Stochastic proton heating by kinetic-Alfvén-wave turbulence in moderately high-β plasmas

Abstract

Stochastic heating refers to an increase in the average magnetic moment of charged particles interacting with electromagnetic fluctuations whose frequencies are much smaller than the particles’ cyclotron frequencies. This type of heating arises when the amplitude of the gyroscale fluctuations exceeds a certain threshold, causing particle orbits in the plane perpendicular to the magnetic field to become stochastic rather than nearly periodic. We consider the stochastic heating of protons by Alfvén-wave (AW) and kinetic-Alfvén-wave (KAW) turbulence, which may make an important contribution to the heating of the solar wind. Using phenomenological arguments, we derive the stochastic-proton-heating rate in plasmas in which β_p ∼ 1 − 30, where β_p is the ratio of the proton pressure to the magnetic pressure. (We do not consider the β_p ≳ 30 regime, in which KAWs at the proton gyroscale become non-propagating.) We test our formula for the stochastic-heating rate by numerically tracking test-particle protons interacting with a spectrum of randomly phased AWs and KAWs. Previous studies have demonstrated that at β_p ≲1, particles are energized primarily by time variations in the electrostatic potential and thermal-proton gyro-orbits are stochasticized primarily by gyroscale fluctuations in the electrostatic potential. In contrast, at β_p ≳ 1, particles are energized primarily by the solenoidal component of the electric field and thermal-proton gyro-orbits are stochasticized primarily by gyroscale fluctuations in the magnetic field.

Graphical Abstract

1. Introduction

In the mid-twentieth century several authors published hydrodynamic models of the solar wind that imposed a fixed temperature at the coronal base and took thermal conduction to be the only heating mechanism (e.g., Parker 1958, 1965; Hartle & Sturrock 1968; Durney 1972). These models were unable to explain the high proton temperatures and fast-solar-wind speeds observed at a heliocentric distance r of 1 astronomical unit (au) for realistic values of the coronal temperature and density, indicating that the fast solar wind is heated primarily by some mechanism other than thermal conduction. Parker (1965) and Coleman (1968) proposed that Alfvén waves (AWs) and AW turbulence provide this additional heating. Support for this suggestion can be found in the many spacecraft observations of AW-like turbulence in the solar wind (see Belcher 1971; Tu & Marsch 1995; Bale et al. 2005), remote observations of AW-like fluctuations in the solar corona (see Tomczyk et al. 2007; De Pontieu et al. 2007), and the agreement between AW-driven solar-wind models and solar-wind temperature, density, and flow-speed profiles (Cranmer et al. 2007; Verdini et al. 2010; Chandran et al. 2011; van der Holst et al. 2014).

AWs oscillate at a frequency ω = k_∥v_A, where k_∥ (k_⊥) is the component of the wave vector k parallel (perpendicular) to the background magnetic field, B₀, $v_{\text{A}}=B_0/\sqrt{4\pi n_{\text{p}}m}$ is the Alfvén speed, n_p is the proton number density, and m is the proton mass.^† In AW turbulence, interactions between counter-propagating AWs cause AW energy to cascade from larger to smaller scales. This energy cascade is anisotropic, in the sense that the small-scale AW “eddies,” or wave packets, generated by the cascade vary much more rapidly perpendicular to the magnetic field than along the magnetic field (e.g., Shebalin et al. 1983; Goldreich & Sridhar 1995; Cho & Vishniac 2000; Horbury et al. 2008; Podesta 2013; Chen 2016). As a consequence, within the inertial range (scales larger than the thermal-proton gyroradius ρ_th and smaller than the outer scale or driving scale), ω ≪ Ω, where Ω is the proton cyclotron frequency. At k_⊥ρ_th ∼ 1, the AW cascade transitions to a kinetic-Alfv´en-wave (KAW) cascade (Schekochihin et al. 2009).

Studies of the dissipation of low-frequency (ω ≪ Ω), anisotropic, AW/KAW turbulence based on linear wave damping (e.g., Quataert 1998; Howes et al. 2008) conclude that AW/KAW turbulence leads mostly to parallel heating of the particles (i.e., heating that increases the speed of the thermal motions along B). On the other hand, perpendicular ion heating is the dominant form of heating in the near-Sun solar wind (Esser et al. 1999; Marsch 2006; Cranmer et al. 2009; Hellinger et al. 2013). This discrepancy suggests that AW/KAW turbulence in the solar wind dissipates via some nonlinear mechanism (e.g. Dmitruk et al. 2004; Markovskii et al. 2006; Lehe et al. 2009; Schekochihin et al. 2009; Chandran et al. 2010; Servidio et al. 2011; Lynn et al. 2012; Xia et al. 2013; Kawazura et al. 2018). This suggestion is supported by studies that find a correlation between ion temperatures and fluctuation amplitudes in solar-wind measurements and numerical simulations (e.g., Wu et al. 2013; Hughes et al. 2017; Grošelj et al. 2017; Vech et al. 2018).

In this paper, we consider one such nonlinear mechanism: stochastic heating. In stochastic proton heating, AW/KAW fluctuations at the proton gyroscale have sufficiently large amplitudes that they disrupt the normally smooth cyclotron motion of the protons, leading to non-conservation of the first adiabatic invariant, the magnetic moment (McChesney et al. 1987; Johnson & Cheng 2001a; Chen et al. 2001; Chaston et al. 2004; Fiksel et al. 2009a; Xia et al. 2013). Chandran et al. (2010) used phenomenological arguments to derive an analytical formula for the stochastic-heating rate at β_p ≲ 1, where β_p is the ratio of the proton pressure to the magnetic pressure (see (2.3)). In Section 2 we use phenomenological arguments to obtain an analytic formula for the proton-stochastic-heating rate in low-frequency AW/KAW turbulence when β_p ∼ 1−30. We limit our analysis to β_p ≲ 30, since KAWs become non-propagating at k_⊥ρ_th = 1 at larger β_p values (see Appendix A and Hellinger & Matsumoto (2000); Kawazura et al. (2018); Kunz et al. (2018)). In Section 3 we present results from simulations of test particles interacting with a spectrum of randomly phased AWs/KAWs, which we use to test our analytic formula for the stochastic-heating rate. Throughout this paper, we focus on perpendicular proton heating rather than parallel proton heating. Stochastic heating can in principle augment the parallel proton heating that results from linear damping of AW/KAW turbulence at β_∥p ≳ 1, but we leave a discussion of this possibility to future work.

2. Stochastic proton heating by AW/KAW turbulence at the proton gyroscale

A proton interacting with a uniform background magnetic field B₀ and fluctuating electric and magnetic fields δE and δB undergoes nearly periodic motion in the plane perpendicular to B₀ if δE and δB are sufficiently small or L/ρ is sufficiently large, where L is the characteristic length scale of δE and δB, ρ = v_⊥/Ω is the proton’s gyroradius, v_⊥ is the component of the proton’s velocity v perpendicular to the magnetic field, Ω = qB₀/mc is the proton gyrofrequency, m and q are the proton mass and charge, and c is the speed of light. When (1) the proton’s motion in the plane perpendicular to B₀ is nearly periodic and (2) Ωτ ≫ 1, where τ is the characteristic time scale of δE and δB, the proton’s magnetic moment $\mu =m{v}_{\perp }^2/2B_0$ is almost exactly conserved (Kruskal 1962).

Perpendicular heating of protons (by which we mean a secular increase in the average value of μ) requires that one of the above two conditions for μ conservation be violated. For example, Alfvén/ion-cyclotron waves can cause perpendicular proton heating via a cyclotron resonance if Ωτ ∼ 1 (Hollweg & Isenberg 2002). Alternatively, low-frequency AW/KAW fluctuations can cause perpendicular proton heating if their amplitudes at k_⊥ρ ∼ 1 are sufficiently large that the proton motion in the plane perpendicular to B₀ becomes disordered or “stochastic” (McChesney et al. 1987; Johnson & Cheng 2001b; Chen et al. 2001; Chaston et al. 2004; Fiksel et al. 2009b).

We focus on this second type of heating, stochastic heating, and on “thermal” protons, for which

$$v_{\Vert }~w_{\Vert }{v}_{\perp }~w_{\perp }\rho ~{\rho }_{\text{th}},$$ (2.1)

where $w_{\perp }=\sqrt{2k_{\text{B}}{T}_{\perp \text{p}}/m}$ and $w_{\Vert }=\sqrt{2k_{\text{B}}{T}_{\Vert \text{p}}/m}$ are the perpendicular and parallel thermal speeds, T_⊥p and T_∥p are the perpendicular and parallel proton temperatures, p k_B is Boltzmann’s constant, and ρ_th = w_⊥/Ω is the thermal-proton gyroradius. We restrict our attention to the contribution to the stochastic-heating rate from turbulent AW/KAW fluctuations with

$$\lambda ~{\rho }_{\text{th}}{k}_{\perp }{\rho }_{\text{th}}~1,$$ (2.2)

where λ is the length scale of the fluctuations measured perpendicular to the background magnetic field, and to

$${\beta }_{\text{p}}\equiv \frac{8\pi n{k}_{\text{B}}{T}_{\text{p}}}{B_0^2}~1-30,$$ (2.3)

where

$$T_{\text{p}}=\frac{2T_{\perp \text{p}}+T_{\Vert \text{p}}}{3}.$$ (2.4)

As mentioned above and discussed further in Appendix A, KAWs at k_⊥ρ_th = 1 become non-propagating at significantly larger values of β_p (see also Hellinger & Matsumoto 2000; Kawazura et al. 2018; Kunz et al. 2018). For simplicity, we assume that

$$T_{\perp \text{p}}~T_{\Vert \text{p}}{w}_{\perp }~w_{\Vert },$$ (2.5)

which implies that

$${\beta }_{\text{p}}~\frac{w_{\perp }^2}{v_{\text{A}}^2}~\frac{w_{\Vert }^2}{v_{\text{A}}^2},$$ (2.6)

and that T_e ∼ T_p, where T_e is the electron temperature. We also assume that

$$\delta B_{\rho }\ll B_0,$$ (2.7)

where δB_ρ is the rms amplitude of the magnetic fluctuations with λ ∼ ρ_th, and that the fluctuations are in critical balance (Goldreich & Sridhar 1995), which implies that

$$\frac{\delta v_{\rho }}{{\rho }_{\text{th}}}~\frac{v_{\text{A}}}{l},$$ (2.8)

where l is the correlation length of the gyroscale AW/KAW fluctuations measured parallel to the background magnetic field, and δv_ρ is the rms amplitude of the E × B velocity of the AW/KAW fluctuations with λ ∼ ρ_th. Since the linear and nonlinear time scales are comparable in the critical-balance model, we take the ratios of the amplitudes of different fluctuating variables to be comparable to the ratios that arise for linear AW/KAWs at k_⊥ρ_th ∼ 1, which, given (2.2) and (2.3), implies that

$$\delta B_{\Vert \rho }~\delta B_{\perp \rho }~\delta B_{\rho }\frac{\delta B_{\rho }}{B_0}~\frac{\delta v_{\rho }}{v_{\text{A}}},$$ (2.9)

where δB_∥ρ and δB_⊥ρ are, respectively, the rms amplitudes of the components of the fluctuating magnetic field parallel and perpendicular to B₀ (TenBarge et al. 2012). Equations (2.2) through (2.9) imply that

$$\omega ~\frac{v_{\text{A}}}{l}~\frac{\delta v_{\rho }}{{\rho }_{\text{th}}}~\Omega \frac{\delta v_{\rho }}{w_{\perp }}~\Omega {\beta }_{\text{p}}^{-1/2}\frac{\delta v_{\rho }}{v_{\text{A}}}~\Omega {\beta }_{\text{p}}^{-1/2}\frac{\delta B_{\rho }}{B_0}\ll \Omega .$$ (2.10)

2.1. Stochastic motion perpendicular to the magnetic field

To understand how gyroscale AW/KAW fluctuations modify a proton’s motion, we cannot use the adiabatic approximation (Northrop 1963), which assumes λ ≫ ρ. Nevertheless, we can still define an effective guiding center

$$R=r+\frac{v\times \widehat{b}}{\Omega },$$ (2.11)

where $\widehat{b}=B/B$. This effective guiding center is always a distance ρ from the particle’s position r and is, at any given time, the location about which the particle attempts to gyrate under the influence of the Lorentz force. We find it useful to focus on R rather than r because the motion of R largely excludes the high-frequency cyclotron motion of the proton. Upon taking the time derivative of (2.11) and making use of the relations dr/dt = v and dv/dt = (q/m)(E + v × B/c), we obtain

$$\frac{\text{d}R}{\text{d}t}=v_{\Vert }\widehat{b}+\frac{cE\times B}{B^2}-\frac{v\times \widehat{b}}{\Omega }\frac{1}{B}\frac{\text{d}B}{\text{d}t}+\frac{v}{\Omega }\times \frac{\text{d}\widehat{b}}{\text{d}t},$$ (2.12)

where $v_{\Vert }=v\cdot \widehat{b}$. The perpendicular component of dR/dt,

$${\left(\frac{\text{d}R}{\text{d}t}\right)}_{\perp }=\frac{\text{d}R}{\text{d}t}-\widehat{b}\left(\widehat{b}\cdot \frac{\text{d}R}{\text{d}t}\right)=\left(\widehat{b}\times \frac{\text{d}R}{\text{d}t}\right)\times \widehat{b},$$ (2.13)

can be found by substituting the right-hand side of (2.12) into the right-hand side of (2.13), which yields

$${\left(\frac{\text{d}R}{\text{d}t}\right)}_{\perp }=\frac{cE\times B}{B^2}-\frac{v\times \widehat{b}}{\Omega }\frac{1}{B}\frac{\text{d}B}{\text{d}t}+\frac{v_{\Vert }}{\Omega }\widehat{b}\times \frac{\text{d}\widehat{b}}{\text{d}t}.$$ (2.14)

We now estimate each term on the right-hand side of (2.14). Since we are considering only gyroscale fluctuations^†, we take the first term on the right-hand side of (2.14) to satisfy the relation

$$\left|\frac{cE\times B}{B^2}\right|~\delta v_{\rho }.$$ (2.15)

To estimate the second and third terms on the right-hand side of (2.14), we take

$$v_{\perp }~\left|v_{\Vert }\right|~w_{\perp }~w_{\Vert },$$ (2.16)

which is satisfied by the majority of particles. The time derivative of the field strength along the particle’s trajectory is

$$\frac{\text{d}B}{\text{d}t}=\frac{\partial B}{\partial t}+v_{\perp }\cdot \nabla B+v_{\Vert }\widehat{b}\cdot \nabla B.$$ (2.17)

As outlined above, our assumption of critical balance implies that λ ≪ l and ω ≪ v_⊥/ρ ∼ w_⊥/ρ_th. The second term on the right-hand side of (2.17) is thus much larger than either the first or third terms, and

$$\frac{\text{d}B}{\text{d}t}~\frac{w_{\perp }\delta B_{\Vert \rho }}{{\rho }_{\text{th}}}.$$ (2.18)

The second term on the right-hand side of (2.14) thus satisfies

$$\left|\frac{v\times \widehat{b}}{\Omega }\frac{1}{B}\frac{\text{d}B}{\text{d}t}\right|~\frac{{\rho }_{\text{th}}}{B}\frac{\text{d}B}{\text{d}t}~\frac{w_{\perp }\delta B_{\Vert \rho }}{B_0},$$ (2.19)

which is larger than the first term on the right-hand side of (2.14) by a factor of $~{\beta }_{\text{p}}^{1/2}$, given (2.6) and (2.9).

For the moment, we assume that the second term on the right-hand side of (2.14) is the dominant term; we discuss the third term in more detail below. If the second term is dominant, then

$$\left|{\left(\frac{\text{d}R}{\text{d}t}\right)}_{\perp }\right|~\frac{w_{\perp }\delta B_{\Vert \rho }}{B_0}.$$ (2.20)

During a single cyclotron period 2π/Ω, a proton passes through an order-unity number of uncorrelated gyroscale AW/KAW eddies, and the values of (dR/dt)_⊥ within different gyroscale eddies are uncorrelated. If (dR/dt)_⊥ is small compared to w_⊥, then a proton undergoes nearly circular gyromotion. However, if |(dR/dt)_⊥| is a significant fraction of w_⊥, then a proton and its guiding center will move in an essentially unpredictable way, and the proton’s orbit will become stochastic rather than quasi-periodic. Given (2.9), |(dR/dt)_⊥| is a significant fraction of w_⊥ if the stochasticity parameter

$$\delta \equiv \frac{\delta B_{\rho }}{B_0}$$ (2.21)

is a significant fraction of unity.

We illustrate how the value of δ affects a proton’s motion in figure 1. We compute the particle trajectories shown in this figure by numerically integrating the equations of motion for protons interacting with randomly phased AWs and KAWs. We present the details of our numerical method and more extensive numerical results in Section 3. In the numerical calculation shown in the left panel of figure 1, δ = 0.03, and the proton’s motion in the plane perpendicular to B₀ is quasi-periodic. In the numerical calculation shown in the right panel of figure 1, δ = 0.15, and the proton trajectory is more disordered or random.

Figure 1:

Trajectories of test-particle protons interacting with a spectrum of randomly phased AWs and KAWs for different values of the stochasticity parameter δ defined in (2.21).

We now consider the third term on the right-hand side of (2.14). The instantaneous value of this term is comparable to the instantaneous value of the second term given (2.9) and (2.16), but the third term is less effective at causing guiding-center displacements over time for the following reason. Because of (2.7), the time t_‖ required for v_‖ to change by a factor of order unity is ≫ Ω⁻¹. If we integrate the third term on the right-hand side of (2.14) from t = 0 to t = t_f, where Ω⁻¹ ≪ t_f ≪ t_‖, we can treat v_‖ as approximately constant in (2.14), obtaining

$${\int }_0^{t_{\text{f}}}\frac{v_{\Vert }}{\Omega }\widehat{b}\times \frac{\text{d}\widehat{b}}{\text{d}t}\text{d}t=\frac{v_{\Vert }}{{\Omega }_0}\frac{B_0}{B_0}\times \Delta \widehat{b}$$ (2.22)

to leading order in δB_ρ/B₀, where Ω₀ = qB₀/mc and $\Delta \widehat{b}=\widehat{b}\left(t_{\text{f}}\right)-\widehat{b}(0)$ is the change in $\widehat{b}$. There is, however, no secular change in the value of $\widehat{b}$ at the proton’s location; the magnetic-field unit vector merely undergoes small-amplitude fluctuations about the direction of the background magnetic field. Thus, over time, the guiding-center displacements caused by the third term on the right-hand side of (2.14) are largely reversible and tend to cancel out. The third term is thus less effective than the second term at making proton orbits stochastic.

When the stochasticity parameter δ defined in (2.21) exceeds some threshold, the motion of a thermal proton’s guiding center in the plane perpendicular to B₀ is reasonably approximated by a random walk. To estimate the time step of this random walk, we begin by defining the cyclotron average of (dR/dt)_⊥,

$$v_R(t)\equiv \frac{\Omega }{2\pi }{\int }_{t-\pi /\Omega }^{t+\pi /\Omega }{\left(\frac{\text{d}R}{\text{d}{t}_1}\right)}_{\perp }\text{d}{t}_1.$$ (2.23)

As stated above, during a single cyclotron period a proton’s motion projected onto the plane perpendicular to B₀ carries the proton through an order-unity number of uncorrelated gyroscale AW/KAW “eddies.” For simplicity, we take the amplitude and direction of each vector term on the right-hand side of (2.14) to be approximately constant within any single gyroscale eddy and the values of these vector terms within different eddies to be uncorrelated. This makes v_R approximately equal to the average of some order-unity number of uncorrelated vectors of comparable magnitude. The amplitude of this average is comparable to the instantaneous value of |(dR/dt)_⊥|. Thus, given (2.6), (2.9), and (2.20),

$$v_R~\frac{w_{\perp }\delta B_{\Vert \rho }}{B_0}~{\beta }_{\text{p}}^{1/2}\delta v_{\rho }.$$ (2.24)

Because we are considering the effects of just the gyroscale AW/KAW eddies, v_R decorrelates after the proton’s guiding center has moved a distance ∼ ρ_th in the plane perpendicular to B₀, which takes a time

$$\Delta t~\frac{{\rho }_{\text{th}}}{v_R}.$$ (2.25)

Thermal protons thus undergo spatial diffusion in the plane perpendicular to B₀ with a spatial diffusion coefficient

$$D_{\perp }~\frac{{\rho }_{\text{th}}^2}{\Delta t}~{\beta }_{\text{p}}^{1/2}\delta v_{\rho }{\rho }_{\text{th}}.$$ (2.26)

Given (2.8), (2.16), and (2.24),

$$\Delta t~\frac{{\rho }_{\text{th}}}{{\beta }_{\text{p}}^{1/2}\delta v_{\rho }}~\frac{l}{{\beta }_{\text{p}}^{1/2}{v}_{\text{A}}}~\frac{l}{v_{\Vert }}.$$ (2.27)

The time required for a particle to wander a distance ∼ ρ_th perpendicular to the background magnetic field is thus comparable to the time required for the particle to traverse the parallel dimension of a gyroscale AW/KAW eddy.

2.2. Energy diffusion and heating

The total energy of a proton is given by its Hamiltonian,

$$H=q\Phi +\frac{1}{2m}{\left(p-\frac{q}{c}A\right)}^2,$$ (2.28)

where Φ is the electrostatic potential, p is the canonical momentum, and A is the vector potential. From Hamilton’s equations,

$$\frac{\text{d}H}{\text{d}t}=q\frac{\partial \Phi }{\partial t}-\frac{qv}{c}\cdot \frac{\partial A}{\partial t},$$ (2.29)

where v = m⁻¹(p−qA/c) is the velocity, and the electric field is E = −∇Φ−c⁻¹∂A/∂t. The second term on the right hand side of (2.29) is qv · E_s, where E_s = −c⁻¹∂A/∂t is the solenoidal component of the electric field. Equation (46) of Hollweg (1999) gives the ratio of E_s to the magnitude of the irrotational component of the electric field |∇Φ| for AWs/KAWs with k_⊥ρ_th ≲ 1, ^†

$$\frac{E_{\text{s}}}{|\nabla \Phi |}~\frac{{\beta }_{\text{p}}\omega }{\Omega }.$$ (2.30)

In their treatment of stochastic heating at β_p ≲ 1, Chandran et al. (2010) neglected the second term on the right-hand side of (2.29), because this term makes a small contribution to the heating rate when β_p is small. Here we focus on the effects of E_s and make the approximation that

$$\frac{\text{d}H}{\text{d}t}~qv\cdot E_{\text{s}}.$$ (2.31)

We show in Appendix B that the irrotational part of the electric field contributes less to the heating rate than does the solenoidal part when β_p ≳ 1.

As a proton undergoes spatial diffusion in the plane perpendicular to the background magnetic field, the electromagnetic field at its location resulting from gyroscale AW/KAW fluctuations decorrelates on the time scale Δt given in (2.27). Within each time interval of length ∼ Δt, the proton energy changes by an amount δH (which can be positive or negative), and the values of δH are uncorrelated within successive time intervals of length Δt. As a consequence, the proton undergoes energy diffusion.

To estimate the rms value of δH, which we denote ΔH, we adopt a simple model of a proton’s motion, in which the proton’s complicated trajectory is replaced by a repeating two-step process. In the first step, the proton undergoes circular cyclotron motion in the plane perpendicular to B₀ for a time Δt. In the second step, the proton is instantly translated a distance ρ_th in some random direction perpendicular to B₀.^†

In this simple model, a proton undergoes N ∼ ΩΔt ∼ (v_⊥/ρ_th) × (l/v_k) ∼ l/ρ_th ≫ 1 circular gyrations in the plane perpendicular to B₀ during a time Δt. Integrating (2.31) for a time Δt, we obtain

$$\delta H~q{\int }_0^{\Delta t}v(t)\cdot E_{\text{s}}(r(t),t)\text{d}t,$$ (2.32)

where r(t) is the proton’s position at time t. Since $\Delta t~{\beta }_{\text{p}}^{-1/2}{\rho }_{\text{th}}/\delta v_{\rho }$, when β_p ≳ 1, the time Δt for a particle to diffuse across one set of gyroscale eddies is shorter than or comparable to the linear or nonlinear time scale ρ_th/δv_ρ of those eddies. We thus approximate the right-hand side of (2.32) by setting E_s(r(t),t) = E_s(r(t),0) and rewrite (2.32) as

$$\delta H~qN\oint E_{\text{s}}(r,0)\cdot \text{d}l~qN{\int }_S\nabla \times E(r,0)\cdot \text{d}S~-\frac{qN}{c}{\int }_S\frac{\partial B}{\partial t}(r,0)\cdot \text{d}S,$$ (2.33)

where the line integral is along the proton’s path during one complete circular gyration in the plane perpendicular to B₀, the surface integral is over the circular surface S of radius ρ_th enclosed by the gyration, and we have used Faraday’s Law ∇ × E = (−1/c)∂B/∂t. The surface S is perpendicular to B₀, and dS is anti-parallel to B₀ (antiparallel rather than parallel since q > 0). The rms value of δH thus satisfies the orderof-magnitude relation

$$\Delta H~\frac{qN}{c}{\omega }_{\text{eff}}\delta B_{\Vert \rho }{\rho }_{\text{th}}^2,$$ (2.34)

where

$${\omega }_{\text{eff}}\equiv \frac{\delta v_{\rho }}{{\rho }_{\text{th}}}$$ (2.35)

is the nonlinear frequency of the gyroscale fluctuations. Upon setting q/c = Ωm/B, N = ΩΔt, and ${\rho }_{\text{th}}^2=w_{\perp }^2/{\Omega }^2$ in (2.34), we obtain

$$\Delta H~\frac{m{w}_{\perp }^2}{B}\frac{\delta v_{\rho }}{{\rho }_{\text{th}}}\delta B_{\Vert \rho }\Delta t.$$ (2.36)

Although we are in the process of estimating the rate at which μ changes over long times, our estimate of ΔH is comparable to the value that would follow from μ conservation: $\Delta H~\mu \Delta B~\left(m{w}_{\perp }^2/B\right){\omega }_{\text{eff}}\delta B_{\Vert \rho }\Delta t$, where ΔB ∼ ω_effδB_∥ρΔt is the rms amplitude of the change in the magnetic flux through the proton’s Larmor orbit, divided by πρ², during the time Δt in which the proton is (in our simple two-step model of proton motion) undergoing continuous, circular, cyclotron motion. This correspondence highlights an alternative interpretation of the stochastic-heating process at β_p ≳ 1. In the guiding-center approximation, when $v_{\perp }^2$ increases by some factor because of E_s, the field strength at the particle’s guiding center increases by approximately the same factor, essentially because of Faraday’s law. This proportionality underlies μ conservation. In stochastic heating, the same proportionality is approximately satisfied during a single time interval Δt, but the proton is then stochastically transported to a neighboring set of gyroscale eddies, in which the field strength is not correlated with the field strength at the proton’s original location. The proton thus “forgets” about what happened to the field strength at its original location and gets to keep the energy that it gained without “paying the price” of residing in a higher-field-strength location. In this way, spatial diffusion perpendicular to B breaks the connection between changes to $v_{\perp }^2$ and changes to B that arises in the ρ/λ → 0 limit.

In our simple model, the energy gained by a proton is in the form of perpendicular kinetic energy,

$$K_{\perp }=\frac{m{v}_{\perp }^2}{2},$$ (2.37)

because we neglect the parallel motion of protons. (We do not preclude the possibility of parallel stochastic heating, but we do not consider it further here.) The perpendicular-kinetic-energy diffusion coefficient D_K is thus ∼ ΔH²/Δt, or

$$D_K~\frac{m^2{w}_{\perp }^4}{{\beta }_{\text{P}}^{1/2}}\frac{\delta v_{\rho }}{{\rho }_{\text{th}}}\frac{\delta B_{\rho }^2}{B_0^2},$$ (2.38)

where we have used (2.27) to estimate Δt and (2.9) to set δB_∥ρ ∼ δB_ρ. A single proton undergoing a random walk in energy can gain or lose energy with equal probability during a time Δt. However, if a large number of thermal protons (e.g., with an initially Maxwellian distribution) undergo energy diffusion, then on average more protons will gain energy than lose energy, leading to proton heating. The heating time scale τ_h is the characteristic time for the perpendicular kinetic energy of a thermal proton to double, ${\tau }_{\text{h}}~{\left(m{w}_{\perp }^2\right)}^2/D_K$, and the perpendicular-heating rate per unit mass is Q_⊥ ∼ K_⊥/(mτ_h) ∼ D_K/(mK_⊥), or,

$$Q_{\perp }~{\beta }_{\text{p}}^{1/2}\frac{{\left(\delta v_{\rho }\right)}^3}{{\rho }_{\text{th}}}.$$ (2.39)

To account for the uncertainties introduced by our numerous order-of-magnitude estimates, we multiply the right-hand side of (2.39) by an as-yet-unknown dimensionless constant σ₁. As δv_ρ → 0, dμ/dt decreases faster than any power of δv_ρ (Kruskal 1962). To account for this “exponential” μ conservation in the small-δv_ρ limit, we follow Chandran et al. (2010) by multiplying the right-hand side of (2.39) by the factor exp(−σ₂/δ),

$$Q_{\perp }={\sigma }_1\frac{{\left(\delta v_{\rho }\right)}^3}{{\rho }_{\text{th}}}\sqrt{{\beta }_{\text{p}}}\exp \left(-\frac{{\sigma }_2}{\delta }\right),$$ (2.40)

where σ₂ is another as-yet-unknown dimensionless constant, and δ is defined in (2.21).

For comparison, the stochastic-heating rate per unit mass found by Chandran et al. (2010) when β_p ≲ 1 is

$$Q_{\perp }=c_1\frac{{\left(\delta v_{\rho }\right)}^3}{{\rho }_{\text{th}}}\exp \left(-\frac{c_2}{\epsilon }\right),$$ (2.41)

where

$$\epsilon =\frac{\delta v_{\rho }}{w_{\perp }},$$ (2.42)

and the dimensionless constants c₁ and c₂ serve the same purpose as those in (2.40). As discussed by Chandran et al. (2010) for the case of c₁ and c₂, we expect the constants σ₁ and σ₂ to depend on the nature of the fluctuations. For example, at fixed δv_ρ, we expect stronger heating rates (i.e., larger σ₁ and/or smaller σ₂) from intermittent turbulence than from randomly phased waves (Chandran et al. 2010; Xia et al. 2013; Mallet et al. 2018), because, in intermittent turbulence, most of the heating takes place near coherent structures in which the fluctuations are unusually strong and in which the proton orbits are more stochastic than on average.

2.3. Orbit stochasticity from parallel motion

In Section 2.1, we focused on proton motion perpendicular to B. However, motion along the magnetic field can also produce stochastic motion in the plane perpendicular to B₀ (see, e.g., Hauff et al. 2010). In particular, the perpendicular magnetic fluctuations at the scale of a proton’s gyroradius perturb the direction of $\widehat{b}$. These perturbations, when fed into the first term on the right-hand side of (2.12), $v_{\Vert }\widehat{b}$, cause the proton’s guiding center R to acquire a velocity perpendicular to B₀ of

$$u_{\perp }~v_{\Vert }\times \frac{\delta B_{\perp \rho }}{B_0},$$ (2.43)

where δB_⊥ρ is the component of δB (from gyroscale fluctuations) perpendicular to B₀ at the proton’s location. The value of u_⊥ varies in an incoherent manner in time, with a correlation time ∼ Ω⁻¹. If u_⊥ is a significant fraction of v_⊥, then u_⊥ will cause a proton’s orbit in the plane perpendicular to B₀ to become stochastic. This leads to an alternative high-β_p stochasticity parameter,

$$\tilde{\delta }=\frac{u_{\perp }}{v_{\perp }}=\frac{v_{\Vert }\delta B_{\perp \rho }}{v_{\perp }{B}_0}.$$ (2.44)

As $\tilde{\delta }$ increases towards unity, proton orbits become stochastic. For thermal protons with v_⊥ ∼ v_∥ and ρ ∼ ρ_th, $\tilde{\delta }$ is equivalent to δ in (2.21), which was based upon the parallel magnetic-field fluctuation δB_kρ (even though we set δB_∥ρ ∼ δB_ρ in (2.21))). The contribution of parallel motion to orbit stochasticity thus does not change our conclusions about the rate at which thermal protons are heated stochastically. However, the contribution of parallel motion to orbit stochasticity should be taken into account when considering the ability of stochastic heating to produce superthermal tails, because in AW turbulence the perpendicular (parallel) magnetic fluctuation at perpendicular scale λ, denoted δB_⊥λ (δB_∥λ), is an increasing (decreasing) function of λ when λ is in the inertial range. Orbit stochasticity through the interaction between parallel motion and δB_⊥ρ could thus contribute to the development of superthermal tails when β_p ≳ 1. An investigation of superthermal tails, however, lies beyond the scope of this paper.

3. Numerical Test-Particle Calculations

To test the phenomenological theory developed in Section 2, we numerically track test-particle protons interacting with a spectrum of low-frequency randomly phased AWs and KAWs. The initial particle positions are random and uniformly distributed within a cubical region of volume (100d_p)³, where d_p = v_A/Ω is the proton inertial length. The initial velocity distribution is an isotropic Maxwellian with proton temperature T_p. To trace each particle, we solve the equations of motion,

$$\frac{\text{d}x}{\text{d}t}=v\frac{\text{d}v}{\text{d}t}=\frac{q}{m}\left(E+\frac{v\times B}{c}\right),$$ (3.1)

using the Boris method (Boris 1970) with a time step of 0.01Ω⁻¹.

3.1. Randomly phased waves

The code used to implement the AW/KAW spectrum is similar to the code used by Chandran et al. (2010). The magnetic field is $B=B_0\widehat{z}+\delta B$, where B₀ is constant. We take E and δB to be the sum of the electric and magnetic fields of waves at each of 81 different wave vectors, with two waves of equal amplitude at each wave vector, one with ω/k_z < 0 and the other with ω/k_z > 0. ^† The initial phase of each wave is randomly chosen.

The 81 wave vectors correspond to nine evenly spaced values of the azimuthal angle in k space (in cylindrical coordinates aligned with B₀) at each of nine specific values of k_⊥i : i ∈ [0,...,8]. The values of k_⊥i are evenly spaced in ln(k_⊥)-space, with ln(k_⊥iρ_th) = −4/3 + i/3. The middle three cells, in which i = 3,4, and 5, have a combined width of unity in ln(k_⊥)-space, centred at precisely k_⊥ρ_th = 1. We computationally evaluate δv_ρ and δB_ρ via the rms values of the E × B velocity and δB that result from the waves in just these middle three cells.

There is one value of k_∥ ≡ |k_z| at each k_⊥i, denoted k_∥i. We determine k_∥4 by setting the linear frequency at k_∥4 equal to k_⊥4δv_ρ. At other values of k_⊥, we set

$$\frac{k_{\Vert i}}{k_{\Vert 4}}=\{\begin{array}{ll}{\left(\frac{k_{\perp i}}{k_{\perp 4}}\right)}^{2/3}\hfill & :0⩽i<4\hfill \\ {\left(\frac{k_{\perp i}}{k_{\perp 4}}\right)}^{1/3}\hfill & :4<i⩽8.\hfill \end{array}$$ (3.2)

The exponents 2/3 and 1/3 in (3.2) are chosen to match the scalings in the criticalbalance models of Goldreich & Sridhar (1995) and Cho & Lazarian (2004), respectively. We take the individual wave magnetic-field amplitudes to be proportional to $k_{\perp }^{-1/3}$ and $k_{\perp }^{-1/3}$ for k_⊥𝜌th < 1 and k_⊥𝜌th > 1, respectively, in order to match the same two critical-balance models. (All the waves at the same value of k_⊥i have the same amplitudes.) We determine the wave frequency and relative amplitudes of the different components of the fluctuating electric and magnetic fields using Hollweg’s (1999) two-fluid analysis of linear KAWs, setting

$$\frac{T_{\text{e}}}{T_{\text{p}}}=0.5\frac{v_{\text{A}}}{c}=0.003,$$ (3.3)

where T_e and T_p are the (isotropic) electron and proton temperatures. We do not expect the particular choices in (3.3) to have a large effect on our results, but choose those values to facilitate a direct comparison to the previous numerical results of Chandran et al. (2010). Since we take T_⊥p = T_kp, we set

$$w_{\perp }=w_{\Vert }=w\equiv \sqrt{\frac{2k_{\text{B}}{T}_{\text{p}}}{m}}=v_{\text{A}}\sqrt{{\beta }_{\text{p}}}.$$ (3.4)

3.2. A Note on the Electric Field

Following Lehe et al. (2009), we correct the electric field because the magnetic field (including its fluctuations, i.e., $B=B_0\widehat{z}+\delta B$) in the simulation is not orientated along the z-axis. The simulation, however, equates the parallel and perpendicular components of the electric field to the parallel and perpendicular components of the wave electric field that would arise if the magnetic field were aligned exactly on the z-axis. The result is a numerical addition of perpendicular electric field terms to the parallel electric field, which, in turn, causes non-physical parallel heating. This may be seen in figure 2 of Chandran et al. (2010). To fix this, we replace the sum of the individual wave electric fields described in Section 3.1, which we denote E_wave, with the modified electric field $E=E_{\text{wave}}+\widehat{b}\left(\widehat{z}\cdot E_{\text{wave}}-\widehat{b}\cdot E_{\text{wave}}\right)$.

Figure 2:

The mean square velocity perpendicular to B₀, $\langle v_{\perp }^2\rangle$, as a function of time in two test-particle calculations, each of which tracks 10⁵ protons. The value of the stochasticity parameter δ (defined in (2.21)) is 0.15 in both calculations.

3.3. Perpendicular Heating

We perform numerical test-particle calculations at five different values of β_p, in particular β_p = {0.006,0.01,0.1,1,10}. For each β_p value, we carry out a test-particle calculations for five different values of δ (or, equivalently, five different values of ε). Each calculation returns the value of $\langle v_{\perp }^2\rangle$ and $\langle v_{\Vert }^2\rangle$ as functions of time. We show two examples in figure 2. The slope of the best-fit line for each $\langle v_{\perp }^2(t)\rangle$ curve determines the perpendicular-heating rate per unit mass $Q_{\perp }=(1/2)(\text{d}/\text{d}t)\langle v_{\perp }^2\rangle$, where ⟨···⟩ indicates an average over the 10⁴ or 10⁵ particles in the simulation. (We use more particles in simulations with smaller ε and δ because the heating rates are smaller in these simulations, and the extra particles increase the signal-to-noise ratio.) We fit the $\langle v_{\perp }^2(t)\rangle$ curves during the time interval (t_i,t_f), where t_i = 20π/Ω and t_f is the smaller of the following two values: 10⁴Ω⁻¹ and the time required for $\langle v_{\perp }^2\rangle$ to increase by ≃ 30%. We do not include the first ten cyclotron periods when calculating Q_⊥, because it takes the particles a few cyclotron periods to adjust to the presence of the waves, during which time there is typically strong transient heating. (As figure 2 shows, the test particles undergo parallel heating as well as perpendicular heating, as was found previously by Xia et al. (2013) in simulations of test particles interacting with reduced magnetohydrodynamic turbulence at β_∥p = 1.)

The perpendicular-heating rates in our test-particle calculations are shown in figure 3. The solid-line curves in the two panels on the left correspond to (2.41), with

$$c_1=0.77c_2=\mathrm{0.33.}$$ (3.5)

Figure 3:

Numerical results for the perpendicular-heating rate per unit mass, Q_⊥, for protons interacting with randomly phased AWs and KAWs. Top-Left: β_p < 1, and Q_⊥ normalized by $\Omega v_{\text{A}}^2$. Top right: β_p ⩾1 and Q_⊥ normalized by $\Omega v_{\text{A}}^2$. Bottom left: β_p < 1 and $Q_{\perp }$ normalized by Ωw², where w is the proton thermal speed defined in (3.4); the numerical results for all three β_p values (0.006, 0.01, and 0.1) are within the bars shown. Bottom right: β_p ⩾ 1 and Q_⊥ normalized by Ωw². In the left two panels, the solid lines are plots of (2.41) for the best-fit values of c₁ and c₂ given in (3.5). In the right two panels, the solid lines are plots of (2.40) for the best-fit values of σ₁ and σ₂ in (3.6).

These values are very similar to the values c₁ = 0.75 and c₂ = 0.34 obtained by Chandran et al. (2010) at β_p = 0.006. The solid-line curves in the two panels on the right correspond to (2.40) with

$${\sigma }_1=5.0{\sigma }_2=\mathrm{0.21.}$$ (3.6)

The agreement between our numerical results and (2.40) suggests that the approximations used to derive this equation are reasonable.

The lower-left panel of figure 3 shows that at β_p < 1, Q_⊥/(Ωw²) is a function of ε alone, consistent with the fact that (2.41) can be rewritten in the form

$$\frac{Q_{\perp }}{\Omega w^2}=c_1{\epsilon }^3\exp \left(-\frac{c_2}{\epsilon }\right)(\text{at}{\beta }_{\text{p}}\lesssim 1).$$ (3.7)

The top-right panel of figure 3 shows that at ${\beta }_{\text{p}}⩾1,Q_{\perp }/\left(\Omega v_{\text{A}}^2\right)$ is a function of δ alone, consistent with the fact that (2.40) can be rewritten as

$$\frac{Q_{\perp }}{\Omega v_{\text{A}}^2}={\sigma }_1{\delta }^3\exp \left(-\frac{{\sigma }_2}{\delta }\right)(\text{at}{\beta }_{\text{p}}\gtrsim 1).$$ (3.8)

We note that in our model of randomly phased KAWs (Chandran et al. 2010),

$$\frac{\delta B_{\rho }}{B_0}=0.84\frac{\delta v_{\rho }}{v_{\text{A}}},$$ (3.9)

and thus

$$\delta =\frac{\delta B_{\rho }}{B_0}=0.84\frac{\delta v_{\rho }}{v_{\text{A}}}=0.84{\beta }_{\text{p}}^{1/2}\frac{\delta v_{\rho }}{w_{\perp }}=0.84{\beta }_{\text{p}}^{1/2}\epsilon .$$ (3.10)

As a consequence, if we adopt the best-fit values of σ₁, σ₂, c₁, and c₂, then the value of Q_⊥ at β_p = 1 in (2.40), which matches our test-particle calculations quite well, exceeds the value that would follow from (2.41) at β_p = 1. A similar phenomenon was found by Xia et al. (2013) in numerical simulations of test particles interacting with strong reduced magnetohydrodynamic turbulence.

To obtain a fitting formula that can be used to model stochastic heating at large β_p, small β_p, and β_p ≃ 1, we use (3.4) and (3.10) to rewrite the low-β_p heating rate in (3.7) in terms of δ and v_A. We then add the low-β_p heating rate to the high-β_p heating rate in (3.8), obtaining

$$\frac{Q_{\perp }}{\Omega v_{\text{A}}^2}={\sigma }_1{\delta }^3\exp \left(-\frac{{\sigma }_2}{\delta }\right)+\frac{1.69c_1{\delta }^3}{{\beta }_{\text{p}}^{1/2}}\exp \left(-\frac{0.84c_2{\beta }_{\text{p}}^{1/2}}{\delta }\right).$$ (3.11)

The first term on the right-hand side dominates at β_p ≳ 1 in part because σ₁ ≃ 6.5c₁. The second term on the right-hand side dominates at β_p ≪ 1. Figure 4 shows that (3.11) is consistent with our numerical results. This figure also illustrates how, at fixed δB_ρ/B₀, the stochastic heating rate increases as β_p decreases.

Figure 4:

The data points reproduce the numerical results from figure 3 for β_p = 0.01,0.1,1.0, and 10. The dotted, long-dashed, solid, and short-dashed lines plot (3.11) for, respectively, β_p = 0.01, β_p = 0.1, β_p = 1, and β_p = 10. The solid and short-dashed lines are difficult to distinguish because they are nearly on top of each other.

As mentioned above, stochastic heating becomes more effective as the fluctuations become more intermittent (Xia et al. 2013; Mallet et al. 2018). The randomly phased waves in our test-particle simulations are not intermittent, but gyroscale fluctuations in space and astrophysical plasmas generally are (see, e.g., Mangeney et al. 2001; Carbone et al. 2004; Salem et al. 2009; Chandran et al. 2015; Mallet et al. 2015). Further work is needed to determine how the best-fit constants in (3.5) and (3.6) depend upon the degree of intermittency at the proton gyroradius scale. Until this dependency is determined, some caution should be exercised when applying (3.11) to space and astrophysical plasmas. For reference, Bourouaine & Chandran (2013) found that lowering c₂ to ≃ 0.2 led the heating rate in (2.41) to be consistent with the proton heating rate and fluctuation amplitudes inferred from measurements of the fast solar wind from the Helios spacecraft at r = 0.3 au. However, if c₂ = 0.33, then the heating rate in (2.41) is too weak to explain the proton heating seen in the Helios measurements.

4. Summary

In this paper we use phenomenological arguments to derive an analytic formula for the rate at which thermal protons are stochastically heated by AW/KAW turbulence at k_⊥ρ_th ∼ 1. We focus on β_p ∼ 1 − 30. Smaller values of β_p were considered by Chandran et al. (2010). At larger values of β_p, KAWs at k_⊥ρ_th ∼ 1 become non-propagating, and some of the scalings we have assumed do not apply. At β_p ∼ 1−30, the motion of a proton’s effective guiding center is dominated by the interaction between the proton and gyroscale fluctuations in the magnetic field, whose amplitude is denoted δB_ρ. As δB_ρ/B₀ increases from infinitesimal values towards unity, the proton motion in the plane perpendicular to B₀ becomes random (stochastic), leading to spatial diffusion, and this spatial diffusion breaks the strong correlation between changes in a proton’s perpendicular kinetic energy and the magnetic-field strength at the proton’s location that normally gives rise to magnetic-moment conservation. The interaction between the proton and the electric field then becomes a Markov process that causes the proton to diffuse in energy. This energy diffusion leads to heating. At β_p ∼ 1−30, it is the solenoidal component of the electric field that dominates the heating.

The analytic formula that we derive for the stochastic heating rate Q_⊥ contains two dimensionless constants, σ₁ and σ₂, whose values depend upon the nature of the AW/KAW fluctuations that the proton interacts with (e.g., randomly phased waves or intermittent turbulence). We numerically track test particles interacting with randomly phased AWs and KAWs and find that our analytic formula for Q_⊥ agrees well with the heating rate of these test particles for the choices σ₁ = 5.0 and σ₂ = 0.21. We note that previous work has shown that for fixed rms amplitudes of the gyroscale fluctuations, stochastic heating is more effective when protons interact with intermittent turbulence than when protons interact with randomly phased waves (Chandran et al. 2010; Xia et al. 2013; Mallet et al. 2018). The reason for this is that in intermittent turbulence, most of the heating occurs near coherent structures, in which the fluctuation amplitudes are larger than average and in which the particle orbits are more stochastic than on average.

Our work leaves a number of interesting questions unanswered. Two such questions are how the energy-diffusion coefficient depends on energy at β_p ∼ 1 − 30 and how the proton distribution function evolves in the presence of stochastic heating. (For a discussion of the low-β_p case, see Klein & Chandran (2016).) We also have not addressed the question of how stochastic heating changes as β_p is increased to values ≳ 30 and KAWs at k_⊥ρ_th ∼ 1 become non-propagating, or how the stochastic heating rate for minor ions depends upon minor-ion mass, charge, and average flow speed along B₀ in the proton frame. (For a discussion of the low-β_p case, see Chandran et al. (2013).) Previous studies have compared observationally inferred heating rates in the solar wind with the low-β_kp stochastic-heating rate in (2.41) derived by Chandran et al. (2010), finding quantitative agreement at r = 0.3 au assuming c₂ ≃ 0.2 (Bourouaine & Chandran 2013) and qualitative agreement at r = 1 au (Vech et al. 2017). However, it is not yet clear whether the stochastic heating rate in (2.40) agrees with solar-wind measurements in the large-β_kp regime. In addition, stochastic heating at β_∥p ≳ 1 could trigger temperature-anisotropy instabilities, which could in turn modify the rate(s) of perpendicular (and parallel) proton heating. Future investigations of these questions will be important for determining more accurately the role of stochastic heating in space and astrophysical plasmas.

Appendix A. Non-propagation of KAWs at k_⊥ρ_th ∼1 at high β_p

In figure 5, we compare the AW/KAW dispersion relation from the two-fluid model of Hollweg (1999) and the PLUME hot-plasma dispersion-relation solver (Klein & Howes 2015) for T_e/T_p = 0.5 and v_A/c = 0.003 and for various values of β_p. The PLUME results shown here assume that k_kρ_th = 0.001 and that the proton and electron distributions are Maxwellian. The two-fluid dispersion relation agrees reasonably well with the more accurate PLUME results at β_p ≲ 1. However, at β_p ≳ 1, the PLUME results deviate from the two-fluid theory because of ion damping, which becomes stronger as β_p increases (Howes et al. 2008; Kunz et al. 2018). Starting at β_p ≃ 30 (for T_e/T_p = 0.5, v_A/c = 0.003, and k_∥ρ_p = 0.001), the real part (but not the imaginary part) of the KAW frequency at k_⊥ρ_th = 1 vanishes (i.e., KAWs become damped, non-propagating modes). For larger β_p values, KAWs are non-propagating throughout an interval of k_⊥ρ_th values centered on unity that broadens to both larger and smaller values as β_p increases (Kawazura et al. 2018).

Figure 5:

The KAW dispersion relation from Hollweg’s (1999) two-fluid model (solid lines) and the PLUME hot-plasma dispersion-relation solver (Klein & Howes 2015) (dotted lines) for T_e/T_p = 0.5 and v_A/c = 0.003 for various powers of β_p.

Appendix B. Stochastic heating by the electrostatic potential at β_p ≳ 1

In Section 2 we considered the rms change to a thermal proton’s energy ΔH resulting from the solenoidal component of the electric field E_s during the particle residence time Δt within one set of gyroscale eddies. We also evaluated the contribution of E_s to the stochastic heating rate Q_⊥. Here, we show that the contribution to Q_⊥ from E_s is larger than the contribution from the electrostatic potential Φ when β_p ≳ 1.

We assume that the rms amplitude of the potential part of the electric field at k_⊥ρ_th is comparable to the rms amplitude of the total gyroscale electric-field fluctuation, δE_ρ, which in turn is ∼ δv_ρB₀/c. As discussed by Chandran et al. (2010), the contribution of the time-varying electrostatic potential to ΔH is

$$\Delta H_{\text{potential}}~q{\omega }_{\text{eff}}\Delta {\Phi }_{\rho }\Delta t,$$ (B1)

where ω_eff = δv_ρ/ρ_th (see (2.35)), and

$$q\Delta {\Phi }_{\rho }~q{\rho }_{\text{th}}\delta E_{\rho }~q{w}_{\perp }\times \frac{mc}{q{B}_0}\times \frac{\delta v_{\rho }{B}_0}{c}~m{w}_{\perp }\delta v_{\rho }.$$ (B2)

Since (2.27) gives $\Delta t~{\beta }_{\text{P}}^{-1/2}{\rho }_{\text{th}}/\delta v_{\rho }$,

$${\omega }_{\text{eff}}\Delta t~{\beta }_{\text{p}}^{-1/2}.$$ (B3)

Combining (B1) through (B3), we obtain

$$\Delta H_{\text{potential}}~{\beta }_{\text{p}}^{-1/2}m{w}_{\perp }\delta v_{\rho },$$ (B4)

$$D_{K,\text{potential}}~\frac{{\left(\Delta H_{\text{potential}}\right)}^2}{\Delta t}~\frac{{\beta }_{\text{p}}^{-1}{m}^2{w}_{\perp }^2{\left(\delta v_{\rho }\right)}^2}{{\beta }_{\text{p}}^{-1/2}{\rho }_{\text{th}}/\delta v_{\rho }}~{\beta }_{\text{p}}^{-1/2}{m}^2{w}_{\perp }^2\frac{{\left(\delta v_{\rho }\right)}^3}{{\rho }_{\text{th}}},$$ (B5)

and

$$Q_{\perp \text{,potential}}~\frac{D_{K,\text{potential}}}{m{K}_{\perp }}~{\beta }_{\text{p}}^{-1/2}\frac{{\left(\delta v_{\rho }\right)}^3}{{\rho }_{\text{th}}},$$ (B6)

which is a factor of $~{\beta }_{\text{p}}^{-1}$ smaller than the estimate of $Q_{\perp }$ in (2.39).