LINEAR THEORY OF WEAKLY AMPLIFIED, PARALLEL PROPAGATING, TRANSVERSE TEMPERATURE-ANISOTROPY INSTABILITIES IN MAGNETIZED THERMAL PLASMAS

Kinetic plasma relaxation and turbulence generation processes are responsible for the observed properties of the solar wind plasma which is the only cosmic collision poor plasma accessible to detailed in situ satellite observations (Bale et al. 2009). Although the detailed plasma relaxation processes are not understood, the observed electron and proton distribution functions are close to bi-Maxwellian distributions with different temperatures along and perpendicular to the ordered magnetic field direction. Ten years of Wind/SWE data (Kasper et al. 2002) have demonstrated that the proton and electron temperature anisotropies A = T_⊥/T_∥ are bounded by mirror and fire-hose instabilities (Hellinger et al. 2006) at large values of the parallel plasma beta β_∥ = 8 π nk_BT_∥/B² ⩾ 1. In the parameter plane defined by the temperature anisotropy A = T_⊥/T_∥ and the parallel plasma beta β_∥, stable plasma configuration is possible only within a rhomb-like configuration around β_∥ ≃ 1, whose limits are defined by the threshold conditions for the mirror and fire-hose instabilities. If a plasma would start with parameter values outside this rhomb-like configuration, it immediately would generate fluctuations via the mirror and fire-hose instabilities, which quickly relax the plasma distribution into the stable regime within the rhomb configuration. The plasma parameters of other dilute cosmic plasmas including the interstellar and intracluster medium (Shekochihin et al. 2005) and accretion disks around compact, massive objects (Sharma et al. 2007) are similar to the solar wind plasma, so that the temperature-anisotropy instabilities should also operate in these systems.

The study of linear electromagnetic instabilities in a collisionless, homogeneous, magnetized, electron–proton plasma with temperature anisotropies has a long history; for reviews we refer the interested reader to chapter 7 of the monograph by Gary (1993), as well as Schwartz (1980), Cuperman (1981), and Marsch (2006). Hellinger et al. (2006) considered linear instability calculations for the oblique mirror, fire-hose, and proton cyclotron instabilities, and represented the approximate threshold conditions from the work of Gary et al. (1994, 2001), Gary & Lee (1994), Samsonov et al. (2001), and Pokhotelov et al. (2004) by analytic relations of the form A = 1 + a(β_∥ − β₀)^−b where a, b, and β₀ are fitted parameter values. In order to understand the confinement limits also at small values of the parallel plasma beta β_∥ < 1, here we rigorously analyze the full linear dispersion relation in a collisionless homogenous plasma with anisotropic (A ≠ 1) bi-Maxwellian particle velocity distributions of electrons and protons for electromagnetic fluctuations with wavevectors ($\vec{k}\times \vec{B}=0$) parallel to the uniform background magnetic field $\vec{B}$. With a typical solar wind temperature T = 10⁵ T₅ K, the thermal particle energy k_BT = 8.6 T₅ eV is much less than the electron rest mass m_ec² = 5.11 · 10⁵ eV, so that the use of the nonrelativistic linearized Vlasov/Maxwell equations is justified. For weakly (γ ≪ ω_R) amplified wave solutions, we analytically derive existence and instability conditions. Here, ω_R and γ refer to real and imaginary part of the complex frequency ω = ω_R + ıγ. Our analysis closely follows the recent study (Schlickeiser 2010) for equal-mass pair plasmas—hereafter referred to as paper S. Our investigation is restricted to weakly amplified (γ ≪ ω_R) solutions covering the left-handed (LH) polarized Alfvén proton cyclotron branch and the right-handed (RH) polarized Alfvén Whistler electron cyclotron branch. The corresponding analysis of weakly propagating (ω_R ≪ γ) solutions, including mirror, fire-hose, electron cyctronic, and cold magnetized Weibel fluctuations, is the subject of a subsequent paper. As we will demonstrate below, for the case of parallel ($\vec{k}\times \vec{B}=0$) wavevectors simple analytical threshold conditions for the two branches can be derived in terms of the combined temperature anisotropy A = T_⊥/T_∥, the parallel plasma beta β_∥ = 8 π n_ek_BT_∥/B², the electron–proton mass ratio, and the electron plasma frequency phase speed w = ω_p,e/(kc).

For a nonzero background magnetic field strength the nonrelativistic dispersion relations for RH- and LH-polarized fluctuations with wavevectors $\vec{k}\times \vec{B}=0$ in a thermal electron–proton plasma are (Gary 1993)

$$\begin{eqnarray} D_{\rm RH,LH}(k, \omega) &=& \omega ^2-k^2c^2+\sum _{a=p,e}\omega _{p,a}^2 \nonumber\\ && \times\bigg[{\omega \over \sqrt{2}ku_{a,\parallel } }Z\left({\omega \pm \Omega _a\over \sqrt{2}ku_{a,\parallel } }\right) \nonumber\\ && +{1\over 2}\left(1-A_a\right)Z^{^{\prime }} \left({\omega \pm \Omega _a\over \sqrt{2}ku_{a,\parallel } } \right)\bigg]=0, \end{eqnarray} \tag{ 1 }$$

where we sum over the proton (p) and electron (e) contributions, and where k = |k_∥|. ω_p,e = (4πe²n_e/m_e)^1/2 and ω_p,p = (m_e/m_p)^1/2ω_p,e denote the electron and proton plasma frequencies, respectively. u_a,∥ = (k_BT_a,∥/m_a)^1/2 is the parallel thermal velocity of component a, Ω_a = e_aB/(m_ac) is the nonrelativistic gyrofrequency, and A_a = T_a,⊥/T_a,∥ is the temperature anisotropy of component a, where the directional subscripts refer to directions relative to the background magnetic field. The dispersion relations (1) allow for different values of the proton and electron parallel temperatures and temperature anisotropies.

Z(x) and $Z^{^{\prime }}(x)$ denote the plasma dispersion function (Fried & Conte 1961) and its derivative

$$\begin{eqnarray} &&Z(x)=\pi ^{-1/2}\int _{-\infty }^\infty dt\, {e^{-t^2}\over t-x} \end{eqnarray} \tag{ 2 }$$

with the well-known properties

$$\begin{eqnarray} &&Z^{^{\prime }}(x)=-2\left[1+xZ(x)\right], \end{eqnarray} \tag{ 3 }$$

and

$$\begin{eqnarray} Z(-x) &=& 2\pi ^{1/2}\imath e^{-x^2}-Z(x),\cr Z^{^{\prime }}(-x)&=&4\pi ^{1/2}\imath xe^{-x^2}+Z^{^{\prime }}(x). \end{eqnarray} \tag{ 4 }$$

We will frequently use the asymptotic expansions

$$\begin{eqnarray} &&Z(x)\simeq \imath \pi ^{1/2}e^{-x^2}-\, 2x\left[1-{2x^2\over 3}\right],\quad |x|\ll 1 \end{eqnarray} \tag{ 5 }$$

and

$$\begin{eqnarray} &&Z(x)\simeq \imath \sigma \pi ^{1/2}e^{-x^2}-\, {1\over x}\left[1+{1\over 2x^2}+{3\over 4x^4}\right],\quad |x|\gg 1{,}\; \end{eqnarray} \tag{ 6 }$$

where σ = 0 if ℑ(x)>0, σ = 1 if ℑ(x) = 0, and σ = 2 if ℑ(x) < 0.

Paper S has demonstrated that the analytical analysis is enormously facilitated if we work with phase speeds rather than frequencies. We therefore introduce the complex phase speeds

$$\begin{eqnarray} &&f={\omega \over kc}={\omega _R+\imath \gamma \over kc}=R+\imath S,\;\;\; R={\omega _R\over kc},\;\;\; S={\gamma \over kc}, \end{eqnarray} \tag{ 7 }$$

the plasma frequency phase speed

$$\begin{eqnarray} &&w={\omega _{p,e}\over kc}, \end{eqnarray} \tag{ 8 }$$

and the absolute value of the electron gyrofrequency phase speed

$$\begin{eqnarray} &&b={|\Omega _e|\over kc}, \end{eqnarray} \tag{ 9 }$$

where |Ω_e| = eB/m_ec is the absolute value of the electron gyrofrequency. We also introduce the mass ratio

$$\begin{eqnarray} &&\mu =m_p/m_e=1836, \end{eqnarray} \tag{ 10 }$$

and the dimensionless proton and electron temperatures

$$\begin{eqnarray} &&\Theta _e\equiv \left({2k_BT_{e, \parallel }\over m_ec^2}\right)^{1/2},\;\; \Theta _p\equiv \left({2k_BT_{p, \parallel }\over m_pc^2}\right)^{1/2}. \end{eqnarray} \tag{ 11 }$$

Throughout this work, in the classification of Swanson (1989) we discuss high-density plasmas with ω_p,e ≫ |Ω_e|, corresponding to w ≫ b which applies to nearly all dilute astrophysical plasmas. These plasmas are dense enough that the electron plasma frequency is much larger than the electron gyrofrequency, but small enough that elastic Coulomb collisions can be neglected.

The two dispersion relations (1) then read

$$\begin{eqnarray} 0&=&{D_{\rm RH,LH}(k, f)\over k^2c^2}= \Lambda _{\rm RH,LH}(k, f)=f^2-1\nonumber\\[10pt] &&+{w^2\over \mu }\left[{f\over \Theta _p}Z\left({f\pm {b\over \mu }\over \Theta _p}\right)+ {1\over 2}\left(1-A_p\right)Z^{^{\prime }}\left({f\pm {b\over \mu }\over \Theta _p}\right)\right]\nonumber\\[10pt] &&+w^2\left[{f\over \Theta _e}Z\left({f\mp b\over \Theta _e}\right)+{1\over 2}\left(1-A_e\right)Z^{^{\prime }}\left({f\mp b\over \Theta _e}\right)\right].\nonumber\\ \end{eqnarray} \tag{ 12 }$$

We note the symmetry Λ(−k_∥, f) = Λ(k_∥, f) of both dispersion relations allowing us to consider only positive values of the wavenumber k>0. In the following, we will simplify the analysis by considering only equal parallel temperature plasmas (T_e,∥ = T_p,∥) so that Θ_e = Θ and Θ_p = Θ/μ^1/2. Bale et al. (2009), reporting on the observations of the solar wind plasma properties, do not provide information on any measured differences of T_e,∥ and T_p,∥, so we adopt parallel temperature equality which enormously simplifies the calculations.

The dispersion relations (12) can be separated into real and imaginary parts Λ(R, S) = ℜΛ(R, S) + ıℑΛ(R, S) = 0, implying the two conditions

$$\begin{eqnarray} &&\Re \Lambda (R, S)=0,\, \;\; \Im \Lambda (R, S)=0. \end{eqnarray} \tag{ 13 }$$

In terms of the complex phase speed f = R + ıS the real and imaginary parts of the two dispersion relations (12) read

$$\begin{eqnarray} 0&=& \Re \Lambda _{\rm RH,LH}(R,S)=R^2-S^2-1\cr &&+w^2\Bigg [{R\over \Theta \mu ^{1/2}}\Re Z\Bigg({\mu ^{1/2}\over \Theta }\Bigg[R+\imath S\pm {b\over \mu }\Bigg]\Bigg)\nonumber\\ && + {1-A_p\over 2\mu }\Re Z^{^{\prime }}\Bigg({\mu ^{1/2}\over \Theta }\Bigg[R+\imath S\pm {b\over \mu }\Bigg]\Bigg)\cr &&+ {R\over \Theta }\Re Z\Bigg({R+\imath S\mp b\over \Theta }\Bigg) +{1-A_e\over 2}\Re Z^{^{\prime }}\Bigg({R+\imath S\mp b\over \Theta }\Bigg)\Bigg ]\cr &&-{w^2S\over \Theta }\Bigg[{1\over \mu ^{1/2}}\Im Z\Bigg({\mu ^{1/2}\over \Theta }\Bigg[R+\imath S\pm {b\over \mu }\Bigg]\Bigg) \cr && +\Im Z\Bigg({R+\imath S\mp b\over \Theta }\Bigg)\Bigg] \end{eqnarray} \tag{ 14 }$$

and

$$\begin{eqnarray} 0&=&\Im \Lambda _{\rm RH,LH}(R, S)=2RS \cr &&+\, {w^2\over 2}\left[{1-A_p\over \mu }\Im Z^{^{\prime }}\left({\mu ^{1/2}\over \Theta }\left[R+\imath S\pm {b\over \mu }\right]\right)\right.\cr &&\left. +\, (1-A_e)\Im Z^{^{\prime }}\left({R+\imath S\mp b\over \Theta }\right)\right]\cr &&+\,{w^2S\over \Theta }\left[{1\over \mu ^{1/2}}\Re Z\left({\mu ^{1/2}\over \Theta }\left[R+\imath S\pm {b\over \mu }\right]\right)\right.\cr &&\left.+\, \Re Z\left({R+\imath S\mp b\over \Theta }\right)\right]\cr &&+\,{w^2R\over \Theta }\left[{1\over \mu ^{1/2}}\Im Z\left({\mu ^{1/2}\over \Theta }\left[R+\imath S\pm {b\over \mu }\right]\right)\right.\cr &&\left. +\, \Im Z\left({R+\imath S\mp b\over \Theta }\right)\right]. \end{eqnarray} \tag{ 15 }$$

Here, we consider solutions of the dispersion relations in the weak damping/amplification limit |S| ≪ R. As shown in paper S in this limit the real part of the dispersion relation satisfies

$$\begin{eqnarray} &&\Re \Lambda (R, S=0)=0, \end{eqnarray} \tag{ 16 }$$

whereas the corresponding imaginary part is given by

$$\begin{eqnarray} &&S=-{\Im \Lambda (R, S=0)\over {\partial \Re \Lambda (R, S=0)\over \partial R}}. \end{eqnarray} \tag{ 17 }$$

As before we introduce the parallel plasma beta

$$\begin{eqnarray} &&\beta _{\parallel }={\Theta ^2w^2\over b^2}={8\,\pi \, n_ek_BT_{\parallel }\over B^2}, \end{eqnarray} \tag{ 18 }$$

which expresses the magnetic field strength as

$$\begin{eqnarray} &&b={\Theta w\over \beta _{\parallel }^{1/2}} \end{eqnarray} \tag{ 19 }$$

in terms of the electron plasma frequency phase speed (w), the parallel temperature (Θ), and the parallel plasma beta β_∥. Our restriction to high-density plasmas with ω_p,e ≫ |Ω_e| or w ≫ b then requires to have parallel plasma betas β_∥ ≫ Θ² ≃ 3.4 × 10⁻⁵(T_e,∥/10⁵ K) in the solar wind plasma with typical temperatures T_e,∥ ≃ 10⁵ K.

For weakly damped or amplified fluctuations Equations (14) and (15) read

$$\begin{eqnarray} 0& = &\Re \Lambda _{\rm RH,LH}(R, S=0)=R^2-1\cr && + w^2\bigg [{R\over \Theta }\Re Z\left({R\mp b\over \Theta }\right) +{1-A_e\over 2}\Re Z^{^{\prime }}\left({R\mp b\over \Theta }\right)\cr && + {R\over \Theta \mu ^{1/2}}\Re Z\left({\mu ^{1/2}\over \Theta }\left[R\pm {b\over \mu }\right]\right) \cr && + {1-A_p\over 2\mu }\Re Z^{^{\prime }}\left({\mu ^{1/2}\over \Theta }\left[R\pm {b\over \mu }\right]\right)\bigg ] \end{eqnarray} \tag{ 20 }$$

and

$$\begin{eqnarray} \Im \Lambda _{\rm RH,LH}(R, S=0)&=& {w^2\over 2}\left[{1-A_p\over \mu }\Im Z^{^{\prime }}\left({\mu ^{1/2}\over \Theta }\left[R\pm {b\over \mu }\right]\right)\right.\cr &&\left. + (1-A_e)\Im Z^{^{\prime }}\left({R\mp b\over \Theta }\right)\right]\cr &&+{w^2R\over \Theta }\left[{1\over \mu ^{1/2}}\Im Z\left({\mu ^{1/2}\over \Theta }\left[R\pm {b\over \mu }\right]\right)\right. \cr &&\left. +\, \Im Z\left({R\mp b\over \Theta }\right)\right], \end{eqnarray} \tag{ 21 }$$

which have to be investigated for positive values of R ⩾ 0.

As noted before, these dispersion relations can be further reduced with the asymptotic expansions (5) and (6), depending on the absolute values of the arguments,

$$\begin{eqnarray} &&P_{\pm }(R)={\mu ^{1/2}\over \Theta }\left[|R\pm {b\over \mu }|\right],\;\; E_{\pm }(R)={|R\pm b|\over \Theta }, \end{eqnarray} \tag{ 22 }$$

of the plasma dispersion function and its derivative being small or large compared to unity. Scaling R = bx, corresponding to x = ω_R/|Ω_e|, the absolute values of the arguments (22) in terms of the parallel plasma beta β_∥ read

$$\begin{eqnarray} &&\hspace{-13pt}P_{\pm }(x)={w\mu ^{1/2}\over \beta _{\parallel }^{1/2}}\left[|x\pm {1\over \mu }|\right],\;\; E_{\pm }(x)={w\over \beta _{\parallel }^{1/2}}|x\pm 1|, \end{eqnarray} \tag{ 23 }$$

which are shown in Figure 1 as a function of the normalized real frequency x.

Zoom In Zoom Out Reset image size

For parallel plasma beta values β_∥ ≪ (w²/μ), we note that P₊(x) ≫ 1, E₊(x) ≫ 1 for all values of x, whereas P₋(x) ≫ 1 for x outside the small interval

$$\begin{eqnarray} &&x\notin \left[{1\over \mu }-{\beta _{\parallel }^{1/2}\over w\mu ^{1/2}}, {1\over \mu }+{\beta _{\parallel }^{1/2}\over w\mu ^{1/2}}\right] \end{eqnarray} \tag{ 24 }$$

around the proton cyclotron frequency, and E₋(x) ≫ 1 for x outside the small interval

$$\begin{eqnarray} &&x\notin \left[1-{\beta _{\parallel }^{1/2}\over w}, 1+{\beta _{\parallel }^{1/2}\over w}\right] \end{eqnarray} \tag{ 25 }$$

around the electron cyclotron frequency.

In the following, we limit our analysis to such values of the parallel plasma beta β_∥ ≪ (w²/μ), so that the asymptotic expansion (6) of the plasma dispersion function can be used except near the indicated proton and electron cyclotron frequencies.

Irrespective of the values of P_± and E_±, we note that both asymptotic expansions (5) and (6) yield the same imaginary part of the dispersion relation

$$\begin{eqnarray} \Im \Lambda _{\rm RH,LH}(R, S=0)&=& \pi ^{1/2}w^2{b\over \Theta } \bigg ({1\over \mu ^{1/2}}\left[A_p\left[{R\over b}\pm {1\over \mu }\right]\mp {1\over \mu }\right]\cr &&\times e^{-{{\mu \over \Theta ^2}(R\pm {b\over \mu })^2}} +\left[A_e\left[{R\over b}\mp 1\right]\pm 1\right]\cr &&\times e^{-{(R-b\over \Theta })^2}\bigg). \end{eqnarray} \tag{ 26 }$$

For parallel plasma beta values β_∥ ≪ (w²/μ) the asymptotic expansion (6) yields

$$\begin{eqnarray} &&\Re Z\left({R\pm b\over \Theta }\right)\simeq -{\Theta \over R\pm b}\left[1+{\Theta ^2\over 2(R\pm b)^2}\right], \end{eqnarray} \tag{ 27 }$$

$$\begin{eqnarray} &&\Re Z^{^{\prime }}\left({R\pm b\over \Theta }\right)\simeq +{\Theta ^2\over (R\pm b)^2}\left[1+{3\Theta ^2\over 2(R\pm b)^2}\right], \end{eqnarray} \tag{ 28 }$$

$$\begin{eqnarray} &&\Re Z\left({\mu ^{1/2}\over \Theta }\left[R\pm {b\over \mu }\right]\right)\simeq -{\Theta \over \mu ^{1/2}\big(R\pm {b\over \mu }\big)}\left[1+{\Theta ^2\over 2\mu \big(R\pm {b\over \mu }\big)^2}\right]\nonumber\\ \end{eqnarray} \tag{ 29 }$$

and

$$\begin{eqnarray} &&\Re Z^{^{\prime }}\left({\mu ^{1/2}\over \Theta }\left[R\pm {b\over \mu }\right]\right)\simeq +{\Theta ^2\over \mu \big(R\pm {b\over \mu }\big)^2}\left[1+{3\Theta ^2\over 2\mu \big(R\pm {b\over \mu }\big)^2}\right].\nonumber\\ \end{eqnarray} \tag{ 30 }$$

We then obtain for the real part of the dispersion relations (20) to lowest order in Θ² ≪ 1

$${\selectfont\fontsize{8}{9}{\begin{eqnarray} 0&=&\Re \Lambda _{\rm RH,LH}(R, S=0)=R^2-1-w^2R\left[{1\over R\mp b}+{1\over \mu (R\pm {b\over \mu })}\right]\cr &&+{\Theta ^2w^2\over 2}\left[{1-A_e\over (R\mp b)^2}-{R\over (R\mp b)^3}\right] +{\Theta ^2w^2\over 2\mu ^2}\left[{1-A_p\over \big(R\pm {b\over \mu }\big)^2}-{R\over \big(R\pm {b\over \mu }\big)^3}\right].\nonumber\\ \end{eqnarray}}} \tag{ 31 }$$

With the scaling R = bx and the parallel plasma beta (18) the dispersion relation (31) reads

$$\begin{eqnarray} 0 &=& \Re \Lambda _{\rm RH,LH}(x, S=0)\cr &=& \left[b^2+{(1+\mu)w^2\over (1\pm \mu x)(1\mp x)}\right]x^2-1+{\beta _{\parallel }\over 2}K_{\rm RH,LH}, \end{eqnarray} \tag{ 32 }$$

with

$$\begin{eqnarray} &&K_{\rm RH,LH}={1\over (1\mp x)^3}+{1\over (1\pm \mu x)^3}-{A_e\over (1\mp x)^2}-{A_p\over (1\pm \mu x)^2}.\nonumber\\ \end{eqnarray} \tag{ 33 }$$

In different limits the solutions of Equations (31) and (32) describe Alfvén waves, Whistler waves, cyclotron waves, and electromagnetic light. We consider each case in the next subsections.

4.1. Alfvén Waves at Phase Speeds R ≪ b/μ for Θ² ≪ β_∥ ≪ w²/μ

For phase speeds R ≪ b/μ the dispersion relation (31) simplifies to

$$\begin{eqnarray} 0 &=& \Re \Lambda _{\rm RH,LH}(R, S=0)\cr &=& R^2\left(1+{2w^2(1+\mu)\over b^2} \right)-1+ {\beta _{\parallel }\over 2}\left[2-A_e-A_p \right.\nonumber\\ &&\left.\pm\, {2R\over b}(3-A_e)\mp {2\mu R\over b}(3-A_p)\right]\cr &\simeq & R^2\left(1+{c^2\over V_A^2}\right)-\left[1+\left(A-1\right)\beta _{\parallel }\right], \end{eqnarray} \tag{ 34 }$$

where we introduce the Alfvén speed

$$\begin{eqnarray} &&{(1+\mu)w^2\over b^2}={4\pi n_e(m_e+m_p)c^2\over B^2}={c^2\over V_A^2}, \end{eqnarray} \tag{ 35 }$$

the parallel plasma beta (18), and the combined plasma temperature anisotropy

$$\begin{eqnarray} &&A={A_p+A_e\over 2}. \end{eqnarray} \tag{ 36 }$$

For high-density plasmas V_A ≪ c, the dispersion relation (34) yields the LH- and RH-polarized Alfvén modes with the same phase speed

$$\begin{eqnarray} R\simeq {b\over \sqrt{1+\mu }w}\sqrt{1+(A-1)\beta _{\parallel }} &=& \Theta \sqrt{{1+(A-1)\beta _{\parallel }\over (1+\mu)\beta _{\parallel }}} \nonumber\\ &=&{V_A\over c}\sqrt{1+(A-1)\beta _{\parallel }},\nonumber\\ \end{eqnarray} \tag{ 37 }$$

which either can propagate forward and backward. Note that the condition R ≪ b/μ requires (w²/μ) ≫ 1.

Moreover, the four Alfvén modes only exist for temperature anisotropies such that 1 + (A − 1)β_∥ ⩾ 0 corresponding to

$$\begin{eqnarray} &&A\ge \left(1-{1\over \beta _{\parallel }}\right), \end{eqnarray} \tag{ 38 }$$

which includes the isotropic (A_p = A_e = A = 1) plasma temperature case. For small plasma betas (β_∥ ⩽ 1) the condition (38) is always fulfilled whereas for large plasma betas (β_∥>1) the combined anisotropy A has to be larger than 1 − β⁻¹_∥.

Equation (34) also provides

$$\begin{eqnarray} &&{\partial \Re \Lambda _{\rm RH,LH}(R, S=0)\over \partial R} \simeq 2{c^2\over V_A^2}R, \end{eqnarray} \tag{ 39 }$$

so that for all four modes according to Equations (17) and (26) the growth/damping rate is

$$\begin{eqnarray} S_{\rm RH,LH} &=& {-}{V_A^2\over 2c^2R}\Im \Lambda _{\rm RH,LH}(R, S=0)\cr &&-{\pi ^{1/2}b^3\over 2(1+\mu)\Theta R} \bigg ({1\over \mu ^{1/2}}\left[A_p\left[{R\over b}\pm {1\over \mu }\right]\mp {1\over \mu }\right] \cr &&\times\, e^{-{{\mu \over \Theta ^2}(R\pm {b\over \mu })^2}}+\left[A_e\left[{R\over b}\mp 1\right]\pm 1\right]e^{-\left({R\mp b\over \Theta }\right)^2}\bigg).\nonumber\\ \end{eqnarray} \tag{ 40 }$$

For isotropic (A_p = A_e = 1) plasma temperatures all four Alfvén modes are damped in agreement with Brinca's (1990) general theorem on the electromagnetic stability of isotropic plasma populations. The Alfvén damping rate in the isotropic case is given by

$$\begin{eqnarray} S_{\rm RH,LH}(A_p=A_e=1) &=& {-}{\pi ^{1/2}b^2\over 2(1+\mu)\Theta } \cr &&\times\left[{1\over \mu ^{1/2}}e^{-{{\mu \over \Theta ^2}(R\pm {b\over \mu })^2}}+ e^{-\left({R\mp b\over \Theta }\right)^2}\right].\nonumber\\ \end{eqnarray} \tag{ 41 }$$

In order to drive the Alfvén modes unstable, the growth rate (40) has to be positive requiring that

$${\selectfont\fontsize{9}{10}{\begin{eqnarray} I_{\rm RH,LH}(A_p,A_e,\beta _{\parallel })&\equiv& \pm \left({A_p-1\over \mu ^{3/2}}e^{-{w^2\over \mu \beta _{\parallel }}(1\pm \mu x)^2} -(A_e-1)e^{-{w^2\over \beta _{\parallel }}(1\mp x)^2}\right)\cr &&+x\left[{A_p\over \mu ^{1/2}}e^{-{w^2\over \mu \beta _{\parallel }}(1\pm \mu x)^2} +A_ee^{-{w^2\over \beta _{\parallel }}(1\mp x)^2}\right]<0. \nonumber\\ \end{eqnarray}}} \tag{ 42 }$$

This instability condition is analyzed further in the next section.

4.2. LH-polarized Alfvén Proton Cyclotron and RH-polarized Alfvén Whistler Electron Cyclotron Branches for β_∥ ≪ 1

Because we are primarily interested in low plasma beta plasmas for completeness we consider the solutions of the full dispersion relation (32) for subluminal solutions R ≪ 1 and small parallel plasma beta β_∥ ≪ 1:

$$\begin{eqnarray} &&\hspace{-8pt}0=\Re \Lambda _{\rm RH,LH}(x, S=0)\simeq {(1+\mu)w^2x^2\over 1\pm (\mu -1)x-\mu x^2}-1 \end{eqnarray} \tag{ 43 }$$

yielding the quadratic equation

$$\begin{eqnarray} &&[(1+\mu)w^2+\mu ]x^2\mp (\mu -1)x=1. \end{eqnarray} \tag{ 44 }$$

Because μ = 1836 ≫ 1 this equation is well approximated by

$$\begin{eqnarray} &&(1+w^2)x^2\mp x={1\over \mu } \end{eqnarray} \tag{ 45 }$$

with the solutions

$$\begin{eqnarray} &&x_{\rm RH,LH}={1\over 2(1+w^2)}\left[\sqrt{1+{4(1+w^2)\over \mu }}\pm 1\right], \end{eqnarray} \tag{ 46 }$$

covering the well-known (e.g., Swanson 1989) LH-polarized Alfvén proton cyclotron and RH-polarized Alfvén Whistler electron cyclotron branches for parallel propagation, as the appropriate limits for small and large values of w demonstrate.

For $w\ll \sqrt{(\mu /4)-1}=21.4$, corresponding to large wavenumbers k ≫ ω_p,e/21.4c, we obtain

$$\begin{eqnarray} &&x_{\rm RH}\simeq {1\over \mu }+{1\over 1+w^2}\simeq {1\over 1+w^2} \end{eqnarray} \tag{ 47 }$$

and

$$\begin{eqnarray} &&\hspace{-1.5pc}x_{\rm LH}\simeq {1\over \mu }\left[1-{1+w^2\over \mu }\right]. \end{eqnarray} \tag{ 48 }$$

The solution (48), subject to the constraint (24), represents the LH-polarized proton cyclotron waves.

For values of w ≪ 1 the solution (47) represents the RH-polarized electron cyclotron waves subject to the constraint (25) with the dispersion relation

$$\begin{eqnarray} &&x_{\rm RH}\simeq 1-w^2, \end{eqnarray} \tag{ 49 }$$

whereas for intermediate values 1 ≪ w ≪ 21.4 the solution (47) reduces to

$$\begin{eqnarray} &&x_{\rm RH}\simeq {1\over w^2}, \end{eqnarray} \tag{ 50 }$$

representing the RH-polarized Whistler waves (R ≃ b/w²).

For large values of $w\gg \sqrt{(\mu /4)-1}=21.4$, corresponding to small wavenumbers k ≪ ω_p,e/21.4c, we obtain for the solution (46)

$$\begin{eqnarray} &&x_{\rm RH,LH}\simeq {1\over \sqrt{\mu (1+w^2)}}\simeq {1\over \mu ^{1/2}w}, \end{eqnarray} \tag{ 51 }$$

which agrees with the Alfvén wave solutions (37) in the limit β_∥ ≪ 1.

Equation (43) for small plasma beta β_∥ ≪ 1 also provides

$$\begin{eqnarray} {\partial \Re \Lambda _{\rm RH,LH}(R, S=0)\over \partial R}&\simeq& {(1+\mu)w^2x\over b}{2\pm (\mu -1)x\over [1\pm (\mu -1)x-\mu x^2]^2}\cr &=&{2\pm (\mu -1)x\over b(1+\mu)w^2x^3}. \end{eqnarray} \tag{ 52 }$$

We note that for all solutions (47)–(51), Equation (52) is positive. According to Equations (17) and (26) the growth/damping rate then is

$$\begin{eqnarray} S_{\rm RH,LH}&=& {-}{b(1+\mu)w^2x^3\over 2\pm (\mu -1)x}\Im \Lambda _{\rm RH,LH}(R, S=0)\cr & = & -{\pi ^{1/2}(1+\mu)b^2w^4x^3\over \Theta [2\pm (\mu -1)x]} \bigg ({1\over \mu ^{1/2}}\left[\! A_p\! \left[x\pm {1\over \mu }\right]\mp {1\over \mu }\right]\nonumber\\ &&\times e^{-{{\mu b^2\over \Theta ^2}(x\pm {1\over \mu })^2}} +\left[A_e(x\mp 1)\pm 1\right]e^{-{b^2(x\mp 1)^2\over \Theta ^2}}\bigg), \end{eqnarray} \tag{ 53 }$$

which generalizes the Alfvénic growth/damping rate (40) to non-Alfvénic modes. Apart from the different notation in phase speeds, the rate (53) agrees with Equation (7.1.8) of Gary (1993). For isotropic (A_p = A_e = 1) plasma temperatures all modes of the RH and LH branches, including the cyclotron and Whistler modes, are damped in agreement with Brinca's general theorem. We notice that, in order to drive the cyclotron and Whistler modes unstable, the growth rate (53) has to be positive, requiring again the earlier derived instability condition (42).

4.3. Electromagnetic Waves at Large Frequencies R ≫ b + Θ

For large frequencies R ≫ b + Θ the dispersion relation (31) reduces to

$$\begin{eqnarray} 0&=&\Re \Lambda _{\rm RH,LH}(R, S=0)\simeq R^2-1-\left(1+{1\over \mu }\right)w^2 \mp w^2{b\over R} \nonumber\\ && -{w^2b^2\over R^2}\left[1+{1\over \mu }+{\Theta ^2\over 2b^2}\left(A_e+{A_p\over 4\mu ^2}\right)\right]\cr &\simeq & R^2-1-\left(1+{1\over \mu }\right)w^2 \end{eqnarray} \tag{ 54 }$$

and

$$\begin{eqnarray} 0 &=& \Im \Lambda _{\rm RH,LH}(R, S=0)\simeq \pi ^{1/2}w^2{R\over \Theta }\cr &&\times \left({A_p\over \mu ^{1/2}}e^{-\mu R^2/\Theta ^2}+A_ee^{-R^2/\Theta ^2}\right), \end{eqnarray} \tag{ 55 }$$

so that for both polarizations to lowest order in (b/R)² we obtain the dispersion relation of electromagnetic waves

$$\begin{eqnarray} &&R^2\simeq 1+\left(1+{1\over \mu }\right)w^2 \end{eqnarray} \tag{ 56 }$$

with the same damping rate

$$\begin{eqnarray} &&S=-{\pi ^{1/2}w^2\over 2\Theta }\left({A_p\over \mu ^{1/2}}e^{-\mu R^2/\Theta ^2}+A_ee^{-R^2/\Theta ^2}\right). \end{eqnarray} \tag{ 57 }$$

In order to drive the LH-polarized Alfvén proton cyclotron and the RH-polarized Alfvén Whistler electron cyclotron branches unstable, the condition (42) has to be fulfilled which, after multiplication with μ^3/4, reads

$$\begin{eqnarray} &&\pm \left({A_p-1\over \mu ^{3/4}}e^{-X_p^2}-(A_e-1)\mu ^{3/4}e^{-X_e^2}\right)\cr &&\quad + x\mu ^{1/2}\left[{A_p\over \mu ^{1/4}}e^{-X_p^2}+A_e\mu ^{1/4}e^{-X_e^2}\right] \cr && =\, {\pm} \left(e^{-{3\over 4}\ln \mu +\ln (A_p-1)-X_p^2}-e^{{3\over 4}\ln \mu +\ln (A_e-1)-X_e^2}\right)\cr &&\quad + x\mu ^{1/2}\left(e^{-{1\over 4}\ln \mu +\ln A_p-X_p^2}+e^{{1\over 4}\ln \mu +\ln A_e-X_e^2}\right)<0, \end{eqnarray} \tag{ 58 }$$

with

$$\begin{eqnarray} &&X_p={w \over (\mu \beta _{\parallel })^{1/2}}(1\pm \mu x),\;\; X_e={w \over \beta _{\parallel }^{1/2}}(1\mp x). \end{eqnarray} \tag{ 59 }$$

Both brackets in Equation (58) can be further reduced using the identities

$$\begin{eqnarray} &&e^{a_1+a_2}-e^{a_1-a_2}=2e^{a_1}\sinh (a_2),\;\; e^{c_1+c_2}-e^{c_1-a_c}=2e^{c_1}\sinh (c_2). \nonumber\\ \end{eqnarray} \tag{ 60 }$$

From the first bracket we identify

$$\begin{eqnarray} a_1 &=& \ln \left[(A_p-1)(A_e-1)\right]^{1/2}-{X_p^2+X_e^2\over 2}, \cr a_2 &=& {-}{3\over 4}\ln \mu -\ln \left[{A_e-1\over A_p-1}\right]^{1/2}-W_{\pm }, \end{eqnarray} \tag{ 61 }$$

where

$${\selectfont\fontsize{9}{10}{\begin{eqnarray} &&W_{\pm }(x)={X_p^2-X_e^2\over 2}={w^2\over 2\beta _{\parallel }}\left[\pm 4x+(\mu -1)\left(x^2-{1\over \mu }\right)\right]. \end{eqnarray}}} \tag{ 62 }$$

From the second bracket we determine

$$\begin{eqnarray} c_1 &=& \ln \left[A_pA_e\right]^{1/2}-{X_p^2+X_e^2\over 2}, \cr c_2 &=& {-}{1\over 4}\ln \mu -\ln \left[{A_e\over A_p}\right]^{1/2}-W_{\pm }. \end{eqnarray} \tag{ 63 }$$

With sinh(−a₂) = −sinh(a₂) and cosh(−c₂) = cosh(c₂) condition (58) becomes

$$\begin{eqnarray} &&\pm (A_p-1)^{1/2}(A_e-1)^{1/2}\sinh \left[{3\over 4}\ln \mu +\ln \left[\!{A_e-1\over A_p-1}\right]^{1/2}\!\!{+}W_{\pm }\right]\cr &&\quad -A_p^{1/2}A_e^{1/2}x\mu ^{1/2}\cosh \left[{1\over 4}\ln \mu +\ln \left[{A_e\over A_p}\right]^{1/2}+W_{\pm }\right]\cr &&=\pm \left(\mu ^{3/4}(A_e-1)-\mu ^{-3/4}(A_p-1)\right)\cosh W_{\pm }\cr &&\quad \pm \left(\mu ^{3/4}(A_e-1)+\mu ^{-3/4}(A_p-1)\right)\sinh W_{\pm }\cr &&\quad -x\mu ^{1/2}\left(\mu ^{1/4}A_e+\mu ^{-1/4}A_p\right)\cosh W_{\pm }\cr &&\quad -x\mu ^{1/2}\left(\mu ^{1/4}A_e-\mu ^{-1/4}A_p\right)\sinh W_{\pm }>0, \end{eqnarray} \tag{ 64 }$$

which yields the general instability condition for the two branches in the form

$$\begin{eqnarray} &\pm \{ (A_e-A_p)[(\mu ^{3/2}+1)+(\mu ^{3/2}-1)\tanh (W_{\pm }]\cr &\quad+(A_e+A_p-2)[(\mu ^{3/2}-1)+(\mu ^{3/2}+1)\tanh (W_{\pm }]\}/ \cr &\quad \{x\mu ((A_e+A_p)[\mu ^{1/2}+1+(\mu ^{1/2}-1)\tanh (W_{\pm })]\cr &\quad +(A_e-A_p)[\mu ^{1/2}-1+(\mu ^{1/2}+1)\tanh (W_{\pm })])\}\nonumber\\ &{>}1. \end{eqnarray} \tag{ 65 }$$

Inserting the general dispersion relation for small plasma betas (46) for x and W_±(x) then provides the general instability conditions for the LH-polarized Alfvén proton cyclotron and the RH-polarized Alfvén Whistler electron cyclotron branches, which to the best of our knowledge has not been derived before.

For equal electron and proton temperature anisotropies (A_p = A_e = A₀) the condition (65) reduces to

$$\begin{eqnarray} &\pm \left(1-{1\over A_0}\right)\left[(\mu ^{3/2}-1)+(\mu ^{3/2}+1)\tanh (W_{\pm }\right]\nonumber\\ &\quad {>}\,\mu x\left[\mu ^{1/2}+1+(\mu ^{1/2}-1)\tanh (W_{\pm })\right]. \end{eqnarray} \tag{ 66 }$$

Note that for pair plasmas (μ = 1) the condition (66) reduces to Equation (S-66) in paper S.

Instead of working with the general dispersion relations (46) used in the general instability condition (65), we will discuss different ranges of the plasma frequency phase speed w, where the dispersion relations (46) reduce to simpler limits, as demonstrated in Section 4.2. For w ≫ 21.4 we can use the Alfvenic relation (51), whereas for small values of w ≪ 21.4 we can use the cyclotron and Whistler relations (48)–(50), respectively. We are particularly interested in applying our results to the solar wind observations of Bale et al. (2009), which, as we demonstrate in the next section, cover the phase speed range 90 ⩽ w ⩽ 330.

Before investigating the individual modes, we further inspect the instability condition (66). The instability conditions holds for plasma betas β_∥ ≪ w²/μ, implying w²/2β_∥ ≫ μ/2 = 918. Consequently, the argument of the tanh -function

$$\begin{eqnarray} &&W_{\pm }={w^2\over 2\beta _{\parallel }}f_{\pm }(x),\;\; f_{\pm }(x)=\pm 4x+(\mu -1)\left(x^2-{1\over \mu }\right)\nonumber\\ \end{eqnarray} \tag{ 67 }$$

is much larger than unity for f_±(x) ≫ 2/μ.

The function f₊(x) increases monotonically from its smallest negative value f₊(0) = −(1 − μ⁻¹) = −0.9995 and becomes zero at

$$\begin{eqnarray} &&x_{0+}={\mu +1\over (\mu -1)\mu ^{1/2}}\left[1-{2\,\mu ^{1/2}\over \mu +1}\right]=0.95\,\mu ^{-1/2}\simeq \mu ^{-1/2}. \nonumber\\ \end{eqnarray} \tag{ 68 }$$

Likewise, the function f₋(x) attains its negative minimum value −1.0016 at x_E = 2/(μ-1) = 0.0011 and becomes zero at

$$\begin{eqnarray} &&x_{0-}={\mu +1\over (\mu -1)\mu ^{1/2}}\left[1+{2\,\mu ^{1/2}\over \mu +1}\right]=1.05\,\mu ^{-1/2}\simeq \mu ^{-1/2}.\nonumber\\ \end{eqnarray} \tag{ 69 }$$

Hence, only for values of x = R/b inside a small interval around μ^−1/2, which for RH-polarized fluctuations lies in the Whistler phase speed range, the functions f_±(x) are smaller than 2/μ. Both functions are well approximated by

$$\begin{eqnarray} &&f_{\pm }(x)\simeq f_0(x)=\mu x^2-1, \end{eqnarray} \tag{ 70 }$$

which is negative in the range of Alfvén and proton cyclotron waves and positive in the range of electron cyclotron waves, respectively.

The solar wind magnetic fluctuations measured by Bale et al. (2009) near 1 AU have wavenumbers

$$\begin{eqnarray} &&k\simeq \alpha /\rho _p \end{eqnarray} \tag{ 71 }$$

with α = 0.56 ± 0.32 and the thermal proton gyroradius ρ_p = 4.23 ×  10⁶ T^1/2₅B⁻¹₄ cm, where we adopt an interplanetary magnetic field value B = 10⁻⁴ B₄ G and a temperature T = 10⁵ T₅ K. With the solar wind particle density n_e = 10²n₂ cm⁻³, we find that the plasma frequency phase speed (8)

$$\begin{eqnarray} &&w={79.5\over \alpha }{(T_5n_2)^{1/2}\over B_4}={142\over 1\pm 0.57}{(T_5n_2)^{1/2}\over B_4} \end{eqnarray} \tag{ 72 }$$

covers the interval

$$\begin{eqnarray} &&90{(T_5n_2)^{1/2}\over B_4}\le w\le 330{(T_5n_2)^{1/2}\over B_4}. \end{eqnarray} \tag{ 73 }$$

The electron gyrofrequency phase speed (9)

$$\begin{eqnarray} &&b=3.13\cdot 10^{-3}w{B_4\over n_2^{1/2}} \end{eqnarray} \tag{ 74 }$$

is indeed much smaller than the electron plasma frequency phase speed w justifying the high-density plasma approximation w ≫ b.

Because of the observed large value (72) of the plasma frequency phase speed only the LH- and RH-polarized Alfvén wave instabilities can operate in the w-interval (73). We therefore concentrate our analysis of the instability condition (66) on this case.

In terms of the proton plasma frequency phase speed

$$\begin{eqnarray} &&h={w\over \mu ^{1/2}}={w\over 43}, \end{eqnarray} \tag{ 75 }$$

the Bale et al. (2009) measurements cover the interval

$$\begin{eqnarray} &&2.1{(T_5n_2)^{1/2}\over B_4}\le h\le 7.7{(T_5n_2)^{1/2}\over B_4}. \end{eqnarray} \tag{ 76 }$$

In the range of Alfvén waves x ≪ μ⁻¹ the functions f_±(x)≃ −1 are negative and practically constant, so that the instability condition (66) becomes

$$\begin{eqnarray} &\mp \left(1-{1\over A_0}\right)\left[(\mu ^{3/2} +1)\tanh \left[{w^2\over 2\beta _{\parallel }}\right]-(\mu ^{3/2} -1)\right]\nonumber\\ &\quad {>}\,x\,\mu \left[(\mu ^{1/2}+1)-(\mu ^{1/2}-1)\tanh \left[{w^2\over 2\beta _{\parallel }}\right]\right]. \end{eqnarray} \tag{ 77 }$$

Because β_∥ ≪ w²/μ the argument of the tanh -function

$$\begin{eqnarray} &&{w^2\over 2\beta _{\parallel }}\gg {\mu /2}=918\gg 1, \end{eqnarray} \tag{ 78 }$$

we approximate

$$\begin{eqnarray} &&\tanh \left[{w^2\over 2\beta _{\parallel }}\right]\simeq 1. \end{eqnarray} \tag{ 79 }$$

We then find for condition (77)

$$\begin{eqnarray} &&\mp \left(1-{1\over A_0}\right)>\mu x, \end{eqnarray} \tag{ 80 }$$

where the parallel plasma beta dependence and additional A₀-dependences enter via the solution x(A₀, β_∥) of the dispersion relation (32).

For b ≪ w and A_p = A_e = A₀ the dispersion relation (32) reads

$$\begin{eqnarray} &&0=\Re \Lambda _{\rm RH,LH}(x, S=0)\simeq {(1+\mu)w^2x^2\over (1\pm \mu x)(1\mp x)}-1+{\beta _{\parallel }\over 2}K_{\rm RH,LH}, \nonumber\\ \end{eqnarray} \tag{ 81 }$$

with

$${\selectfont\fontsize{9}{10}{\begin{eqnarray} &&K_{\rm RH, LH}={1\over (1\mp x)^3}+{1\over (1\pm \mu x)^3}-A_0\left[{1\over (1\mp x)^2}+{1\over (1\pm \mu x)^2}\right]. \nonumber\\ \end{eqnarray}}} \tag{ 82 }$$

For x ≪ μ⁻¹ we approximate K_RH,LH ≃ 2(1 − A₀) and

$$\begin{eqnarray} &&\Re \Lambda _{\rm RH,LH}(x, S=0)\simeq (1+\mu)w^2x^2-1-\beta _{\parallel }(A_0-1)=0, \nonumber\\ \end{eqnarray} \tag{ 83 }$$

yielding the solution (37) and

$$\begin{eqnarray} &&\hspace{-8pt} \mu x\simeq {\mu ^{1/2}\over w}\sqrt{1+(A_0-1)\beta _{\parallel }}={1\over h}\sqrt{1+(A_0-1)\beta _{\parallel }}, \end{eqnarray} \tag{ 84 }$$

in terms of the proton plasma frequency phase speed (75).

7.1. LH-polarized Alfvén Waves

Inserting Equation (84), we obtain for the instability condition of LH-polarized Alfvén waves (80)

$$\begin{eqnarray} &&1-{1\over A_0}>{1\over h}\sqrt{1+(A_0-1)\beta _{\parallel }}, \end{eqnarray} \tag{ 85 }$$

which can only be fulfilled for A₀>1. Condition (85) is equivalent to

$$\begin{eqnarray} &&\beta _{\parallel }<{1\over A_0-1}\left[h^2\left({A_0-1\over A_0}\right)^2-1\right], \end{eqnarray} \tag{ 86 }$$

which is shown as the upper dashed curve in Figure 2 calculated for the value w = 142, corresponding to h = 3.3, representing the Bale et al. (2009) measurements.

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For values of h>1 the limiting plasma beta value of condition (86) is zero for A₀ = h/(h − 1) = 1.43 for h = 3.3. Substituting y = A₀/(A₀ − 1)>1 yields for the right-hand side of Equation (86)

$$\begin{eqnarray} &&F_L(y)=\left(y-1\right)\left({h^2\over y^2}-1\right), \end{eqnarray} \tag{ 87 }$$

and for its first derivative

$$\begin{eqnarray} &&{dF_L\over dy}={2h^2\over y^3}-{h^2\over y^2}-1. \end{eqnarray} \tag{ 88 }$$

The function F_L(y) exhibits one extreme value at y_E = ht, where

$$\begin{eqnarray} &&t^3+t={2\over h}, \end{eqnarray} \tag{ 89 }$$

yielding

$$\begin{eqnarray} &&y_E=h^{2/3}\left(\left[\sqrt{1+{h^2\over 27}}+1\right]^{1/3}-\left[\sqrt{1+{h^2\over 27}}-1\right]^{1/3}\right). \nonumber\\ \end{eqnarray} \tag{ 90 }$$

For the case h = 3.3 shown in Figure 2 we obtain y_E = 1.61, corresponding to A₀ = 2.64, and the maximum plasma beta F_max = 1.95, which agree with the plot for LH-polarized Alfvén waves shown in Figure 2.

7.2. RH-polarized Alfvén Waves

Likewise, inserting Equation (84) we obtain for the instability condition of RH-polarized Alfvén waves (80)

$$\begin{eqnarray} &&{1\over A_0}-1>{1\over h}\sqrt{1-(1-A_0)\beta _{\parallel }}, \end{eqnarray} \tag{ 91 }$$

which can only be fulfilled for A₀ < 1 and requires the existence condition (38) A₀>1 − β⁻¹_∥. Condition (91) is equivalent to

$$\begin{eqnarray} &&\beta _{\parallel }>{1\over 1-A_0}\left[1-h^2\left({1-A_0\over A_0}\right)^2\right], \end{eqnarray} \tag{ 92 }$$

which is shown as the lower full curve in Figure 2 calculated for the value h = 3.3, representing the Bale et al. (2009) measurements.

The limiting plasma beta value of condition (92) is zero for A₀ = h/(h + 1) = 0.77 for h = 3.3. Substituting y = A₀/(1 − A₀)>0 yields for the right-hand side of Equation (92)

$$\begin{eqnarray} &&F_R(y)=\left(y+1\right)\left(1-{h^2\over y^2}\right). \end{eqnarray} \tag{ 93 }$$

The first derivative

$$\begin{eqnarray} &&{dF_R\over dy}={2h^2\over y^3}+{h^2\over y^2}+1 \end{eqnarray} \tag{ 94 }$$

is positive for all values of y, so that the function F_R(y) is monotonically increasing, in agreement with the plot for RH-polarized Alfvén waves shown in Figure 2.

The properties of the weakly amplified LH- and RH-polarized Alfvén mode are summarized in Tables 1 and 2, respectively.

Table 1. Properties of Weakly Amplified LH-polarized Alfvén Wave Mode

Real phase speed range	R ≪ b/μ = wΘ/(μβ1/2∥)
Parallel plasma beta range	Θ2 ≪ β∥ ≪ h2 = w2/μ
Dispersion relation	$x=R/b={1\over \mu ^{1/2}w}\sqrt{1+(A_0-1)\beta _{\parallel }}$
Existence condition	w ≫ μ1/2 = 43 corresponding to h ≫ 1
Instability conditions	A0>1 and $\beta _{\parallel }<{1\over A_0-1}\left[h^2\left({A_0-1\over A_0}\right)^2-1\right]$

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Table 2. Properties of Weakly Amplified RH-polarized Alfvén Wave Mode

Real phase speed range	R ≪ b/μ = wΘ/(μβ1/2∥)
Parallel plasma beta range	Θ2 ≪ β∥ ≪ h2 = w2/μ
Dispersion relation	$x=R/b={1\over \mu ^{1/2}w}\sqrt{1+(A_0-1)\beta _{\parallel }}$
Existence condition	w ≫ μ1/2 = 43 corresponding to h ≫ 1
Instability conditions	A0 < 1 and $\beta _{\parallel }>{1\over 1-A_0}\left[1-h^2\left({1-A_0\over A_0}\right)^2\right]$

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We emphasize that the Alfvenic instability threshold diagram shown in Figure 2 is controlled by a single observed plasma parameter: the proton plasma frequency phase speed h = w/μ^1/2. The limiting anisotropy values for small plasma betas β_∥ → 0 as well as the maximum plasma beta value for LH-polarized Alfvén waves (see Equation (90)) are fixed by the value of h.

In the range of LH-polarized proton cyclotron waves x ⩽ μ⁻¹ the functions f₋(x) ≃ −1 are negative and practically constant, so that here the instability condition (80) also holds:

$$\begin{eqnarray} &&1-{1\over A_0}>\mu x, \end{eqnarray} \tag{ 95 }$$

where again the parallel plasma beta dependence and additional A₀-dependences enter via the solution x(A₀, β_∥) of the dispersion relation (81). Because x>0 the proton cyclotron instability condition (95) can only be fulfilled for A₀>1.

Substituting s = 1 − μx, the condition (95) becomes

$$\begin{eqnarray} &&A_0>{1\over s}. \end{eqnarray} \tag{ 96 }$$

We now need solutions of the LH dispersion relation (83) for values of s ≪ 1 but, due to the constraint (24), much larger than s ≫ β^1/2_∥/h, i.e.,

$$\begin{eqnarray} &&{\beta _{\parallel }^{1/2}\over h}\ll s\ll 1. \end{eqnarray} \tag{ 97 }$$

Because of the smallness of s the LH dispersion relation (83) can be approximated as

$$\begin{eqnarray} &&{h^2\over s}\left[1+{\beta _{\parallel }\over 2\,h^2\,s^2}\right]=1+{\beta _{\parallel }(A_0-1)\over 2}+{\beta _{\parallel }A_0\over 2\,s^2}. \end{eqnarray} \tag{ 98 }$$

Because of the left inequality in Equation (97), we neglect the second term in the first bracket in Equation (98) to obtain

$$\begin{eqnarray} &&{h^2\over s}\simeq 1+{\beta _{\parallel }(A_0-1)\over 2}+{\beta _{\parallel }A_0\over 2\,s^2} \end{eqnarray} \tag{ 99 }$$

with the solution

$$\begin{eqnarray} &&s={h^2\over 2+\beta _{\parallel }(A_0-1)}\left[1+\sqrt{1-{\beta _{\parallel }A_0[2+\beta _{\parallel }(A_0-1)]\over h^4}}\right], \nonumber\\ \end{eqnarray} \tag{ 100 }$$

which in the limit β_∥ → 0 correctly reproduces solution (48). This solution exists provided that

$$\begin{eqnarray} &&\beta _{\parallel }A_0[2+\beta _{\parallel }(A_0-1)]\le h^4. \end{eqnarray} \tag{ 101 }$$

For β_∥A₀[2 + β_∥(A₀ − 1)] ≪ h⁴, which corresponds to the first constraint

$$\begin{eqnarray} &&A_0\ll {1\over \beta _{\parallel }}\left[\sqrt{\left({\beta _{\parallel }-2\over 2}\right)^2+h^4}+{\beta _{\parallel }-2\over 2}\right], \end{eqnarray} \tag{ 102 }$$

the solution (101) simplifies to

$$\begin{eqnarray} &&s\simeq {2h^2\over 2+\beta _{\parallel }(A_0-1)}. \end{eqnarray} \tag{ 103 }$$

The restrictions (97) on s require two additional constraints

$$\begin{eqnarray} &&1-{2(1-h^2)\over \beta _{\parallel }}\ll A_0\ll 1+{2\over \beta _{\parallel }^{3/2}}\left[h^3-\beta _{\parallel }^{1/2}\right], \end{eqnarray} \tag{ 104 }$$

which requires β_∥ ≪ h⁶. Then using the solution (103) we obtain for the instability condition (96)

$$\begin{eqnarray} &&A_0>{2-\beta _{\parallel }\over 2\,h^2-\beta _{\parallel }}. \end{eqnarray} \tag{ 105 }$$

The requirement A₀>1 then implies small proton plasma frequency phase speeds h < 1, and consequently also small values of β_∥ < h⁶ ≪ 1. In this case, the left-hand side of the constraint (104) is negative and therefore automatically fulfilled. Its right-hand side yields

$$\begin{eqnarray} &&A_0\ll {2\,h^3\over \beta _{\parallel }^{3/2}}. \end{eqnarray} \tag{ 106 }$$

Moreover, Equations (102) and (105) simplify to

$$\begin{eqnarray} &&A_0\ll {1\over \beta _{\parallel }}\left[\sqrt{1+h^4}-1\right]\simeq {h^4\over 2\beta _{\parallel }} \end{eqnarray} \tag{ 107 }$$

and

$$\begin{eqnarray} &&A_0>{1\over 2\,h^2}\left[1+{\beta _{\parallel }\over 2\,h^2}(1-\,h^2)\right]. \end{eqnarray} \tag{ 108 }$$

At plasma betas β_∥ < h⁶ the constraint (106) is stronger than the constraint (107), so that we are left with

$$\begin{eqnarray} &&{1\over 2\,h^2}\left[1+{\beta _{\parallel }\over 2\,h^2}(1-h^2)\right]<A_0\ll {2\,h^3\over \beta _{\parallel }^{3/2}} \end{eqnarray} \tag{ 109 }$$

as instability conditions for proton cyclotron waves subject to h < 1 and β_∥ < h⁶ < 1. The final conditions (109) are shown in Figure 3 for a value of h = 2^−1/2. Because of the required small value h < 1 this instability cannot operate in the regime (76) observed by Bale et al. (2009).

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The properties of the weakly amplified proton cyclotron mode are summarized in Table 3.

Table 3. Properties of Weakly Amplified LH-polarized proton Cyclotron Mode

Real phase speed range	R ≪ b
Dispersion relation	$x=R/b={1\over \mu }[1-{2\,h^2\over 2+\beta _{\parallel }(A_0-1)}]$
Existence condition	h = w/μ1/2 < 1
Parallel plasma beta range	Θ2 ≪ β∥ ≪ h6
Instability conditions	${1\over 2\,h^2}\left[1+{\beta _{\parallel }\over 2\,h^2}(1-h^2)\right]<A_0\ll {2\,h^3\over \beta _{\parallel }^{3/2}}$

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In the range of electron cyclotron waves μ^−1/2 ≪ x < 1 the function f₊(x) ≃ μx² ≫ 1 is positive and much larger than unity. The instability condition (66) together with W₊ = w²f₊(x)/(2β_∥) = μw²x²/(2β_∥) then becomes

$$\begin{eqnarray} &&\left(1-{1\over A_0}\right)\left[(\mu ^{3/2} -1)+(\mu ^{3/2} +1)\tanh \left[{\mu w^2x^2\over 2\beta _{\parallel }}\right]\right]\cr &&\quad {>}\,x\mu \left[(\mu ^{1/2}+1)+(\mu ^{1/2}-1)\tanh \left[{\mu w^2x^2\over 2\beta _{\parallel }}\right]\right]. \end{eqnarray} \tag{ 110 }$$

Again the argument of the tanh -function is very large compared to unity so that

$$\begin{eqnarray} &&\tanh \left[{\mu w^2x^2\over 2\beta _{\parallel }}\right]\simeq \tanh \left[{w^2\over 2\beta _{\parallel }}\right]\simeq 1, \end{eqnarray} \tag{ 111 }$$

yielding for the instability condition (110)

$$\begin{eqnarray} &&A_0>{1\over 1-x}, \end{eqnarray} \tag{ 112 }$$

where again the parallel plasma beta dependence and additional A₀-dependences enter via the solution x(A₀, β_∥) of the dispersion relation (81). Because x < 1 the electron cyclotron instability condition (112) can only be fulfilled for A₀>1.

Substituting y = 1 − x we need solutions of the RH dispersion relation (83) for values of y ≪ 1 but, due to the constraint (25), much larger than y ≫ β^1/2_∥/w, i.e.,

$$\begin{eqnarray} &&{\beta _{\parallel }^{1/2}\over w}\ll y\ll 1. \end{eqnarray} \tag{ 113 }$$

Because of the smallness of y the RH dispersion relation (83) can be approximated as

$$\begin{eqnarray} &&{w^2\over y}\left[1+{\beta _{\parallel }\over 2w^2y^2}\right]=1+{\beta _{\parallel }A_0\over 2y^2}. \end{eqnarray} \tag{ 114 }$$

Because of the left inequality in Equation (113), we neglect the second term in the first bracket in Equation (114) to obtain

$$\begin{eqnarray} &&{w^2\over y}\simeq 1+{\beta _{\parallel }A_0\over 2y^2} \end{eqnarray} \tag{ 115 }$$

with the solution

$$\begin{eqnarray} &&y={w^2\over 2}\left[1+\sqrt{1-{2\beta _{\parallel }A_0\over w^4}}\right], \end{eqnarray} \tag{ 116 }$$

which in the limit β_∥ → 0 correctly reproduces solution (49). This solution exists provided that

$$\begin{eqnarray} &&2\beta _{\parallel }A_0\le w^4. \end{eqnarray} \tag{ 117 }$$

For

$$\begin{eqnarray} &&A_0\ll {w^4\over 2\beta _{\parallel }} \end{eqnarray} \tag{ 118 }$$

the solution (116) simplifies to

$$\begin{eqnarray} &&y\simeq w^2\left[1-{\beta _{\parallel }A_0\over 2w^2}\right]. \end{eqnarray} \tag{ 119 }$$

The restrictions (113) on s require w² ≪ 1 and β_∥ ≪ w⁸. Then using the solution (117) we obtain for the instability condition (112)

$$\begin{eqnarray} &&A_0w^2\left[1-{\beta _{\parallel }A_0\over 2w^2}\right]>1, \end{eqnarray} \tag{ 120 }$$

or with β_∥ ≪ w⁶

$$\begin{eqnarray} &&A_0>{w^4\over \beta _{\parallel }}\left[1-\sqrt{1-{2\beta _{\parallel }\over w^6}}\right]\simeq {1\over w^2}\left[1+{\beta _{\parallel }\over 2w^6}\right]. \end{eqnarray} \tag{ 121 }$$

The dual instability requirements subject to w ≪ 1 and β_∥ ≪ w⁶ ≪ 1

$$\begin{eqnarray} &&{1\over w^2}\left[1+{\beta _{\parallel }\over 2w^6}\right]<A_0\ll {w^4\over 2\beta _{\parallel }} \end{eqnarray} \tag{ 122 }$$

are shown in Figure 4 for a value of w = 0.1. Because of the required small value w ≪ 1 this instability cannot operate in the regime (72) observed by Bale et al. (2009).

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The properties of the weakly amplified RH-polarized electron cyclotron modes are summarized in Table 4.

Table 4. Properties of Weakly Amplified RH-polarized Electron Cyclotron Mode

Real phase speed range	bμ−1/2 ≪ R ≪ b
Dispersion relation	$x=R/b=1-w^2\left[1-{\beta _{\parallel }A_0\over 2w^2}\right]$
Existence condition	w ≪ 1
Parallel plasma beta range	Θ2 ≪ β∥ ≪ w6
Instability condition	${1\over w^2}\left[1+{\beta _{\parallel }\over 2w^6}\right]<A_0\ll {w^4\over 2\beta _{\parallel }}$

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We rigorously studied the dispersion relations of weakly amplified fluctuations with wavevectors $\vec{k}\times \vec{B}_0=0$ in an anisotropic bi-Maxwellian magnetized proton–electron plasma by the appropriate Taylor expansion of the plasma dispersion function. Apparently for the first time, for equal parallel electron and proton temperatures, a general analytical instability condition (65) is derived that holds for different values of the electron (A_e) and proton (A_p) temperature anisotropies. We determine the conditions for which the weakly amplified LH-polarized Alfvén proton cyclotron and RH-polarized Alfvén Whistler electron cyclotron branches can be excited. For different regimes of the electron plasma frequency phase speed w = ω_p,e/(kc) these branches reduce to the RH- and LH-polarized Alfvén waves, RH-polarized proton, and LH-polarized electron cyclotron modes. The properties of the individual modes are summarized in Tables 1–4. Analytic instability threshold conditions are derived in terms of the combined temperature anisotropy A = T_⊥/T_∥, the parallel plasma beta β_∥ = 8 π n_ek_BT_∥/B² and the electron plasma frequency phase speed w = ω_p,e/(kc) for each mode.

For large values of w ≫ 21.4, corresponding to small wavenumbers k ≪ ω_p,e/21.4 c, RH- and LH-polarized Alfvén waves are excited in different regions of the temperature anisotropy (A) versus parallel plasma beta (β_∥) parameter plane. For appropriate temperature anisotropies, RH-polarized low and high phase speed Whistler waves are excited for intermediate values of 6.5 ≪ w ≪ 21.4 and 1 ≪ w ≪ 6.5, respectively. At values w ≪ 21.4 LH-polarized proton cyclotron waves can be excited, whereas RH-polarized electron cyclotron waves can be excited at small values of w ⩽ 1. In agreement with Brinca's (1990) general theorem on the electromagnetic stability of isotropic plasma populations none of these modes can be excited for isotropic plasma distributions (A_e = A_p = 1).

We apply the results of our instability study to the observed solar wind magnetic turbulence (Bale et al. 2009), corresponding to values of 90 ⩽ w ⩽ 330. According to the existence conditions of the different instabilities, only the LH and RH-polarized Alfvén wave instabilities can operate here. The Alfvénic instability threshold conditions are controlled by a single observed plasma parameter: the proton plasma frequency phase speed h = w/μ^1/2. All asymptotic and characteristic values of the threshold anisotropies and the characteristic plasma beta values are solely determined by the value of h. The Alfvénic instability diagram explains well the observed confinement limits at small parallel plasma beta values in the solar wind. It remains to be investigated in future work how weakly propagating and/or obliquely propagating weakly amplified modes add to this diagram.

This work was supported by the Deutsche Forschungsgemeinschaft through grants Schl 201/19-1 and Schl 201/21-1.