RESONANT INTERACTIONS BETWEEN PROTONS AND OBLIQUE ALFVÉN/ION-CYCLOTRON WAVES IN THE SOLAR CORONA AND SOLAR FLARES

Most models for the origin of the fast solar wind require substantial heating of the protons in order to explain the large solar-wind speeds and proton temperatures observed at 1 AU. One of the possible heating mechanisms that has received considerable attention in the literature is resonant cyclotron interactions with Alfvén/ion-cyclotron (A/IC) waves. An appealing feature of cyclotron-heating models is that they naturally lead to an increase in the speed of ion thermal motions in directions perpendicular to the background magnetic field (Vedonov et al. 1961; Abraham-Shrauner & Feldman 1977; Hollweg & Turner 1978), thereby offering a possible explanation for the observation of large perpendicular ion temperatures in the solar corona and solar wind (Kohl et al. 1998; Hollweg 1999a, 1999b, 1999c, 1999d; Cranmer et al. 1999; Antonucci et al. 2000; Marsch & Tu 2001; Hollweg & Isenberg 2002). In order for A/IC waves to resonate with coronal ions, the wave frequencies need to be very close to the ions' gyrofrequencies, in the kHz range. Although such high-frequency waves are not directly observed, they may be produced by magnetic reconnection in the photosphere and chromosphere (Axford & McKenzie 1992; Ruzmaikin & Berger 1998; Sturrock 1999), plasma instabilities (Markovskii & Hollweg 2002, 2004; Markovskii 2007), and/or turbulence (Isenberg & Hollweg 1983; Tu et al. 1984; Marsch 1991; Chandran 2005, 2008; Luo & Melrose 2006; Yoon & Fang 2008).

A/IC waves may also play an important role in solar flares. In flares, compressible MHD turbulence is excited by the large-scale flows induced by the magnetic tension of reconnected magnetic field lines. Within these turbulent flows, wave energy cascades to small scales (Yan & Lazarian 2004; Petrosian et al. 2006; Yan et al. 2008). Turbulent, small-scale A/IC waves may then stochastically accelerate ions to large energies (Eichler 1979; Miller & Roberts 1995; Liu et al. 2004, 2006).

Most of the work that has been done on A/IC waves in the corona and solar flares has worked in the approximation that the wave vector k is parallel to the background magnetic field B₀ ("slab symmetry"). We refer to waves with k parallel to B₀ as "slab waves." When protons in a low-β plasma interact with slab A/IC waves (where β = 8πp/B²₀ is the ratio of thermal to magnetic pressure), they diffuse along a set of nested, closed contours in the v_⊥ − v_∥ plane defined by the equation η(v_⊥, v_∥) = constant, where v_⊥ and v_∥ are the components of the proton velocity parallel and perpendicular to B₀, and the approximate functional form of η is given below in Equation (19). If wave–particle interactions are sufficiently strong, the proton distribution function f becomes almost constant along these η = constant surfaces, and proton damping of slab waves becomes very weak. (Vedonov et al. 1961; Rowlands et al. 1966; Kennel & Engelmann 1966; Kennel & Petschek 1966). To investigate the effects of this "quasilinear relaxation" in coronal holes, Isenberg et al. (2001) and Isenberg (2001, 2004) developed the "kinetic-shell model," in which wave pitch-angle scattering by slab waves causes f to relax toward a function of η on a short timescale. Protons then evolve on a longer timescale in response to the magnetic-mirror force, gravity, and the electric field required to maintain charge neutrality. (Similar ideas were also used in the work of Galinsky & Shevchenko (2000).) It is only this longer-timescale evolution that allows protons to absorb energy ongoingly from the A/IC waves, since slab A/IC waves are not damped when f = f(η). Upon taking into account wave dispersion, Isenberg (2004) found that the kinetic-shell model results in only weak acceleration of protons, and that the dissipation of slab A/IC waves is unable to explain the heating and acceleration of protons in the fast solar wind.

Although the assumption that k is parallel to B₀ leads to useful simplifications in theoretical calculations, the candidate sources for A/IC waves in the solar corona and solar flares produce waves with a distribution of wavevector directions. In this paper, we therefore consider oblique A/IC waves, for which the angle θ between k and B₀ is nonzero. Before presenting our new results, we describe some general results from quasilinear theory (Section 2) and review the properties of resonant interactions between protons and slab A/IC waves (Section 3). In Section 4, we argue that if the evolution of f is dominated by cyclotron–resonant interactions between protons and high-frequency A/IC waves having a range of θ values, then wave–particle interactions initially make f so anisotropic that the plasma becomes unstable to A/IC waves with θ = 0. We argue that the resulting amplification of θ = 0 waves subsequently causes the angular distribution of A/IC waves to become sharply peaked around θ = 0 at the large wavenumbers at which A/IC waves resonate with protons, and that wave pitch-angle scattering then causes f to become approximately constant along surfaces of constant η. We present a graphical argument to explain why oblique waves are damped when f = f(η), and in Section 5 we analytically calculate the approximate value of the proton contribution to the damping rate of oblique A/IC waves, assuming f = f(η). In Section 6, we analytically calculate the electron contribution to the damping rate of oblique A/IC waves, assuming Maxwellian electrons, and in Section 7 we discuss the possible consequences of our results for proton heating from the dissipation of high-frequency A/IC-wave turbulence in the solar corona.

Our analysis suggests that proton heating by oblique, high-frequency A/IC waves is qualitatively different than proton heating by A/IC waves with θ = 0. If θ = 0 for all the A/IC waves, then wave pitch-angle scattering causes the protons to evolve toward a state in which the damping of these slab A/IC waves vanishes. As discussed above, it is in large part because of this that slab A/IC waves are unable to produce the heating and acceleration of the fast solar wind in Isenberg's (2004) kinetic-shell model. In contrast, if protons are heated by resonant–cyclotron interactions with oblique A/IC waves, then f does not evolve toward a state in which the damping of these waves ceases. Instead, it again evolves to a state in which f is approximately a function of η, and the oblique A/IC waves continue to be damped by protons. Because of this difference, oblique high-frequency A/IC waves can in principle be more effective at heating protons than A/IC waves with θ = 0. Although the proton–cyclotron damping rate of oblique A/IC waves when f = f(η) is significantly less than the damping rate that arises for Maxwellian protons, we find that it is still much larger than the electron contribution to the damping rate at large wavenumbers and moderate obliquities. As a result, in at least some circumstances most of the heating power that results from the dissipation of oblique A/IC waves will be deposited into the protons, even when f = f(η).

We note that we consider only the idealized case of a proton–electron plasma in order to simplify the analysis. It would be of interest in a future study to account for helium, which significantly changes the properties of the linear wave dispersion relation at the frequencies at which cyclotron resonance occurs (see, e.g., Hollweg & Isenberg (2002), and references therein). Possible consequences of the altered dispersion relation for the acceleration of ³He in solar flares have been studied by Liu et al. (2004, 2006).

In a collisionless, non-relativistic plasma, the distribution function or phase-space density f(x, v, t) of a particle species with charge q and mass m evolves according to the Vlasov equation,

$$\begin{equation} \frac{\partial f}{\partial t} + {\bf v} \cdot \nabla f + \frac{q}{m} \left({\bf E} + \frac{1}{c}\, {\bf v} \times {\bf B} \right) \cdot \nabla _v f = 0, \end{equation} \tag{ 1 }$$

where E and B are the electric and magnetic fields, and ∇_v is a gradient with respect to the velocity v. For most cases of interest, an analytic solution of the coupled Vlasov and Maxwell equations is not possible. On the other hand, in the case of a uniform plasma perturbed by small-amplitude, weakly damped waves, Equation (1) can be solved approximately using a standard technique known as quasilinear theory (Yakimenko 1963; Kennel & Engelmann 1966; Stix 1992). When this theory is applied to a plasma pervaded by a uniform background magnetic field ${\bf B}_0 = B_0 {\bf \hat{z}}$, and when there is no background electric field, each particle's orbit is approximated as the unperturbed helix that would arise in the absence of wave excitation. A first-order perturbation to f is then calculated, and the right-hand side of Equation (1) is averaged over a large volume V, yielding the following equation for the (averaged) distribution function (Kennel & Engelmann 1966),

$$\begin{eqnarray} \frac{\partial f}{\partial t} &=& \lim _{V \rightarrow \infty } \sum _{n=-\infty }^{\infty } \frac{\pi q^2 }{m^2} \int \frac{ d^3 k}{(2\pi)^3 V v_\perp } G\, v_\perp \delta (\omega _{kr} - k_\parallel v_\parallel - n \Omega)\nonumber\\ &&\times |\psi _{n,k}|^2 G \,f, \end{eqnarray} \tag{ 2 }$$

where v_∥ (v_⊥) is the component of v parallel (perpendicular) to B₀,

$$\begin{equation} G \equiv \left(1 - \frac{k_\parallel v_\parallel }{\omega _{kr}}\right) \frac{\partial }{\partial v_\perp } + \frac{k_\parallel v_\perp }{ \omega _{kr}} \frac{\partial }{\partial v_\parallel }, \end{equation} \tag{ 3 }$$

ω_kr is the real part of the wave frequency at wave vector k, Ω = qB₀/mc is the cyclotron frequency, k_∥ (k_⊥) is the component of k parallel (perpendicular) to B₀,

$$\begin{equation} \psi _{n,k} = \frac{1}{\sqrt{2}}[ E_{k,{\rm r}} e^{i\phi } J_{n+1}(\sigma) + E_{k,{ l}} e^{-i\phi } J_{n-1}(\sigma)] + \frac{v_\parallel }{v_\perp } E_{kz}J_n(\sigma), \end{equation} \tag{ 4 }$$

J_n is the Bessel function of nth order, σ = k_⊥v_⊥/Ω, ϕ is the azimuthal angle in wavenumber space, $E_{k,{\rm r}} = (E_{kx}-iE_{ky})/\sqrt{2}$, $E_{k, {l}} = (E_{kx}+iE_{ky})/\sqrt{2}$, and E_k is the Fourier transform of the electric field, given by ∫E(x, t)exp(−ik · x) d³x.¹

Although Equation (2) is itself difficult to solve in many cases, two important results can be obtained even when the full solution for f is not available. These results are illustrated in Figure 1. First, for particles with parallel velocity v_∥, the delta function in Equation (2) restricts the integral to waves satisfying the resonance condition

$$\begin{equation} \omega _{kr} - k_\parallel v_\parallel = n \Omega, \end{equation} \tag{ 5 }$$

where n is any integer. The left-hand side of Equation (5) is the Doppler-shifted wave frequency seen in the frame of reference moving along B₀ at velocity v_∥. If this Doppler-shifted frequency is an integer multiple of Ω, then the effects of the wave's electromagnetic field on the particle add coherently over successive wave periods, and the wave has a large effect on the particle.

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Second, at each point in velocity space the operator G is proportional to a derivative in a particular direction in the v_∥ − v_⊥ plane. This direction is along a semi-circular curve of constant particle energy K as measured in a frame of reference moving at the wave phase velocity along the background magnetic field ${\bf \hat{z}}\omega _{kr}/k_\parallel$, where

$$\begin{equation} K = \frac{m}{2}\left[ \left(v_\parallel - \frac{\omega _{kr}}{k_\parallel }\right)^2 + v_\perp ^2 \right]. \end{equation} \tag{ 6 }$$

Because the operator G appears twice, Equation (2) implies that each wave causes resonant particles to diffuse in velocity space along a curve of constant K. If particles throughout a broad range of v_∥ interact with a spectrum of waves at different k_∥ and ω_kr, but all the waves have the same value of ω_kr/k_∥, then the particles can diffuse a substantial distance in velocity space while conserving K. The reason that K is conserved in this case is the following. In the reference frame moving at velocity ${\bf \hat{z}}\omega _{kr}/k_\parallel$, the magnetic field is independent of time, and Faraday's law thus implies that ∇ × E = 0, so that E = −∇Φ, where Φ is the electrostatic potential. As a consequence, the change in a particle's kinetic energy is equal and opposite to the change in the particle's electrostatic potential energy. Since there is no background electric field in this reference frame, the variations in Φ are bounded and small. The particle's kinetic energy in this reference frame thus undergoes only small-amplitude oscillations and does not increase or decrease secularly in time.

We consider Alfvén/ion-cyclotron (A/IC) waves in a low-β, proton–electron plasma of density ρ₀ pervaded by a uniform magnetic field ${\bf B}_0 = B_0 \hat{z}$. We assume that the real part of the wave frequency, ω_kr, is given by the cold-plasma dispersion relation (Stix 1992, Chapter 2, Equation (21); Hollweg 2000),

$$\begin{equation} \frac{k_n^4 \cos ^2 \theta }{w^4} - \frac{k_n^2(1 + \cos ^2 \theta)}{w^2(1-w^2)} + \frac{1}{1-w^2} = 0, \end{equation} \tag{ 7 }$$

where

$$\begin{equation} w = \frac{\omega _{kr}}{\Omega _p}, \end{equation} \tag{ 8 }$$

Ω_p is the proton cyclotron frequency,

$$\begin{equation} k_n = \frac{k v_{\rm A}}{\Omega _p}, \end{equation} \tag{ 9 }$$

$v_{\rm A}=B_0/\sqrt{4\pi \rho _0}$ is the Alfvén speed, and θ is the angle between k and B₀. In Equation (7), we have neglected terms proportional to v²_A/c², m_e/m_p, and (m_e/m_p)tan²θ, where m_e and m_p are the electron and proton masses.

The root of Equation (7) corresponding to A/IC waves with θ = 0 (i.e., "slab waves") is given by

$$\begin{equation} w = \pm \left(\frac{2 k_n^2 + k_n^4 -k_n^3\sqrt{4 + k_n^2}}{2}\right)^{\!\!\!1/2}. \end{equation} \tag{ 10 }$$

For k_n ≫ 1, the right-hand side of Equation (10) can be expanded in powers of k⁻²_n to yield

$$\begin{equation} w = \pm \left(1 - \frac{1}{k_n^2} + \frac{2}{k_n^4} + \dots \right). \end{equation} \tag{ 11 }$$

Through order k⁻⁴_n, Equation (11) is equivalent to the approximate solution w ≃ 1 − (k²_n + 2)⁻¹ obtained by Hollweg (1999e).

When only waves with θ = 0 are present, the only nonvanishing terms in the sum on the right-hand side of Equation (2) are the n = 1 and n = −1 terms. This is because J_n(0) = 0 unless n = 0, and also because we have set m_e/m_p → 0, which implies that E_kz = 0. In Figure 2, we plot the A/IC wave frequency in Equation (10) (solid lines) as well as the approximate expression for ω_kr given in Equation (11) (dashed lines). The dotted lines are plots of nΩ_p + k_∥v_∥ for n = ±1 and v_∥ = ∓0.03v_A. For simplicity, we focus on positive k_∥; the discussion and plots are unchanged if one sets k_∥ → −k_∥, ω_kr → −ω_kr, and n → −n. For each (nonzero) value of v_∥ there is a unique value of |k_∥|, denoted k_∥,res, that satisfies the resonance condition for protons (i.e., Equation (5) with Ω = Ω_p). This "resonant wavenumber" corresponds to the intersection of the dotted and solid lines.² As illustrated in the figure, protons with positive v_∥ resonate with A/IC waves with negative parallel phase velocity ω_kr/k_∥, while protons with v_∥ < 0 resonate with A/IC waves with ω_kr/k_∥>0 (Dusenbery & Hollweg 1981).

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It can also be seen from Figure 2 that protons with very large |v_∥| resonate with waves with small |k_∥|, for which |ω_kr/k_∥| ∼ v_A. On the other hand, as |v_∥| decreases, k_∥,res increases and |ω_kr/k_∥,res| decreases. The condition |v_∥| ≪ v_A implies that

$$\begin{equation} k_{n\parallel, {\rm res}}\, \equiv \, \frac{k_{\parallel, {\rm res}} v_A}{\Omega _p} \, \gg \,1. \end{equation} \tag{ 12 }$$

In this limit, Equations (5) and (11) can be used to express k_n∥,res as an asymptotic series in powers of |v_∥|/v_A,

$$\begin{equation} k_{n\parallel, {\rm res}} = \left| \frac{v_\parallel }{v_{\rm A}}\right|^{-1/3} \left(1 - \frac{2}{3}\left|\frac{v_\parallel }{v_{\rm A}}\right|^{2/3} + \dots \right). \end{equation} \tag{ 13 }$$

(The first term in this series was used as an approximation by Hollweg (1999e).) Equations (5), (11), and (13) can then be used to express the parallel phase velocity v_ph = ω_kr/k_∥ of the resonant waves as an asymptotic series in v_∥/v_A:

$$\begin{equation} v_{\rm ph}(v_\parallel) = \pm v_{\rm A}^{2/3}|v_\parallel |^{1/3} \left(1 - \frac{1}{3} \left|\frac{v_\parallel }{v_{\rm A}}\right|^{2/3} + \dots \right), \end{equation} \tag{ 14 }$$

where the sign of v_ph(v_∥) is the opposite of the sign of v_∥.

As discussed in Section 2 and illustrated in Figure 1, a small-amplitude, weakly damped wave causes particles satisfying the resonance condition in Equation (5) to diffuse in velocity space along semi-circular paths of constant energy K as measured in the reference frame moving at velocity ${\bf \hat{z}}\omega _{kr}/k_\parallel$. When many different waves are present, particles with a given velocity typically resonate with more than one wave at the same time, and the constant-K curves for the different resonant waves do not coincide. On the other hand, when only slab waves are present, each proton resonates with waves with only a single value of |k_∥|. In this case, proton diffusion in the v_∥ − v_⊥ plane is one dimensional, in the sense that at each point in the v_∥ − v_⊥ plane protons diffuse along a single characteristic curve, and not in the direction perpendicular to this curve. We call these characteristic curves "scattering contours." Quantitatively, the equation describing these scattering contours can be written

$$\begin{equation} \eta (v_\perp, v_\parallel) = \mbox{constant}, \end{equation} \tag{ 15 }$$

where η is any function that satisfies the equation G(η) = 0 for waves with ω_kr/k_∥ = v_ph(v_∥); i.e.,

$$\begin{equation} \left[1 - \frac{v_\parallel }{v_{\rm ph}(v_\parallel)}\right] \frac{\partial \eta }{\partial v_\perp } + \frac{ v_\perp }{v_{\rm ph}(v_\parallel)}\frac{\partial \eta }{\partial v_\parallel } = 0. \end{equation} \tag{ 16 }$$

Equation (16) is solved by setting

$$\begin{equation} \eta = v_\perp ^2 + h(v_\parallel), \end{equation} \tag{ 17 }$$

where h satisfies the differential equation (Rowlands et al. 1966; Gendrin 1968)

$$\begin{equation} \frac{dh}{dv_\parallel } = 2[v_\parallel - v_{\rm ph}(v_\parallel) ]. \end{equation} \tag{ 18 }$$

For |v_∥| ≪ v_A, we can use Equations (14) and (18) to solve for h as an asymptotic series in powers of |v_∥|/v_A, from which we obtain

$$\begin{equation} \eta = v_\perp ^2 + \frac{3v_{\rm A}^{2/3}|v_\parallel |^{4/3}}{2} + \frac{2v_\parallel ^2}{3} + \dots \end{equation} \tag{ 19 }$$

(A parametric solution for the η = constant surfaces was obtained by Isenberg & Lee (1996).) The shape of the η = 0.075v²_A contour is plotted (solid line) in Figure 3. In this plot, v₀=-0.08v_A and v₁ = −0.02v_A, and v_ph(v₀) and v_ph(v₁) are obtained from Equation (14). At v_∥ = v₀, the constant-η contour is perpendicular to the dotted-line connecting the contour to the point [0, v_ph(v₀)], reflecting the point made in Section 2, that the pitch-angle scattering at v₀ occurs at constant energy as measured in the reference frame moving at velocity v_ph(v₀) along B₀. Similarly, at v_∥ = v₁, the constant-η contour is perpendicular to the dotted line connecting the contour to the point [0, v_ph(v₁)].

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If protons interact with a broad spectrum of A/IC waves, all with θ = 0, and if the proton distribution function f is initially Maxwellian, then the protons will diffuse in velocity space along the η = constant scattering contours, resulting in a net increase in average value of v²_⊥. The energy gained by protons is removed from the waves, causing the waves to damp. However, in the limit of strong wave pitch-angle scattering, f will eventually become constant along the scattering contours, the protons will stop gaining energy from the waves, and the wave damping will cease (Kennel & Petschek 1966).

In this section, we argue that if the evolution of the proton distribution is dominated by resonant interactions with A/IC waves, and if the A/IC wave vectors are distributed over a range of directions, then the proton distribution function will (at least initially) become so anisotropic that A/IC waves with θ = 0 become unstable and grow. As in the previous section, we assume that ω_kr is given by the cold-plasma dispersion relation and take the proton distribution to be confined to highly sub-Alfvénic parallel velocities, so that resonant wave–particle interactions arise only for k²_n∥ ≫ 1 and |w| ≃ 1, where

$$\begin{equation} k_{n \parallel } = \frac{k_\parallel v_{\rm A}}{\Omega _p}. \end{equation} \tag{ 20 }$$

In this limit, the solution to Equation (7) can be expressed as an asymptotic series in powers of |k_n∥|⁻²,

$$\begin{equation} w = \pm \left[ 1 - \frac{(1 + \cos ^2\theta)}{2 k_{n \parallel }^2} + \cdots \ \right], \end{equation} \tag{ 21 }$$

In general, for A/IC waves with nonzero θ, one has to take into account wave–particle resonances with |n| ≠ 1 in Equation (5). However, for A/IC waves and protons with |v_∥| ≪ v_A, these resonances occur only for extremely large wavenumbers, |k_n∥| ∼ v_A/|v_∥|, while the n = ±1 resonances arise for smaller wavenumbers, |k_n∥| ∼ (v_A/|v_∥|)^1/3. We assume that the power spectrum of A/IC waves decreases rapidly with increasing |k_n∥| at |k_n∥| ≳ (v_A/|v_∥|)^1/3 due to wave damping. We thus neglect resonances with |n| ≠ 1. Using Equations (5) and (21) with n = ±1, we find that the value of |k_n∥| for which A/IC waves with a given θ resonate with protons with a given v_∥ is, to leading order in |v_∥|/v_A,

$$\begin{equation} k_{n\parallel,{\rm res}} = \left[\frac{v_{\rm A}(1 + \cos ^2\theta)}{2|v_\parallel |} \right]^{1/3}. \end{equation} \tag{ 22 }$$

The parallel phase velocity ω_kr/k_∥ of the waves that are resonant with particles of parallel velocity v_∥ is then, to leading order in |v_∥|/v_A,

$$\begin{equation} v_{\rm ph} (v_\parallel, \theta) = \pm \left(\frac{2 v_{\rm A}^2 |v_\parallel |}{1 + \cos ^2\theta } \right)^{1/3}. \end{equation} \tag{ 23 }$$

The resonance condition requires that the sign of v_ph be the opposite of the sign of v_∥, as for waves with θ = 0. In this section we use a different notation than in Equation (14), in that we now take v_ph to be a function of both v_∥ and θ.

The key point to emerge from Equation (23) is that |v_ph| is an increasing function of θ at fixed v_∥ for 0 < θ < 90°. One of the consequences of this point is illustrated in the top panel of Figure 4. In this panel, we take the proton distribution to be constant along the scattering contours corresponding to the slab waves. The scattering contour for η = 0.075v²_A is shown as a solid-line curve. Protons at v_∥ = v₀ that are scattered by oblique A/IC waves with some nonzero value of θ will scatter along the dashed-line trajectory, which corresponds to constant energy as measured in a reference frame moving at velocity v_ph(v₀, θ) along the magnetic field. (We note that we have artificially increased v_ph(v₀, θ) relative to v_ph(v₀, 0) in both panels of Figure 4 to make the figure easier to read.) If we take f to be a decreasing function of η, then there will be a net diffusive flux of protons upward along this dashed line, resulting in an increase in particle energy and damping of oblique waves.

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A second consequence of Equation (23) is illustrated in the bottom panel of Figure 4. Here, we take the proton distribution to be constant along the scattering contours or "kinetic shells" corresponding to waves with a single nonzero value of θ. One of these contours is now drawn with a solid line. Protons at v_∥ = v₀ interacting with A/IC waves with θ = 0 will scatter along the short-dashed line in this panel, which locally corresponds to an η = constant curve, with η defined in Equation (19). If we take the proton distribution to decrease as one moves to shells that are farther from the origin, then there will be a net diffusive flux of protons downward along the short-dashed line in the bottom panel of Figure 4. In this case, the protons will lose energy, and waves with θ = 0 will be amplified.

If some mechanism such as plasma instabilities or a turbulent cascade generates high-frequency A/IC waves with a range of θ values, and if the form and evolution of the proton distribution function are dominated by wave–particle interactions, then interactions involving waves with nonzero θ will initially act to make the constant-f contours somewhat steeper in the v_∥ − v_⊥ plane than the η = constant scattering contours of the slab waves. This in turn will lead to the amplification of waves with θ = 0 and cause the angular distribution of the waves at large k_n that resonate with the protons to become sharply peaked around θ = 0. Wave–particle interactions will subsequently become dominated by waves with θ = 0, f will become approximately constant along surfaces of constant η, and oblique waves will be damped. Numerical evidence supporting these arguments has recently been presented by Pongkitiwanichakul et al. (2010). We note that the damping of oblique waves coupled with the amplification of slab waves represents a kinetic mechanism for transporting wave energy toward θ = 0. In the next section, we analytically calculate γ_k assuming f = f(η).

In this section, we consider a low-β proton–electron plasma in which f = f(η), where f is the proton distribution function and η is given by Equation (19). We then calculate the proton contribution to the damping rate of A/IC waves with θ ≠ 0. We denote the proton and electron contributions to the damping rate γ^(p)_k and γ^(e)_k, respectively. We assume that θ = 0 waves are present with both positive and negative values of ω_kr/k_∥, so that the scattering contours depicted in Figure 3 are filled for both positive and negative v_∥. We assume that the total damping rate at wavevector k,

$$\begin{equation} \gamma _k = \gamma _k^{\rm (p)} + \gamma _k^{\rm (e)}, \end{equation} \tag{ 24 }$$

satisfies the inequality

$$\begin{equation} |\gamma _k| \ll |\omega _{kr} - n\Omega _p| \end{equation} \tag{ 25 }$$

for all n. If f were Maxwellian, this assumption would break down throughout a range of wavevectors in which |ω_kr| is close to Ω_p. (Stawicki et al. 2001; Gary & Borovsky 2004) However, because f = f(η), γ^(p)_k vanishes at all k at θ = 0, and is ≪|ω_kr − nΩ_p| at all k and for all n if θ is sufficiently small. It is not clear a priori that Equation (25) is valid at all values of θ, but the results of our calculation are consistent with Equation (25), at least for θ ⩽ 45°.³

When Equation (25) is satisfied, the proton contribution to the damping rate is (see Kennel & Wong 1967)

$$\begin{eqnarray} \frac{\gamma _k^{(\rm p)}}{|\omega _{kr}|} &=& \frac{\pi }{8N} \left|\frac{\omega _{kr}}{k_\parallel }\right| \left(\frac{\omega _{pi}}{\omega _{kr}}\right)^2 \sum _{n=-\infty }^{\infty } \int _0^\infty dv_\perp \, v_\perp ^2 \nonumber\\ &&\times \int _{-\infty }^{\infty } dv_\parallel \, \frac{|\psi _{n,k}^{(p)}|^2 \,G(f)}{W_k}\;\; \delta \!\!\left(v_\parallel - \frac{\omega _{kr} - n\Omega _{p}}{k_\parallel }\right),\quad \ \ \ \end{eqnarray} \tag{ 26 }$$

where N is the proton number density, ω²_pi = 4πNe²/m_p is the proton plasma frequency, m_p and e are the proton mass and charge, G is defined in Equation (3),

$$\begin{equation} W_k = \frac{1}{16\pi } \left[ {\bf B}_k^\ast \cdot {\bf B}_k + {\bf E}_k^\ast \cdot \frac{\partial }{\partial \omega } (\omega {\bf \epsilon }_h) \cdot {\bf E}_k \right], \end{equation} \tag{ 27 }$$

E_k and B_k are the spatial Fourier transforms of the electric and magnetic fields, _h is the hermitian part of the dielectric tensor, which we approximate using the dielectric tensor for a cold plasma, and the frequency in Equation (27) is evaluated at ω = ω_kr. The quantity ψ^(p)_n,k is equal to the quantity ψ_n,k defined in Equation (4) with Ω set equal to Ω_p. Given the Fourier transform convention described following Equation (4) and in footnote 1, the wave energy (electromagnetic plus kinetic) per unit volume is lim_V→∞ (2π)⁻³ V⁻¹∫2W_kd³k (see Equations (20) and (67) in Stix 1992, Chapter 4), and its rate of change is lim_V→∞ (2π)⁻³ V⁻¹∫4γ_kW_kd³k. Adding this to the rate of change of the proton energy per unit volume 0.5m_p∫v²(∂f/∂t) d³v with ∂f/∂t given by Equation (2), and integrating by parts, one finds that the sum of the wave energy and proton energy lim_V→∞ (2π)⁻³ V⁻¹∫2W_kd³k + 0.5m_p∫v²f d³v remains constant. As in Section 3, we neglect terms of order m_e/m_p when determining the dispersion relation and polarization properties of the waves. There is thus no parallel electric field fluctuation.

In general, for oblique waves, one has to include all the terms in the sum over n in Equation (26). However, we take the typical proton parallel velocity, v_∥,th, to be ≪v_A. As shown in Equation (22), the resonance condition for n = ±1 can be satisfied by the bulk of the protons for k_n∥ ∼ (v_A/v_∥,th)^1/3. In contrast, the resonance condition for |n| ≠ 1 can only be satisfied by the bulk of the protons for much larger wavenumbers, k_n∥ ∼ v_A/v_∥,th, as discussed in the previous section. Moreover, we find that the damping rate decreases rapidly with increasing k_n∥ for k_n∥ ≳ (v_A/v_∥,th)^1/3. Anticipating our discussion below, this is primarily because as k_n∥ increases, ω_kr → Ω_p, and the wave energy becomes increasingly dominated by the kinetic energy of the oscillating proton fluid velocity, so that |E_kx|²/W_k decreases. The damping caused by resonant protons interacting with the wave electric field then becomes energetically less important, and the damping rate drops. We thus focus on proton damping at wavenumbers k_n∥ ≪ v_A/v_∥,th, for which the n = ±1 resonances dominate, and neglect resonances with |n| ≠ 1. On the other hand, we take advantage of the fact that k²_n∥,res ∼ (v_A/|v_∥|)^2/3 ≫ 1 and calculate γ^(p)_k as an asymptotic series in powers of (1 − w), or equivalently k⁻²_n∥, keeping only the leading-order term. Since we treat k²_n∥ as large, and since we neglect the |n| ≠ 1 resonances which arise at k_n∥ ≳ v_A/v_∥,th, our analysis is restricted to the wavenumber range

$$\begin{equation} 1 \ll k_{n \parallel }^2 \ll \left(\frac{v_{\rm A}}{v_{\parallel, {\rm th}}}\right)^2. \end{equation} \tag{ 28 }$$

Without loss of generality, we consider waves with ω_kr>0 and k_∥>0, which resonate with protons with v_∥ < 0. (The value of |γ^(p)_k/ω_kr| is unchanged if we take ω_kr/k_∥ < 0 since the proton distribution is symmetric in v_∥.) The damping rate is independent of ϕ, and so we set ϕ = 0, k_y = 0, and k_x = k_⊥. To leading order in (1 − w), the polarization of A/IC waves is given by (Stix 1992, Chapter 2, Equation (28))

$$\begin{equation} E_{ky} = -i \cos ^2(\theta) E_{kx}. \end{equation} \tag{ 29 }$$

The recurrence relations 2nJ_n(x)/x = J_n−1(x) + J_n+1(x) and 2J'_n(x) = J_n−1(x) − J_n+1(x), together with Equation (29), imply that

$$\begin{equation} |\psi _{1,k}^{(p)}|^2 = |E_{kx}|^2 \, F\!\left(\frac{k_\perp v_\perp }{\Omega _p}, \theta \right), \end{equation} \tag{ 30 }$$

where

$$\begin{eqnarray} &&F(x,\theta) = \frac{\left[ J_1(x)\right]^2}{x^2} + \frac{2 J_1(x) J_1^\prime (x) \cos ^2 \theta }{x} + [ J_1^\prime (x)]^2\cos ^4 \theta,\nonumber\\ \end{eqnarray} \tag{ 31 }$$

and J'_n(x) = dJ_n(x)/dx. Using Equation (21) above and Equation (18) from Chapter 1 of Stix (1992), and neglecting terms proportional to m_e/m_p, we find that to leading order in k⁻²_n∥,

$$\begin{equation} W_k = \frac{ \omega _{pi}^2 k_{n \parallel }^{4} |E_{kx}|^2}{8\pi \Omega _p^2}. \end{equation} \tag{ 32 }$$

(The derivation of Equation (32) is described in Section 6.) Equation (21) implies that to leading order in k⁻²_n∥,

$$\begin{equation} v_{\parallel, {\rm res}} \equiv \frac{\omega _{kr} - \Omega _p}{k_\parallel } = - \: \frac{v_{\rm A} s}{ k_{n \parallel }^3}, \end{equation} \tag{ 33 }$$

where

$$\begin{equation} s = \frac{1 + \cos ^2 \theta }{2}. \end{equation} \tag{ 34 }$$

After setting f = f(η), we obtain

$$\begin{equation} \int _{- \infty }^\infty dv_\parallel \, G(f)\; \delta (v_\parallel - v_{\parallel, {\rm res}}) = 2 v_\perp (1 - s^{1/3})\left. \frac{df}{d\eta }\right|_{\eta = \eta _{\rm res}}, \end{equation} \tag{ 35 }$$

where

$$\begin{equation} \eta _{\rm res} = v_\perp ^2 + \frac{3 v_{\rm A}^2 s^{4/3}}{2 k_{n \parallel }^4}. \end{equation} \tag{ 36 }$$

Since we have taken ω_kr>0, we can neglect the n=−1 term in Equation (26), and keep only the n = 1 term. Substituting Equations (30) through (35) into Equation (26), we obtain

$$\begin{eqnarray} &&\frac{\gamma _k^{\rm (p)}}{\Omega _p} = \frac{2\pi ^2 v_{\rm A} (1-s^{1/3})}{N k_{n \parallel }^5} \int _0^\infty dv_\perp v_\perp ^3 F\left(\frac{k_\perp v_\perp }{\Omega _p}, \theta \right) \left. \frac{df}{d\eta }\right|_{\eta = \eta _{\rm res} }.\nonumber\\ \end{eqnarray} \tag{ 37 }$$

As an example, we set

$$\begin{equation} f(v_\perp, v_\parallel) = c_1 e^{-\eta /\eta _0} \end{equation} \tag{ 38 }$$

with η₀ ≪ v²_A. The condition N = ∫d³vf then implies that to leading order in η₀/v²_A,

$$\begin{equation} c_1 = \left(\frac{2}{3}\right)^{\!\!\!1/4}\!\!\!\! \frac{ N v_{\rm A}^{1/2}}{\pi \, \Gamma (3/4)\, \eta _0^{7/4}}, \end{equation} \tag{ 39 }$$

where Γ is the Gamma function. For this proton distribution function, Equation (37) and the Bessel-function identities on pp. 257–258 of Stix (1992) imply that

$$\begin{eqnarray} &&\frac{\gamma _k^{\rm (p)}}{\Omega _p} = - \,\left(\frac{2}{3}\right)^{\!\!\!1/4}\!\! \frac{\pi v_{\rm A}^{3/2} (1 - s^{1/3})H\left(\lambda, \theta \right)}{2\, \Gamma (3/4)\,\eta _0^{3/4} k_{n \parallel }^5} \;\exp \!\left[ - \frac{3v_{\rm A}^2 s^{4/3}}{2 \eta _0\,k_{n \parallel }^4 }\right],\nonumber\\ \end{eqnarray} \tag{ 40 }$$

where

$$\begin{equation} \lambda = \frac{k_\perp ^2 \eta _0}{2 \Omega _p^2} \end{equation} \tag{ 41 }$$

is roughly the square of k_⊥ρ_p for the bulk of the protons, ρ_p is the proton gyroradius,

$$\begin{eqnarray} H(\lambda, \theta) &=& e^{-\lambda }\left\lbrace \frac{I_1(\lambda)}{\lambda } - 2\left[ I_1(\lambda) - I_1^\prime (\lambda)\right]\cos ^2\theta \right.\nonumber\\ &&\left.+ \left[ \frac{I_1(\lambda)}{\lambda } + 2\lambda I_1(\lambda) - 2\lambda I_1^\prime (\lambda)\right]\cos ^4\theta \right\rbrace,\qquad \end{eqnarray} \tag{ 42 }$$

I₁(λ) = (λ/2)(1 +  λ²/8 + ⋅ ⋅ ⋅) is the modified Bessel function of order 1, and I'₁(λ) = dI₁(λ)/dλ.

To simplify our results further, we consider the limit

$$\begin{equation} \tan ^2\theta \ll \frac{v_{\rm A}^2}{\eta _0 k_{n \parallel }^2}. \end{equation} \tag{ 43 }$$

For the values of k_n∥ at which γ^(p)_k is maximal, i.e., k_n∥ ∼ v^1/2_A/η^1/4₀, the inequality in Equation (43) is satisfied at most values of θ but is not satisfied when θ is very close to 90°. Equation (43) implies that

$$\begin{equation} \lambda \ll 1. \end{equation} \tag{ 44 }$$

In this limit, to leading order in λ, H(λ, θ) = 2s², and

$$\begin{equation} \frac{\gamma _k^{\rm (p)}}{\Omega _p} = - \,\left(\frac{2}{3}\right)^{\!\!\!1/4}\!\! \frac{\pi v_{\rm A}^{3/2} s^2(1 - s^{1/3})}{ \Gamma (3/4)\,\eta _0^{3/4} k_{n \parallel }^5} \;\exp \!\left[- \frac{3v_{\rm A}^2 s^{4/3}}{2 \eta _0\,k_{n \parallel }^4 }\right]. \end{equation} \tag{ 45 }$$

For fixed |θ|, the maximum value of |γ^(p)_k|, denoted γ^(p)_max, occurs at k_n∥ = k_n∥,max, where

$$\begin{equation} k_{n\parallel, {\rm max}} = \left(\frac{6 v_{\rm A}^2 s^{4/3}}{5 \eta _0}\right)^{1/4}, \end{equation} \tag{ 46 }$$

and is given by

$$\begin{equation} \frac{\gamma _{\rm max}^{\rm (p)}}{\Omega _p} = \frac{5^{5/4} \pi e^{-5/4} \eta _0^{1/2}(s^{1/3} - s^{2/3})}{2 (3^{3/2}) \Gamma (3/4) v_{\rm A}}. \end{equation} \tag{ 47 }$$

We define the perpendicular and parallel thermal velocities v_⊥,th and v_∥,th such that f(v_⊥,th, 0) = f(0, v_∥,th) = (1/e)f(0, 0). Equations (19) and (38) then imply that v_⊥,th = η^1/2₀ and v_∥,th = (2η₀/3v^2/3_A)^3/4, where in this last expression we have neglected the last term in Equation (19) since it is small. As a numerical example, we take v_⊥,th = 200 km s^-1 and v_A = 2000 km s^-1 and plot |γ^(p)_k|/ω_kr at θ = 45°, 30°, and 10° in Figure 5, and γ^(p)_max/Ω_p in Figure 6. For these parameters, v_∥,th = 47 km s^-1. For comparison, we also plot the electron damping rate, which is discussed in Section 6. As can be seen in Figure 5, the argument of the exponentials in Equations (40) and (45) causes the damping rate to drop rapidly as k_n∥ is decreased below ∼v^1/2_A/η^1/4₀ ≃ 3, because in this case there are exponentially few protons moving fast enough along the magnetic field to satisfy the resonance condition. On the other hand, γ^(p)_k is a rapidly decreasing function of k_n∥ at k_n∥ ≳ v^1/2_A/η^1/4₀, primarily because γ^(p)_k is proportional to the energy in the electric field of the resonant waves divided by the wave energy, and |E_kx|²/W_k ∝ k⁻⁴_n∥ at large k_n∥ since the wave energy becomes dominated by proton kinetic energy as ω_kr → Ω_p.

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In the Appendix, we calculate the next higher-order term in powers of k⁻²_n in the asymptotic expansion of γ^(p)_k in the limit that θ ≪ 1. This higher-order correction can be viewed as an estimate of the error in our leading-order result.

In this section, we calculate the electron contribution to the damping rate of oblique A/IC waves, γ^(e)_k. We consider wavenumbers satisfying the inequality k_∥v_te ≪ Ω_e, where Ω_e and v_te are the electron gyrofrequency and thermal speed. Since, k_∥v_te ≪ Ω_e and ω_kr ≪ Ω_e, electrons interact with A/IC waves only through the n = 0 "Landau resonance" in Equation (5). In general, this resonance leads to two types of wave–particle interactions: Landau damping and "transit-time damping." In Landau damping, particles are pushed along the magnetic field by the wave electric field E_kz, whereas in transit-time damping particles are pushed along field lines by the μ∇B force of the waves. In our analysis, we neglect terms of order m_e/m_p when determining the polarization properties of the waves. As a result, E_kz = 0 in our calculation, and there is no Landau damping. Gary & Borovsky (2004) showed that the contributions to γ^(e)_k from electron transit-time damping and Landau damping have the same order of magnitude for θ ≲ 45°, but that electron Landau damping becomes much stronger than electron transit-time damping as θ is increased from ∼45° toward 90°. The expression we derive for γ^(e)_k is thus only a reasonable approximation of the actual electron damping rate for θ ≲ 45°, as discussed further below.

We assume that |γ_k/ω_kr| ≪ 1, so that the electron contribution to the damping rate is (Kennel & Wong 1967)

$$\begin{eqnarray} &&\frac{\gamma _k^{\rm (e)}}{|\omega _{kr}|} = \frac{\pi \omega _{pe}^2 \, \mbox{sgn}(\omega _{kr}/k_\parallel)}{8N \omega _{kr}^2 W_k} \int _0^\infty dv_\perp v_\perp ^3 \big|\psi _{0,k}^{(e)}\big|^2 \left. \frac{\partial f_e}{\partial v_\parallel }\right|_{v_\parallel = \omega _{kr}/k_\parallel },\nonumber\\ \end{eqnarray} \tag{ 48 }$$

where N and W_k are defined in the previous section, ω²_pe = 4πNe²/m_e is the square of the electron plasma frequency, sgn(x) ≡ x/|x|, v_⊥ and v_∥ are the velocity components perpendicular and parallel to B₀,

$$\begin{equation} f_e = N \left(\frac{m_e}{2\pi k_B T_e}\right)^{3/2} \exp \left(- \frac{m_e v^2}{2 k_B T_e}\right) \end{equation} \tag{ 49 }$$

is the Maxwellian electron distribution function, and T_e is the electron temperature. The quantity ψ^(e)_n,k is equal to the quantity ψ_n,k defined in Equation (4) with Ω set equal to Ω_e.

To determine the dispersion relation and wave polarization properties, we use the cold-plasma approximation. The cold-plasma dispersion relation in the A/IC frequency range is given by Equation (7), which neglects terms of order v²_A/c², m_e/m_p, and (m_e/m_p)tan²θ. The A/IC frequency is obtained by solving Equation (7) for w² and choosing the minus sign in the quadratic formula, which gives

$$\begin{eqnarray} w^2 &=& \frac{k_n^2}{2}\big\lbrace k_n^2 \cos ^2\theta + 1 + \cos ^2\theta \nonumber\\ &&- \big[ k_n^4 \cos ^4 \theta + 2k_n^2 \cos ^2(\theta) (1+\cos ^2\theta) + \sin ^4 \theta \big]^{1/2}\big\rbrace.\nonumber\\ \end{eqnarray} \tag{ 50 }$$

The damping rate is independent of the azimuthal angle ϕ in k-space, and so we take ϕ = 0 and ${\bf k} = k_x \hat{\bf x} + k_z \hat{\bf z}$. The transverse components of the electric-field fluctuation are then related by the equation (Stix 1992)

$$\begin{equation} \frac{iE_{kx}}{E_{ky}} = \frac{\big[c^2 k_n^2/\big(v_{\rm A}^2 w^2\big)\big] - S}{D}, \end{equation} \tag{ 51 }$$

where S and iD are the xx and yx components of the dielectric tensor. The general definitions of S and D are given in Equations (19) through (21) from Chapter 1 of Stix (1992). Neglecting terms of order m_e/m_p, v²_A/c², and (m_e/m_p)tan²θ, we find that for A/IC waves,

$$\begin{equation} S = \frac{c^2}{v_{\rm A}^2 (1-w^2)}, \end{equation} \tag{ 52 }$$

and

$$\begin{equation} D = \frac{c^2 w}{v_{\rm A}^2 (w^2 -1)}. \end{equation} \tag{ 53 }$$

Equations (51) through (53) can be rearranged to yield

$$\begin{equation} E_{ky} = i g E_{kx}, \end{equation} \tag{ 54 }$$

where

$$\begin{equation} g = \frac{w^3}{k_n^2 (w^2-1) + w^2}. \end{equation} \tag{ 55 }$$

Equations (4) and (55) imply that

$$\begin{equation} |\psi _{0,k}^{(e)}|^2 = \frac{g^2 k_\perp ^2 v_\perp ^2 |E_{kx}|^2}{4 \Omega _e^2}, \end{equation} \tag{ 56 }$$

where we have made use of the inequality k_⊥v_⊥/Ω_e ≪ 1 and employed the approximation that J₁(x) ≃ x/2 when x ≪ 1. To calculate W_k in Equation (27), we use the cold-plasma approximation for the dielectric tensor,

$$\begin{equation} {\bf \epsilon }_h = \left(\begin{array}{@{}ccc@{}} S & -iD & 0 \\ iD & S & 0 \\ 0 & 0 & P \end{array} \right), \end{equation} \tag{ 57 }$$

where S and D are given in Equations (52) and (53), and P ≃ −ω²_pe/ω²_kr. Faraday's law implies that B*_k · B_k = c²k²_n|E_kx|²(g² + cos²θ)/(v²_Aw²). Equations (27), (54), and (57) then imply that

$$\begin{eqnarray} &&W_k = \frac{c^2 |E_{kx}|^2}{16 \pi v_{\rm A}^2} \left[ \frac{(1+w^2)(1+g^2) - 4 wg}{(1-w^2)^2} + \frac{k_n^2(g^2 + \cos ^2\theta)}{w^2} \right].\nonumber\\ \end{eqnarray} \tag{ 58 }$$

Equation (32) follows from Equation (58) in the limit that k²_n∥ ≫ 1. After substituting Equations (56) and (58) into Equation (48), we find that

$$\begin{eqnarray} &&\frac{\gamma _k^{\rm (e)}}{|\omega _{kr}|} = - \left(\frac{\pi \beta _e m_e}{ m_p}\right)^{1/2} \!\left(\frac{k_\perp ^2 v_{\rm A}^2}{\Omega _p^2} \right)\! R(k_n, \theta) \exp \left(- \frac{m_e w^2}{\beta _e m_p k_{n\parallel }^2}\right),\nonumber\\ \end{eqnarray} \tag{ 59 }$$

where

$$\begin{equation} \beta _e = \frac{8\pi N k_B T_e}{B_0^2}, \end{equation} \tag{ 60 }$$

and

$$\begin{eqnarray} &&R(k_n,\theta) \nonumber\\ &&\quad = \frac{g^2 (1-w^2)^2 |w/k_{n\parallel }|}{w^2(1+w^2)(1+g^2) - 4 w^3 g + (1-w^2)^2 k_n^2 (g^2 + \cos ^2\theta)}.\nonumber\\ \end{eqnarray} \tag{ 61 }$$

The exponential in Equation (59) implies that when β_e ≪ m_e/m_p, the damping rate becomes exponentially small at k_n∥ ≲ 1 where w/k_n∥ ∼ 1 because there are extremely few electrons with v_∥ ≃ ±v_A.

Equation (59) can be compared with Equation (5) of Gary & Borovsky (2004),

$$\begin{equation} \frac{\gamma _k^{\rm (e)}}{|\omega _{kr}|} = - 0.35 \left(\frac{m_e\beta _e}{m_p}\right)^{1/2} \left(\frac{k_\perp v_{\rm A}}{\Omega _p}\right)^2, \end{equation} \tag{ 62 }$$

which was obtained by fitting numerical results for the damping rate at k_∥ ≃ k_d/3, where k_d ≃ Ω_p/v_A is the wavenumber corresponding to the rapid onset of proton cyclotron damping. Gary & Borovsky (2004) argued that Equation (62) is approximately valid for most wavenumbers satisfying k_∥ < k_d, which is consistent with Figure 4 of Cranmer & van Ballegooijen (2003), which suggests that γ^(e)_k is independent of k_∥ for |k_∥| ≪ k_d. Our results, however, suggest that γ^(e)_k departs from the scaling in Equation (62) as k_∥ is increased from k_d/3 to k_d (at least when the proton distribution function satisfies f = f(η) so that |γ^(p)_k| ≪ |ω_kr|), and thus we take Equation (62) to apply only for k_∥ < k_d/3 ≃ Ω_p/(3v_A).⁴ In the limit |k_n∥| ≪ sin²θ, R(k_n, θ) ≃ (cos⁴θ)/(2sin⁴θ), and Equation (59) becomes

$$\begin{equation} \frac{\gamma _k^{\rm (e)}}{ |\omega _{kr}|} \simeq - \left(\frac{\pi \beta _e m_e}{ m_p}\right)^{1/2} \left(\frac{k_n^2 \cos ^4 \theta }{2\sin ^2 \theta } \right) \exp \left(- \frac{m_e }{\beta _e m_p }\right). \end{equation} \tag{ 63 }$$

We emphasize that Equation (63) is not the proper limit of Equation (59) as θ → 0, since Equation (63) assumes that sin²θ ≫ |k_n∥|. If β_e>m_e/m_p, then the exponential in Equation (63) is ∼1, and Equation (63) is similar to Equation (62) for θ ≃ 45°. However, the damping rates in these two equations scale differently with θ. For θ significantly larger than 45°, our result for γ^(e)_k is smaller than that of Gary & Borovsky (2004), and we do not view our result as reliable for this range of θ because of our neglect of Landau damping and finite-Larmor-radius effects on the wave properties. For θ < 45°, we find that the electron damping rate is larger than in the analysis of Gary & Borovsky (2004). Although their result is based on numerical solutions to the full hot-plasma dispersion relation, it is not clear that their result is to be preferred at small θ, because almost all of the numerical data used to obtain Equation (62) corresponded to wave vectors satisfying θ>45°. We thus take Equation (59) to be reasonably accurate for θ < 45°.

Equation (59) implies that as k_n∥ increases above ∼1, the electron contribution to the damping rate decreases. As in the case of the proton cyclotron damping rate, this is primarily because |E_kx|²/W_k decreases with increasing |k_n∥|, since the the wave energy becomes dominated by proton kinetic energy as ω_kr → Ω_p. The values of γ^(e)_k/|ω_kr| in Equation (59) at θ = 45°, 30°, and 10° are plotted in Figure 5 for β_e = 0.01. For comparison, the result of Gary & Borovsky (2004) in Equation (62) is plotted in these figures as well. It can be seen that at these values of θ the maximum value of |γ^(p)_k/ω_kr| is a factor of ∼10 − 40 larger than the maximum value of |γ^(e)_k/ω_kr| from Equation (59). We note that we are comparing the damping rate for an "evolved" proton distribution to the damping rate for an "un-evolved" electron distribution. It is possible that the electron distribution will evolve in such a way that the electron damping rate of A/IC waves is reduced relative to the case of Maxwellian electrons, in particular, by flattening (i.e., reducing |∂f_e/∂v_∥|) at |v_∥| < v_A.

If nonlinear wave–wave interactions cause A/IC-wave energy to cascade to larger k_∥, then this energy cascade will be truncated by wave damping once it reaches a wavenumber at which γ_kτ_k ≳ 1, where τ_k is the energy cascade time, i.e., the timescale on which energy at wavenumber k is transferred to wavenumber 2k. In the solar corona and solar flares, δB_k/B₀ is very small for k_n>1, where δB_k is the rms amplitude of the magnetic field fluctuation at scale k⁻¹. As a result, high-frequency A/IC waves at k_n>1 and |ω| ∼ Ω_p are weakly turbulent, with τ_k ≫ ω⁻¹_kr. (Zakharov et al. 1992) Thus, although γ^(p)_k in Equation (37) is ≪ω_kr, γ^(p)_kτ_k may nevertheless exceed 1 near the maximum (in k_n) of γ^(p)_k. If it does, then proton damping is sufficient to terminate the cascade. As shown in Figure 5, the maximum electron damping rate is much less than the maximum proton damping rate, at least for moderately oblique waves. If the proton damping rate is sufficient to truncate the cascade but the electron damping rate is not, then most of the turbulent power will go into proton heating, at least for A/IC waves with θ ≲ 45°. On the other hand, if the turbulence amplitude is so small that γ^(e)_kτ_k>1 at k_n∥ ≲ 1, then the cascade would be truncated by electron damping at k_n∥ < 1 before it reaches sufficiently high k_n∥ that proton cyclotron damping becomes important.

A thorough description of the above scenario is, however, beyond the scope of this paper. One complication is that the dissipation scenario just described involves the amplification of A/IC waves with θ = 0, and the processes that drain A/IC-wave energy at θ = 0 from the system (e.g., propagation out of the system, cascade, phase mixing (Heyvaerts & Priest 1983), or damping) need to be identified and modeled. Another source of uncertainty concerns the value of τ_k at the scales of interest in collisionless plasmas such as the solar corona and solar flares. Recent studies have begun to address the cascade of Alfvén-wave energy to large k_∥ at frequencies ≪Ω_p (Chandran 2005, 2008; Luo & Melrose 2006) as well as the way in which slow magnetosonic waves can cause A/IC waves with θ = 0 to cascade to larger k_∥ (Yoon & Fang 2008). However, the cascade of A/IC-wave energy to larger k_∥ at k_∥ ≳ Ω_p/v_A in collisionless plasmas, in which slow waves are strongly damped, is still not well understood.

In this paper, we consider resonant interactions between protons and oblique A/IC waves for the case in which the proton distribution function f is strongly modified by wave pitch-angle scattering. We argue that wave–particle interactions involving oblique waves cause f to become unstable to waves with θ = 0, where θ is the angle between the wave vector and the background magnetic field. The resulting amplification of A/IC waves with θ = 0 causes the angular distribution of A/IC waves to become sharply peaked about θ = 0 at the large wavenumbers at which protons resonate with A/IC waves. Wave–particle interactions then cause f to become approximately constant along the set of closed, nested scattering contours corresponding to waves with θ = 0, which are defined by the equation η(v_⊥, v_∥) = constant, with the approximate value of η given in Equation (19). Once f = f(η), oblique A/IC waves continue to be damped by protons. The damping of oblique waves coupled with the amplification of θ = 0 waves represents a kinetic mechanism for the spectral transport of wave energy toward θ = 0. Because the plasma does not relax to a state in which proton damping of oblique A/IC waves ceases, oblique high-frequency A/IC waves can be significantly more effective at heating protons than A/IC waves with θ = 0. In sections 5 and 6 we analytically calculate the approximate values of the proton and electron contributions to the damping rate of oblique A/IC waves, assuming Maxwellian electrons and f = f(η). As shown in Figure 5, the peak proton damping rate is much larger than the peak electron damping rate for moderately oblique waves. As a result, in at least some circumstances most of the heating power that results from the dissipation of moderately oblique, high-frequency A/IC waves will be deposited into the protons.

This work was supported in part by DOE Grant DE-FG02-07-ER46372, NSF Grant AGS-0851005, NSF-DOE Grants AST-0613622 and AGS-1003451, and NASA Grants NNX07AP65G, NNX08AH52G, NNX09AJ80G, and NNX07AI14G.

In this Appendix, we calculate γ^(p)_k as an asymptotic series in both θ and k⁻²_n, keeping only the leading-order term in θ, the leading-order term in k⁻²_n, and the first-order correction in k⁻²_n. By calculating the first-order correction in k⁻²_n, and viewing this correction as an estimate of the error in the leading-order result, we can gain some insight into how accurate Equations (40) and (45) are for parameters of interest, at least for small θ.

The solution to Equation (7) can be expressed as the asymptotic series,

$$\begin{eqnarray} &&w = \pm \left\lbrace 1 - \frac{1}{(k_{n \parallel })^2} + \frac{2}{(k_{n \parallel })^4} + \theta ^2\left[ \frac{1}{2(k_{n \parallel })^2} - \frac{2}{(k_{n \parallel })^4}\right] + \cdots \ \right\rbrace.\nonumber\\ \end{eqnarray} \tag{ A1 }$$

At θ = 0, E_ky = −iE_kx exactly at all k_n for A/IC waves (Stix 1992). To leading order in θ, |ψ^(p)_1,k|² = |E_kx|² to all orders in k⁻¹_n, and

$$\begin{equation} W_k = \frac{\omega _{pi}^2 |E_{kx}|^2 k_n^{4} }{8\pi \Omega _p^2} \;\left(1 + \frac{5}{k_n^2} + \cdots \ \right). \end{equation} \tag{ A2 }$$

Equation (A1) implies that

$$\begin{eqnarray} v_{\parallel, {\rm res}} &\equiv& \frac{\omega _{kr} - \Omega _p}{k_\parallel } \nonumber\\ &=& - \frac{v_{\rm A}}{(k_{n \parallel })^3} \left\lbrace 1 - \frac{2}{(k_{n \parallel })^2} - \theta ^2 \left[\frac{1}{2} - \frac{2}{(k_{n \parallel })^2}\right] + \cdots \ \right\rbrace.\nonumber\\ \end{eqnarray} \tag{ A3 }$$

After setting f = f(η), we obtain

$$\begin{eqnarray} &&\int _{\infty }^\infty dv_\parallel \, G(f)\;\; \delta (v_\parallel - v_{\parallel, {\rm res}}) \nonumber\\ &&\quad = \frac{\theta ^2 v_\perp }{3} \left[1 + \frac{1}{3(k_{n \parallel })^2} + \cdots \ \right] \left. \frac{df}{d\eta }\right|_{\eta = \eta _{\rm res}}, \end{eqnarray} \tag{ A4 }$$

where η_res = η[v_⊥, v_∥,res(k_n∥, 0)]. Noting that to lowest order in θ, k_n and k_n∥ are interchangeable, we finally obtain

$$\begin{eqnarray} &&\frac{\gamma _k^{\rm (p)}}{\omega _{kr}} = \frac{\pi ^2 \theta ^2 v_{\rm A}}{3N k_n^5} \left(1 - \frac{11}{3k_n^2} + \cdots \ \right) \int _0^\infty dv_\perp v_\perp ^3 \left. \frac{df}{d\eta }\right|_{\eta = \eta _{\rm res}}.\nonumber\\ \end{eqnarray} \tag{ A5 }$$

As an example, we again set

$$\begin{equation} f(v_\perp, v_\parallel) = c_1 e^{-\eta /\eta _0} \end{equation} \tag{ A6 }$$

with η₀ ≪ v²_A. The condition N = ∫d³vf then implies that

$$\begin{eqnarray} &&c_1 = \left(\frac{2}{3}\right)^{\!\!\!1/4}\!\!\!\! \frac{ N v_{\rm A}^{1/2}}{\pi \, \Gamma (3/4)\, \eta _0^{7/4}} \left[ 1 + \left(\frac{2}{3}\right)^{\!\!5/2} \! \frac{\ \Gamma (9/4) \: \eta _0^{1/2}}{\Gamma (3/4) \: v_{\rm A}} + \cdots \right],\nonumber\\ \end{eqnarray} \tag{ A7 }$$

where Γ is the Gamma function, and the dots (⋅⋅⋅) represent higher-order corrections in powers of η^1/2₀/v_A, which we neglect. Using Equations (A6) and (A7) in Equation (A5), we obtain

$$\begin{equation} \frac{\gamma _k^{\rm (p)}}{|\omega _{kr}|} = - \,\left(\frac{2}{3}\right)^{\!\!\!1/4}\!\! \frac{\pi \theta ^2 v_{\rm A}^{3/2}\,(1+\delta)}{6\, \Gamma (3/4)\,\eta _0^{3/4} k_n^5} \;\exp \!\left[-\,\frac{3v_{\rm A}^2}{2 \eta _0\,k_n^4 } \right], \end{equation} \tag{ A8 }$$

where

$$\begin{equation} \delta = -\frac{11}{3k_n^2} + \left(\frac{2}{3}\right)^{\!\!\!5/2} \frac{\Gamma (9/4)\: \eta _0^{1/2}}{\Gamma (3/4)\: v_{\rm A}} + \frac{10 v_{\rm A}^2}{3 \eta _0 \, k_n^6}. \end{equation} \tag{ A9 }$$

Since the peak damping rates occur for $k_n \sim \sqrt{v_{\rm A}}/\sqrt[4]{\eta _0}$, we treat k_n and $\sqrt{v_{\rm A}}/\sqrt[4] {\eta _0}$ as being of equivalent order. As in the previous section, we take v_⊥,th = 200 km s^-1 and v_A = 2000 km s^-1 and then plot the value of γ^(p)_k for these parameters in Figure 7. As seen in the figure, our results for γ^(p)_k at small θ are not changed very much when we go from keeping just the first term in powers of k⁻²_n in the asymptotic series for γ^(p)_k to keeping both the first and second terms in this series.

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