INTERPRETING MAGNETIC VARIANCE ANISOTROPY MEASUREMENTS IN THE SOLAR WIND

At scales of k_⊥ρ_i ≪ 1 and k_∥d_i ≪ 1 in the free solar wind, the Alfvén wave is the dominantly observed wave mode (Tu & Marsch 1995; Bruno & Carbone 2005; Horbury et al. 2008; Podesta 2009; Sahraoui et al. 2010; Podesta & Gary 2011), where ρ_i = v_ti/Ω_i is the ion (proton) gyroradius, Ω_i = eB₀/m_ic is the proton gyrofrequency, $d_i = c / \omega _{{\rm pi}} = \rho _i / \sqrt{\beta _i}$ is the ion inertial length, and ω²_pi = 4πn_ie²/m_i is the ion plasma frequency. The turbulent energy cascade at these scales has been thoroughly explored in the literature, where the one-dimensional perpendicular magnetic energy is observed to scale as $E_{B_\perp } \propto k_\perp ^{-\alpha }$ with a wavevector anisotropy k_∥∝k^ξ_⊥. When the turbulence is in critical balance—i.e., the nonlinear cascade rate is of order the linear frequency—the theoretical values for α and ξ are expected to be α = 5/3 and ξ = 2/3 for the model of Goldreich–Sridhar (Goldreich & Sridhar 1995) or α = 3/2 and ξ = 1/2 for the dynamic alignment model (Boldyrev 2005, 2006). Which model is correct is not completely settled. Solar wind observations typically show a magnetic field spectrum with α ≃ 5/3 and total (kinetic plus magnetic) energy spectrum with α ≃ 3/2 (Luo & Wu 2010; Podesta & Borovsky 2010; Wicks et al. 2010; Boldyrev et al. 2011; Chen et al. 2011a, 2011b), while MHD turbulence simulations suggest the dynamic alignment model may be more correct (Mason et al. 2006, 2008). For the purpose of modeling the energy cascade, we assume the Goldreich–Sridhar model for simplicity. The choice of model does not significantly affect the behavior of the $\mathcal {A}_m$.

The properties of the MHD Alfvén root are well understood. However, the MHD solution of the Alfvén root is incompressible since δB_∥ = 0, suggesting the Alfvénic $\mathcal {A}_m$ is unbounded. The full collisionless VM solution of the Alfvén root in the inertial range has a small but non-vanishing δB_∥ which increases with k_⊥, thereby causing the $\mathcal {A}_m$ to decrease with k_⊥. The VM solution for the Alfvén root (black) $\mathcal {A}_m$ and dispersion relation are plotted against k_⊥ρ_i in Figures 2(a) and (b) for β_i = 0.01, 0.1, 1, and 10 (dash-dotted, dotted, solid, and dashed, respectively), and β_i = v²_ti/v²_A and v²_A = B₀²/4πm_in is the Alfvén speed. The solution assumes an inertial range spectral anisotropy as described by the Goldreich–Sridhar model, k_∥ = k^1/3_ik^2/3_⊥, where k_i is the isotropic outer-scale of the turbulence and is taken to be k_iρ_i = 10⁻⁴, consistent with in situ solar wind measurements at 1 AU (Howes et al. 2008).

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Continuing along the critical balance cascade to kinetic scales naturally produces spectral anisotropy with k_⊥ ≫ k_∥. So, we next consider the Alfvén root with k_⊥ρ_i > 1 and k_∥d_i < 1. At these scales, the Alfvén wave transitions into the KAW. The KAW is dispersive, damped, and much more compressible than the Alfvén wave. The dispersive nature of the root steepens the magnetic energy spectrum to α = 7/3 in the undamped case (Howes et al. 2008; Schekochihin et al. 2009), while the inclusion of damping steepens the spectrum further (Howes et al. 2011b, 2011c; TenBarge & Howes 2012a). Damping here refers to collisionless wave–particle interactions, primarily ion transit time damping on the parallel magnetic field that peaks at ion scales for β_i ≳ 1 (Barnes 1966; Quataert 1998) and electron Landau damping on the parallel electric field that peaks at electron scales (Howes et al. 2006; Schekochihin et al. 2009). The spectral anisotropy of the KAW is assumed to scale as k_∥∝k^1/3_⊥ (Galtier 2006; Howes et al. 2008; Schekochihin et al. 2009; TenBarge & Howes 2012b).

The increased compressibility of the KAW can be seen at scales k_⊥ρ_i > 1 in Figure 2(a), where the KAW $\mathcal {A}_m$ is seen to have a β_i dependent plateau. The KAW $\mathcal {A}_m$ in the dissipation range also depends upon the ion-to-electron temperature ratio, T_i/T_e. An analytical form for the KAW $\mathcal {A}_m$ can be derived (see Appendix A.3) within the framework of electron-reduced MHD (ERMHD) developed in Schekochihin et al. (2009),

$$\begin{equation} \mathcal {A}_m = \frac{2 + \beta _i (1+T_e/T_i)}{\beta _i (1+T_e/T_i)}. \end{equation} \tag{ 1 }$$

The KAW $\mathcal {A}_m$ has no wavenumber dependence and thus plateaus at a value determined by β_i and T_i/T_e. The β_i and temperature ratio dependence of the KAW $\mathcal {A}_m$ derived from VM theory is plotted in Figure 3, where the value for the $\mathcal {A}_m$ is averaged across the plateau at k_⊥ρ_i ∈ [3, 4].

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The behavior of the Alfvén root becomes more complicated for scales k_∥d_i > 1. When k_∥d_i > 1 and k_⊥ρ_i ≲ 1, the Alfvén root becomes the Alfvén ion cyclotron mode. This mode is strongly cyclotron damped and characterized by a left-handed magnetic helicity and left-handed electric field polarization. Because of the strong damping and in situ solar wind observations suggesting this mode are less common than the KAW (Podesta & Gary 2011; He et al. 2012), we do not consider it further. When k_∥d_i ≳ 1 and k_⊥ρ_i ≳ 1, the behavior of the Alfvén root has not been thoroughly explored. The ion Bernstein wave (IBW; Stix 1992) is conventionally assumed to couple to the KAW at the ion gyrofrequency; however, the KAW root that exists for k_∥d_i < 1 may continue undamped at higher frequencies (Howes et al. 2008; Sahraoui et al. 2011; Klein et al. 2012a). Since the behavior in this region of wavenumber space is uncertain, we do not attempt to describe it here.

The majority of in situ solar wind observations of the inertial range suggest the components responsible for most of the measured free solar wind compressibility are pressure balanced structures (PBSs; Tu & Marsch 1995; Bruno & Carbone 2005); however, linear PBSs are degenerate with the k_∥ = 0, non-propagating limit of slow waves (Tu & Marsch 1995; Kellogg & Horbury 2005). For simplicity, we classify linear PBSs as slow modes. The association of the compressible portion of the solar wind with PBSs is due to the measured anti-correlation of the thermal and magnetic pressure (Burlaga & Ogilvie 1970) or anti-correlation of density and magnetic field magnitude (Vellante & Lazarus 1987; Roberts 1990) at inertial range scales. Recent analyses (Howes et al. 2011a; Klein et al. 2012b) exploring the more telling anti-correlation of density and parallel magnetic field suggest that the compressible portion of the solar wind is primarily composed of propagating slow modes rather than PBSs. These analyses suggest that on average Alfvén modes comprise ∼90% of the energy and slow modes ∼10% of the energy.

The notion that the warm, T_i/T_e ≃ 1, β_i ≃ 1 propagating slow mode exists in the solar wind defies conventional wisdom that the slow mode is strongly damped in such plasmas (Barnes 1966), so we here elucidate this point. Theory (Goldreich & Sridhar 1997; Lithwick & Goldreich 2001; Schekochihin et al. 2009) and numerical simulations (Cho & Lazarian 2002, 2003) suggest that the slow mode does not have its own active turbulent cascade; rather, the slow mode is passively cascaded by the Alfvénic turbulence. The slow mode damping rate is proportional to the parallel wavenumber, γ_S∝k_∥S, and the strong damping of the slow mode suggests γ_S/ω_S ≃ 1, where k_∥S, γ_S, and ω_S are the parallel wavenumber, linear damping rate, and frequency of the slow mode. However, the passive cascade of the slow mode implies that the slow modes are cascaded by the Alfvén cascade on the Alfvén timescale, ω_A = k_∥Av_A. Therefore, at a given k*_⊥, the parallel wavenumber of the slow mode compared to the parallel wavenumber of the Alfvén mode determines the strength of the damping relative to the cascade rate: at a particular scale, k*_⊥, a slow mode with k*_∥S ≪ k_∥A* can be passively cascade before being damped since the slow mode damping rate will be smaller than the Alfvén cascade time, γ*_S/ω*_A ≪ 1.

For slow waves in the MHD limit, $\mathcal {A}_m = k_\parallel ^2 / k_\perp ^2$ (see Appendix A.1 for the derivation of this equation), and the VM solution does not deviate significantly from the MHD solution until finite Larmor radius effects become dominant at k_⊥ρ_i ≃ 0.1. The VM solution for the slow mode $\mathcal {A}_m$ (blue) is plotted in Figure 2(a) assuming a passive cascade of slow waves with k_∥S∝k^2/3_⊥S—a more anisotropic cascade decreases the $\mathcal {A}_m$ below that in the figure but does not alter the qualitative behavior. The $\mathcal {A}_m$ for a PBS, i.e., k_∥ = 0 slow mode, is identically zero since δB_⊥ = 0 for these modes.

Since the slow mode is strongly damped at large k_∥ unless T_i ≪ T_e, we will exclude a discussion of the behavior of this root for k_∥d_i > 1.

At inertial range scales, the fast mode is well described by MHD, whose solution provides an $\mathcal {A}_m$ identical to that of slow waves: $\mathcal {A}_m = k_\parallel ^2 / k_\perp ^2$. However, the distribution of fast waves in wavenumber space is not as well constrained as that of slow modes since the fast mode is not strongly damped for parallel propagation. Further, compressible MHD turbulence simulations indicate that the fast mode is cascaded isotropically in wavenumber space (Cho & Lazarian 2003). Therefore, a turbulent cascade of fast modes has an $\mathcal {A}_m$ that can take on all possible values, and the measured average value of the $\mathcal {A}_m$ will depend sensitively on the distribution of fast modes in wavenumber space. To represent the fast mode turbulence, we plot in Figure 2(a) the $\mathcal {A}_m$ for a fast wave (red) with $\theta _{kB} = \arccos {({\bm k}\cdot {\bm B}_0 / |{\bm k}| |{\bm B}_0|)} = 3^\circ$. We choose this value as representative because the average $\mathcal {A}_m$ of an isotropic distribution of fast modes will be dominated by those modes with θ_kB ≃ 0 and the largest measured $\mathcal {A}_m$ in a large ensemble of solar wind data is ∼500 (Smith et al. 2006).

In the dissipation range at scales k_∥d_i > 1, the fast mode transitions to a parallel whistler (k_⊥ρ_i < 1) or an oblique whistler (k_⊥ρ_i > 1) wave with ω/Ω_i > 1. The whistler mode is well described by the electron MHD (EMHD) equations (Kingsep et al. 1990), which describes phenomena on scales kd_i > 1. Note that one must take care when applying the EMHD equations, because EMHD is only valid for T_i ≪ T_e. For scales k_⊥d_i > 1 and k_∥d_i < 1, EMHD describes the cold ion limit of KAWs and not whistler waves (Ito et al. 2004; Hirose et al. 2004; Howes 2009; Schekochihin et al. 2009). Solving the EMHD equations provides $\mathcal {A}_m = 1 + 2 k_\parallel ^2 / k_\perp ^2$ (see Appendix A.2), which is a good approximation of the VM solution and can again attain all values between 1 and infinity and is sensitively dependent upon the wavenumber distribution. Therefore, if the propagation angle or whistler wavenumber distribution does not change from that of the inertial range fast modes, the average $\mathcal {A}_m$ will approximately double in the dissipation range. However, theory (Galtier & Bhattacharjee 2003; Cho & Lazarian 2004; Narita & Gary 2010) and simulation (Dastgeer et al. 2000; Cho & Lazarian 2004; Svidzinski et al. 2009; Saito et al. 2010) suggest that the cascade of whistler waves is highly anisotropic in the same sense as critically balanced KAWs, k_∥∝k^1/3_⊥. The anisotropic nature of the dissipation range cascade suggests that the average $\mathcal {A}_m$ of a distribution of whistler waves will decrease rapidly to a β_i independent value $\mathcal {A}_m \sim 1$ as the cascade progresses to smaller scales. The whistler portion of the fast modes plotted in Figure 2(a) follows the critical balance prediction described above, with θ_kB = 3° up to k_∥d_i = 1 and then k_∥∝k^1/3_⊥. Note that the $\mathcal {A}_m$ of whistler waves has a very weak β_i and ion-to-electron temperature ratio dependence. The apparent β_i dependence of the whistler branch in Figure 2(a) is due to the fast-whistler break point being at $k_\parallel d_i = k_\parallel \rho _i / \sqrt{\beta _i}$; therefore, the β_i dependence in the figure would vanish if the x-axis were normalized to the ion inertial length.

For completeness, the fast mode to whistler transition $\mathcal {A}_m$ is shown for several different propagation angles, θ_kB, in Figure 4(a). The $\mathcal {A}_m$ agrees well with the predictions above, except in the θ_kB = 75° case. For such highly oblique angles, the fast mode transitions to an IBW: at scales of k_⊥ρ_i ≳ 1 and k_∥d_i ≲ 1, the fast mode transitions to the electrostatic IBW with frequency approximately equal to an integer multiple of the ion cyclotron frequency (Stix 1992; Li & Habbal 2001; Howes 2009). The transition to the IBW is clear in the fast/whistler wave dispersion relations in Figure 4(b) corresponding for the same angles as the $\mathcal {A}_m$ presented in Figure 4(a). Although the IBW is electromagnetic during the transition from electromagnetic fast or Alfvén modes, IBWs are dominantly electrostatic (Stix 1992). The properties of the transition electromagnetic IBW have not been fully explored in the literature, so we will not consider the potential contribution of IBWs to the dissipation range $\mathcal {A}_m$.

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The wavenumber distribution of whistler waves generated in a local instability such as the parallel firehose (Gary et al. 1998) and whistler anisotropy instabilities is not easily determined and depends upon the instability, so we will not consider them further.

The assumptions that only slow modes with k_∥ ≫ k_⊥ are weakly damped and approximately parallel fast modes dominate the inertial range fast mode $\mathcal {A}_m$ lead to asymptotically small and large values of the slow and fast mode $\mathcal {A}_m$s, respectively. Due to the asymptotically large and small values of the component $\mathcal {A}_m$s, we can estimate the value of $\mathcal {A}_{mT}$ in the inertial range provided AC ≠ 1 and FS ≠ 1 by using the $\mathcal {A}_m$ of each mode from Figure 2(a) to estimate the compressibilities, which have only a very weak β_i dependence: C_⊥A ≃ 1, C_∥A ≪ 1, C_⊥F ≃ 1, C_∥F ≪ 1, C_⊥S ≪ 1, and C_∥S ≃ 1. Therefore, in the inertial range,

$$\begin{equation} \mathcal {A}_{mT} \simeq \frac{{\rm AC} + (1-{\rm AC}) {\rm FS}}{(1-{\rm AC})(1-{\rm FS})}. \end{equation} \tag{ 5 }$$

Note that whether the slow modes are in fact propagating slow modes or PBSs does not affect the estimate because C_⊥S is asymptotically small in either case. Therefore, the inertial range $\mathcal {A}_m$ depends only on the ratios of Alfvén to total energy and the fast to total compressible energy ratio and is unable to differentiate between propagating slow modes and PBSs. Note that for observed average solar wind values of AC ≃ 0.9 and FS ≲ 0.1 (Howes et al. 2011a; Klein et al. 2012b), the $\mathcal {A}_m$ can be further reduced to $\mathcal {A}_{mT} \simeq {\rm AC} / (1-{\rm AC})$. We take as fiducial solar wind values AC = 0.9 and FS = 0.1, or equivalently 90% Alfvén, 9% slow, and 1% fast wave energy. Although we take FS = 0.1, there is very little quantitative difference between FS = 0.1 and 0 (see Figure 5).

Plotted in green in Figure 2(a) is a single mixture of the three modes representing fiducial solar wind values of AC = 0.9 and FS = 0.1. Clearly, the $\mathcal {A}_m$ has virtually no β_i or wavenumber dependence in the inertial range, as expected from the above analysis. To explore the dependence of the $\mathcal {A}_m$ on fluctuation composition, we plot in Figure 5 the $\mathcal {A}_{mT}$ given by Equation (5). To confirm the quality of the estimate given by Equation (5), we plot in Figure 6 the $\mathcal {A}_{mT}$ given by Equation (2) (thick) and the estimate given by Equation (5) (thin). The figure demonstrates that Equation (5) provides a good estimate for the total $\mathcal {A}_m$ in the inertial range.

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Also highlighted by the figure is the degeneracy of the $\mathcal {A}_m$, since different mixtures of modes can replicate nearly identical $\mathcal {A}_m$ in the inertial range. This implies that the $\mathcal {A}_m$ when used alone is not a good metric for differentiating modes in the inertial range; however, the $\mathcal {A}_m$ can be a useful secondary metric. For instance, when used together with density-parallel magnetic field correlations to identify the proportion of fast to slow wave energy, the $\mathcal {A}_m$ can provide an estimate of the Alfvén to compressible energy proportion, thereby quantifying the total population of the solar wind.

The behavior of the $\mathcal {A}_m$ in the transition between the inertial and dissipation ranges and in the dissipation range is markedly different from the inertial range. Unlike the inertial range, where the asymptotically large and small values of the Alfvén, fast, and slow waves leads to an $\mathcal {A}_m$ that is controlled by the mode fractions AC and FS, the KAW dominates the dissipation range $\mathcal {A}_m$ for typical solar wind parameters (see Figure 2(a) for k_⊥ρ_i > 1, where the $\mathcal {A}_m$ for the mixture closely follows the $\mathcal {A}_m$ for KAWs).

The strong β_i and T_i/T_e dependence of the KAW $\mathcal {A}_m$ implies that for certain values, namely, β_i ≳ 1, the whistler and KAW dissipation range $\mathcal {A}_m$ become approximately degenerate (see Figure 2(a) for k_⊥ρ_i > 1). However, the behavior through the transition range for the Alfvén to KAW and fast to whistler mode transitions differs considerably. For example, the curves in Figure 6 with (AC, FS) = (0.9, 0.1) (solid black) and (0.1, 0.9) (short dash-dotted black) are nearly identical in the inertial and dissipation ranges, but exhibit very different behavior across the transition range. For this reason, the $\mathcal {A}_m$ is best presented as a function of wavenumber rather than as a quantity averaged over a band of wavenumbers, as has often been done in the literature (Smith et al. 2006, 2012; Hamilton et al. 2008).

In the solar wind, the mean magnetic field direction is constantly changing and can sweep through a wide range of angles over a period of tens of minutes. As such, defining $\delta B_\parallel = {\bm B}_0 \cdot \delta {\bm B}$ becomes questionable since ${\bm B}_0$ is averaged over long, global periods relative to the rapidly fluctuating field—this is especially problematic when measuring in the dissipation range when δB_∥ is even smaller and more rapidly fluctuating. The poor definition of parallel and perpendicular in the global analysis will pollute the parallel energy with perpendicular energy, thereby decreasing the measured solar wind $\mathcal {A}_m$. Therefore, a local analysis employing a local mean magnetic field (Cho & Vishniac 2000; Maron & Goldreich 2001; Horbury et al. 2008; Podesta 2009) must be employed to accurately measure δB_∥. The local analysis has the added advantage of being able to differentiate between different θ_VB measurements, where θ_VB is the angle between the mean magnetic field and the solar wind flow velocity. This can be helpful because the $\mathcal {A}_m$ will have different inertial/dissipation range breakpoints for KAWs and whistlers when plotted against k_∥ or k_⊥.

The conventional $\mathcal {A}_m$ is constructed by separately calculating the perpendicular and parallel fluctuating magnetic energies and finding their quotient,

$$\begin{equation} \mathcal {A}_m = \frac{\langle |\delta B_\perp |^2\rangle }{\langle |\delta B_\parallel |^2\rangle }. \end{equation} \tag{ 6 }$$

However, a similar measure could be constructed from the normalized perpendicular and parallel energies, $\langle |\delta B_\perp |^2 / |\delta {\bm B}|^2 \rangle$ and $\langle |\delta B_\parallel |^2 / |\delta {\bm B}|^2 \rangle$,

$$\begin{equation} \mathcal {A}_m = \frac{\langle |\delta B_\perp |^2 / |\delta {\bm B}|^2 \rangle }{\langle |\delta B_\parallel |^2 / |\delta {\bm B}|^2 \rangle }. \end{equation} \tag{ 7 }$$

This measure has the physically motivated advantage of avoiding possible small denominators due to small δB_∥. Another possible measure of the $\mathcal {A}_m$ is

$$\begin{equation} \mathcal {A}_m = \langle |\delta B_\perp |^2 / |\delta B_\parallel |^2\rangle. \end{equation} \tag{ 8 }$$

This method has the conceptual advantage of maximizing any local effect of the mean magnetic field since it averages the $\mathcal {A}_m$ at each point rather than separately averaging the energies. However, the expression 〈|δB_⊥|²/|δB_∥|²〉 will be dominated by those terms with small denominators, which could make it an unphysical measure of the $\mathcal {A}_m$.

To explore the effect different averaging procedures have on the $\mathcal {A}_m$, we employ synthetic spacecraft data. The details for constructing the synthetic spacecraft data are discussed in Klein et al. (2012b), so we here only detail the relevant parameters. A three-dimensional box is populated with a spectrum of all three wave modes, where the proportion of each mode is given by AC = 0.9 and FS = 0.1, corresponding to 90% Alfvén, 9% slow, and 1% fast wave energy. The Alfvén and slow modes satisfy critical balance, with all modes less than the critical balance envelope equally populated. The fast modes are isotropically populated from θ_kB = 5°–85°, with equal energy at each angle. To represent solar wind turbulence, the phase of the wave modes is randomized. The data are sampled by advecting the turbulence past a stationary "spacecraft" with velocity v = 10v_A at a fixed angle with respect to the mean magnetic field and fixed sampling rate.

Figure 7 presents the three different methods for calculating the $\mathcal {A}_m$. The $\mathcal {A}_m$ is averaged across the interval k_⊥ρ_i ∈ [0.003, 0.03] with θ_vB = 85°, where θ_vB is the sampling angle analogous to that in solar wind. The sampling angle, θ_vB, was varied from 5°–85° and found to have no significant effect on the $\mathcal {A}_m$ in the inertial or dissipation range, but the sampling angle does alter the behavior in the transition region.

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For AC = 0.9 and FS = 0.1, the predicted $\mathcal {A}_m$ from linear theory is $\mathcal {A}_{mT} \simeq 10$. From the figure, it is clear that the conventional definition of the $\mathcal {A}_m$, Equation (6), (solid) computed from a spectrum of wave modes agrees best with linear theory and is the one we will continue to employ. The averaged anisotropy, Equation (7), (dotted) approach is a poor measure of the $\mathcal {A}_m$ because at any given point in space, the superposition of a collection of wave modes can lead to anomalously small values of δB_∥ due to cancellation. While the definition based upon normalized energies, Equation (8), (dashed) differs from the linear prediction, it captures the correct qualitative behavior and may be a safer measure to use in some circumstances.

Here, we present results from a collection of fully nonlinear gyrokinetic turbulence simulations performed with β_i = 0.1 and 1, T_i/T_e = 1, and a realistic mass ratio, m_i/m_e = 1836. The simulations were performed using the Astrophysical Gyrokinetics Code, AstroGK (Numata et al. 2010). All runs presented herein are driven with an oscillating Langevin antenna coupled to the parallel vector potential at the simulation domain scale (TenBarge et al. 2012). Relevant parameters for the simulations are given in Table 1, where = ρ_i/L₀ ≪ 1 is the gyrokinetic expansion parameter, A₀ is the antenna amplitude, and ν_s is the collision frequency of species s. The expansion parameter sets the parallel simulation domain elongation and is determined by assuming critical balance with an outer-scale k_iρ_i = 10⁻⁴: $\epsilon = \hat{k}_i^{1/3} \hat{k}_{\perp 0}^{2/3} (1 + \hat{k}_{\perp 0}^{5/3}) / (1 + \hat{k}_{\perp 0}^2)$, where $\hat{k} = k \rho _i$ and subscript naught indicates simulation domain scale quantities. The antenna amplitude is chosen to satisfy critical balance at the domain scale so that the simulations all represent critically balanced, strong turbulence.

Table 1. Parameters for AstroGK Simulations Used in Figures 8, 9(a), and 9(b)

Run	k⊥ρi	k∥ρi/		A0/ρiB0	νi/Ωi	νe/Ωi
β0.1I	[0.05, 1.05]	[1, 64]	0.006	500	0.001	0.03
β0.1T	[0.2, 8.4]	[1, 64]	0.016	35	0.003	0.05
β0.1D	[1, 42]	[1, 64]	0.046	1	0.02	0.4
β1I	[0.05, 1.05]	[1, 64]	0.006	500	0.001	0.0005
β1T	[0.2, 4.2]	[1, 64]	0.016	30	0.008	0.03
β1D	[1, 42]	[1, 64]	0.046	1	0.04	0.5
β1ED	[5, 105]	[1, 16]	0.081	0.2	0.2	0.5

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An example instantaneous one-dimensional perpendicular magnetic energy spectrum composed by overlaying the four β_i = 1 simulations is plotted in Figure 8—the spectrum is produced via conventional Fourier analysis techniques. The inertial range simulation (red) has a spectral index ∼ − 3/2, which steepens (green) to −2.8 (blue) before rolling off exponentially (cyan) as the electron collisionless dissipation becomes increasingly strong. The dissipation range simulations β1D and β1ED have been explored in detail (Howes et al. 2011c; TenBarge & Howes 2012a). The full spectrum agrees well with recent solar wind observations (Alexandrova et al. 2011).

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Figures 9(a) and (b) overlay the $\mathcal {A}_m$ for the β_i = 0.1 and 1 AstroGK simulations. The $\mathcal {A}_m$ is produced by computing, via conventional Fourier techniques, the perpendicular and parallel energy with respect to the global mean magnetic field. Due to the restrictions of gyrokinetics (Howes et al. 2006) and the method of driving, the simulations are almost purely populated by Alfvén/KAWs. Therefore, we plot as a dashed line in each figure the VM linear solution for the Alfvén root. The agreement between the linear prediction and the nonlinear simulation is excellent up to the point of strong damping at k_⊥ρ_i ∼ 10, where agreement with linear theory is expected to breakdown. The good agreement supports the validity of using linear theory to describe the $\mathcal {A}_m$ of fully nonlinear turbulence.

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Note that in constructing Figures 9(a) and (b) a global mean magnetic field has been used. This is justified, because in typical numerical simulations, the global mean field direction is fixed and $|\delta {\bm B}| / B_0 \ll 1$. Although a local mean field can be defined in the same sense as in the solar wind, the global mean field provides a relatively accurate measure of the $\mathcal {A}_m$ in numerical simulations of turbulence with a fixed global mean field direction and $|\delta {\bm B}| / B_0 \ll 1$.

Smith et al. (2006) present the most comprehensive collection of the solar wind inertial range $\mathcal {A}_m$ made to date, incorporating 960 data intervals recorded by the ACE spacecraft. The data are divided into periods of open field lines and magnetic clouds (Burlaga et al. 1981). They find that the $\mathcal {A}_m$ is proportional to a power of β_i or $|\delta {\bm B}| / B_0$ but are unable to identify which is the source of the functional relationship since β_i and $|\delta {\bm B}|$ are themselves positively correlated in the solar wind (Grappin et al. 1990). They follow the identification of the $\mathcal {A}_m$ relationship with a discussion of possible sources for the relationship based on a variety of considerations, some of which contradict the discussion in Section 2 of this paper. Smith et al. (2006) do not consider the superposition of modes that has been shown to exist in the solar wind, neglect the contribution of slow modes due to the belief that they are completely damped outside of local excitation, consider Alfvén waves to be completely incompressible ($\mathcal {A}_m = \infty$), consider PBSs but assert that the β_i dependence of δB_∥ of PBSs could contribute the β_i dependent $\mathcal {A}_m$ despite $\mathcal {A}_{m{{\rm PBS}}} = 0$ for all β_i.

The findings of Smith et al. (2006) seem to be in direct contradiction to the conclusions drawn in Section 3.1; however, there are possible reasons for the apparent discrepancy. As noted in Smith et al. (2006), the proton temperature and fluctuating magnetic field are positively correlated. It has also been shown that the proton temperature and bulk solar wind speed are positively correlated (Burlaga & Ogilvie 1973; Newbury et al. 1998). Therefore, the measured $\mathcal {A}_m$ might depend on any or a combination of β_i, $|\delta {\bm B}|$, or v_SW. If the underlying dependence is actually on v_SW, then the observed variation of the $\mathcal {A}_m$ could be from different types of solar wind launched from different regions of the Sun having a variable population of Alfvén to compressible components. For instance, slower wind may have evolved to a more Alfvénic state because fast modes would have had time to be dissipated in shocks and slow modes could have collisionlessly damped, leaving a primarily Alfvénic and PBS population. Since no measure other than $\mathcal {A}_m$ binned by β_i or $|\delta {\bm B}| / B_0$ was used in Smith et al. (2006), it is difficult to draw a firm conclusion.

Hamilton et al. (2008) extend the study of the same data set employed in Smith et al. (2006) to the dissipation range, defined to be 0.3 Hz ⩽f ⩽ 0.8 Hz. They find that dissipation range fluctuations are less anisotropic than those in the inertial range. Although not discussed in Hamilton et al. (2008), this result is consistent with the mixtures of wave modes discussed in this paper. The linear predictions of KAW $\mathcal {A}_m$ presented in Figure 3 for T_i/T_e = 1 pass through the core of the measured $\mathcal {A}_m$ for open magnetic field data for the measured $\mathcal {A}_m$ in the dissipation range presented in the lower panel of Figure 8 of Hamilton et al. (2008). Further, the KAW $\mathcal {A}_m$ for T_i/T_e = 0.1 and T_i/T_e = 1 bound the core of the low β_i magnetic cloud data for the measured dissipation range $\mathcal {A}_m$ in Hamilton et al. (2008). The temperature ratio is not provided for the data set in Hamilton et al. (2008); however, magnetic clouds are typically characterized by low proton temperatures (Osherovich et al. 1993; Richardson et al. 1997), so T_i/T_e < 1 for the cloud data is expected. Gary & Smith (2009) also noted that the β_i > 0.1 portion of the same data set is well fit by the KAW solution; however, the temperature ratio dependence was not explored. Gary & Smith (2009) suggest the shallow slope at low β_i might be suggestive of a whistler component. Another possible explanation for some of the very low values of the dissipation range $\mathcal {A}_m$ in Hamilton et al. (2008) is the use of a global measurement of the magnetic field, which will tend to decrease the measured $\mathcal {A}_m$, especially in the dissipation range. Also, due to the limited high frequency information available from ACE measurements f_Nyquist ≃ 2 Hz (Smith et al. 1998), the region defined as the dissipation range corresponds to the transition region between the inertial and dissipation ranges.

Smith et al. (2012) return again to the same ACE data set but focus on β_i > 1 intervals and augment the set with 29 additional high β_i ACE measurements. As such, this data set suffers from the same high frequency limitations noted above for the Hamilton et al. (2008) analysis. Smith et al. (2012) suggest that the measured dissipation range $\mathcal {A}_m$ is consistent with KAWs, but the inertial range $\mathcal {A}_m$ cannot be explained by a purely Alfvénic population. This observation is partially used to conclude that the Alfvén/KAW model cannot explain solar wind measurements. However, their $\mathcal {A}_m$ measurements agree well with a solar wind population dominated by Alfvén/KAWs with a small component of slow waves. Therefore, we find no contradiction between the ACE $\mathcal {A}_m$ measurements and the Alfvén mode dominant $\mathcal {A}_m$ model constructed herein.

Using STEREO measurements, He et al. (2012) employ a variant of the $\mathcal {A}_m$, δB_∥/δB_⊥, as a secondary metric to the magnetic helicity to differentiate between KAWs and whistlers, which both have right-handed helicity for oblique propagation (Howes & Quataert 2010). However, they consider fast modes propagating at fixed θ_kB = 60°, 80°, and 89°, and only the θ_kB = 60° root connects to an oblique whistler. The other two oblique roots connect to IBWs, which is made clear in the upper right panel of Figure 3 of He et al. (2012), where the dispersion relation for these two roots do not extend above ω/Ω_i = 1. Due to this mischaracterization of the roots, they mistakenly find the oblique whistler root $\mathcal {A}_m \ll 1$. As noted in Section 2.3, the whistler (fast mode for k_∥d_i > 1) $\mathcal {A}_m > 1$ for all propagation angles and β_i and is thus of the same orientation as KAW and does not provide a useful secondary metric to the helicity.

Using Cluster measurements, Salem et al. (2012) use magnetic compressibility, which is related to the $\mathcal {A}_m$ in Equation (3), as a secondary metric to the ratio of the electric to magnetic field, $|\delta {\bm E}| / |\delta {\bm B}|$, which is degenerate between KAWs and whistlers. Although the dissipation range asymptotic behavior of the fixed propagation angle KAWs and the whistlers at certain angles is similar, the transition from the inertial range to dissipation range for the two modes is different, and it is the transition region that is used to differentiate the two modes. Salem et al. (2012) find the Cluster data to be most consistent with a spectrum of KAWs.

Although much of the work does not directly address the $\mathcal {A}_m$, some recent solar wind analyses of the magnetic field have moved toward three dimensions (Chen et al. 2011a; Podesta & TenBarge 2012; Wicks et al. 2012). Presenting the full three-dimensional structure of the magnetic field is of obvious potential benefit; however, there are complications due to geometrical and sampling effects of the fluctuations being advected past the spacecraft (Turner et al. 2011; Wicks et al. 2012). To elucidate the complications, we follow Turner et al. (2011) and choose a coordinate system with B₀ in the z-direction and V_sw in the y-direction. Under Taylor's hypothesis, the measured components of magnetic energy in the perpendicular plane are then

$$E_{x,y} (\omega)= \frac{1}{8\pi } \int d^3 {\bm k}|\delta {\bm B}_{x,y}|^2 \delta (\omega - {\bm k}\cdot {\bm V}_{{\rm sw}}).$$

If a scaling between k_∥ and k_⊥ is assumed, then

$$E_x(\omega) = \int d^2{\bm k}_\perp E_{2D}({\bm k}_\perp)\delta (\omega - k_y V_{{\rm sw}}) \cos ^2{(\phi)}$$

$$E_y(\omega) = \int d^2{\bm k}_\perp E_{2D}({\bm k}_\perp)\delta (\omega - k_y V_{{\rm sw}}) \sin ^2{(\phi)},$$

where $E_{2D}({\bm k}_\perp) = C k_\perp ^{-\xi -1}$ is the two-dimensional spectrum, ξ is the one-dimensional scaling exponent, and tan (ϕ) = k_y/k_x. Turner et al. (2011) find that E_y/E_x = 1/ξ, implying a nonaxisymmetric energy distribution in the perpendicular plane. This complication is obviated by measuring $\mathcal {A}_m$ in the standard way, since

$$\begin{eqnarray*} E_\perp (\omega) &=& E_x(\omega) + E_y(\omega) \\ &=& \int d^2{\bm k}_\perp E_{2D}({\bm k}_\perp)\delta (\omega - k_y V_{{\rm sw}}) \end{eqnarray*}$$

avoids the angular sampling issue altogether.

To compare the analysis method outlined in Sections 2 and 3 to solar wind measurements, we choose a five-day interval, 2008 February 12 06:00 to February 17 06:00, of magnetic field data captured by the STEREO A spacecraft. This interval is 1 of 20 similar intervals analyzed by Podesta & TenBarge (2012), where details concerning the data selection and analysis can be found. So, we here summarize only the most pertinent aspects of the measurement and analysis. The interval was chosen because it represents an interval of high-speed wind, which typically satisfies the assumption that the solar wind flow direction is in the heliocentric radial direction, ${\bm R}$. Also, the interval is sufficiently long to include a statistically large sample of periods during which the mean magnetic field, ${\bm B}_0$, is nearly perpendicular to the radial direction, which in practice includes the range 84° < θ_VB < 96°. Focusing on these orthogonal periods is helpful because it facilitates easier comparison to theory since the measured wavenumber in the solar wind corresponds more closely to the perpendicular wavenumber used in linear theory.

The magnetic field data were analyzed using wavelet techniques to determine the local mean magnetic field, ${\bm B}_0(\tau)$, at a given scale τ (Horbury et al. 2008; Podesta 2009). The parallel and perpendicular magnetic energies are then given by $\delta B_\parallel (\tau)^2 = | \hat{{\bm B}}_0(\tau) \cdot \delta {\bm B}(\tau) |^2$ and $\delta B_\perp (\tau)^2 = |\delta {\bm B}(\tau)|^2 - \delta B_\parallel (\tau)^2$, where $\hat{{\bm B}}_0(\tau) = {\bm B}_0(\tau) / B_0(\tau)$.

The proton plasma parameters for the interval were supplied by the PLASTIC instrument (Galvin et al. 2008). The relevant proton plasma parameters are V_p = 657 km s⁻¹, n_p = 2.6 cm⁻³, T_p = 1.7 × 10⁵ K, β_p = 0.65, where all quantities are averaged over the entire interval except β_p whose median is taken because of its high variability. Electron thermal data are unavailable due to problems with the IMPACT Solar Wind Electron Analyzer on board STEREO (Fedorov et al. 2011), so we assume the electron temperature to be the average for fast wind streams found by Newbury et al. (1998), T_e = 1.4 × 10⁵ K. This assumption for the electrons implies T_p/T_e ≃ 1 for the data interval.

To convert spacecraft-frame frequency, f_sc, to wavenumber, we employ Taylor's hypothesis, which states that $2 \pi f_{{\rm sc}} = \omega _{{\rm sc}} = \omega _p + {\bm k}\cdot {\bm V}_{{\rm sw}}$, where ω_p is the plasma rest-frame frequency. For oblique Alfvénic and KAW fluctuations, $\omega _p \ll {\bm k}\cdot V_{{\rm sw}}$ (Howes et al. 2012; Smith et al. 2012), and we can assume 2πf_sc = kV_sw. To compare with linear theory, we convert spacecraft frame frequency to wavenumber using kρ_i = 2πρ_if_sc/V_sw, where ρ_i is computed from the average solar wind proton quantities for the measured interval.

In Figure 10, we compare the measured $\mathcal {A}_m$ for this interval (blue) to the $\mathcal {A}_m$ from linear theory with β_i = 0.65 and T_i/T_e = 1 for two different mixtures of wave modes that reproduce the solar wind $\mathcal {A}_m$ in the inertial range: a mixture that is dominantly Alfvénic (black) with (AC, FS) = (0.9, 0.1) and a mixture that is dominantly fast mode (red dashed) with (AC, FS) = (0.1, 0.9). Although both mixtures of linear wave modes reproduce well the behavior in the inertial range, the fast mode dominant construction displays markedly different transition and dissipation range behavior, whereas the Alfvén dominant mixture fits well the slope of the transition range and the asymptotic value in the dissipation range. From this, we can conclude that the measured interval is most likely dominated by Alfvén wave fluctuations at k_⊥ρ_i ≲ 1 and KAW fluctuations for k_⊥ρ_i ≳ 1.

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We speculate that the positive slope in the measured inertial range $\mathcal {A}_m$ may be attributable to a decreasing fraction of slow wave energy due to collisionless damping as the turbulent cascade proceeds to kinetic scales. Also, the approximate factor-of-two difference between the predicted $\mathcal {A}_m$ from linear theory and the solar wind values in the transition and dissipation ranges are likely due to averaging β_i and ρ_i over the entire five-day interval. The $\mathcal {A}_m$ of KAWs has a moderately strong β_i dependence (see Figures 2(a) and 3). For the purposes of comparison, we use a median β_i = 0.65. However, the $\mathcal {A}_m$ of KAWs increases with decreasing β_i, so periods of low β_i will tend to dominate the average $\mathcal {A}_m$ and shift it upward from the prediction based on a median value of β_i = 0.65. Similarly, we use a value for ρ_i based on quantities averaged over 5 days of data to determine the relationship between frequency and wavenumber. The variance associated with the averaged ρ_i could cause an apparent horizontal shift of the data in the transition and dissipation ranges.

For comparison, we also plot in Figure 10 the measured $\mathcal {A}_m$ for the STEREO A data computed via a more conventional global analysis (green). The five-day interval of data was segmented into 120 one-hour intervals. In each one-hour segment, the mean magnetic field was calculated, the data rotated into mean field coordinates, and a Welch windowed FFT was applied to compute the parallel and perpendicular magnetic energies. The parallel and perpendicular magnetic energies were then averaged over the 120 1 hr segments and binned into logarithmic wavenumber bins to smooth the spectra. The global $\mathcal {A}_m$ (green) was then calculated from the binned and averaged magnetic field data. As expected, this method is inappropriate to compare with linear theory and produces a significantly reduced $\mathcal {A}_m$ compared to the wavelet analysis because the parallel magnetic energy is polluted with perpendicular energy, as discussed in Section 4.1.

This analysis of solar wind data highlights the importance of performing a local analysis and the weaknesses and strengths of the $\mathcal {A}_m$ for determining the solar wind fluctuation composition. The $\mathcal {A}_m$ is a poor tool to use in the inertial range due to degeneracies of different wave mode mixtures. However, if the inertial range $\mathcal {A}_m$ is augmented with a second measure that identifies the fast to total compressible energy fraction (FS), such as the density-parallel magnetic field correlation, the $\mathcal {A}_m$ can provide a measure of the Alfvén to total energy fraction (AC). Also, the slope of the $\mathcal {A}_m$ transition range and the asymptotic value in the dissipation range are useful for identifying the solar wind fluctuation composition.

The MHD equations can be written as

$$\begin{equation} \frac{\partial \rho }{\partial t} + \nabla \cdot (\rho {\bm u}) = 0, \end{equation} \tag{ A1 }$$

$$\begin{equation} \rho \frac{d {\bm u}}{dt} = -\nabla p +\frac{1}{4\pi } (\nabla \times {\bm B}) \times {\bm B}, \end{equation} \tag{ A2 }$$

$$\begin{equation} \frac{d}{dt} \left(\frac{p}{\rho ^\gamma }\right) = 0, \end{equation} \tag{ A3 }$$

$$\begin{equation} c {\bm E}+ {\bm u}\times {\bm B}= 0, \end{equation} \tag{ A4 }$$

$$\begin{equation} \nabla \cdot {\bm B}= 0, \end{equation} \tag{ A5 }$$

$$\begin{equation} \frac{\partial {\bm B}}{\partial t} + c \nabla \times {\bm E}= 0. \end{equation} \tag{ A6 }$$

The linear dispersion relation for this system is

$$\begin{equation} \big(\omega ^2 - v_A^2 k_\parallel ^2 \big) \big[\omega ^4 - \omega ^2 k^2 \big(c_s^2 + v_A^2 \big) + k_\parallel ^2 k^2 v_A^2 c_s^2 \big] = 0, \end{equation} \tag{ A7 }$$

where the first term corresponds to the Alfvén root and the two remaining roots correspond to the fast and slow compressible roots. Finally, the eigenfunctions for the system can be expressed as

$$\begin{equation} \big(\omega ^2 - v_A^2 k_\parallel ^2 \big) \frac{\delta B_x}{B_0} = -k_\perp k_\parallel \frac{\omega ^2 \big(c_s^2 + v_A^2 \big) - c_s^2 v_A^2 k_\parallel ^2}{\omega ^2 - c_s^2 k_\parallel ^2} \frac{\delta B_\parallel }{B_0} \end{equation} \tag{ A8 }$$

and

$$\begin{equation} \big(\omega ^2 - v_A^2 k_\parallel ^2 \big) \frac{\delta B_y}{B_0} = 0. \end{equation} \tag{ A9 }$$

The Alfvén root corresponds to ω² = v²_Ak_∥², which implies δB_∥ = 0. Therefore, the $\mathcal {A}_m$ for the MHD Alfvén root is formally infinite.

The fast and slow compressible roots correspond to ω² ≠ v²_Ak_∥², which implies δB_y = 0. Therefore, δB²_⊥ = δB_x² and after manipulation Equation (A8) reduces to

$$\begin{equation} \mathcal {A}_m = \frac{k_\parallel ^2}{k_\perp ^2}. \end{equation} \tag{ A10 }$$

Note that the $\mathcal {A}_m$ for the fast and slow roots can be trivially derived from Equation (A5) since δB_y = 0.

EMHD is valid for T_i ≪ T_e (β_i ≫ 1) and scales kd_i > 1 (Kingsep et al. 1990; Schekochihin et al. 2009) and is most useful for describing the whistler mode. At these scales, the ions decouple from the magnetic field and are treated as being stationary within the framework of EMHD. Therefore, Ohm's law reduces to ${\bm E}= -{\bm u}_e \times B / c$. Inserting this form of Ohm's law into Faraday's law together with ${\bm J}= - n e {\bm u}_e$, we obtain the EMHD equation

$$\begin{equation} \frac{\partial {\bm B}}{\partial t} + \nabla \times \left[ \left(\nabla \times {\bm B}\right) \times \frac{c {\bm B}}{4 \pi n e} \right] = 0. \end{equation} \tag{ A11 }$$

The linear dispersion relation for this equation is

$$\begin{equation} \omega = \pm v_A k_\parallel d_i k, \end{equation} \tag{ A12 }$$

and the eigenfunctions for this system are

$$\begin{equation} \delta B_x = -\frac{k_\parallel }{k_\perp } \delta B_\parallel \end{equation} \tag{ A13 }$$

and

$$\begin{equation} \delta B_y = \pm i \frac{k}{k_\perp } \delta B_\parallel. \end{equation} \tag{ A14 }$$

Thus, the $\mathcal {A}_m$ for the whistler root is

$$\begin{equation} \mathcal {A}_m = \frac{k_\parallel ^2 + k^2}{k_\perp ^2} = 1 + 2 \frac{k_\parallel ^2}{k_\perp ^2}. \end{equation} \tag{ A15 }$$

ERMHD was introduced in Schekochihin et al. (2009) and is the k_⊥ρ_i ≫ 1, m_e/m_i ≪ 1 limit of gyrokinetics. The system generalizes the EMHD equations for low-frequency, anisotropic (k_∥ ≪ k_⊥) fluctuations without assuming incompressibility, and KAWs are described well by ERMHD. The equations of ERMHD are most simply expressed in terms of scalar flux functions

$$\begin{equation} v_A \frac{\delta {\bm B}_\perp }{B_0} = \hat{{\bm z}} \times \nabla _\perp \Psi \end{equation} \tag{ A16 }$$

and

$$\begin{equation} \delta {\bm u}_\perp = \hat{{\bm z}} \times \nabla _\perp \Phi. \end{equation} \tag{ A17 }$$

The ERMHD equations are

$$\begin{equation} \frac{\partial \Psi }{\partial t} = v_A \left(1 + \frac{T_e}{T_i}\right) \hat{{\bm z}}\cdot \nabla \Phi , \end{equation} \tag{ A18 }$$

$$\begin{equation} \frac{\partial \Phi }{\partial t} = -\frac{v_A}{ 2 + \beta _i \left(1 + \frac{T_e}{T_i}\right)} \hat{{\bm z}}\cdot \nabla \left(\rho _i^2 \nabla _\perp ^2 \Psi \right), \end{equation} \tag{ A19 }$$

$$\begin{equation} \frac{\delta B_\parallel }{B_0} = \sqrt{\beta _i} \left(1 + \frac{T_e}{T_i} \right) \frac{\Phi }{\rho _i v_A}, \end{equation} \tag{ A20 }$$

$$\begin{equation} \frac{\delta n_e}{n_0} = - \frac{2}{\sqrt{\beta _i}} \frac{\Phi }{\rho _i v_A}, \end{equation} \tag{ A21 }$$

$$\begin{equation} u_{\parallel e} = - \frac{\rho _i \nabla _\perp ^2 \Psi }{\sqrt{\beta _i}}, \end{equation} \tag{ A22 }$$

where $\hat{{\bm z}} \cdot \nabla *= \partial */ \partial z + (1/v_A) \hat{{\bm z}} \cdot (\nabla _\perp \Psi \times \nabla _\perp *)$. Note that the anisotropy of the system together with $\nabla \cdot {\bm B}= 0$ implies δB_x ≪ δB_z ∼ δB_y, so we need only to determine the eigenfunction for δB_y.

The linear dispersion relation for this system is

$$\begin{equation} \omega = \pm \sqrt{\frac{1+T_e/T_i}{2 + \beta _i (1 + T_e/T_i)}} k_\perp \rho _i k_\parallel v_A \end{equation} \tag{ A23 }$$

and the eigenfunction is

$$\begin{equation} k_\perp \rho _i \Psi = \rho _i v_A \frac{\delta B_y}{B_0} = \pm \sqrt{\left(1 + \frac{T_e}{T_i}\right)\left[2 + \beta _i \left(1 + \frac{T_e}{T_i} \right) \right]} \Phi. \end{equation} \tag{ A24 }$$

Combining Equations (A20) and (A24) yields the $\mathcal {A}_m$ for the KAW:

$$\begin{equation} \mathcal {A}_m = \frac{2 + \beta _i (1+T_e/T_i)}{\beta _i (1+T_e/T_i)}. \end{equation} \tag{ A25 }$$