Interstellar turbulent magnetic field generation by plasma instabilities

Abstract

The maximum magnetic field strength generated by Weibel-type plasma instabilities is estimated for typical conditions in the interstellar medium. The relevant kinetic dispersion relations are evaluated by conducting a parameter study both for Maxwellian and for suprathermal particle distributions showing that micro Gauss magnetic fields can be generated. It is shown that, depending on the streaming velocity and the plasma temperatures, either the longitudinal or a transverse instability will be dominant. In the presence of an ambient magnetic field, the filamentation instability is typically suppressed while the two-stream and the classic Weibel instability are retained.

Appendix: Dispersion relations

For a any given distribution function with a specified anisotropy pattern, the dispersion relations D _ℓ , D _t , and D _⊥ can be evaluated. The resulting equations relating $\omega\in \mathbb {C}$ and k _∥ or $k _{\perp} \in \mathbb {R}$ are usually non-linear and often transcendental. In most cases, a numerical solution is required, even though a series expansions can often lead to reasonable approximative solutions.

It should be mentioned that there are investigations without specifying a distribution function (e.g., Schaefer-Rolffs and Lerche 2006; Tautz and Lerche 2012b), which has shed light on the general behavior of the instability. A comparison of the instability for various distribution functions (Schaefer-Rolffs and Tautz 2008) has shown that the mechanism is indeed robust and does not strongly depend on the precise form of the distribution, as long as the anisotropy clearly dominates over the thermal spread of the particle ensemble.

Furthermore, note that all dispersion relations are valid in the non-relativistic regime only, i.e., for counterstreaming and thermal velocities small compared to the speed of light. For a discussion of relativistic effects see, e.g., Schaefer-Rolffs and Schlickeiser (2005), Tautz and Schlickeiser (2005b), Tautz and Lerche (2012b).

1.1 A.1 Maxwellian distribution

For the Maxwellian distribution from Eq. (3), the temperature- and κ-dependent parameters θ _∥ and θ _⊥ now play the role of the thermal velocities in the sense that, by calculating the first moment of the distribution, the appropriate θ is obtained.

For a plasma consisting of multiple particle species (denoted with the index a), the longitudinal dispersion relation, D _ℓ , reads (Tautz and Schlickeiser 2005a)

$$\begin{aligned} D_\ell =&k^2-\frac{1}{2}\sum _a \biggl(\frac{\omega_{\text {p},a}}{w _\parallel } \biggr)^2 \biggl[Z' \biggl(\frac{\omega-v_0k}{kw _\parallel } \biggr) \\ &{}+Z' \biggl( \frac{\omega +v_0k}{kw _\parallel } \biggr) \biggr]=0, \end{aligned}$$

(12)

where Z′ is the derivative of the plasma dispersion function,

$$ Z(x)=\frac{1}{\sqrt{\pi}} \int _{-\infty }^\infty \mathrm {d} t\;\frac{e^{-t^2}}{t-x}=i\sqrt{\pi}e^{-x^2} \bigl[1+ \text {erf}(ix) \bigr], $$

(13)

where the first form is valid for ℑ(x)>0 only and where erf denotes the error function.

For the transverse dispersion relation, there are two versions if a background magnetic field, B ₀, is present—the left-handed and right-handed modes—which can be expressed as (Tautz and Schlickeiser 2005a)

$$\begin{aligned} D_t^\pm =&\omega^2-c^2k^2- \sum_a\omega_{\text {p},a}^2\mp \frac{1}{2}\sum_a\frac{\omega_{\text {p},a}^2}{w _\parallel } \frac{ \varOmega }{k} \\ &{}\times\biggl[Z \biggl(\frac {\omega-kv_0\pm \varOmega }{kw _\parallel } \biggr)+Z \biggl( \frac{\omega+kv_0\pm \varOmega }{kw _\parallel } \biggr) \biggr] \\ &{}-\frac{1}{4}\sum_a\omega_{\text {p},a}^2 \biggl(\frac{w _\perp }{w _\parallel } \biggr)^2 \biggl[Z' \biggl( \frac{\omega-kv_0\pm \varOmega }{kw _\parallel } \biggr) \\ &{}+Z' \biggl(\frac {\omega+kv_0\pm \varOmega }{kw _\parallel } \biggr) \biggr]. \end{aligned}$$

(14)

Note that, due to the linear factor Ω in the terms containing Z(…), the dispersion relation is greatly simplified for an unmagnetized plasma, i.e., where B ₀=0.

For perpendicular wave propagation, the dispersion relation for the ordinary-wave mode reads (Tautz and Schlickeiser 2006)

$$\begin{aligned} D _\perp =&\omega^2-c^2k^2+ \sum_a\omega_{\text {p},a}^2+\sum _a\omega_{\text {p},a}^2 \frac{w _\parallel ^2+2v_0^2}{w _\perp ^2} \\ &{}\times \biggl[1-{}_2F_2\biggl( \frac{1}{2},1;1+\frac{\omega}{ \varOmega },1-\frac{\omega}{ \varOmega };-\frac{k^2w _\perp ^2}{ \varOmega ^2} \biggr) \biggr], \end{aligned}$$

(15)

where ₂ F ₂(a,b;c,d;z) is the generalized hypergeometric function.

1.2 A.2 Suprathermal distribution

For the kappa-type distribution function that includes particles forming a so-called supra-thermal tail, the longitudinal dispersion relation reads (Lazar et al. 2008)

$$\begin{aligned} D_\ell =&k^2+\sum_a \frac{\omega_{\text {p},a}^2}{\theta _\parallel ^2} \biggl[2-\frac {1}{\kappa}+\frac{\omega-kv_0}{k\theta _\parallel } Z_\kappa \biggl(\frac{\omega -kv_0}{k\theta _\parallel } \biggr) \\ &{}+\frac{\omega+kv_0}{2k\theta _\parallel } Z_\kappa \biggl( \frac{\omega+kv_0}{2k\theta _\parallel } \biggr) \biggr], \end{aligned}$$

(16)

where one must take care not to confuse κ (the power-law index in the distribution function) with k (the wavenumber). The modified plasma dispersion function (Summers and Thorne 1991) is given through

$$ Z_\kappa(x)=\frac{1}{\sqrt{\pi\kappa}} \frac{ \varGamma (\kappa)}{ \varGamma (\kappa -1/2)} \int _{-\infty }^\infty \mathrm {d} t \;\frac{ (1+x^2/\kappa )^{-(\kappa+1)}}{t-x}, $$

(17)

where Γ(z) is the Gamma function. Again, Eq. (17) is valid for ℑ(x)>0 only.

For the transverse dispersion relation, the form that has been derived by Lazar et al. (2008) is given as

$$\begin{aligned} D_t^\pm =&\omega^2-c^2k^2- \sum_a\omega_{\text {p},a}^2\mp \frac{1}{2}\sum_a\frac{\omega_{\text {p},a}^2}{\theta _\parallel } \frac{ \varOmega }{k} \\ &{}\times \biggl[\tilde{Z}_\kappa \biggl(\frac{\omega-kv_0\pm \varOmega }{k\theta _\parallel } \biggr)+\tilde{Z}_\kappa \biggl(\frac{\omega+kv_0\pm \varOmega }{k\theta _\parallel } \biggr) \biggr] \\ &{}+\frac{1}{2}\sum_a\omega_{\text {p},a}^2 \biggl(\frac{\theta _\perp }{\theta _\parallel } \biggr)^2 \biggl[2+\frac{\omega-kv_0\pm \varOmega }{k\theta _\parallel } \\ &{}\times\tilde{Z}_\kappa \biggl(\frac{\omega-kv_0\pm \varOmega }{k\theta _\parallel } \biggr) \\ &{}+\frac{\omega+kv_0\pm \varOmega }{k\theta _\parallel } \tilde{Z}_\kappa \biggl(\frac{\omega+kv_0\pm \varOmega }{k\theta _\parallel } \biggr) \biggr], \end{aligned}$$

(18)

where, for the plasma dispersion function, a new form has been introduced as

$$ \tilde{Z}_\kappa(x)=\frac{1}{\sqrt{\pi\kappa}} \frac{ \varGamma (\kappa)}{ \varGamma (\kappa-1/2)} \int _{-\infty }^\infty \mathrm {d} t\; \frac{ (1+x^2/\kappa )^{-\kappa}}{t-x}, $$

(19a)

which is related to Z _κ (x) in Eq. (17) as

$$ \tilde{Z}_\kappa(x)= \biggl(1+\frac{x^2}{\kappa} \biggr)Z_\kappa(x)+ \frac {x}{\kappa} \biggl(1-\frac{1}{2\kappa} \biggr). $$

(19b)

In general, the ordinary-wave mode for perpendicular wave propagation would involve a rather tedious integral. Therefore, Lazar et al. (2010) used the large-wavelength limit, in which case the dispersion relation can be written in simplified form as

$$\begin{aligned} D _\perp (k R_{\mathrm{L}} \ll1) \approx&\omega^2-c^2k^2-\sum _a\omega_{\text {p},a}^2 \\ &{}-k^2 \sum_a\frac{\omega_{\text {p},a}^2v_0^2}{\omega^2- \varOmega ^2} \biggl[1+ \biggl( \frac {w _\perp }{v_0} \biggr)^2 \biggr], \end{aligned}$$

(20)

which agrees with the corresponding expansion of the dispersion relation for the case of a Maxwellian distribution function, Eq. (15). In the opposite limit of small wavelengths, the dispersion relation reads

$$\begin{aligned} D _\perp (k R_{\mathrm{L}} \gg1) \approx&\omega^2-c^2k^2-\sum _a\omega_{\text {p},a}^2 \\ &{}+\sum _a\omega_{\text {p},c}^2 \biggl( \frac{\theta _\parallel }{\theta _\perp } \biggr)^2 \biggl[1+ \biggl(2-\frac{1}{\kappa} \biggr)\frac{v_0^2}{\theta _\parallel ^2} \biggr]. \end{aligned}$$

(21)

A discussion of the applicability and numerical solutions connecting the two limiting cases has been given by Lazar et al. (2010).