ζ_r used in Equation (4) and plotted in Figure 2, we assumed the electrons are cold and a fraction $n_{\rm {cr}}/n_i$ of them are drifting with the cosmic rays. As long as v_D/v_e is sufficiently large, the electrons are unstable not only to the electromagnetic streaming instability considered here, but also to the much faster growing electrostatic instabilities discussed in Section 2 (see the Appendix). On the other hand, if the electrons were all drifting at speed $n_{\rm {cr}}v_D/v_i$ both the electromagnetic and electrostatic instabilities would be stabilized.

Assuming the two peaked electron distribution, it can be shown that the instability growth rate is bounded above by $\omega _{\rm {ci}}(n_{\rm {cr}}v_D/n_iv_i)$. This is generally less than ω_wice defined in Equation (14), showing that these very short wavelength instabilities are not as important as the thermally modified or standard Bell instabilities. At even shorter wavelengths, such that kr_e>1, the derivation of ζ_r becomes invalid. In view of our uncertainty about the electron distribution function, we have pursued the hot ion case no further.

3.1. Instability Regimes

We begin by summarizing the different regimes of the streaming instability as functions of magnetic field strength B and cosmic ray flux $n_{\rm {cr}}v_D$. These regimes were introduced without proof in Section 1 and depicted schematically in Figure 1. The criteria used to delineate these regimes are approximate, but as Figures 4(a) and (b) indicate, the transitions between regimes are fairly sharp.

According to Equation (1), the condition that the Bell instability be nonresonant, i.e., that $k_{\rm Bell}r_{\rm {cr}}> 1$, is

$$\begin{equation} B < B_S\equiv 8.7\times 10^{-7}(n_{\rm {cr}}v_D)^{1/2}, \end{equation} \tag{ 16 }$$

where here and below $n_{\rm {cr}}v_D$ is given in units of cm⁻² s⁻¹ and B is in G. The condition that thermal effects modify the Bell instability such that the wavelength of the fastest growing mode is at k ∼ k_wice (Equation (9)) rather than k ∼ k_Bell (Equation (13)) is

$$\begin{equation} B < B_M\equiv 2.3\times 10^{-9}T^{1/3}n_i^{1/6}(n_{\rm {cr}}v_D)^{1/3}. \end{equation} \tag{ 17 }$$

The condition for the thermally modified Bell instability to be nonresonant is $k_{\rm wice}r_{\rm {cr}}> 1$. At the same time, the thermal ions must be magnetized at k = k_wice; k_wicer_i < 1. These conditions limit $n_{\rm {cr}}v_D$ to the range

$$\begin{equation} n_i\frac{v_i^4}{c^3} < n_{\rm {cr}}v_D < n_i v_i, \end{equation} \tag{ 18 }$$

or numerically

$$\begin{equation} 3\times 10^{-16}n_iT^2 < n_{\rm {cr}}v_D < 10^4 n_iT^{1/2}. \end{equation} \tag{ 19 }$$

Finally, the condition that the ions be magnetized at k = k_Bell, k_Bellr_i < 1, is $n_{\rm {cr}}v_D < n_i v_A^2/v_i$, or

$$\begin{equation} B > B_T\equiv 5\times 10^{-10}T^{1/4}(n_{\rm {cr}}v_D)^{1/2}. \end{equation} \tag{ 20 }$$

Equations (16)–(20) are plotted on the ($n_{\rm {cr}}v_D$, B) plane in Figures 6 and 7. Because Equations (17)–(20) depend on n_i and T, we give two versions of the plot. Figure 6 represents the interstellar medium; n_i = 1 cm⁻³, T = 10⁴ K. Figure 7 represents the intracluster medium; n_i = 10⁻³ cm⁻³, T = 10⁷ K.

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For $n_{\rm {cr}}v_D < n_iv_i^4/c^3$, B_S < B_M. This is also the condition $k_{\rm wice}r_{\rm {cr}}< 1$ (Equation (19)) and is represented by the leftmost vertical line on the plot. To the left of this line, there can be no nonresonant instability; since B_M>B_S the instability would take the thermally modified form, but for B>B_M the field is too large and for B < B_M the flux is too low. Streaming instability exists, but it is resonant. In the cold plasma limit, the Bell instability exists for any B < B_S, no matter how small.

For $n_{\rm {cr}}v_D > n_i v_i^4/c^3$, B_M < B_S. For B>B_S, the field is too large for nonresonant instability (this is the case in the interstellar medium, away from cosmic ray sources). For B < B_S, the Bell instability operates in standard form as long as B exceeds B_M and B_T. Although B_M < B_T is theoretically possible, it requires $n_{\rm {cr}}v_D > n_i v_i$, which is rather extreme. If we confine ourselves to $n_{\rm {cr}}v_D < n_i v_i$ (represented by the rightmost vertical line, defined by k_wicer_i = 1) then the Bell instability operates for B_M < B < B_S. For B < B_M, there is nonresonant instability as long as $n_{\rm {cr}}v_D$ is to the left of the vertical line. To the right of this line, the ions are unmagnetized. As we have argued, in this case the instability is controlled primarily by the electrons, and for kr_i ∼ 1, ion cyclotron damping is strong.

In summary, nonresonant instabilities exist in the range of cosmic ray fluxes given by Equation (19). When B is between B_M, defined in Equation (17), and B_S, defined in Equation (16), the maximum growth rate is independent of B. When B < B_M the instability is thermally modified, occurs at longer wavelength, and grows at a rate proportional to B. This is also shown in Figure 8, which is a contour plot of the maximum growth rate in Equation (4) as a function of cosmic ray flux and magnetic field. Toward the lower-right of the plot, one can see the maximum growth rate decreasing, downward, with decreasing B, but relatively constant above an approximately diagonal line in ($n_{\rm {cr}} v_D$,B) space from (10⁻⁸ cm⁻² s⁻¹, 10⁻¹⁰ G) to (10^4.4 cm⁻² s⁻¹, 10^−5.8 G).

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3.2. Astrophysical Settings

Nonresonant instability driven by cosmic ray streaming has emerged as a strong candidate for amplification of magnetic fields in environments such as strong shock waves, where the cosmic ray flux is large (Bell 2004). When the flux is high enough and/or the magnetic field is low enough, that Equation (1) is violated, the nonresonant instability replaces the classical resonant streaming instability as the dominant electromagnetic instability generated by cosmic rays. Although the instability scale length predicted by linear theory is small even compared to the cosmic ray gyroradius $r_{\rm {cr}}$, nonlinear simulations suggest that as the amplitude of the instability grows it generates fluctuations at larger scales. This can increase the energy to which particles are accelerated in shocks.

Cosmic ray acceleration and magnetic field growth are both of interest in a variety of environments, including young galaxies which may be actively forming stars but have not yet built up magnetic fields, shocks in galaxy clusters, and the intergalactic medium at large. In this paper, we have carried out a parameter study of nonresonant instabilities including ion thermal effects. We solved the full dispersion relation (4) numerically and verified that a simple analytical approximation, Equation (11), is quite accurate in the wavenumber regime of interest. We corroborated the criterion of Reville et al. (2008a) for when ion gyroviscosity reduces the instability growth rate and shifts it to longer wavelength. We showed that ion cyclotron damping cuts off the instability at short wavelengths and argued that at wavelengths short enough that the ions are unmagnetized the instability depends only on the electron distribution function, the prediction of which is beyond the scope of this paper.

The joint requirements that the instability wavelength be much less than the cosmic ray gyroradius but much more than the thermal ion gyroradius limits the range of fluxes which excite nonresonant instability to $n_i v_i^4/c^3 < n_{\rm {cr}}v_D < n_i v_i$. In practical terms, this range is large and accommodates most cases of interest. Within the unstable range, there is a "strong field" regime in which all streaming instabilities are resonant, an "intermediate" regime in which nonresonant instability in the form derived by Bell dominates, and a "weak field" regime in which the instability is thermally modified. In the Bell regime, the maximum growth rate is independent of B but in the thermally modified regime it depends linearly on B. Young galactic supernova remnants are generally in the intermediate regime unless the ambient medium is hot and rarefied (like the interior of a superbubble), in which case the instability is weakened. Generally, if B is in the nanogauss range, the growth rates are fast enough for nonresonant instability to be a potential source of magnetic field amplification in weakly magnetized interstellar and intergalactic gas. At much lower field strengths, the instability is too slow to be of interest, but other instabilities, such as Weibel modes, could be an important ingredient in magnetogenesis (Medvedev et al. 2006).

Although nonresonant instabilities amplify magnetic fields on rather small scales—much smaller than the eddy scales characteristic of interstellar and intergalactic turbulence—they should not be ignored in discussions of magnetogenesis. Because only the right circularly polarized modes are unstable, nonresonant instabilities are a source of magnetic helicity on scales at which the background magnetic field is coherent. Magnetic helicity is thought to be a key ingredient in the growth of large-scale magnetic fields from small-scale fluctuations (Pouquet et al. 1976). Cosmic ray generated fluctuations could be important in driving an inverse cascade of magnetic power to longer wavelengths and could prevent the pileup of power at short wavelengths that currently confounds interstellar and intergalactic dynamo theories.

We acknowledge useful discussions with P. Blasi, J. Kirk, and B. Reville, and comments by the referee. Support was provided by NSF grants AST 0507367, PHY 0821899, and AST 0907837 to the University of Wisconsin.

Here we briefly consider the constraints on the electron distribution function f_e.

One way or another, the cosmic ray current must be canceled: an uncompensated cosmic ray current $en_{\rm {cr}}v_D$ flowing in a channel of width L_pc measured in parsecs generates a magnetic field $B\sim 0.5n_{\rm {cr}}v_DL_{pc}$ G. Even the galactic flux of 10⁻² cm⁻² s⁻¹ with L_pc = 1 would generate a 5 mG field. Since cosmic rays are ion dominated, thermal electrons must cancel their flux.

When an electron beam drifts with respect to the bulk plasma, it can excite rapidly growing electrostatic instabilities which tend to re-distribute electron momentum and bring the system to a state of marginal stability. Langmuir waves (also called plasma oscillations) with wavenumber ω_pe/v (ω_pe is the electron plasma frequency (4πn_ee²/m_e)^1/2) are destabilized if ∂f_e/∂v>0. If the beam and bulk electrons have the same temperature T_e and the beam velocity v_b much exceeds the electron thermal velocity $v_e\equiv \sqrt{2k_BT_e/m_e}$ (which is necessary for instability if the beam density n_b is much less than the bulk density n_e, the case of interest here), then the requirement for stability is approximately

$$\begin{equation} \frac{v_b}{v_e} e^{-v_b^2/v_e^2} > \frac{n_b }{n_e } \end{equation} \tag{ A1 }$$

(e.g., Krall & Trivelpiece 1973). In shock acceleration, it is sometimes assumed n_b/n_e ∼ 10⁻⁵; according to Equation (A1), stability then requires v_b/v_e < 3.5. Assuming T_e = T_i in the upstream plasma, beams associated with shocks of Mach number $M<3.5\sqrt{m_i/m_e}$ are stable while shocks at higher M are unstable. In a 10⁴ K gas, the stability boundary is at about 1500 km s⁻¹. Thus, while the Langmuir instability is a constraint for very fast shocks, it is probably irrelevant for older supernova remnants, and in galaxy cluster accretion shocks or galactic wind termination shocks, where the background gas is hot and the Mach numbers are expected to be moderate. The fluxes associated with cosmic ray escape from galaxies are probably also electrostatically stable.

Therefore, it appears that f_e is not determined by stability considerations alone, but depends on other factors such as the history of the system and the source of cosmic rays.