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 Reference Northrop1963), which assumes $\unicode[STIX]{x1D706}\gg \unicode[STIX]{x1D70C}$ . Nevertheless, we can still define an effective guiding centre

(2.11) $$\begin{eqnarray}\boldsymbol{R}=\boldsymbol{r}+\frac{\boldsymbol{v}\times \hat{\boldsymbol{b}}}{\unicode[STIX]{x1D6FA}},\end{eqnarray}$$

where $\hat{\boldsymbol{b}}=\boldsymbol{B}/B$ . This effective guiding centre is always a distance  $\unicode[STIX]{x1D70C}$ from the particle’s position  $\boldsymbol{r}$ 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 $\boldsymbol{R}$ rather than  $\boldsymbol{r}$ because the motion of  $\boldsymbol{R}$ largely excludes the high-frequency cyclotron motion of the proton. Upon taking the time derivative of (2.11) and making use of the relations $\text{d}\boldsymbol{r}/\text{d}t=\boldsymbol{v}$ and $\text{d}\boldsymbol{v}/\text{d}t=(q/m)(\boldsymbol{E}+\boldsymbol{v}\times \boldsymbol{B}/c)$ , we obtain

(2.12) $$\begin{eqnarray}\frac{\text{d}\boldsymbol{R}}{\text{d}t}=v_{\Vert }\hat{\boldsymbol{b}}+\frac{c\boldsymbol{E}\times \boldsymbol{B}}{B^{2}}-\frac{\boldsymbol{v}\times \hat{\boldsymbol{b}}}{\unicode[STIX]{x1D6FA}}\frac{1}{B}\frac{\text{d}B}{\text{d}t}+\frac{\boldsymbol{v}}{\unicode[STIX]{x1D6FA}}\times \frac{\text{d}\hat{\boldsymbol{b}}}{\text{d}t},\end{eqnarray}$$

where $v_{\Vert }=\boldsymbol{v}\boldsymbol{\cdot }\hat{\boldsymbol{b}}$ . The perpendicular component of $\text{d}\boldsymbol{R}/\text{d}t$ ,

(2.13) $$\begin{eqnarray}\left(\frac{\text{d}\boldsymbol{R}}{\text{d}t}\right)_{\bot }=\frac{\text{d}\boldsymbol{R}}{\text{d}t}-\hat{\boldsymbol{b}}\left(\hat{\boldsymbol{b}}\boldsymbol{\cdot }\frac{\text{d}\boldsymbol{R}}{\text{d}t}\right)=\left(\hat{\boldsymbol{b}}\times \frac{\text{d}\boldsymbol{R}}{\text{d}t}\right)\times \hat{\boldsymbol{b}},\end{eqnarray}$$

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

(2.14) $$\begin{eqnarray}\left(\frac{\text{d}\boldsymbol{R}}{\text{d}t}\right)_{\bot }=\frac{c\boldsymbol{E}\times \boldsymbol{B}}{B^{2}}-\frac{\boldsymbol{v}\times \hat{\boldsymbol{b}}}{\unicode[STIX]{x1D6FA}}\frac{1}{B}\frac{\text{d}B}{\text{d}t}+\frac{v_{\Vert }}{\unicode[STIX]{x1D6FA}}\hat{\boldsymbol{b}}\times \frac{\text{d}\hat{\boldsymbol{b}}}{\text{d}t}.\end{eqnarray}$$

We now estimate each term on the right-hand side of (2.14). Since we are considering only gyroscale fluctuations,Footnote ² the first term on the right-hand side of (2.14) satisfies the relation

(2.15) $$\begin{eqnarray}\left|\frac{c\boldsymbol{E}\times \boldsymbol{B}}{B^{2}}\right|\sim \unicode[STIX]{x1D6FF}v_{\unicode[STIX]{x1D70C}}.\end{eqnarray}$$

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

(2.16) $$\begin{eqnarray}v_{\bot }\sim |v_{\Vert }|\sim w_{\bot }\sim w_{\Vert },\end{eqnarray}$$

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

(2.17) $$\begin{eqnarray}\frac{\text{d}B}{\text{d}t}=\frac{\unicode[STIX]{x2202}B}{\unicode[STIX]{x2202}t}+\boldsymbol{v}_{\bot }\boldsymbol{\cdot }\unicode[STIX]{x1D735}B+v_{\Vert }\hat{\boldsymbol{b}}\boldsymbol{\cdot }\unicode[STIX]{x1D735}B.\end{eqnarray}$$

As outlined above, our assumption of critical balance implies that $\unicode[STIX]{x1D706}\ll l$ and $\unicode[STIX]{x1D714}\ll v_{\bot }/\unicode[STIX]{x1D70C}\sim w_{\bot }/\unicode[STIX]{x1D70C}_{\text{th}}$ . The second term on the right-hand side of (2.17) is thus much larger than either the first or third terms, and

(2.18) $$\begin{eqnarray}\frac{\text{d}B}{\text{d}t}\sim \frac{w_{\bot }\unicode[STIX]{x1D6FF}B_{\Vert \unicode[STIX]{x1D70C}}}{\unicode[STIX]{x1D70C}_{\text{th}}}.\end{eqnarray}$$

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

(2.19) $$\begin{eqnarray}\left|\frac{\boldsymbol{v}\times \hat{\boldsymbol{b}}}{\unicode[STIX]{x1D6FA}}\frac{1}{B}\frac{\text{d}B}{\text{d}t}\right|\sim \frac{\unicode[STIX]{x1D70C}_{\text{th}}}{B}\frac{\text{d}B}{\text{d}t}\sim \frac{w_{\bot }\unicode[STIX]{x1D6FF}B_{\Vert \unicode[STIX]{x1D70C}}}{B_{0}},\end{eqnarray}$$

which is larger than the first term on the right-hand side of (2.14) by a factor of ${\sim}\unicode[STIX]{x1D6FD}_{\text{p}}^{1/2}$ , 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

(2.20) $$\begin{eqnarray}\left|\left(\frac{\text{d}\boldsymbol{R}}{\text{d}t}\right)_{\bot }\right|\sim \frac{w_{\bot }\unicode[STIX]{x1D6FF}B_{\Vert \unicode[STIX]{x1D70C}}}{B_{0}}.\end{eqnarray}$$

During a single cyclotron period $2\unicode[STIX]{x03C0}/\unicode[STIX]{x1D6FA}$ , a proton passes through an order-unity number of uncorrelated gyroscale AW/KAW eddies, and the values of $(\text{d}\boldsymbol{R}/\text{d}t)_{\bot }$ within different gyroscale eddies are uncorrelated. If $(\text{d}\boldsymbol{R}/\text{d}t)_{\bot }$ is small compared to  $w_{\bot }$ , then a proton undergoes nearly circular gyromotion. However, if $|(\text{d}\boldsymbol{R}/\text{d}t)_{\bot }|$ is a significant fraction of  $w_{\bot }$ , then a proton and its guiding centre will move in an essentially unpredictable way, and the proton’s orbit will become stochastic rather than quasi-periodic. Given (2.9), $|(\text{d}\boldsymbol{R}/\text{d}t)_{\bot }|$ is a significant fraction of  $w_{\bot }$ if the stochasticity parameter

(2.21) $$\begin{eqnarray}\unicode[STIX]{x1D6FF}\equiv \frac{\unicode[STIX]{x1D6FF}B_{\unicode[STIX]{x1D70C}}}{B_{0}}\end{eqnarray}$$

is a significant fraction of unity.

We illustrate how the value of  $\unicode[STIX]{x1D6FF}$ 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 § 3. In the numerical calculation shown in figure 1(a), $\unicode[STIX]{x1D6FF}=0.03$ , and the proton’s motion in the plane perpendicular to  $\boldsymbol{B}_{0}$ is quasi-periodic. In the numerical calculation shown in figure 1(b), $\unicode[STIX]{x1D6FF}=0.15$ , 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  $\unicode[STIX]{x1D6FF}$ 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-centre displacements over time for the following reason. Because of (2.7), the time $t_{\Vert }$ required for $v_{\Vert }$ to change by a factor of order unity is $\gg \unicode[STIX]{x1D6FA}^{-1}$ . If we integrate the third term on the right-hand side of (2.14) from $t=0$ to $t=t_{\text{f}}$ , where $\unicode[STIX]{x1D6FA}^{-1}\ll t_{\text{f}}\ll t_{\Vert }$ , we can treat $v_{\Vert }$ as approximately constant in (2.14), obtaining

(2.22) $$\begin{eqnarray}\int _{0}^{t_{\text{f}}}\frac{v_{\Vert }}{\unicode[STIX]{x1D6FA}}\hat{\boldsymbol{b}}\times \frac{\text{d}\hat{\boldsymbol{b}}}{\text{d}t}\,\text{d}t=\frac{v_{\Vert }}{\unicode[STIX]{x1D6FA}_{0}}\frac{\boldsymbol{B}_{0}}{B_{0}}\times \unicode[STIX]{x0394}\hat{\boldsymbol{b}}\end{eqnarray}$$

to leading order in $\unicode[STIX]{x1D6FF}B_{\unicode[STIX]{x1D70C}}/B_{0}$ , where $\unicode[STIX]{x1D6FA}_{0}=qB_{0}/mc$ and $\unicode[STIX]{x0394}\hat{\boldsymbol{b}}=\hat{\boldsymbol{b}}(t_{\text{f}})-\hat{\boldsymbol{b}}(0)$ is the change in $\hat{\boldsymbol{b}}$ . There is, however, no secular change in the value of $\hat{\boldsymbol{b}}$ 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-centre 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  $\unicode[STIX]{x1D6FF}$ defined in (2.21) exceeds some threshold, the motion of a thermal proton’s guiding centre in the plane perpendicular to  $\boldsymbol{B}_{0}$ is reasonably approximated by a random walk. To estimate the time step of this random walk, we begin by defining the cyclotron average of $(\text{d}\boldsymbol{R}/\text{d}t)_{\bot }$ ,

(2.23) $$\begin{eqnarray}\boldsymbol{v}_{R}(t)\equiv \frac{\unicode[STIX]{x1D6FA}}{2\unicode[STIX]{x03C0}}\int _{t-\unicode[STIX]{x03C0}/\unicode[STIX]{x1D6FA}}^{t+\unicode[STIX]{x03C0}/\unicode[STIX]{x1D6FA}}\left(\frac{\text{d}\boldsymbol{R}}{\text{d}t_{1}}\right)_{\bot }\,\text{d}t_{1}.\end{eqnarray}$$

As stated above, during a single cyclotron period, a proton’s motion projected onto the plane perpendicular to  $\boldsymbol{B}_{0}$ 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  $\boldsymbol{v}_{R}$ 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 $|(\text{d}\boldsymbol{R}/\text{d}t)_{\bot }|$ . Thus, given (2.6), (2.9), and (2.20),

(2.24) $$\begin{eqnarray}v_{R}\sim \frac{w_{\bot }\unicode[STIX]{x1D6FF}B_{\Vert \unicode[STIX]{x1D70C}}}{B_{0}}\sim \unicode[STIX]{x1D6FD}_{\text{p}}^{1/2}\unicode[STIX]{x1D6FF}v_{\unicode[STIX]{x1D70C}}.\end{eqnarray}$$

Because we are considering the effects of just the gyroscale AW/KAW eddies, $\boldsymbol{v}_{R}$ decorrelates after the proton’s guiding centre has moved a distance  ${\sim}\unicode[STIX]{x1D70C}_{\text{th}}$ in the plane perpendicular to  $\boldsymbol{B}_{0}$ , which takes a time

(2.25) $$\begin{eqnarray}\unicode[STIX]{x0394}t\sim \frac{\unicode[STIX]{x1D70C}_{\text{th}}}{v_{R}}.\end{eqnarray}$$

Thermal protons thus undergo spatial diffusion in the plane perpendicular to $\boldsymbol{B}_{0}$ with a spatial diffusion coefficient

(2.26) $$\begin{eqnarray}D_{\bot }\sim \frac{\unicode[STIX]{x1D70C}_{\text{th}}^{2}}{\unicode[STIX]{x0394}t}\sim \unicode[STIX]{x1D6FD}_{\text{p}}^{1/2}\unicode[STIX]{x1D6FF}v_{\unicode[STIX]{x1D70C}}\unicode[STIX]{x1D70C}_{\text{th}}.\end{eqnarray}$$

Given (2.8), (2.16) and (2.24),

(2.27) $$\begin{eqnarray}\unicode[STIX]{x0394}t\sim \frac{\unicode[STIX]{x1D70C}_{\text{th}}}{\unicode[STIX]{x1D6FD}_{\text{p}}^{1/2}\unicode[STIX]{x1D6FF}v_{\unicode[STIX]{x1D70C}}}\sim \frac{l}{\unicode[STIX]{x1D6FD}_{\text{p}}^{1/2}v_{\text{A}}}\sim \frac{l}{v_{\Vert }}.\end{eqnarray}$$

The time required for a particle to wander a distance  ${\sim}\unicode[STIX]{x1D70C}_{\text{th}}$ 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,

(2.28) $$\begin{eqnarray}H=q\unicode[STIX]{x1D6F7}+\frac{1}{2m}\left(\boldsymbol{p}-\frac{q}{c}\boldsymbol{A}\right)^{2},\end{eqnarray}$$

where $\unicode[STIX]{x1D6F7}$ is the electrostatic potential, $\boldsymbol{p}$ is the canonical momentum and $\boldsymbol{A}$ is the vector potential. From Hamilton’s equations,

(2.29) $$\begin{eqnarray}\frac{\text{d}H}{\text{d}t}=q\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D6F7}}{\unicode[STIX]{x2202}t}-\frac{q\boldsymbol{v}}{c}\boldsymbol{\cdot }\frac{\unicode[STIX]{x2202}\boldsymbol{A}}{\unicode[STIX]{x2202}t},\end{eqnarray}$$

where $\boldsymbol{v}=m^{-1}(\boldsymbol{p}-q\boldsymbol{A}/c)$ is the velocity, and the electric field is $\boldsymbol{E}=-\unicode[STIX]{x1D735}\unicode[STIX]{x1D6F7}-c^{-1}\unicode[STIX]{x2202}\boldsymbol{A}/\unicode[STIX]{x2202}t$ . The second term on the right-hand side of (2.29) is $q\boldsymbol{v}\boldsymbol{\cdot }\boldsymbol{E}_{\text{s}}$ , where $\boldsymbol{E}_{\text{s}}=-c^{-1}\unicode[STIX]{x2202}\boldsymbol{A}/\unicode[STIX]{x2202}t$ is the solenoidal component of the electric field. Equation (46) of Hollweg (Reference Hollweg1999) gives the ratio of $E_{\text{s}}$ to the magnitude of the irrotational component of the electric field  $|\unicode[STIX]{x1D735}\unicode[STIX]{x1D6F7}|$ for AWs/KAWs with $k_{\bot }\unicode[STIX]{x1D70C}_{\text{th}}\lesssim 1$ ,Footnote ³

(2.30) $$\begin{eqnarray}\frac{E_{\text{s}}}{|\unicode[STIX]{x1D735}\unicode[STIX]{x1D6F7}|}\sim \frac{\unicode[STIX]{x1D6FD}_{\text{p}}\unicode[STIX]{x1D714}}{\unicode[STIX]{x1D6FA}}.\end{eqnarray}$$

In their treatment of stochastic heating at $\unicode[STIX]{x1D6FD}_{\text{p}}\lesssim 1$ , Chandran et al. (Reference Chandran, Li, Rogers, Quataert and Germaschewski2010) neglected the second term on the right-hand side of (2.29), because this term makes a small contribution to the heating rate when $\unicode[STIX]{x1D6FD}_{\text{p}}$ is small. Here we focus on the effects of $\boldsymbol{E}_{\text{s}}$ and make the approximation that

(2.31) $$\begin{eqnarray}\frac{\text{d}H}{\text{d}t}\sim q\boldsymbol{v}\boldsymbol{\cdot }\boldsymbol{E}_{\text{s}}.\end{eqnarray}$$

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 $\unicode[STIX]{x1D6FD}_{\text{p}}\gtrsim 1$ .

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  $\unicode[STIX]{x0394}t$ given in (2.27). Within each time interval of length  ${\sim}\unicode[STIX]{x0394}t$ , the proton energy changes by an amount $\unicode[STIX]{x1D6FF}H$ (which can be positive or negative), and the values of  $\unicode[STIX]{x1D6FF}H$ are uncorrelated within successive time intervals of length  $\unicode[STIX]{x0394}t$ . As a consequence, the proton undergoes energy diffusion.

To estimate the r.m.s. value of  $\unicode[STIX]{x1D6FF}H$ , which we denote $\unicode[STIX]{x0394}H$ , 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  $\boldsymbol{B}_{0}$ for a time  $\unicode[STIX]{x0394}t$ . In the second step, the proton is instantly translated a distance  $\unicode[STIX]{x1D70C}_{\text{th}}$ in some random direction perpendicular to  $\boldsymbol{B}_{0}$ .Footnote ⁴

In this simple model, a proton undergoes $N\sim \unicode[STIX]{x1D6FA}\unicode[STIX]{x0394}t\sim (v_{\bot }/\unicode[STIX]{x1D70C}_{\text{th}})\times (l/v_{\Vert })\sim l/\unicode[STIX]{x1D70C}_{\text{th}}\gg 1$ circular gyrations in the plane perpendicular to  $\boldsymbol{B}_{0}$ during a time  $\unicode[STIX]{x0394}t$ . Integrating (2.31) for a time  $\unicode[STIX]{x0394}t$ , we obtain

(2.32) $$\begin{eqnarray}\unicode[STIX]{x1D6FF}H\sim q\int _{0}^{\unicode[STIX]{x0394}t}\boldsymbol{v}(t)\boldsymbol{\cdot }\boldsymbol{E}_{\text{s}}(\boldsymbol{r}(t),t)\,\text{d}t,\end{eqnarray}$$

where $\boldsymbol{r}(t)$ is the proton’s position at time  $t$ . Since $\unicode[STIX]{x0394}t\sim \unicode[STIX]{x1D6FD}_{\text{p}}^{-1/2}\unicode[STIX]{x1D70C}_{\text{th}}/\unicode[STIX]{x1D6FF}v_{\unicode[STIX]{x1D70C}}$ , when $\unicode[STIX]{x1D6FD}_{\text{p}}\gtrsim 1$ , the time  $\unicode[STIX]{x0394}t$ for a particle to diffuse across one set of gyroscale eddies is shorter than or comparable to the linear or nonlinear time scale $\unicode[STIX]{x1D70C}_{\text{th}}/\unicode[STIX]{x1D6FF}v_{\unicode[STIX]{x1D70C}}$ of those eddies. We thus approximate the right-hand side of (2.32) by setting $\boldsymbol{E}_{\text{s}}(\boldsymbol{r}(t),t)=\boldsymbol{E}_{\text{s}}(\boldsymbol{r}(t),0)$ and rewrite (2.32) as

(2.33) $$\begin{eqnarray}\unicode[STIX]{x1D6FF}H\sim qN\oint \boldsymbol{E}_{\text{s}}(\boldsymbol{r},0)\boldsymbol{\cdot }\text{d}\boldsymbol{l}\sim qN\int _{S}\unicode[STIX]{x1D735}\times \boldsymbol{E}(\boldsymbol{r},0)\boldsymbol{\cdot }\text{d}\boldsymbol{S}\sim -\frac{qN}{c}\int _{S}\frac{\unicode[STIX]{x2202}\boldsymbol{B}}{\unicode[STIX]{x2202}t}(\boldsymbol{r},0)\boldsymbol{\cdot }\text{d}\boldsymbol{S},\end{eqnarray}$$

where the line integral is along the proton’s path during one complete circular gyration in the plane perpendicular to  $\boldsymbol{B}_{0}$ , the surface integral is over the circular surface  $S$ of radius  $\unicode[STIX]{x1D70C}_{\text{th}}$ enclosed by the gyration, and we have used Faraday’s law $\unicode[STIX]{x1D735}\times \boldsymbol{E}=(-1/c)\unicode[STIX]{x2202}\boldsymbol{B}/\unicode[STIX]{x2202}t$ . The surface  $S$ is perpendicular to  $\boldsymbol{B}_{0}$ , and $\text{d}\boldsymbol{S}$ is anti-parallel to  $\boldsymbol{B}_{0}$ (anti-parallel rather than parallel since $q>0$ ). The r.m.s. value of $\unicode[STIX]{x1D6FF}H$ thus satisfies the order-of-magnitude relation

(2.34) $$\begin{eqnarray}\unicode[STIX]{x0394}H\sim \frac{qN}{c}\unicode[STIX]{x1D714}_{\text{eff}}\unicode[STIX]{x1D6FF}B_{\Vert \unicode[STIX]{x1D70C}}\unicode[STIX]{x1D70C}_{\text{th}}^{2},\end{eqnarray}$$

where

(2.35) $$\begin{eqnarray}\unicode[STIX]{x1D714}_{\text{eff}}\equiv \frac{\unicode[STIX]{x1D6FF}v_{\unicode[STIX]{x1D70C}}}{\unicode[STIX]{x1D70C}_{\text{th}}}\end{eqnarray}$$

is the nonlinear frequency of the gyroscale fluctuations. Upon setting $q/c=\unicode[STIX]{x1D6FA}m/B$ , $N=\unicode[STIX]{x1D6FA}\unicode[STIX]{x0394}t$ and $\unicode[STIX]{x1D70C}_{\text{th}}^{2}=w_{\bot }^{2}/\unicode[STIX]{x1D6FA}^{2}$ in (2.34), we obtain

(2.36) $$\begin{eqnarray}\unicode[STIX]{x0394}H\sim \frac{mw_{\bot }^{2}}{B}\frac{\unicode[STIX]{x1D6FF}v_{\unicode[STIX]{x1D70C}}}{\unicode[STIX]{x1D70C}_{\text{th}}}\unicode[STIX]{x1D6FF}B_{\Vert \unicode[STIX]{x1D70C}}\unicode[STIX]{x0394}t.\end{eqnarray}$$

Although we are in the process of estimating the rate at which $\unicode[STIX]{x1D707}$ changes over long times, our estimate of $\unicode[STIX]{x0394}H$ is comparable to the value that would follow from $\unicode[STIX]{x1D707}$ conservation: $\unicode[STIX]{x0394}H\sim \unicode[STIX]{x1D707}\unicode[STIX]{x0394}B\sim (mw_{\bot }^{2}/B)\unicode[STIX]{x1D714}_{\text{eff}}\unicode[STIX]{x1D6FF}B_{\Vert \unicode[STIX]{x1D70C}}\unicode[STIX]{x0394}t$ , where $\unicode[STIX]{x0394}B\sim \unicode[STIX]{x1D714}_{\text{eff}}\unicode[STIX]{x1D6FF}B_{\Vert \unicode[STIX]{x1D70C}}\unicode[STIX]{x0394}t$ is the r.m.s. amplitude of the change in the magnetic flux through the proton’s Larmor orbit, divided by $\unicode[STIX]{x03C0}\unicode[STIX]{x1D70C}^{2}$ , during the time  $\unicode[STIX]{x0394}t$ 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 $\unicode[STIX]{x1D6FD}_{\text{p}}\gtrsim 1$ . In the guiding-centre approximation, when $v_{\bot }^{2}$ increases by some factor because of  $\boldsymbol{E}_{\text{s}}$ , the field strength at the particle’s guiding centre increases by approximately the same factor, essentially because of Faraday’s law. This proportionality underlies $\unicode[STIX]{x1D707}$ conservation. In stochastic heating, the same proportionality is approximately satisfied during a single time interval  $\unicode[STIX]{x0394}t$ , but the proton is then stochastically transported to a neighbouring 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  $\boldsymbol{B}$ breaks the connection between changes to  $v_{\bot }^{2}$ and changes to  $B$ that arises in the $\unicode[STIX]{x1D70C}/\unicode[STIX]{x1D706}\rightarrow 0$ limit.

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

(2.37) $$\begin{eqnarray}K_{\bot }=\frac{mv_{\bot }^{2}}{2},\end{eqnarray}$$

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 ${\sim}\unicode[STIX]{x0394}H^{2}/\unicode[STIX]{x0394}t$ , or

(2.38) $$\begin{eqnarray}D_{K}\sim \frac{m^{2}w_{\bot }^{4}}{\unicode[STIX]{x1D6FD}_{\text{p}}^{1/2}}\frac{\unicode[STIX]{x1D6FF}v_{\unicode[STIX]{x1D70C}}}{\unicode[STIX]{x1D70C}_{\text{th}}}\frac{\unicode[STIX]{x1D6FF}B_{\unicode[STIX]{x1D70C}}^{2}}{B_{0}^{2}},\end{eqnarray}$$

where we have used (2.27) to estimate $\unicode[STIX]{x0394}t$ and (2.9) to set $\unicode[STIX]{x1D6FF}B_{\Vert \unicode[STIX]{x1D70C}}\sim \unicode[STIX]{x1D6FF}B_{\unicode[STIX]{x1D70C}}$ . A single proton undergoing a random walk in energy can gain or lose energy with equal probability during a time $\unicode[STIX]{x0394}t$ . 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 $\unicode[STIX]{x1D70F}_{\text{h}}$ is the characteristic time for the perpendicular kinetic energy of a thermal proton to double, $\unicode[STIX]{x1D70F}_{\text{h}}\sim (mw_{\bot }^{2})^{2}/D_{K}$ , and the perpendicular-heating rate per unit mass is $Q_{\bot }\sim K_{\bot }/(m\unicode[STIX]{x1D70F}_{\text{h}})\sim D_{K}/(mK_{\bot })$ , or,

(2.39) $$\begin{eqnarray}Q_{\bot }\sim \unicode[STIX]{x1D6FD}_{\text{p}}^{1/2}\frac{(\unicode[STIX]{x1D6FF}v_{\unicode[STIX]{x1D70C}})^{3}}{\unicode[STIX]{x1D70C}_{\text{th}}}.\end{eqnarray}$$

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  $\unicode[STIX]{x1D70E}_{1}$ . As $\unicode[STIX]{x1D6FF}v_{\unicode[STIX]{x1D70C}}\rightarrow 0$ , $\text{d}\unicode[STIX]{x1D707}/\text{d}t$ decreases faster than any power of  $\unicode[STIX]{x1D6FF}v_{\unicode[STIX]{x1D70C}}$ (Kruskal Reference Kruskal1962). To account for this ‘exponential’ $\unicode[STIX]{x1D707}$ conservation in the small- $\unicode[STIX]{x1D6FF}v_{\unicode[STIX]{x1D70C}}$ limit, we follow Chandran et al. (Reference Chandran, Li, Rogers, Quataert and Germaschewski2010) by multiplying the right-hand side of (2.39) by the factor $\exp (-\unicode[STIX]{x1D70E}_{2}/\unicode[STIX]{x1D6FF})$ ,

(2.40) $$\begin{eqnarray}Q_{\bot }=\unicode[STIX]{x1D70E}_{1}\frac{(\unicode[STIX]{x1D6FF}v_{\unicode[STIX]{x1D70C}})^{3}}{\unicode[STIX]{x1D70C}_{\text{th}}}\sqrt{\unicode[STIX]{x1D6FD}_{\text{p}}}\exp \left(-\frac{\unicode[STIX]{x1D70E}_{2}}{\unicode[STIX]{x1D6FF}}\right),\end{eqnarray}$$

where $\unicode[STIX]{x1D70E}_{2}$ is another as-yet-unknown dimensionless constant, and $\unicode[STIX]{x1D6FF}$ is defined in (2.21).

For comparison, the stochastic-heating rate per unit mass found by Chandran et al. (Reference Chandran, Li, Rogers, Quataert and Germaschewski2010) when $\unicode[STIX]{x1D6FD}_{\text{p}}\lesssim 1$ is

(2.41) $$\begin{eqnarray}Q_{\bot }=c_{1}\frac{(\unicode[STIX]{x1D6FF}v_{\unicode[STIX]{x1D70C}})^{3}}{\unicode[STIX]{x1D70C}_{\text{th}}}\exp \left(-\frac{c_{2}}{\unicode[STIX]{x1D700}}\right),\end{eqnarray}$$

where

(2.42) $$\begin{eqnarray}\unicode[STIX]{x1D700}=\frac{\unicode[STIX]{x1D6FF}v_{\unicode[STIX]{x1D70C}}}{w_{\bot }},\end{eqnarray}$$

and the dimensionless constants $c_{1}$ and $c_{2}$ serve the same purpose as those in (2.40). As discussed by Chandran et al. (Reference Chandran, Li, Rogers, Quataert and Germaschewski2010) for the case of $c_{1}$ and  $c_{2}$ , we expect the constants $\unicode[STIX]{x1D70E}_{1}$ and  $\unicode[STIX]{x1D70E}_{2}$ to depend on the nature of the fluctuations. For example, at fixed $\unicode[STIX]{x1D6FF}v_{\unicode[STIX]{x1D70C}}$ , we expect stronger heating rates (i.e. larger $\unicode[STIX]{x1D70E}_{1}$ and/or smaller $\unicode[STIX]{x1D70E}_{2}$ ) from intermittent turbulence than from randomly phased waves (Chandran et al. Reference Chandran, Li, Rogers, Quataert and Germaschewski2010; Xia et al. Reference Xia, Perez, Chandran and Quataert2013; Mallet et al. Reference Mallet, Klein, Chandran, Groselj, Hoppock, Bowen, Salem and Bale2018), 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 § 2.1, we focused on proton motion perpendicular to  $\boldsymbol{B}$ . However, motion along the magnetic field can also produce stochastic motion in the plane perpendicular to  $\boldsymbol{B}_{0}$ (see, e.g. Hauff et al. Reference Hauff, Jenko, Shalchi and Schlickeiser2010). In particular, the perpendicular magnetic fluctuations at the scale of a proton’s gyroradius perturb the direction of  $\hat{\boldsymbol{b}}$ . These perturbations, when fed into the first term on the right-hand side of (2.12), $v_{\Vert }\hat{\boldsymbol{b}}$ , cause the proton’s guiding centre  $\boldsymbol{R}$ to acquire a velocity perpendicular to  $\boldsymbol{B}_{0}$ of

(2.43) $$\begin{eqnarray}\boldsymbol{u}_{\bot }\sim v_{\Vert }\times \frac{\unicode[STIX]{x1D6FF}\boldsymbol{B}_{\bot \unicode[STIX]{x1D70C}}}{B_{0}},\end{eqnarray}$$

where $\unicode[STIX]{x1D6FF}\boldsymbol{B}_{\bot \unicode[STIX]{x1D70C}}$ is the component of  $\unicode[STIX]{x1D6FF}\boldsymbol{B}$ (from gyroscale fluctuations) perpendicular to  $\boldsymbol{B}_{0}$ at the proton’s location. The value of $\boldsymbol{u}_{\bot }$ varies in an incoherent manner in time, with a correlation time ${\sim}\unicode[STIX]{x1D6FA}^{-1}$ . If $u_{\bot }$ is a significant fraction of  $v_{\bot }$ , then $\boldsymbol{u}_{\bot }$ will cause a proton’s orbit in the plane perpendicular to  $\boldsymbol{B}_{0}$ to become stochastic. This leads to an alternative high- $\unicode[STIX]{x1D6FD}_{\text{p}}$ stochasticity parameter,

(2.44) $$\begin{eqnarray}\tilde{\unicode[STIX]{x1D6FF}}=\frac{u_{\bot }}{v_{\bot }}=\frac{v_{\Vert }\unicode[STIX]{x1D6FF}B_{\bot \unicode[STIX]{x1D70C}}}{v_{\bot }B_{0}}.\end{eqnarray}$$

As $\tilde{\unicode[STIX]{x1D6FF}}$ increases towards unity, proton orbits become stochastic. For thermal protons with $v_{\bot }\sim v_{\Vert }$ and $\unicode[STIX]{x1D70C}\sim \unicode[STIX]{x1D70C}_{\text{th}}$ , $\tilde{\unicode[STIX]{x1D6FF}}$ is equivalent to $\unicode[STIX]{x1D6FF}$ in (2.21), which was based upon the parallel magnetic-field fluctuation $\unicode[STIX]{x1D6FF}B_{\Vert \unicode[STIX]{x1D70C}}$ (even though we set $\unicode[STIX]{x1D6FF}B_{\Vert \unicode[STIX]{x1D70C}}\sim \unicode[STIX]{x1D6FF}B_{\unicode[STIX]{x1D70C}}$ 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  $\unicode[STIX]{x1D706}$ , denoted $\unicode[STIX]{x1D6FF}B_{\bot \unicode[STIX]{x1D706}}$ ( $\unicode[STIX]{x1D6FF}B_{\Vert \unicode[STIX]{x1D706}}$ ), is an increasing (decreasing) function of  $\unicode[STIX]{x1D706}$ when $\unicode[STIX]{x1D706}$ is in the inertial range. Orbit stochasticity through the interaction between parallel motion and $\unicode[STIX]{x1D6FF}B_{\bot \unicode[STIX]{x1D70C}}$ could thus contribute to the development of superthermal tails when $\unicode[STIX]{x1D6FD}_{\text{p}}\gtrsim 1$ . An investigation of superthermal tails, however, lies beyond the scope of this paper.

3.1 Randomly phased waves

The code used to implement the AW/KAW spectrum is similar to the code used by Chandran et al. (Reference Chandran, Li, Rogers, Quataert and Germaschewski2010). The magnetic field is $\boldsymbol{B}=B_{0}\hat{\boldsymbol{z}}+\unicode[STIX]{x1D6FF}\boldsymbol{B}$ , where $B_{0}$ is constant. We take $\boldsymbol{E}$ and $\unicode[STIX]{x1D6FF}\boldsymbol{B}$ 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 $\unicode[STIX]{x1D714}/k_{z}<0$ and the other with $\unicode[STIX]{x1D714}/k_{z}>0$ .Footnote ⁵ The initial phase of each wave is randomly chosen.

The 81 wave vectors correspond to nine evenly spaced values of the azimuthal angle in $\boldsymbol{k}$ space (in cylindrical coordinates aligned with  $\boldsymbol{B}_{0}$ ) at each of nine specific values of $k_{\bot i}:i\in [0,\ldots ,8]$ . The values of $k_{\bot i}$ are evenly spaced in $\ln (k_{\bot })$ -space, with $\ln (k_{\bot i}\unicode[STIX]{x1D70C}_{\text{th}})=-4/3+i/3$ . The middle three cells, in which $i=3,4$ and $5$ , have a combined width of unity in $\ln (k_{\bot })$ -space, centred at precisely $k_{\bot }\unicode[STIX]{x1D70C}_{\text{th}}=1$ . We computationally evaluate $\unicode[STIX]{x1D6FF}v_{\unicode[STIX]{x1D70C}}$ and $\unicode[STIX]{x1D6FF}B_{\unicode[STIX]{x1D70C}}$ via the r.m.s. values of the $\boldsymbol{E}\times \boldsymbol{B}$ velocity and $\unicode[STIX]{x1D6FF}\boldsymbol{B}$ that result from the waves in just these middle three cells.

There is one value of $k_{\Vert }\equiv |k_{z}|$ at each $k_{\bot i}$ , denoted $k_{\Vert i}$ . We determine $k_{\Vert 4}$ by setting the linear frequency at $k_{\Vert 4}$ equal to $k_{\bot 4}\unicode[STIX]{x1D6FF}v_{\unicode[STIX]{x1D70C}}$ . At other values of $k_{\bot }$ , we set

(3.2) $$\begin{eqnarray}\frac{k_{\Vert i}}{k_{\Vert 4}}=\left\{\begin{array}{@{}l@{}}\displaystyle \left(\frac{k_{\bot i}}{k_{\bot 4}}\right)^{2/3}:0\leqslant i<4\quad \\ \displaystyle \left(\frac{k_{\bot i}}{k_{\bot 4}}\right)^{1/3}:4<i\leqslant 8.\quad \end{array}\right.\end{eqnarray}$$

The exponents $2/3$ and $1/3$ in (3.2) are chosen to match the scalings in the critical-balance models of Goldreich & Sridhar (Reference Goldreich and Sridhar1995) and Cho & Lazarian (Reference Cho and Lazarian2004), respectively. We take the individual wave magnetic-field amplitudes to be proportional to $k_{\bot }^{-1/3}$ and $k_{\bot }^{-2/3}$ for $k_{\bot }\unicode[STIX]{x1D70C}_{\text{th}}<1$ and $k_{\bot }\unicode[STIX]{x1D70C}_{\text{th}}>1$ , respectively, in order to match the same two critical-balance models. (All the waves at the same value of $k_{\bot i}$ 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 (Reference Hollweg1999) two-fluid analysis of linear KAWs, setting

(3.3a,b ) $$\begin{eqnarray}\frac{T_{\text{e}}}{T_{\text{p}}}=0.5\quad \frac{v_{\text{A}}}{c}=0.003,\end{eqnarray}$$

where $T_{\text{e}}$ and $T_{\text{p}}$ 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. (Reference Chandran, Li, Rogers, Quataert and Germaschewski2010). Since we take $T_{\bot \text{p}}=T_{\Vert \text{p}}$ , we set

(3.4) $$\begin{eqnarray}w_{\bot }=w_{\Vert }=w\equiv \sqrt{\frac{2k_{\text{B}}T_{\text{p}}}{m}}=v_{\text{A}}\sqrt{\unicode[STIX]{x1D6FD}_{\text{p}}}.\end{eqnarray}$$

3.2 A note on the electric field

Following Lehe et al. (Reference Lehe, Parrish and Quataert2009), we correct the electric field because the magnetic field (including its fluctuations, i.e.  $\boldsymbol{B}=B_{0}\hat{\boldsymbol{z}}+\unicode[STIX]{x1D6FF}\boldsymbol{B}$ ) 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. (Reference Chandran, Li, Rogers, Quataert and Germaschewski2010). To fix this, we replace the sum of the individual wave electric fields described in § 3.1, which we denote  $\boldsymbol{E}_{\text{wave}}$ , with the modified electric field $\boldsymbol{E}=\boldsymbol{E}_{\text{wave}}+\hat{\boldsymbol{b}}(\hat{\boldsymbol{z}}\boldsymbol{\cdot }\boldsymbol{E}_{\text{wave}}-\hat{\boldsymbol{b}}\boldsymbol{\cdot }\boldsymbol{E}_{\text{wave}})$ .

3.3 Perpendicular heating

We perform numerical test-particle calculations at five different values of  $\unicode[STIX]{x1D6FD}_{\text{p}}$ , in particular $\unicode[STIX]{x1D6FD}_{\text{p}}=\{0.006,0.01,0.1,1,10\}$ . For each $\unicode[STIX]{x1D6FD}_{\text{p}}$ value, we carry out test-particle calculations for five different values of $\unicode[STIX]{x1D6FF}$ (or, equivalently, five different values of  $\unicode[STIX]{x1D700}$ ). Each calculation returns the values of $\langle v_{\bot }^{2}\rangle$ 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 $\langle v_{\bot }^{2}(t)\rangle$ curve determines the perpendicular-heating rate per unit mass $Q_{\bot }=(1/2)(\text{d}/\text{d}t)\langle v_{\bot }^{2}\rangle$ , where $\langle \,\cdots \,\rangle$ indicates an average over the $10^{4}$ or $10^{5}$ particles in the simulation. (We use more particles in simulations with smaller $\unicode[STIX]{x1D700}$ and $\unicode[STIX]{x1D6FF}$ because the heating rates are smaller in these simulations, and the extra particles increase the signal-to-noise ratio.) We fit the $\langle v_{\bot }^{2}(t)\rangle$ curves during the time interval $(t_{\text{i}},t_{\text{f}})$ , where $t_{\text{i}}=20\unicode[STIX]{x03C0}/\unicode[STIX]{x1D6FA}$ and $t_{\text{f}}$ is the smaller of the following two values: $10^{4}\unicode[STIX]{x1D6FA}^{-1}$ and the time required for $\langle v_{\bot }^{2}\rangle$ to increase by $\simeq 30\,\%$ . We do not include the first ten cyclotron periods when calculating $Q_{\bot }$ , 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. (Reference Xia, Perez, Chandran and Quataert2013) in simulations of test particles interacting with reduced magnetohydrodynamic turbulence at $\unicode[STIX]{x1D6FD}_{\Vert \text{p}}=1$ .)

Figure 2. The mean square velocity perpendicular to $\boldsymbol{B}_{0}$ , $\langle v_{\bot }^{2}\rangle$ , as a function of time in two test-particle calculations, each of which tracks $10^{5}$ protons. The value of the stochasticity parameter $\unicode[STIX]{x1D6FF}$ (defined in (2.21)) is 0.15 in both calculations.

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

(3.5a,b ) $$\begin{eqnarray}c_{1}=0.77,\quad c_{2}=0.33.\end{eqnarray}$$

These values are very similar to the values $c_{1}=0.75$ and $c_{2}=0.34$ obtained by Chandran et al. (Reference Chandran, Li, Rogers, Quataert and Germaschewski2010) at $\unicode[STIX]{x1D6FD}_{\text{p}}=0.006$ . The solid-line curves in (b,d) on the right correspond to (2.40) with

(3.6a,b ) $$\begin{eqnarray}\unicode[STIX]{x1D70E}_{1}=5.0,\quad \unicode[STIX]{x1D70E}_{2}=0.21.\end{eqnarray}$$

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

Figure 3. Numerical results for the perpendicular-heating rate per unit mass,  $Q_{\bot }$ , for protons interacting with randomly phased AWs and KAWs. (a) $\unicode[STIX]{x1D6FD}_{\text{p}}<1$ , and $Q_{\bot }$ normalized by $\unicode[STIX]{x1D6FA}v_{\text{A}}^{2}$ . (b) $\unicode[STIX]{x1D6FD}_{\text{p}}\geqslant 1$ and $Q_{\bot }$ normalized by $\unicode[STIX]{x1D6FA}v_{\text{A}}^{2}$ . (c) $\unicode[STIX]{x1D6FD}_{\text{p}}<1$ and $Q_{\bot }$ normalized by $\unicode[STIX]{x1D6FA}w^{2}$ , where $w$ is the proton thermal speed defined in (3.4); the numerical results for all three $\unicode[STIX]{x1D6FD}_{\text{p}}$ values (0.006, 0.01, and 0.1) are within the bars shown. (d) $\unicode[STIX]{x1D6FD}_{\text{p}}\geqslant 1$ and $Q_{\bot }$ normalized by $\unicode[STIX]{x1D6FA}w^{2}$ . In (a,c) the solid lines are plots of (2.41) for the best-fit values of  $c_{1}$ and $c_{2}$ given in (3.5). In (b,d) the solid lines are plots of (2.40) for the best-fit values of $\unicode[STIX]{x1D70E}_{1}$ and $\unicode[STIX]{x1D70E}_{2}$ in (3.6).

Figure 3(c) shows that at $\unicode[STIX]{x1D6FD}_{\text{p}}<1$ , $Q_{\bot }/(\unicode[STIX]{x1D6FA}w^{2})$ is a function of  $\unicode[STIX]{x1D700}$ alone, consistent with the fact that (2.41) can be rewritten in the form

(3.7) $$\begin{eqnarray}\frac{Q_{\bot }}{\unicode[STIX]{x1D6FA}w^{2}}=c_{1}\unicode[STIX]{x1D700}^{3}\exp \left(-\frac{c_{2}}{\unicode[STIX]{x1D700}}\right)\quad (\text{at }\unicode[STIX]{x1D6FD}_{\text{p}}\lesssim 1).\end{eqnarray}$$

Figure 3(b) shows that at $\unicode[STIX]{x1D6FD}_{\text{p}}\geqslant 1$ , $Q_{\bot }/(\unicode[STIX]{x1D6FA}v_{\text{A}}^{2})$ is a function of  $\unicode[STIX]{x1D6FF}$ alone, consistent with the fact that (2.40) can be rewritten as

(3.8) $$\begin{eqnarray}\frac{Q_{\bot }}{\unicode[STIX]{x1D6FA}v_{\text{A}}^{2}}=\unicode[STIX]{x1D70E}_{1}\unicode[STIX]{x1D6FF}^{3}\exp \left(-\frac{\unicode[STIX]{x1D70E}_{2}}{\unicode[STIX]{x1D6FF}}\right)\quad (\text{at }\unicode[STIX]{x1D6FD}_{\text{p}}\gtrsim 1).\end{eqnarray}$$

We note that in our model of randomly phased KAWs (Chandran et al. Reference Chandran, Li, Rogers, Quataert and Germaschewski2010),

(3.9) $$\begin{eqnarray}\frac{\unicode[STIX]{x1D6FF}B_{\unicode[STIX]{x1D70C}}}{B_{0}}=0.84\frac{\unicode[STIX]{x1D6FF}v_{\unicode[STIX]{x1D70C}}}{v_{\text{A}}},\end{eqnarray}$$

and thus

(3.10) $$\begin{eqnarray}\unicode[STIX]{x1D6FF}=\frac{\unicode[STIX]{x1D6FF}B_{\unicode[STIX]{x1D70C}}}{B_{0}}=0.84\frac{\unicode[STIX]{x1D6FF}v_{\unicode[STIX]{x1D70C}}}{v_{\text{A}}}=0.84\unicode[STIX]{x1D6FD}_{\text{p}}^{1/2}\frac{\unicode[STIX]{x1D6FF}v_{\unicode[STIX]{x1D70C}}}{w_{\bot }}=0.84\unicode[STIX]{x1D6FD}_{\text{p}}^{1/2}\unicode[STIX]{x1D700}.\end{eqnarray}$$

As a consequence, if we adopt the best-fit values of $\unicode[STIX]{x1D70E}_{1}$ , $\unicode[STIX]{x1D70E}_{2}$ , $c_{1}$ and  $c_{2}$ , then the value of $Q_{\bot }$ at $\unicode[STIX]{x1D6FD}_{\text{p}}=1$ in (2.40), which matches our test-particle calculations quite well, exceeds the value that would follow from (2.41) at $\unicode[STIX]{x1D6FD}_{\text{p}}=1$ . A similar phenomenon was found by Xia et al. (Reference Xia, Perez, Chandran and Quataert2013) 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  $\unicode[STIX]{x1D6FD}_{\text{p}}$ , small  $\unicode[STIX]{x1D6FD}_{\text{p}}$ and $\unicode[STIX]{x1D6FD}_{\text{p}}\simeq 1$ , we use (3.4) and (3.10) to rewrite the low- $\unicode[STIX]{x1D6FD}_{\text{p}}$ heating rate in (3.7) in terms of $\unicode[STIX]{x1D6FF}$ and $v_{\text{A}}$ . We then add the low- $\unicode[STIX]{x1D6FD}_{\text{p}}$ heating rate to the high- $\unicode[STIX]{x1D6FD}_{\text{p}}$ heating rate in (3.8), obtaining

(3.11) $$\begin{eqnarray}\frac{Q_{\bot }}{\unicode[STIX]{x1D6FA}v_{\text{A}}^{2}}=\unicode[STIX]{x1D70E}_{1}\unicode[STIX]{x1D6FF}^{3}\exp \left(-\frac{\unicode[STIX]{x1D70E}_{2}}{\unicode[STIX]{x1D6FF}}\right)+\frac{1.69c_{1}\unicode[STIX]{x1D6FF}^{3}}{\unicode[STIX]{x1D6FD}_{\text{p}}^{1/2}}\exp \left(-\frac{0.84c_{2}\unicode[STIX]{x1D6FD}_{\text{p}}^{1/2}}{\unicode[STIX]{x1D6FF}}\right).\end{eqnarray}$$

The first term on the right-hand side dominates at $\unicode[STIX]{x1D6FD}_{\text{p}}\gtrsim 1$ in part because $\unicode[STIX]{x1D70E}_{1}\simeq 6.5c_{1}$ . The second term on the right-hand side dominates at $\unicode[STIX]{x1D6FD}_{\text{p}}\ll 1$ . Figure 4 shows that (3.11) is consistent with our numerical results. This figure also illustrates how, at fixed $\unicode[STIX]{x1D6FF}B_{\unicode[STIX]{x1D70C}}/B_{0}$ , the stochastic-heating rate increases as $\unicode[STIX]{x1D6FD}_{\text{p}}$ decreases.

As mentioned above, stochastic heating becomes more effective as the fluctuations become more intermittent (Xia et al. Reference Xia, Perez, Chandran and Quataert2013; Mallet et al. Reference Mallet, Klein, Chandran, Groselj, Hoppock, Bowen, Salem and Bale2018). 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. Reference Mangeney, Salem, Veltri, Cecconi and Warmbein2001; Carbone et al. Reference Carbone, Bruno, Sorriso-Valvo and Lepreti2004; Salem et al. Reference Salem, Mangeney, Bale and Veltri2009; Chandran, Schekochihin & Mallet Reference Chandran, Schekochihin and Mallet2015; Mallet, Schekochihin & Chandran Reference Mallet, Schekochihin and Chandran2015). 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 (Reference Bourouaine and Chandran2013) found that lowering $c_{2}$ to $\simeq 0.2$ 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 $r=0.3\text{ au}$ . However, if $c_{2}=0.33$ , then the heating rate in (2.41) is too weak to explain the proton heating seen in the Helios measurements.

Figure 4. The data points reproduce the numerical results from figure 3 for $\unicode[STIX]{x1D6FD}_{\text{p}}=0.01,0.1,1.0$ and 10. The dotted, long-dashed, solid and short-dashed lines plot (3.11) for, respectively, $\unicode[STIX]{x1D6FD}_{\text{p}}=0.01$ , $\unicode[STIX]{x1D6FD}_{\text{p}}=0.1$ , $\unicode[STIX]{x1D6FD}_{\text{p}}=1$ and $\unicode[STIX]{x1D6FD}_{\text{p}}=10$ . The solid and short-dashed lines are difficult to distinguish because they are nearly on top of each other.