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The Astrophysical Journal, 742:41 (11pp), 2011 November 20 C 2011.

doi:10.1088/0004-637X/742/1/41

The American Astronomical Society. All rights reserved. Printed in the U.S.A.

EFFECT OF DIFFERENTIAL FLOW OF ALPHA PARTICLES ON PROTON PRESSURE ANISOTROPY INSTABILITIES IN THE SOLAR WIND John J. Podesta and S. Peter Gary Los Alamos National Laboratory, Los Alamos, NM 87545, USA; [email protected] Received 2011 June 1; accepted 2011 September 14; published 2011 November 3

ABSTRACT In the solar wind, when the effects of proton–proton Coulomb collisions are negligible, alpha particles usually flow faster than the protons in such a way that the differential alpha–proton flow velocity Vd = Vα − Vp is on the order of the Alfvén speed, is directed away from the Sun, and is nearly aligned with the local mean magnetic field. When this differential flow is taken into account, solutions of the hot plasma dispersion relation show that for the parallel propagating electromagnetic ion cyclotron (EMIC) instability driven by the proton temperature anisotropy T⊥p > Tp , the maximum growth rate occurs in the +V d direction and for the parallel firehose instability driven by the opposite proton temperature anisotropy Tp > T⊥p , the maximum growth rate occurs in the −V d direction. Thus, the EMIC instability preferentially generates left circularly polarized Alfvén-ion-cyclotron waves propagating away from the Sun and the parallel firehose instability preferentially generates right circularly polarized magnetosonic-whistler waves propagating toward the Sun with the maximum growth rates occurring for frequencies on the order of the proton cyclotron frequency and wavenumbers on the order of the proton inertial length. Because of the Doppler shift caused by the motion of the solar wind, both types of waves are left circularly polarized in the spacecraft frame for observations taken when the local mean magnetic field is collinear with the solar wind flow velocity. Theoretical investigation of these instabilities also shows that regions of parameter space exist where the unstable waves are generated propagating unidirectionally such as, for the EMIC instability for example, when the temperature anisotropy is small |(T⊥p /Tp ) − 1| < 1. Taken together, the above properties can explain the origin of parallel propagating electromagnetic waves recently observed near the proton inertial length in high-speed solar wind. The observed waves are most likely produced in situ by these instabilities. A remarkable property of the proposed mechanism that may be of practical importance is that the magnetic helicity of the unstable waves has the same sign no matter whether the proton temperature anisotropy (Tp⊥ /Tp ) − 1 is positive or negative. Key words: instabilities – interplanetary medium – solar wind – turbulence – waves Online-only material: color figures are left- or right-hand polarized in the plasma frame which is necessary to uniquely identify the waves. This is because the frequencies and phase speeds of parallel propagating ion cyclotron waves and magnetosonic-whistler waves are quantitatively similar to each other in the solar wind near 1 AU where, as will be shown herein for both types of waves, ω ∼ Ωp , k c/ωpp ∼ 1 and vph ∼ vA . In this study, we shall simply acknowledge that one or the other type of wave is present in the data and try to explain how such waves may be generated in situ. The observations show that the waves usually propagate within 5◦ or 10◦ of the direction of the local mean magnetic field and that σm ≈ −1 in magnetic sectors directed away from the Sun and σm ≈ +1 in magnetic sectors directed toward the Sun. This not only indicates that the waves are left-hand polarized in the spacecraft frame, but because |σm | ≈ 1 the waves must be approximately circularly polarized and must propagate nearly unidirectionally. If for a given type of wave, say EMIC waves, comparable power exists in waves propagating both parallel and anti-parallel to the local mean magnetic field, then observations would show |σm | ≈ 0. Thus, the mechanism that generates the observed waves must have the following two properties: (1) waves with a particular state of circular polarization in the plasma frame must propagate nearly unidirectionally and (2) if they are EMIC waves, then they must propagate away from the Sun in the plasma frame and, if they are magnetosonicwhistler waves, then they must propagate toward the Sun in the plasma frame. The purpose of this study is to show that waves with these properties are naturally generated in situ by proton temperature anisotropies when one takes into account

1. INTRODUCTION The first solar wind measurements of the reduced magnetichelicity spectrum σm (k, θ ) as a function of the angle θ between the local mean magnetic field and the flow direction of the solar wind have recently been performed by He et al. (2011) and Podesta & Gary (2011). These measurements reveal the existence of two distinct populations of electromagnetic fluctuations at scales on the order of the proton thermal gyroradius k⊥ ρp = 1 and the proton inertial length k c/ωpp = 1; both of these scales are equal when β⊥p = 1 which is typical of solar wind plasma near 1 AU (all the symbols used in this paper are defined at the end of the Introduction). One population of fluctuations is seen at oblique angles, centered about an angle of 90◦ , and is believed to consist of obliquely propagating waves, perhaps kinetic Alfvén waves (KAWs). A separate population of fluctuations seen at angles near 0◦ is believed to consist of electromagnetic ion cyclotron (EMIC) waves propagating away from the Sun or magnetosonic-whistler waves (also known as whistler waves) propagating toward the Sun along the local mean magnetic field, or both. In this study, we focus exclusively on the parallel propagating waves and show that they may be generated in situ through the EMIC instability or the parallel firehose instability driven by proton temperature anisotropies in the solar wind plasma. The parallel propagating waves observed by He et al. (2011) and Podesta & Gary (2011) exhibit a left-hand sense of polarization in the spacecraft frame. Unfortunately, by using magnetic field data alone it is difficult to determine whether these waves 1

The Astrophysical Journal, 742:41 (11pp), 2011 November 20

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the differential streaming of alpha particles relative to the protons. In general, the parallel firehose instability may become active when Tp > T⊥p and, for parallel propagation, the EMIC instability may become active when T⊥p > Tp . The parallel firehose instability generates right circularly polarized (RCP) magnetosonic-whistler waves and the EMIC instability generates left circularly polarized (LCP) Alfvén-ion-cyclotron waves. In the solar wind, the maximum growth rates for both of these instabilities occur at wavenumbers on the order of the proton inertial length k c/ωpp ∼ 1 and at frequencies on the order of the proton cyclotron frequency ω/Ωp ∼ 1. Another electromagnetic instability that is believed to be active in the solar wind when T⊥p > Tp and βp 1 is the mirror instability; see, for example, Southwood & Kivelson (1993), Pokhotelov et al. (2004), and references therein. This is a low frequency mode, ω Ωp , with a maximum growth rate at k⊥ k . For typical solar wind conditions at 1 AU, the growth rate of the mirror instability is roughly of the same order of magnitude as that of the EMIC instability. However, the normalized magnetic helicity σm is close to zero for the mirror mode (Gary 1992, Figure 7). Therefore, the mirror mode cannot explain the signals with |σm | 1 observed near parallel propagation by He et al. (2011) and Podesta & Gary (2011). Under certain circumstances, a differential flow of alpha particles with respect to the protons can significantly change the growth rates of both the parallel firehose and EMIC instabilities. In collisionally young solar wind near 1 AU, the alpha–proton drift velocity V d = V α − V p points away from the Sun, is parallel to the magnetic field, and has a magnitude on the order of the local Alfvén speed vA (Marsch et al. 1982; Kasper et al. 2008). We show here that the alpha–proton drift V d causes the EMIC instability to preferentially generate left circularly polarized Alfvén-ion-cyclotron waves propagating in the +V d direction (away from the Sun) and it causes the parallel firehose instability to preferentially generate RCP magnetosonic-whistler waves propagating in the −V d direction (toward the Sun). The different growth rates for propagation parallel and anti-parallel to V d have been investigated for the firehose instability by Hellinger & Trávn´ıcˇ ek (2006) but, to our knowledge, it has not been studied for the parallel EMIC instability in any detail. The asymmetric growth rates of these two instabilities are believed to be the key to explaining the origin of parallel propagating waves observed in the solar wind near the proton inertial scale k c/ωpp = 1 by He et al. (2011) and Podesta & Gary (2011). The parallel fluctuations observed by He et al. (2011) and Podesta & Gary (2011) exhibit similar properties to the Alfvénion-cyclotron and/or magnetosonic-whistler waves first detected in the solar wind by Behannon (1976) and further investigated by Tsurutani et al. (1994) and Jian et al. (2009, 2010). The occurrence of these waves often coincides with an enhancement in the power of magnetic field fluctuations near k c/ωpp = 1 that may be weak or strong. Such an enhancement has also been noted in wavelet spectra taken when the local mean magnetic field is parallel to the solar wind velocity (Podesta 2009; Wicks et al. 2010) and similar observations obtained using Fourier transform techniques have been discussed previously in the solar wind literature. The parallel propagating waves observed by He et al. (2011) and Podesta & Gary (2011) may or may not be related to the enhanced plasma density fluctuations at the proton Larmor radius scale investigated by Neugebauer (1975, 1976) and later attributed to the Tp > T⊥p firehose instability

by Neugebauer et al. (1978). It has been suggested by Hollweg (1999) that the enhanced density fluctuations at the Larmor radius scale may instead be caused by KAWs with k⊥ k (see also Chandran et al. 2009) which is supported by the work of Leamon et al. (1998, 1999), Bale et al. (2005), and others. Thermal pressure anisotropy instabilities have long been believed to regulate ion temperature anisotropies in the solar wind and to prevent large pressure anisotropies from developing. Even before the existence of the solar wind was confirmed by spacecraft measurements in the early 1960s, Parker (1958, 1963) had indicated the potential importance of these instabilities for the regulation of pressure anisotropies in the solar wind. Many details of these kinetic processes have since been investigated within the framework of the collisionless Vlasov–Maxwell kinetic theory. The linear Vlasov–Maxwell theory of temperature anisotropy instabilities is of fundamental importance in both laboratory and space plasmas and has been studied, for example, by Weibel (1959), Harris (1961), Sagdeev & Shafranov (1961), Stix (1962), Kennel & Petschek (1966), Scharer & Trivelpiece (1967), Kennel & Scarf (1968), Cuperman & Landau (1969), Scharer (1969), Hollweg & Völk (1970), Eviatar & Schulz (1970), Landau & Cuperman (1973), Davidson & Ogden (1975), Gary et al. (1976), Abraham-Shrauner & Feldman (1977), Neugebauer et al. (1978), Hall (1979), Otani (1988), Thorne & Summers (1991a), Thorne & Summers (1991b), Gary (1992), Gary et al. (1993), Xue et al. (1993), McKean et al. (1993), McKean et al. (1994), Gary & Lee (1994), Xue et al. (1996b), Xue et al. (1996c), Xue et al. (1996a), Gary et al. (1997), Gary et al. (1998), Mace (1998), Gary et al. (2000), Gomberoff & Valdivia (2003), Dasso et al. (2003), Hellinger & Trávn´ıcˇ ek (2005), Hellinger & Trávn´ıcˇ ek (2006), Matteini et al. (2006), Hellinger & Trávn´ıcˇ ek (2008), Lazar & Poedts (2009), and Lazar et al. (2011). This list is far from exhaustive. Theoretical and experimental studies with special emphasis on solar wind observations and comparisons between theory and observation include the work of Leubner (1979), Dum et al. (1980), Leubner & Viñas (1986), Gary et al. (2001), Gary et al. (2002), Kasper et al. (2002), Kasper et al. (2003), Gary et al. (2003), Marsch et al. (2004), Gary et al. (2005), Gary et al. (2006), Marsch et al. (2006), Hellinger et al. (2006), Matteini et al. (2007), and Bale et al. (2009). Review papers that touch upon various aspects of the theory and its relation to solar wind observations were provided by Scarf (1970), Hollweg (1975), Dobrowolny & Moreno (1977), Feldman (1979), Schwartz (1980), Cuperman (1981), Marsch (1991), Gary (1993), Marsch (2006), Marsch (2010), Pierrard & Lazar (2010), Araneda et al. (2011), and Matteini et al. (2011). Studies of electromagnetic proton beam driven instabilities in the solar wind have also been performed, including instabilities that occur when both a proton beam and a proton temperature anisotropy are present. These studies have been reviewed by Gary (1991); see also Marsch & Livi (1987), Daughton & Gary (1998), Daughton et al. (1999), Li & Habbal (2000), and references therein. Electrostatic instabilities form another class of kinetic processes in the solar wind, but such instabilities arise only under the atypical solar wind condition Te Tp (see, for example, Gary 1993, Chapter 3) and are not considered here. In relation to the work presented here, it is of interest to note that a minor population of alpha particles can have significant effects on the behavior of proton pressure anisotropy instabilities, even without differential streaming of the alpha particles, as shown by Scarf & Fredricks (1968), Gomberoff & 2


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Cuperman (1977), and others. Recently, Matteini et al. (2011) have reanalyzed Figure 1 in Hellinger et al. (2006) in which the maximum growth rates of various plasma instabilities as functions of T⊥p /Tp and βp are overplotted on histograms showing, for slow solar wind ( 0, Boltzmann’s constant is kB , and the speed of light in vacuum is c = 1/(ε0 μ0 )1/2 , where ε0 is the permittivity of free space and μ0 is the permeability of free space (SI units). The ambient magnetic field is B 0 = B0 eˆ z , where eˆ z is the unit vector in the z-direction and B0 > 0. The proton Alfvén speed is vA = B0 /(μ0 np mp )1/2 . The cyclotron frequency of particle species s in radians per second is Ωs = qs B0 /ms (s = p, α, e), where qs and ms are the charge and mass of species s and Ωs retains the algebraic sign of the charge so that Ωp > 0, Ωα > 0, and Ωe < 0. The plasma frequency of species s in radians per second is ωps = (ns qs2 /ε0 ms )1/2 . The average flow velocity of particle species s is V s = V s eˆ z , where the algebraic sign of Vs can be positive or negative; and V d ≡ V α − V p is the relative flow velocity of the alpha particles with respect to the protons, also called the drift velocity. For the linear theory considered here, the electromagnetic field perturbations are plane waves of the form Re{E 0 exp[i(k · x − ωt)]}, where E 0 is the complex amplitude, the wavevector k = (kx , ky , kz ) is real valued, k = kz is the component of the wavevector in the direction of B 0 , the perpendicular wavenumber is k⊥ = (kx2 + ky2 )1/2 , the complex frequency in radians per second is ω = ωr + iγ , and γ is the growth rate. By convention ωr 0. Parallel propagating waves are left or right circularly polarized when the electric field vector at the origin rotates in the same direction as the proton or electron gyromotion, respectively. The equilibrium particle distribution functions are assumed to be bi-Maxwell distributions, Equation (11), and the temperatures of species s parallel and perpendicular to B 0 are Ts and T⊥s , respectively. The total temperature of species s is Ts = (2T⊥s + Ts )/3. The particle thermal speeds are defined by vs = (2kB Ts /ms )1/2 and v⊥s = (2kB T⊥s /ms )1/2 . The plasma beta of species s is defined as β⊥s = ns kB T⊥s /(B02 /2μ0 ), βs = ns kB Ts /(B02 /2μ0 ) and the total beta is βs = (2β⊥s + βs )/3.

significantly more complex and instabilities can be sensitive to small changes in the functional form of the distribution functions. Nevertheless, the instability analysis performed here provides physical insight that is useful for the interpretation of solar wind measurements. For completeness, the characteristics of proton distribution functions in the solar wind are briefly summarized in this section. More complete details can be found in the works by Marsch (1991), Feldman & Marsch (1997), Kasper et al. (2003), Marsch (2006), Kasper et al. (2008), and Marsch (2010). This section may be skipped without loss of continuity. At heliocentric distances beyond a few solar radii, Coulomb collisions can sometimes be neglected. However, the Coulomb collision time can vary by several orders of magnitude and the passage of just a few collision times during the transit of the solar wind from the corona to 1 AU can significantly thermalize the proton distribution functions (Kasper et al. 2008). The average number of proton–proton Coulomb collisions between the corona and 1 AU is known as the collisional age of the plasma. The protons are said to be collisionally young when their collisional age is much less than 1 and collisionally old when their collisional age is much greater than 1. Near 1 AU, low-speed wind (less than 400 or 500 km s−1 ) is usually collisionally old and the firehose condition Tp > Tp⊥ is found to occur more often than the reverse inequality Tp < Tp⊥ . On the other hand, high-speed streams are usually collisionally young and the thermal core of the proton distribution often satisfies Tp⊥ > Tp . Nevertheless, it is the entire distribution function that must be considered for the evaluation of plasma instabilities and the presence of a secondary proton beam, which is often observed in both low- and high-speed streams, can yield a different ratio Tp⊥ /Tp for the distribution function as a whole. Dum et al. (1980), Leubner & Viñas (1986), and Marsch (1991) have found that the condition for the EMIC instability is often satisfied for the whole proton distribution function in intermediate- and high-speed streams, independently of the presence of a proton beam. In collisionally young solar wind at 1 AU, the alpha particles are usually hotter than the protons by a factor of Tα /Tp 4, whereas in collisionally old parcels of solar wind plasma Tα /Tp 1 (Kasper et al. 2008). The thermal core of the alpha particle distribution functions is generally more isotropic than the protons, with Tα /Tα⊥ ∼ 1.4, on average, at 1 AU (Marsch 2010). The feature of primary interest for our study is the differential streaming of the average alpha particle velocity relative to the average proton velocity (Neugebauer 1981; Marsch et al. 1981, 1982; Neugebauer et al. 1994, 1996; Steinberg et al. 1996; Kasper et al. 2008). The alpha–proton drift velocity V d = V α − V p is observed to be directed outward, away from the Sun, and parallel to the magnetic field B. Typically, Vd 0.6vA in the collisionally young solar wind near 1 AU and Vd 0 in collisionally old solar wind (Marsch et al. 1982; Kasper et al. 2008). 3. DISPERSION RELATIONS Plasma instabilities are investigated using the linear Vlasov–Maxwell kinetic theory (Krall & Trivelpiece 1973; Davidson 1983). Consider a homogeneous fully ionized plasma consisting of electrons and two ion species, protons and alpha particles, permeated by a constant magnetic field B 0 = B0 eˆ z , B0 > 0. The plasma is charge neutral meaning that

2. PROTON DISTRIBUTION FUNCTIONS IN THE SOLAR WIND In this study, bi-Maxwell particle distribution functions are employed to simplify the mathematical analysis, even though it is well known that distribution functions in the solar wind are

np + 2nα = ne ; 3

(1)


Podesta & Gary

the symbols employed here are all defined in the Introduction. The plasma frame of reference is defined such that the total momentum vanishes, np mp Vp + nα mα Vα + ne me Ve = 0,

(Gary 1993, Equation (5.1.14)) where Im(ω) > 0 is assumed to ensure causality and as a shortcut to performing the Laplace transform in time. In Equation (10), the dispersion relation for the right-hand polarized modes is obtained by choosing the upper sign “ + ” in “±” and that for the left-hand polarized modes is obtained by choosing the lower sign “−.” Attention shall be restricted to bi-Maxwell distribution functions of the form 2 ms ms ms v⊥ exp − f0s (v⊥ , v ) = 2π kB T⊥s 2kB T⊥s 2π kB Ts 2 ms (v − Vs ) . (11) × exp − 2kB Ts

(2)

and the net current is zero, np Vp + 2nα Vα − ne Ve = 0,

(3)

where Vp , Vα , and Ve are the average flow velocities of the respective species along the direction of the magnetic field. Introducing the relative drift velocity of the alpha particles with respect to the protons, Vd = Vα −Vp , the previous two equations can be used to express the flow velocities of the different species in terms of the one variable Vd as follows: 1 Vd , Vα = (4) 1+x x Vp = − Vd , (5) 1+x 2mp 1 nα Vd , (6) Ve = − ne mp + me 1+x where nα mα + 2me . (7) x= np mp + me This completes the specification of the equilibrium state. Small amplitude perturbations propagating parallel to B 0 can be decomposed into three linearly independent components: electrostatic modes in which the wave magnetic field vanishes and the electric field E is parallel to B 0 , and two transverse electromagnetic modes in which E is perpendicular to B 0 and either right-hand circularly polarized (rotating in the direction of the electron cyclotron motion) or left-hand circularly polarized (rotating in the direction of the proton cyclotron motion). To analyze these perturbations, the distribution functions are written as the sum of an equilibrium part plus a small perturbation, fs = f0s + f1s , and likewise for the electric and magnetic fields. The linearized Vlasov–Maxwell equations are then solved using Fourier and Laplace transform techniques (Krall & Trivelpiece 1973; Davidson 1983). The resulting dispersion relation for the electrostatic modes is 2 ∞ ωps ∂F0s 1 1− dv = 0, Im(ω) > 0, 2 k −∞ v − (ω/k ) ∂v s

The reduced distribution function in the parallel direction (9) is then ms ms (v − Vs )2 F0s (v ) = . (12) exp − 2π kB Ts 2kB Ts For purposes of numerical calculations it is necessary to express the dispersion relation (10) in terms of the plasma dispersion function Z(ξ ). To do this, consider the two terms in square brackets on the right-hand side of Equation (10). The first of these contains the factor ∂f0s /∂v⊥ and the second contains ∂f0s /∂v . With the substitution of the distribution function (11) into (10), the integral over the second of these two terms is performed using the identity ∞ kB T⊥s 3 π v⊥ f0s dv⊥ = F0s (13) ms 0 to obtain 2 ωps s

ω2

∞

−∞

k kB T⊥s ∂F0s . ω − k v ± Ωs ms ∂v

dv

(14)

Carrying out the derivative, this can be written as −

2 ωps T⊥s ∞ k (v − Vs ) dv F0s 2 ω Ts −∞ ω − k v ± Ωs s

(15)

or, equivalently, 2 ωps ω − k Vs ± Ωs T⊥s ∞ − dv − 1 F0s . ω2 Ts −∞ ω − k v ± Ωs s

(8) where the sum is over all plasma species, k = k eˆ z is the wavevector, k is real, and F0s is the reduced distribution function ∞ F0s (v ) = 2π f0s (v⊥ , v ) v⊥ dv⊥ . (9)

(16)

The first term in square brackets on the right-hand side of Equation (10) is simplified by means of the relation ∞ 2 ∂f0s π v⊥ dv⊥ = −F0s (17) ∂v⊥ 0

0

The dispersion relation (8) can be used to analyze streaming instabilities in plasmas containing protons and alphas. The focus of the present study will be on the dispersion relation for the two electromagnetic modes given by ∞ 2 ∞ k c 2 ωps ω 2 1− + π 2 v⊥ dv⊥ dv ω ω ω − k v ± Ωs 0 −∞ s k v ∂f0s k v⊥ ∂f0s × 1− + = 0, (10) ω ∂v⊥ ω ∂v

and, therefore, that term may be written as 2 ωps

∞

ω − k v F0s ω − k v ± Ωs

(18)

ω − k v ±Ωs =1− , ω − k v ± Ωs ω − k v ± Ωs

(19)

−

s

ω2

−∞

dv

or, using the identity

4


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correct symmetry. In particular, if Vs = 0 for all values of s, then the solution of the dispersion relation yields the same frequency ω = ωr +iγ for both +k and −k since Z0 (−ξs , −k ) = −Z0 (ξs , k ). This important symmetry property is not satisfied by the dispersion relations (34)–(43) in Brambilla (1998). For real valued k , the solutions of Equation (27) also obey the symmetry property ω(k , Vs ) = ω(−k , −Vs ). This is a simple consequence of the fact that under the substitutions k → −k and Vs → −Vs one also has ξs → −ξs . Hence, if the propagation direction and the direction of the drift velocities are all reversed, then the frequency ωr + iγ is unchanged.

it takes the form −

2 ωps s

ω2

1−

∞ −∞

dv

±Ωs ω − k v ± Ωs

F0s .

(20)

The combination of the two terms (16) and (20) is equal to the sum over s in Equation (10) and, consequently, the dispersion relation (10) now becomes 2 k c 2 ωps T⊥s 1− + − 1 ω ω2 Ts s T⊥s T⊥s (ω − k Vs ) ± Ωs − 1 Gs = 0, (21) − Ts Ts where

Gs =

∞

−∞

F0s (v ) dv , ω − k v ± Ωs

Im(ω) > 0.

4. RESULTS The dispersion relation (27) is solved by first changing to a set of dimensionless variables in which the normalized frequency, wavenumber, and velocity are defined as ω = ω/Ωp , k = kc/ωpp , and v = v/vA . For convenience, the primes are omitted from now on. Next, Equation (27) is multiplied through by (ω/k c)2 and f (z) is defined to be the resulting function on the left-hand side, where z = ω/k is the normalized phase speed. Hence, for a given real value of k , the dispersion relation is f (z) = 0. By plotting iso-contours of the quantity 1/|f (z)| in the complex z-plane, the approximate locations of the roots can be found by visual inspection. Starting with an approximate value of the root, Newton’s method is then used to more accurately compute the root to nine significant figures. Only roots with ωr > 0 are of interest. The entire procedure requires fast and accurate evaluation of the plasma dispersion function Z(ξ ) throughout the complex plane. This is achieved using the efficient algorithm developed by Gautschi (1970); see also Poppe & Wijers (1990). All the calculations discussed in this section assume a proton Alfvén speed vA = 60 km s−1 or, equivalently, c/vA = 5000. Throughout this section, the wavevector k is strictly parallel to B 0 and, for simplicity, the electrons and alpha particles are both assumed to have isotropic distributions: T⊥e /Te = 1 and T⊥α /Tα = 1.

(22)

With the substitution of the distribution function (12), a change of variables may be used to show that Gs = −Z0 (ξs± )/k vs , where Z(ξ ), k > 0, (23) Z0 (ξ, k ) = k < 0, Z(ξ ) − 2iπ 1/2 exp(−ξ 2 ), Z(ξ ) is the standard plasma dispersion function (defined below), ξs± =

ω − k Vs ± Ωs , k vs

(24)

and the thermal speed is vs = (2kB Ts /ms )1/2 . The plasma dispersion function is defined by the integral Z(ξ ) =

1 π 1/2

∞ −∞

e−x dx x−ξ 2

(25)

4.1. EMIC Instability

for Im(ξ ) > 0 and by analytic continuation for Im(ξ ) < 0. By virtue of the identity Z(ξ ) + Z(−ξ ) = 2iπ 1/2 exp(−ξ 2 ), it follows that +Z(ξ ), k > 0, Z0 (ξ, k ) = (26) −Z(−ξ ), k < 0.

When the proton temperature is anisotropic with Tp⊥ > Tp , the plasma may be unstable to the generation of left circularly polarized Alfvén-ion-cyclotron waves. For typical solar wind parameters at 1 AU, the maximum growth rate occurs at wavenumbers kc/ωpp ∼ 1 and frequencies ω/Ωp ∼ 1 as shown below. To investigate the effects of a differential flow of alpha particles with respect to the protons the dispersion relation (27) is solved using plasma parameters representative of high-speed wind near 1 AU: nα /np = 0.04, βp = 1, βe = 0.5, T⊥p /Tp = 2.5 (representing the thermal core of the distribution), Tα = 4Tp , c/vA = 5000, and V d = 0.6v A , where V d points in the direction of B 0 . Solutions for the real part of the frequency and the growth rate shown in Figure 1 indicate that the waves are unstable in the wavenumber band 0.4 k c/ωpp 0.9 and that the maximum growth rate is larger for waves propagating in the direction of V d , that is, for k · V d > 0, than for waves propagating in the opposite direction k · V d < 0. In Figure 1, the maximum growth rate for waves with k · V d > 0 is γm /Ωp 5.65 × 10−2 and the maximum growth rate for waves propagating in the opposite direction k · V d < 0 is γm /Ωp 3.19 × 10−2 . The ratio of the maximum growth rate for waves propagating parallel to V d to that for waves propagating anti-parallel to V d is approximately 1.77.

Hence, the electromagnetic dispersion relation (21) takes the final form 2 T⊥s k c 2 ωps 1− + − 1 ω ω2 Ts s

± T⊥s ± ±Ωs = 0. (27) ξ − + Z0 ξs Ts s k vs To obtain the dispersion relation for the right-hand polarized mode, the “ + ” sign must be chosen in place of “±” in both Equations (24) and (27). For the left-hand polarized mode, the “−” sign must be chosen in place of “±” in both Equations (24) and (27). The dispersion relation (27) is identical to Equations (11–2) and (11–3) in Stix (1992) but different from Equations (34)–(43) in Brambilla (1998) where the function Z0 (ξ ) is incorrectly replaced by the function Z(ξ ). The use of the function Z0 (ξ ) is necessary for the solutions to have the 5


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T /T

EMIC Instability

p⊥

0.6 0.4 β = 1, β = 1/2 p

0.2

e

Tα = 4Tp 0

p||

= 2.5,

V = 0.6 V d

A

0.06 Growth Rate γ / Ωp

Frequency ω / Ωp

0.8

0.2

0.4

0.6

0.8

d

0.02

k⋅Vd < 0

0 −0.02 −0.04

1

k⋅V > 0

0.04

0.2

0.4

|kc/ω |

0.6

0.8

1

|kc/ω |

pp

pp

Figure 1. Frequency and growth rate of left circularly polarized EMIC waves driven unstable by a proton temperature anisotropy with T⊥p /Tp = 2.5. The alpha–proton drift velocity is Vd = 0.6vA . The electrons and alpha particles both have isotropic distributions with βp = 1, βe = 1/2, and Tα = 4Tp . The maximum growth rates are approximately 0.057 for k · V d > 0 (blue curve) and 0.032 for k · V d < 0 (red curve). (A color version of this figure is available in the online journal.)

1.5

−3

x 10

βp = 1, βe = 1/2 Tα = 4Tp

1 0.5 RCP Mode

0

0.2

0.4

0.6

0.8

T /T p||

p⊥

= 2.5, V = 0.6 V d

10

Growth Rate γ / Ωp

Frequency ω / Ω

p

Parallel Firehose Instability

A

k⋅V < 0 d

5 0 k⋅V > 0

−5

d

−10

1

0.2

0.4

|kc/ωpp|

0.6

0.8

1

|kc/ωpp| −3

15

β = 1, β = 1/2 p

e

T =T α

p

1 0.5 RCP Mode

0

0.2

0.4

k⋅V < 0

p

1.5

Growth Rate γ / Ω

Frequency ω / Ω

p

x 10

0.6

0.8

d

10 5

k⋅V > 0 d

0 −5

1

0.2

|kc/ω |

0.4

0.6

0.8

1

|kc/ω |

pp

pp

Figure 2. Frequency and growth rate of right circularly polarized (RCP) magnetosonic-whistler waves driven unstable by a proton temperature anisotropy with Tp /T⊥p = 2.5, Vd = 0.6vA , βp = 1, and βe = 1/2. The electrons and alpha particles both have isotropic distributions. In the upper two plots Tα = 4Tp and in the lower two plots Tα = Tp . The results for k · V d > 0 are drawn in blue and the results for k · V d < 0 are drawn in red. (A color version of this figure is available in the online journal.)

Repeating the calculations in Figure 1 for Tα = Tp instead of Tα = 4Tp shows that the growth rates in Figure 1 change very little and, therefore, they are relatively insensitive to changes in the isotropic pressure of the alpha particles. This is not true, however, for the firehose instability as is shown in the next section. It should also be noted that the asymmetry in the growth rates in Figure 1 is independent of the direction of B 0 and depends solely on the direction of V d . This follows from the identity ω(−k , −Vs ) = ω(k , Vs ) demonstrated in Section 3. It is also of interest to note that the pressure anisotropy based on the total plasma pressure in Figure 1 is β⊥s P⊥ = s

1.65 (28) P s βs

which is less than the proton temperature ratio 2.5. Here, the ratio P⊥ /P has been evaluated means of the relations β⊥s =

3βs , 2 + Ts /T⊥s

(29)

βs =

3βs , 1 + 2T⊥s /Ts

(30)

nα Tα βp . np Tp

(31)

and βα = 6


Podesta & Gary

4.2. Firehose Instability Maximum Growth Rate γ / Ωp

When the proton temperature is anisotropic with Tp > Tp⊥ , the plasma can be unstable to the generation of RCP magnetosonic-whistler waves. For typical solar wind parameters at 1 AU and for wavenumbers k c/ωpp ∼ 1, the left-hand polarized Alfvén-ion-cyclotron wave is stable when Tp > Tp⊥ and, consequently, it will not be discussed further. Figure 2 shows results for the real part of the frequency ωr and the growth rate γ for the parameters nα /np = 0.04, βp = 1, βe = 0.5, T⊥p /Tp = 0.4, c/vA = 5000, and V d = 0.6v A . The upper two plots show the solutions for Tα = 4Tp and the lower two plots show the solutions for Tα = Tp . The calculations in Figure 2 indicate that the waves are unstable in the wavenumber band 0.4 k c/ωpp 0.9 and that the maximum growth rate is larger for waves propagating anti-parallel to V d . Interestingly, this directional asymmetry is just the opposite of that found for the ion cyclotron instability in the previous section. As with the EMIC instability, the asymmetry in the growth rates for the firehose instability is independent of the direction of B 0 and depends solely on the direction of V d by virtue of the identity ω(−k , −Vs ) = ω(k , Vs ). The firehose instability is found to be more sensitive than the EMIC instability to variations in the isotropic pressure of the alpha particles. When the alpha and proton temperatures are equal, Tα = Tp , the modes with k · V d < 0 and k · V d > 0 are both growing as seen in the lower right-hand panel in Figure 2. However, when the temperature of the alpha particles increases to Tα = 4Tp , the mode with k · V d < 0 (red curve) is still growing while the mode with k · V d > 0 (blue curve) is damped as is seen in the upper right-hand panel in Figure 2. In this case, the instability will generate waves propagating in just one direction. Unidirectional wave generation is of importance for explaining the observations of parallel propagating waves by He et al. (2011) and Podesta & Gary (2011) as discussed in the Introduction. In the upper right-hand panel in Figure 2, the maximum growth rate is γm /Ωp 1.12 × 10−2 . In the lower right-hand panel in Figure 2, the maximum growth rates are γm /Ωp 1.19 × 10−2 and 0.67 × 10−2 ; and the ratio of the maximum growth rates for propagation anti-parallel and parallel to V d is 1.78. Comparison of Figures 1 and 2 shows that for the same values of beta, βp = 1 and βe = 0.5, the firehose instability with Tp /T⊥p = 2.5 has a much lower growth rate than the EMIC instability with T⊥p /Tp = 2.5. However, the growth rates are still large enough to be important for in situ wave generation in the solar wind. The pressure anisotropy based on the total plasma pressure in Figure 2 is P s βs =

1.75. (32) P⊥ s β⊥s

Parallel Firehose Instability 0.015 0.014 T /T = 2.5 ||p ⊥ p 0.013 0.012 k⋅V < 0 0.011 d 0.01 0.009 k⋅V > 0 0.008 d 0.007 0.006 0.005 0 0.2 0.4 0.6 0.8 Drift Velocity Vd / VA

1

Maximum Growth Rate γ / Ωp

EMIC Instability 0.06 0.05

T⊥p/T||p = 2.5 k⋅Vd > 0

0.04 k⋅V < 0 d

0.03 0.02 0

0.2 0.4 0.6 0.8 Drift Velocity V / V d

1

A

Figure 3. Maximum growth rate as a function of the drift velocity Vd = Vα −Vp for parallel propagation k · V d > 0 (blue curves) and anti-parallel propagation k · V d < 0 (red curves). Top: the firehose instability with Tp /T⊥p = 2.5; Bottom: the EMIC instability with T⊥p /Tp = 2.5. The plasma parameters used in this figure are nα /np = 0.04, βp = 1, βe = 0.5, Tα = Tp , T⊥e = Te , T⊥α = Tα , and c/vA = 5000. (A color version of this figure is available in the online journal.)

Figure 3. For the EMIC instability, the difference between the growth rates for parallel and anti-parallel propagation reaches a maximum at approximately Vd = 0.45vA and then decreases slowly after that. These trends do, of course, depend on the plasma parameters and the behavior shown here may change significantly for other choices of parameters. However, the tendency for unstable waves to be generated progressively more unidirectionally as Vd increases may be relevant for understanding the origin of parallel propagating waves observed in the solar wind by He et al. (2011) and Podesta & Gary (2011) since these observations appear to be characterized by nearly unidirectional propagation.

In applications, it is of interest to know how the maximum growth rate depends on the drift velocity of the alpha particles. The results of one such calculation are shown in Figure 3. Note that the maximum growth rates for parallel and antiparallel propagation become equal as Vd → 0 as is required by symmetry. The most important feature to note about Figure 3 is that as Vd increases, the absolute value of the difference between the growth rates for parallel and anti-parallel propagation also increases, at least for small values of Vd . Over this range, wave generation by the instability becomes progressively more unidirectional as Vd increases. For the firehose instability this trend continues throughout the range of drift speeds shown in

4.3. Growth Rate versus T⊥p /Tp When all the other parameters are held constant, the maximum growth rate of the EMIC instability is an increasing function of the proton temperature ratio T⊥p /Tp > 1, the free energy source that drives the instability, and, similarly, the maximum growth rate of the firehose instability is an increasing function of the inverse temperature ratio Tp /T⊥p > 1. Solutions of the dispersion relation showing the behavior of the maximum growth rate as a function of the proton temperature 7


Podesta & Gary

EMIC Instability, V = 0.6 V d

Firehose Instability, V = 0.6 V

A

d

A

0.35 0.16

0.3

p

k⋅Vd > 0

0.2

γ /Ω

k⋅V < 0

0.15

m

γ /Ω m p

0.25

d

0.1

d

0.08

k⋅Vd > 0

0.04 LCP

0.05 0

k⋅V < 0

0.12

2

4

6

8

RCP

0

10

2

4

T⊥p/T||p 6

5

γm (kVd > 0) γm (kVd < 0)

m

>

γ< / γm

γm / γm

10

Firehose Instability

5

3

8

6 EMIC Instability

4

6

T||p/T⊥p

3

2

2

1

1 2

4

6

8

10

2

T /T ⊥p

γm (kVd < 0) γm (kVd > 0)

4

4

6

8

10

T /T

||p

||p

⊥p

Figure 4. Maximum growth rate γm as a function of the proton temperature ratio for the EMIC instability (upper left) and the firehose instability (upper right) assuming parallel propagation and a relative alpha–proton drift speed Vd = 0.6vA . The growth rates for outward propagation, k · V d > 0, are drawn in blue and those for inward propagation, k · V d < 0, are drawn in red. The lower panels show the ratio of the maximum growth rates for inward and outward propagation. For the calculations shown here: nα /np = 0.04, βp = 1, βe = 0.5, Tα = Tp , T⊥e = Te , T⊥α = Tα , and c/vA = 5000. (A color version of this figure is available in the online journal.)

ratio are displayed in the upper panels in Figure 4. In these calculations, all the plasma parameters except for the proton temperature ratio are held constant; the differential flow speed is Vd = 0.6vA . For left circularly polarized modes generated by the EMIC instability the outward propagating waves, k·V d > 0, have a positive maximum growth rate for all temperature ratios shown, 1.5 T⊥p /Tp 10. However, for the inward propagating waves, k · V d < 0, the maximum growth rate falls to zero for small temperature ratios 1.5 T⊥p /Tp 2 (more specifically, γ is either approximately zero or negative). Therefore, over this range of small to moderate temperature ratios, the waves generated by the EMIC instability are purely unidirectional. That is, only outward propagating waves are generated in this case. For RCP modes generated by the firehose instability, both outward propagating waves with k · V d > 0 and inward propagating waves with k · V d < 0 have positive maximum growth rates over the entire range of proton temperature ratios shown in Figure 4, 1.5 Tp /T⊥p 10. However, the results in Section 4.1 and Figure 2 suggest that an increase in the (isotropic) pressure of the alpha particles will cause the growth rates of the outward propagating waves to fall below zero over some range of the parameter Tp /T⊥p . Hence, when Tα = 4Tp it is expected that the waves generated by the firehose instability will be unidirectional over some range of values of the proton temperature ratio. The lower plots in Figure 4 show the ratio of the maximum growth rates of outward and inward propagating modes re-

stricted to the range of proton temperature ratios where the growth rates are positive for both directions of propagation. For the EMIC instability, this is the range 2 T⊥p /Tp 10 and, for the firehose instability, this is the entire range of the numerical calculations: 1.5 Tp /T⊥p 10. For both the EMIC and firehose instabilities, the ratios shown in the lower plots in Figure 4 are rapidly decreasing functions of the temperature ratio that quickly approach unity as the temperature ratio increases. Consequently, large asymmetries (ratios) between the growth rates of outward and inward propagating modes only occur when the temperature ratio is small and, coincidentally, the growth rates of the individual modes is small. When the driving is large, the growth rate of the instability is large but the asymmetry in the growth rates of outward and inward propagating modes is small. Thus, nearly unidirectional wave generation occurs primarily when the proton temperature ratio is of order unity, although in this range the growth rates are relatively small. Interestingly, these are the conditions most likely to be found in high-speed solar wind. 4.4. Beta Dependence The beta dependence of the maximum growth rate is investigated by varying βp over the range 0.1 < βp < 10 while holding the ratio βp /βe = 2 constant. The results are shown in Figure 5. As is well known, the maximum growth rate of the firehose instability decreases rapidly for βp < 1 and this instability is, therefore, not important at low βp . On the other hand, the 8


Podesta & Gary

EMIC Instability 10

−1

Firehose Instability 10

k⋅V > 0

−1

k⋅Vd < 0

10

p

−2

γm / Ω

γm / Ω

p

d

k⋅V < 0 d

10

10 10

−2

k⋅V > 0 d

−3

−3

10

−4

LCP

10

RCP

−4

0.1

10 1

10

βp

−5

0.1

1

10

βp

γ (kV 0)

LCP

2.5

RCP

d

6

m

1

β

10

p

4

d

m

1.5

1 0.1

5

3

m

2

d

m

d

γ (kV >0) / γ (kV 0 are drawn in blue and those for inward propagation k · V d < 0 are drawn in red. The lower panels show the ratio of the maximum growth rates for inward and outward propagation. For the calculations shown here: nα /np = 0.04, βp = 1, βe = 0.5, Tα = Tp , T⊥e = Te , T⊥α = Tα , and c/vA = 5000. (A color version of this figure is available in the online journal.)

density of alpha particles, nα /np 1, having a differential velocity Vd that is large compared to the velocities of the resonant protons should have no effect on the resonant proton interactions responsible for the mirror instability and, consequently, the growth rate of the instability should not be affected by the alpha particles. The frequency of the instability, however, will be affected by the relative motion of the proton rest frame relative to the rest frame of the plasma. In the proton rest frame the frequency of the unstable mode is ω ≈ 0 (Southwood & Kivelson 1993). Because the proton rest frame is moving with velocity Vp relative to the plasma rest frame defined in Section 2, the plasma rest frame is moving with velocity V = −Vp relative to the proton rest frame and, therefore, the Doppler-shifted frequency in the plasma rest frame will be

maximum growth rate of the EMIC instability is large enough to be of practical importance in the solar wind throughout the entire range 0.1 < βp < 10, at least for the plasma parameters examined here. Note that for the EMIC instability the maximum growth rate in the +V d direction is greater than the maximum growth rate in the −V d direction throughout the entire range of βp shown in Figure 5. However, the ratio of the maximum growth rates is on the order of two and does not vary much, as shown in the lower left-hand plot. For the firehose instability the maximum growth rate in the −V d direction is greater than the maximum growth rate in the +V d direction throughout the entire range 0.1 < βp < 10. In general, as shown in the lower two plots in Figure 5, the waves generated by the instabilities tend to become more unidirectional as βp decreases. This effect is more pronounced for the firehose instability. However, for βp < 1 the growth rates of the firehose instability decrease too rapidly for this effect to be of any practical significance. For the EMIC instability, the unstable waves are far from unidirectional at any βp for the plasma parameters examined here. For plasmas with a κ-distribution, however, a significant increase in unidirectionality for the EMIC instability is found for small βp (Podesta 2011).

ω = ω − k · V ≈ k Vp .

(33)

These expectations have been verified through numerical solution of the hot plasma dispersion relation. The code employed for this purpose is different from the one described earlier in this paper: it allows propagation at any angle θ with respect to the magnetic field B 0 as well as drifts of the different particle species along B 0 . The parameters used in the calculations are nα /np = 0.04, V d = 0.6v A , Tp⊥ /Tp = 2.5, Tα = Te = Tp , βp = 1, and c/vA = 5000. The frequency and growth rate of the mirror instability were investigated as a function of the angle of propagation θ for the wavenumber kc/ωpp = 0.5. The frequencies and growth rates were found to be the same for the angles θ and π + θ ,

4.5. Mirror Instability The mirror instability is caused by resonant wave–particle interactions with protons having small parallel velocities, v ≈ 0 (Southwood & Kivelson 1993). The introduction of a small 9


Podesta & Gary

inward propagating right circularly polarized magnetosonicwhistler waves, respectively, and this holds independently of the direction of the IMF since the preferred propagation directions are determined solely by the direction of the alpha–proton flow velocity V d which is almost always directed outward, away from the Sun. In the spacecraft frame, since |ω/k | Vsw , it follows from Taylor’s frozen flow approximation that both of these waves will produce a left-hand magnetic-helicity signature in agreement with the observations (see, for example, the Appendix in Smith et al. 1983). This appears to be the most logical explanation for the observed magnetic-helicity signals reported near parallel propagation by He et al. (2011) and Podesta & Gary (2011). The critical reader may point out that the growth rates of these instabilities are slow compared to the proton cyclotron period, γ /Ωp ∼ 0.01, and may also be slow compared to the eddy turnover time of the turbulence at the same length scales. In that case, the growth of the instability may be thwarted by the turbulence. Two remarks are in order. First, at the kinetic scales of interest the fluctuations are well within the so-called dissipation range where MHD physics breaks down and kinetic physics takes over. Since the timescales of the turbulence are not known accurately, such arguments have no basis at the present time. Second, our results for the growth rate are based on a normal mode analysis and it is possible that an analysis based on the non-modal approach will produce significantly faster growth rates (see, for example, Camporeale et al. 2009, 2010; Camporeale 2011) since the underlying linear operator for the linearized Vlasov–Maxwell system in magnetized plasmas is probably not normal (Podesta 2010). This is an interesting avenue of investigation for future work. To extend the work initiated in this study, it is of interest to further explore how the asymmetry of the maximum growth rates of solar wind instabilities depends on the large number of parameters in the problem and how nonlinear effects, such as quasilinear saturation of the temperature anisotropy instabilities and possibly mode coupling effects, may affect the evolution of distribution functions in the solar wind. Although the present study has only considered instabilities for parallel propagating modes, related instabilities for obliquely propagating modes, such as the oblique firehose instability, are also expected to have maximum growth rates that are asymmetric under change in sign of k · V d . This asymmetry may help explain, at least in part, the right-hand magnetic-helicity signatures at oblique angles of propagation observed by He et al. (2011) and Podesta & Gary (2011). One remark should be made regarding the relationship of the results reported here to previous work. The asymmetry of the maximum growth rate for the EMIC instability caused by a field-aligned beam of alpha particles also occurs in the presence of a field-aligned proton beam when the thermal core of the proton distribution satisfies Tp⊥ > Tp . Such an asymmetry was reported by Gary & Schriver (1987) (see also Gary 1991), who found that in the presence of a proton beam the maximum growth rate of the EMIC instability occurs in the −V d direction rather than in the +V d direction as was found for the proton-driven instability in the present study. This discrepancy is unexpected on physical grounds and requires further investigation.

implying no asymmetry of the growth rate under reversal of the propagation direction. The maximum growth rate was approximately γ /Ωp 0.034 and occurred at an angle of 60◦ , roughly. Furthermore, the relation (33) was also found to hold. The normalized magnetic helicity of the most unstable mode is computed as σm =

k Im[E · (k × E ∗ )] , |k × E|2

(34)

where E is the electric field perturbation; or, choosing a coordinate system with eˆ z in the direction of B 0 and k = (kx , 0, kz ), σm =

2k Im(kz Ex∗ Ey + kx Ey∗ Ez ) |k × E|2

.

(35)

This formula is equivalent to the expression derived by Gary (1986). For the most unstable mode, numerical solutions of the hot plasma dispersion relation with Vd = 0.6vA yield σm = 0.002 for propagation at both θ = 60◦ and θ = 180◦ + 60◦ . Hence, in the presence of a differential flow of alpha particles, σm is near zero for the mirror mode, at least for the representative solar wind parameters used here. Previous results showing σm = 0 for the mirror mode were obtained by Gary (1992) in the simpler case of an electron–proton plasma without any differential flow. 5. CONCLUSIONS When there exists a weak differential streaming of alpha particles relative to the protons, |V d | vA , the maximum growth rates for the parallel firehose instability and the EMIC instability are not symmetric (even) functions of the parallel wavenumber k. The maximum growth rate for the EMIC instability occurs in the +V d direction and the maximum growth rate for the parallel firehose instability occurs in the −V d direction. Thus, in the solar wind between 0.3 AU and 1 AU where the alpha–proton drift velocity often takes values on the order of the Alfvén speed, the parallel firehose instability preferentially generates magnetosonic-whistler waves propagating toward the Sun along the interplanetary magnetic field (IMF) and the EMIC instability preferentially generates Alfvén-ion-cyclotron waves propagating away from the Sun along the IMF. Moreover, these preferred directions of propagation are independent of the direction of the IMF. Another important result of this study is that for both instabilities there exist regions of parameter space in which the waves generated by the instabilities propagate only in one direction or, alternatively, the waves propagate predominantly in one direction. Approximately unidirectional wave generation is necessary to explain the observations of He et al. (2011) and Podesta & Gary (2011) as discussed in the Introduction. Thus, linear wave theory suggests a simple interpretation for the left-hand magnetic-helicity signatures observed for parallel propagating waves near the proton inertial scale in the studies by He et al. (2011) and Podesta & Gary (2011). Proton temperature anisotropies in the solar wind are thought to be regulated by the parallel firehose and EMIC instabilities—although the oblique firehose and mirror instabilities may also play some role; see, for example, Bale et al. (2009) and Matteini et al. (2011), and references therein. Because of the existence of a differential flow between alphas and protons, the EMIC and parallel firehose instabilities will preferentially generate outward propagating left circularly polarized EMIC waves or

Helpful discussions with Joe Borovsky, Chuck Smith, and Phil Isenberg are gratefully acknowledged. This research was supported by the NASA Solar and Heliospheric Physics Program and the NSF SHINE Program. 10


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