2.1 Transport via magnetic pitch-angle diffusion

The pitch-angle diffusion coefficient in mean free, sub-Larmor-scale magnetic turbulence is a known function of statistical parameters. It may be obtained by considering that the electron’s pitch angle experiences only a slight deflection, ${\it\delta}{\it\alpha}_{B}$ , over a single magnetic correlation length. Consequently, the ratio of the change in the electron’s transverse momentum, ${\rm\Delta}p_{t}$ , to its initial momentum, $p$ , is ${\it\delta}{\it\alpha}_{B}\approx {\rm\Delta}p_{t}/p\sim e({\it\delta}B/c){\it\lambda}_{B}/{\it\gamma}m_{e}v$ , since ${\rm\Delta}p_{t}\sim F_{L}{\it\tau}_{B}$ – where $\boldsymbol{F}_{L}=(e/c)\,\boldsymbol{v}\times {\bf\delta}\boldsymbol{B}$ is the transverse Lorentz force and ${\it\tau}_{B}\sim {\it\lambda}_{B}/v$ is the time to transit ${\it\lambda}_{B}$ . The subsequent deflection will be in a random direction, because the field is uncorrelated over scales greater than ${\it\lambda}_{B}$ . As for any diffusive process, the mean squared pitch angle grows linearly with time. Thus, the diffusion coefficient appears as Keenan & Medvedev (Reference Keenan and Medvedev2013), Keenan et al. (Reference Keenan, Ford and Medvedev2015):

(2.3) $$\begin{eqnarray}D_{{\it\alpha}{\it\alpha}}\equiv \frac{\langle {\it\alpha}^{2}\rangle }{t}=\frac{{\it\lambda}_{B}}{{\it\gamma}^{2}c\langle {\it\beta}_{B}^{2}\rangle ^{1/2}}\langle {\it\Omega}_{{\it\delta}B}^{2}\rangle ,\end{eqnarray}$$

where ${\it\alpha}$ is the electron deflection angle (pitch angle) with respect to the electron’s initial direction of motion, $\langle {\it\beta}_{B}^{2}\rangle ^{1/2}$ is an appropriate ensemble-average over the (transverse) electron velocities, and

(2.4) $$\begin{eqnarray}{\it\bf\Omega}_{{\it\delta}B}\equiv \frac{e{\it\delta}\boldsymbol{B}}{m_{e}c}.\end{eqnarray}$$

In general, the pitch-angle diffusion will be path dependent, owing to the dependence on the magnetic correlation length, ${\it\lambda}_{B}$ . To properly treat the correlation length, we must introduce the two-point autocorrelation tensor of the magnetic fluctuations (Keenan et al. Reference Keenan, Ford and Medvedev2015),

(2.5) $$\begin{eqnarray}\unicode[STIX]{x1D619}^{ij}(\boldsymbol{r},t)\equiv \langle {\it\delta}B^{i}(\boldsymbol{x},{\it\tau}){\it\delta}B^{j}(\boldsymbol{x}+\boldsymbol{r},{\it\tau}+t)\rangle _{\boldsymbol{ x},{\it\tau}},\end{eqnarray}$$

with the path and time-dependent correlation length tensor defined as:

(2.6) $$\begin{eqnarray}{\it\lambda}_{B}^{ij}(\hat{\boldsymbol{r}},t)\equiv \int _{0}^{\infty }\!\frac{\unicode[STIX]{x1D619}^{ij}(\boldsymbol{r},t)}{\unicode[STIX]{x1D619}^{ij}(0,0)}\,\text{d}r.\end{eqnarray}$$

To evaluate this expression, we must consider the physics involved. In magnetic deflections, only the component of the magnetic field transverse to the particle velocity is involved in the acceleration. Thus, for magnetic fields, we only consider fields transverse to the direction of motion. In contrast, electric fields will have a longitudinal and transverse correlation length. The former is important for energy diffusion – whereas, the latter governs pitch-angle diffusion, since transverse deflections do no work.

Evaluation of (2.6) can be very difficult in a general case. If we make some simplifying assumptions about the magnetic turbulence, however, we may evaluate (2.6) exactly. If the transit time of a particle over a correlation length is shorter than the field variability time scale, then we can treat the magnetic field as static. Additionally, assuming statistical homogeneity and isotropy permits us to use a simple expression for the correlation tensor.

The pitch-angle diffusion coefficient, under these simplifying assumptions, has been derived previously (Keenan et al. Reference Keenan, Ford and Medvedev2015). We repeat those results here. The magnetic correlation length assumes the form (Keenan et al. Reference Keenan, Ford and Medvedev2015):

(2.7) $$\begin{eqnarray}{\it\lambda}_{B}=\frac{3{\rm\pi}}{8}\frac{\displaystyle \int _{0}^{\infty }\!k|{\it\delta}\boldsymbol{B}_{k}|^{2}\,\text{d}k}{\displaystyle \int _{0}^{\infty }\!k^{2}|{\it\delta}\boldsymbol{B}_{k}|^{2}\,\text{d}k},\end{eqnarray}$$

where $|{\it\delta}B_{k}|^{2}$ is the spectral distribution of the fluctuation magnetic field in Fourier $k$ -space. Thus, the pitch-angle diffusion coefficient, for electrons moving through sub-Larmor-scale (isotropic/homogeneous) magnetic turbulence, is:

(2.8) $$\begin{eqnarray}D_{{\it\alpha}{\it\alpha}}=\frac{3{\rm\pi}}{8}\sqrt{\frac{3}{2}}\frac{\displaystyle \int _{0}^{\infty }\!k|{\it\delta}\boldsymbol{B}_{k}|^{2}\,\text{d}k}{\displaystyle \int _{0}^{\infty }\!k^{2}|{\it\delta}\boldsymbol{B}_{k}|^{2}\,\text{d}k}\frac{\langle {\it\Omega}_{{\it\delta}B}^{2}\rangle }{{\it\gamma}^{2}c{\it\beta}},\end{eqnarray}$$

where we have assumed a mono-energetic distribution of electrons with velocity, ${\it\beta}$ .

In our derivation, we have considered the average square displacement in the pitch angle of an ensemble of electrons – with respect to the initial direction of motion of each representative electron. This should be contrasted with the usual convention which specifies a pitch angle with respect to the mean magnetic field; i.e. the reference direction is the same for all electrons.

However, since the assumed magnetic turbulence is statistically homogeneous and isotropic, the deflection angle, ${\it\alpha}$ , for each electron may be chosen with respect to an arbitrary axis. When a mean field is present, the parallel component of the velocity is unaffected by $\boldsymbol{B}_{0}$ . Consequently, without loss of generality, we can define ${\it\alpha}$ as the conventional pitch angle – i.e. the angle of the velocity vector with respect to the mean (ambient) magnetic field.

In reality, nevertheless, the turbulence will likely be anisotropic. This is certainly the case for Whistler turbulence. For this reason, our assumption of isotropy must be questioned. Nonetheless, the general expression, equation (2.3) holds for anisotropic turbulence as long as the correlation length is properly specified along the coordinate direction of interest. The regular component of the particle motion – i.e. the gyro motion about the mean magnetic field – must also be subtracted out for directions other than along $\boldsymbol{B}_{\mathbf{0}}$ .

For the sake of computational and analytical simplicity, we choose to consider isotropic turbulence in this work, thereby allowing us to consider only the diffusion coefficient along the axis of the mean magnetic field – which is representative of the diffusion as a whole. We leave more realistic anisotropic cases to future studies.

2.2 Energy diffusion in small-scale electric turbulence

As mentioned previously, random electric fields may induce transport via energy diffusion. Although diffusive energy transport in electromagnetic turbulence has long been a topic of investigation (Stix Reference Stix and Thomas1992), energy diffusion in strictly sub-Larmor-scale electromagnetic fields has yet to be – to the best of our knowledge – explored. This topic has proved to be richly complicated, so we have limited ourselves to a particularly simple regime.

To begin, we must consider the time scales involved. There are two such characteristic time scales: the acceleration time, ${\it\tau}_{E}^{l}$ and the electric field autocorrelation time, ${\it\tau}_{ac}$ . The latter time scale characterizes the temporal inhomogeneity of the electric field. Diffusive (energy) transport may arise, not only from spatial stochasticity in the electric field, but temporal randomness as well.

The former quantity, ${\it\tau}_{E}^{l}$ , characterizes the spatial stochasticity. This is the time required to transit an electric field correlation length, ${\it\lambda}_{E}^{l}$ – with the ‘ $l$ ’ superscript indicating the longitudinal transit time; i.e. the time required to transverse a longitudinal electric correlation length, ${\it\lambda}_{E}^{l}$ , which is along the direction of motion. Assuming that $a_{{\it\lambda}}{\it\tau}_{E}^{l}\ll v_{E}$ , where $a_{{\it\lambda}}$ is the acceleration over ${\it\lambda}_{E}^{l}$ and $v_{E}$ is the component of the electron velocity parallel to the electric field, the transit time is:

(2.9) $$\begin{eqnarray}{\it\tau}_{E}^{l}\sim \frac{{\it\lambda}_{E}^{l}}{v_{E}}.\end{eqnarray}$$

While transiting a single correlation length, the electron is subject to a nearly uniform electric field. These accelerations are uncorrelated on a spatial scale dictated by the electric field correlation length.

The diffusion regime we will explore will consider spatial diffusion to be the dominant process, i.e.

(2.10) $$\begin{eqnarray}{\it\tau}_{E}^{l}\ll {\it\tau}_{ac}.\end{eqnarray}$$

Furthermore, to ensure that the energy change is random on the time scale of consideration, we require that:

(2.11) $$\begin{eqnarray}{\it\tau}_{E}^{l}\ll t.\end{eqnarray}$$

Next, an equation for the electron energy, $W_{e}$ , may be obtained directly from the Lorentz force equation of motion. It is:

(2.12) $$\begin{eqnarray}\frac{\text{d}W_{e}}{\text{d}t}=e\left(\boldsymbol{v}\boldsymbol{\cdot }\boldsymbol{E}\right).\end{eqnarray}$$

Since the electron energy changes over the characteristic time scale, ${\it\tau}_{E}^{l}$ , we may write:

(2.13) $$\begin{eqnarray}\frac{{\rm\Delta}W_{{\it\lambda}}}{{\it\tau}_{E}^{l}}\sim ev_{E}E.\end{eqnarray}$$

If the random process is, indeed, diffusive:

(2.14) $$\begin{eqnarray}D_{WW}\equiv \frac{\langle W_{e}^{2}\rangle }{t}.\end{eqnarray}$$

Thus:

(2.15) $$\begin{eqnarray}D_{WW}\sim \frac{\left({\rm\Delta}W_{{\it\lambda}}\right)^{2}}{{\it\tau}_{E}^{l}}\sim e^{2}v_{E}E^{2}{\it\lambda}_{E}^{l},\end{eqnarray}$$

where we have used (2.9). With the usual assumptions of statistical homogeneity/isotropy and an initially mono-energetic distribution of electrons, we may write the energy diffusion coefficient, thus:

(2.16) $$\begin{eqnarray}D_{WW}=\sqrt{{\textstyle \frac{1}{3}}}e^{2}\langle E^{2}\rangle v{\it\lambda}_{E}^{l}.\end{eqnarray}$$

This result may be contrasted with the temporal, i.e. resonant, energy diffusion coefficient. The physics of this type of diffusion may be understood by considering the so-called quasilinear energy diffusion coefficient. Typically, a velocity space or momentum space diffusion coefficient is desired in the quasilinear regime – since these are connected to the transport coefficients in the Fokker–Planck equation (Schlickeiser Reference Schlickeiser1994; Giacalone & Jokipii Reference Giacalone and Jokipii1999). In our case however, the energy diffusion coefficient is of greater interest, since it is directly proportional to the electric correlation length – a key turbulent statistical parameter.

Nevertheless, the physical intuition behind the quasilinear energy and velocity/momentum diffusion coefficients is the same. Following Stix (Reference Stix and Thomas1992), we will consider only small corrections to the electron’s initial velocity – hence, we will assume the zero-order trajectory:

(2.17) $$\begin{eqnarray}\boldsymbol{r}(t)=\boldsymbol{v}t+\boldsymbol{r}_{0},\end{eqnarray}$$

where $\boldsymbol{r}_{0}$ is the electron’s initial position. Let us suppose that the electric field assumes a simple sinusoidal profile, i.e. 

(2.18) $$\begin{eqnarray}\boldsymbol{E}(\boldsymbol{x},t)=\boldsymbol{E}_{0}\cos (\boldsymbol{k}\boldsymbol{\cdot }\boldsymbol{x}-{\it\Omega}t).\end{eqnarray}$$

Thus, using (2.12) and (2.18), we have:

(2.19) $$\begin{eqnarray}\frac{\text{d}W_{e}}{\text{d}t}=e\left(\boldsymbol{v}\boldsymbol{\cdot }\boldsymbol{E}_{0}\right)\cos (\boldsymbol{k}\boldsymbol{\cdot }\boldsymbol{v}t+\boldsymbol{k}\boldsymbol{\cdot }\boldsymbol{r}_{0}-{\it\Omega}t).\end{eqnarray}$$

Integrating (2.19), averaging over all possible initial positions and squaring the result, gives the energy variance:

(2.20) $$\begin{eqnarray}\langle {\rm\Delta}{W_{e}}^{2}\rangle =\left[\frac{e\left(\boldsymbol{v}\boldsymbol{\cdot }\boldsymbol{E}_{0}\right)}{\left({\it\Omega}-\boldsymbol{k}\boldsymbol{\cdot }\boldsymbol{v}\right)}\right]^{2}\sin ^{2}\left[\frac{\left({\it\Omega}-\boldsymbol{k}\boldsymbol{\cdot }\boldsymbol{v}\right)t}{2}\right].\end{eqnarray}$$

Finally, with ${\it\Omega}t\gg 1$ , we may employ the relation (Stix Reference Stix and Thomas1992):

(2.21) $$\begin{eqnarray}\sin ^{2}\left[\frac{\left({\it\Omega}-\boldsymbol{k}\boldsymbol{\cdot }\boldsymbol{v}\right)t}{2}\right]\sim {\rm\pi}{\it\delta}\left({\it\Omega}-\boldsymbol{k}\boldsymbol{\cdot }\boldsymbol{v}\right),\end{eqnarray}$$

Thus the (quasilinear) diffusion coefficient is:

(2.22) $$\begin{eqnarray}D_{WW}^{res.}\equiv \frac{\langle {\rm\Delta}{W_{e}}^{2}\rangle }{t}\sim {\rm\pi}\left[\frac{e\left(\boldsymbol{v}\boldsymbol{\cdot }\boldsymbol{E}_{0}\right)}{\left({\it\Omega}-\boldsymbol{k}\boldsymbol{\cdot }\boldsymbol{v}\right)}\right]^{2}{\it\delta}\left({\it\Omega}-\boldsymbol{k}\boldsymbol{\cdot }\boldsymbol{v}\right).\end{eqnarray}$$

In general, turbulence will contain a spectrum of waves; hence, an integration of (2.22) over $|\boldsymbol{E}_{\boldsymbol{k},{\it\Omega}}|^{2}$ is required to produce the complete diffusion equation.

Nevertheless, much can be gathered by examining the functional form of this simplified expression. For example, owing to the ${\it\delta}\left({\it\Omega}-\boldsymbol{k}\boldsymbol{\cdot }\boldsymbol{v}\right)$ dependence, only particles that are in resonance with the wave participate in the diffusive process.

Significantly, the quasilinear diffusion equation derived here applies to non-magnetized plasmas. When an ambient magnetic field, $\boldsymbol{B}_{0}$ , is present, the resonance condition generalizes to Stix (Reference Stix and Thomas1992) and Schlickeiser (Reference Schlickeiser1994):

(2.23) $$\begin{eqnarray}{\it\Omega}-k_{\Vert }v_{\Vert }=n{\it\Omega}_{ce}/{\it\gamma},\end{eqnarray}$$

where ${\it\Omega}_{ce}\equiv eB_{0}/m_{e}c$ is the non-relativistic gyro frequency, the parallel direction is along the ambient (mean) magnetic field and $n$ is an integer. Typically, only the $n=0,\pm 1,\pm 2$ resonances need to be considered for Whistler waves; however, sufficiently high-energy electrons may have access to the higher-order resonances (Chang et al. Reference Chang, Ni, Bortnik, Zhou, Zhao, Li and Gu2014).

Equation (2.23) may, alternatively, be expressed as:

(2.24) $$\begin{eqnarray}\frac{{\it\Omega}}{kv}-\cos ({\it\theta}_{k})\cos ({\it\alpha})=\frac{n}{kr_{L0}^{rel.}},\end{eqnarray}$$

where $r_{L0}^{rel.}={\it\gamma}r_{L0}$ , $r_{L0}=eB_{0}/m_{e}c$ is the non-relativistic Larmor radius in the mean magnetic field, ${\it\theta}_{k}$ is the angle between $\boldsymbol{k}$ and $\boldsymbol{B}_{\mathbf{0}}$ and ${\it\alpha}$ is the conventional pitch angle. If we suppose that the magnetic field is small scale, i.e.  $kr_{L0}^{rel.}\gg 1$ , then we have:

(2.25) $$\begin{eqnarray}\frac{{\it\Omega}}{kv}-\cos ({\it\theta}_{k})\cos ({\it\alpha})\simeq 0.\end{eqnarray}$$

The implication of (2.25) is that cyclotron resonances (i.e.  $|n|>0$ ) are not important for electrons moving through sub-Larmor-scale fields. However, we have, thus far, not ruled out the Landau resonant interaction (i.e.  $n=0$ ).

To proceed, we require a dispersion relation, ${\it\Omega}_{r}(\boldsymbol{k})$ . For cold Whistler waves with wavelengths much greater than the electron skin depth, ${\it\Omega}_{r}=k^{2}c^{2}{\it\Omega}_{ce}/{\it\omega}_{pe}^{2}\cos ({\it\theta}_{k})$ – where ${\it\omega}_{pe}$ is the electron plasma frequency. Thus, we may write:

(2.26) $$\begin{eqnarray}\left(\frac{kc}{{\it\omega}_{pe}}\right)\left(\frac{{\it\Omega}_{ce}}{{\it\omega}_{pe}}\right)\simeq {\it\beta}_{\Vert }.\end{eqnarray}$$

However, ${\it\omega}_{pe}\gg {\it\Omega}_{ce}$ and $kc\ll {\it\omega}_{pe}$ in the regime of interest. Therefore, ${\it\beta}_{\Vert }\sim 0$ . This means that the Landau resonance ( $n=0$ ) for these test electrons favours the central part of the test particle distribution function, $f_{e}$ ; i.e. near ${\it\beta}_{\Vert }\sim 0$ , which implies that $\partial f_{e}/\partial v_{\Vert }\sim 0$ . Thus, these electrons do not appreciatively contribute to the growth or damping of the electromagnetic fluctuations, and thus their energy exchange with the waves is minimal.

Moreover, since ${\it\Omega}t\gg 1$ , this quasilinear diffusion via a Landau resonance occurs on a much greater time scale than ${\it\tau}_{E}^{l}$ . For this reason, the non-resonant energy diffusion coefficient – (2.15) – is much greater than the Landau equivalent; i.e. (2.22).

2.3 Particle transport in magnetized plasmas with electric fluctuations

As mentioned previously, the combined effect of electric and magnetic fields can lead to fairly complicated particle dynamics. Particle drifts, for example, involving both the electric and magnetic fields, should be considered. Here, we present two realizations of the drift phenomenon. We will subsequently argue that these effects are of negligible importance for diffusion in small-scale fields.

In § 2.1, we argued that sub-Larmor-scale magnetic fluctuations result in trajectories that occupy the small deflection angle regime. For this reason, the ‘guiding centre approximation’, that underlies the drift theory, breaks down. Consequently, the notions of curvature drift and Grad-B drift lose all meaning in this regime.

Nonetheless, a magnetized plasma contains a large-scale magnetic field – which is, by construction, ‘super-Larmor scale’. Hence, drifts that involve the electric field and the ambient (mean) field are, in principle, important to consider.

The first of these that we will explore is the, so called, E cross B drift. We will, once more, assume a sinusoidal electric field. In this case, however, we assume that an ambient magnetic field, $\boldsymbol{B}_{0}$ , is present. Furthermore, we suppress the time dependence. The solution for an arbitrary electric field is well known, provided that $kr_{L0}\ll 1$ . The constant drift velocity, in this case, is Chen (Reference Chen1984):

(2.27) $$\begin{eqnarray}\boldsymbol{v}_{\boldsymbol{E}\times \boldsymbol{B}}=c\left(1+\frac{1}{4}r_{L0}^{2}{\rm\nabla}^{2}\right)\frac{\boldsymbol{E}\times \boldsymbol{B}_{0}}{B_{0}^{2}}.\end{eqnarray}$$

The second term, i.e. that which involves the Laplacian operator, is a correction known as the finite-Larmor-radius effect. When $kr_{L0}\gg 1$ , the Larmor radius is much larger than the field wavelength. In this case, the particle is acted upon by the electric field on a time scale much shorter than the gyro period. Consequently, the drift approximation is not appropriate for small-scale fields, since the perturbation is implicitly assumed to act on a time scale of many gyro periods.

A similar drift phenomenon occurs when we consider the time dependence of the electric field. Assuming that ${\it\Omega}^{2}\ll {\it\Omega}_{ce}^{2}$ , the particle will drift with velocity (Chen Reference Chen1984):

(2.28) $$\begin{eqnarray}\boldsymbol{v}_{p}=\pm \frac{c}{{\it\Omega}_{ce}B_{0}}\frac{\text{d}\boldsymbol{E}}{\text{d}t}.\end{eqnarray}$$

The quantity, $\boldsymbol{v}_{p}$ , is known as the polarization drift velocity. Similarly, the small-scale processes – by construction – occur on time scales much shorter than ${\it\Omega}^{-1}$ . Hence, the polarization drift time scale will be far greater than ${\it\tau}_{E}^{l}$ . For this reason, polarization drift is not significant on the time scales of immediate interest.

In the next subsection, we will consider the case of small-scale energy diffusion in isotropic, small-scale Whistler turbulence.

2.4 Energy diffusion in small-scale Whistler turbulence

Next, to evaluate (2.16), we consider a concrete example of electromagnetic turbulence in a magnetized plasma. Whistler-mode turbulence in a cold plasma admits a simple dispersion relation (Sazhin Reference Sazhin1993):

(2.29) $$\begin{eqnarray}{\it\Omega}_{r}(k)={\it\Omega}_{ce}\frac{k^{2}c^{2}}{k^{2}c^{2}+{\it\omega}_{pe}^{2}}\cos ({\it\theta}_{k}),\end{eqnarray}$$

where ${\it\theta}_{k}\in (0,{\rm\pi}/2)$ . We will assume a (nearly) steady state, so that the instability is nonlinearly saturated; that is the instability growth rate, ${\it\Omega}_{i}$ , is much less than all relevant frequency scales, and thus it is negligible. This treatment assumes that the turbulence is linear, i.e.  ${\it\delta}B\ll B_{0}$ . We will further assume that:

(2.30) $$\begin{eqnarray}\frac{{\it\gamma}v}{{\it\Omega}_{ce}}>{\it\lambda}_{B},\end{eqnarray}$$

where ${\it\Omega}_{ce}/{\it\gamma}$ is the relativistic gyro frequency.

Equation (2.30) implies that ${\it\rho}\gg 1$ , since ${\it\delta}B\ll B_{0}$ – thus, the test electrons are sub-Larmor scale with respect to the fluctuation magnetic field, ${\bf\delta}\boldsymbol{B}$ .

It is worth mentioning that, formally, the cold plasma approximation requires that $kv/{\it\Omega}_{ce}\ll 1$ (Verkhoglyadova, Tsurutani & Lakhina Reference Verkhoglyadova, Tsurutani and Lakhina2010). This condition would imply that the electron population is super-Larmor scale with respect to the magnetic field, since ${\it\lambda}_{B}\sim k_{B}^{-1}$ , where $k_{B}$ is the wavenumber of the dominant wave mode. For this reason, our model implicitly presupposes the existence of a cold population of super-Larmor-scale electrons which support the Whistler modes. Consequently, our test particles will be comprised of a hot, albeit smaller, population of sub-Larmor-scale electrons. This situation may be approximately realized by the super-halo electron population (Lin Reference Lin1998; Wang et al. Reference Wang, Yang and He2015), as it propagates through the colder solar wind turbulence, which appears to include small-scale kinetic-Alfvén and Whistler-wave modes (Che, Goldstein & Viñas Reference Che, Goldstein and Viñas2014).

An examination of (2.29) reveals that ${\it\Omega}_{r}(k)\ll {\it\Omega}_{ce}$ in the regime where $kc\ll {\it\omega}_{pe}$ . Restricting ourselves to this regime motivates the introduction of a new parameter, which we call the skin number. It is:

(2.31) $$\begin{eqnarray}{\it\chi}\equiv d_{e}{\it\lambda}_{B}^{-1},\end{eqnarray}$$

where $d_{e}\equiv c/{\it\omega}_{pe}$ , is the electron skin depth. Thus, the regime of interest is characterized by ${\it\chi}\ll 1$ .

It is noteworthy that, in principle, the test electron velocities may be large enough so that ${\it\Omega}_{ce}/{\it\gamma}\sim {\it\Omega}_{r}$ . By restricting the electron velocities to the mildly relativistic regime, we may safely presuppose that the field-variability time, ${\it\Omega}_{r}^{-1}$ , is sufficiently greater than the time to transit a magnetic correlation length, thus permitting the static field treatment for the magnetic field and avoiding the wave–particle resonance treatment.

Next, in the ${\it\chi}\ll 1$ regime, the electric field perpendicular to $\boldsymbol{B}_{0}$ is much greater than the component parallel to the ambient magnetic field; i.e.  $E_{\bot }\gg E_{\Vert }$ (Sazhin Reference Sazhin1993). Furthermore, it can be shown that in the frame moving along the direction of $\boldsymbol{B}_{0}$ with velocity equal to the parallel phase velocity, $v_{ph}^{\Vert }\equiv {\it\Omega}_{r}/k_{\Vert }$ , the perpendicular electric field is approximately zero (Sazhin Reference Sazhin1993). Consequently, this allows us, via Lorentz transformation of the electromagnetic fields, to relate the magnetic spectral distribution to the electric distribution. It is, thus:

(2.32) $$\begin{eqnarray}|\boldsymbol{E}_{\boldsymbol{k}}|^{2}\approx |\boldsymbol{E}_{\boldsymbol{ k}}^{\bot }|^{2}\approx {\it\beta}_{ph}^{2}|{\it\delta}\boldsymbol{B}_{\boldsymbol{ k}}^{\bot }|^{2},\end{eqnarray}$$

where $\bot$ refers to the spectrum perpendicular to the mean magnetic field, and

(2.33) $$\begin{eqnarray}{\it\beta}_{ph}\equiv \frac{v_{ph}^{\Vert }}{c}=\frac{{\it\Omega}_{r}(\boldsymbol{k})}{k_{\Vert }c}.\end{eqnarray}$$

Given isotropic/homogeneous magnetic turbulence:

(2.34) $$\begin{eqnarray}|{\it\delta}\boldsymbol{B}_{\boldsymbol{k}}^{\bot }|^{2}=|{\it\delta}\boldsymbol{B}_{k}|^{2}\cos ^{2}({\it\theta}_{k}).\end{eqnarray}$$

This relation then allows us to express $\langle E^{2}\rangle$ in terms of the magnetic field as:

(2.35) $$\begin{eqnarray}\langle E^{2}\rangle =\frac{2}{3}\frac{\displaystyle \int {\it\beta}_{ph}^{2}|{\it\delta}\boldsymbol{B}_{k}|^{2}\,\text{d}\boldsymbol{k}}{\displaystyle \int |{\it\delta}\boldsymbol{B}_{k}|^{2}\,\text{d}\boldsymbol{k}}\langle {\it\delta}B^{2}\rangle .\end{eqnarray}$$

Next, the electric field correlation length may be obtained from the electric field correlation tensor. For isotropic turbulence, one may write the general expression for the Fourier image of the electric field two-point autocorrelation tensor as:

(2.36) $$\begin{eqnarray}{\it\Phi}_{ij}(\boldsymbol{k})=|\boldsymbol{E}_{\boldsymbol{k}}^{t}|^{2}\left({\it\delta}_{ij}-\frac{k_{i}k_{j}}{k^{2}}\right)+|\boldsymbol{E}_{\boldsymbol{k}}^{l}|^{2}\frac{k_{i}k_{j}}{k^{2}}.\end{eqnarray}$$

Isotropy is an approximation here, given the polar asymmetry indicated by (2.34). Using Maxwell’s equations, we may relate the longitudinal, $|\boldsymbol{E}_{\boldsymbol{k}}^{l}|^{2}$ and transverse, $|\boldsymbol{E}_{\boldsymbol{k}}^{t}|^{2}$ distributions to $|{\it\delta}\boldsymbol{B}_{k}|^{2}$ (where longitudinal and transverse are with respect to the wave vector, not the electron velocity). To wit:

(2.37) $$\begin{eqnarray}\left.\begin{array}{@{}l@{}}|\boldsymbol{E}_{\boldsymbol{k}}^{t}|^{2}=\displaystyle \frac{{\it\Omega}_{r}^{2}}{k^{2}c^{2}}|{\it\delta}\boldsymbol{B}_{k}|^{2}\\ |\boldsymbol{E}_{\boldsymbol{k}}^{l}|^{2}=|\boldsymbol{E}_{\boldsymbol{k}}|^{2}-|\boldsymbol{E}_{\boldsymbol{k}}^{t}|^{2}.\\ \end{array}\right\}\end{eqnarray}$$

In the ${\it\chi}\ll 1$ regime, we may substitute (2.32) to express the tensor completely in terms of the magnetic spectrum. The trace of the correlation tensor is then given by:

(2.38) $$\begin{eqnarray}\text{Tr}\left[\stackrel{\leftrightarrow }{{\it\Phi}}(\boldsymbol{k})\right]=2{\it\beta}_{ph}^{2}|{\it\delta}\boldsymbol{B}_{k}|^{2}\cos ^{2}({\it\theta}_{k}).\end{eqnarray}$$

While integrating (2.38) along a selected path, we only consider the component of the electric field parallel to the trajectory, owing to the dot product with velocity in (2.12). This allows us to draw an analogy to the ‘mono-polar’ (magnetic) correlation length considered in Keenan et al. (Reference Keenan, Ford and Medvedev2015) – permitting us to write the expression immediately as:

(2.39) $$\begin{eqnarray}{\it\lambda}_{E}^{l}\equiv {\it\lambda}_{E}^{Tr}(x\hat{x})=\frac{3{\rm\pi}}{4}\frac{\displaystyle \int (v_{ph}^{\Vert })^{2}k|{\it\delta}\boldsymbol{B}_{k}|^{2}\,\text{d}k}{\displaystyle \int (v_{ph}^{\Vert })^{2}k^{2}|{\it\delta}\boldsymbol{B}_{k}|^{2}\,\text{d}k},\end{eqnarray}$$

where the integration path was chosen to be along the $x$ -axis. By comparing (2.7) to (2.39), we see that the electric correlation length differs from the magnetic correlation length only by a factor of a few. For this reason, we may conclude that ${\it\tau}_{E}^{l}$ is less than ${\it\tau}_{ac}\sim {\it\Omega}_{r}^{-1}$ . Consequently, the energy diffusion will be dominated by the electric field’s spatial stochasticity, as per (2.10).

Additionally, ${\it\chi}\ll 1$ and ${\it\Omega}_{ce}\ll {\it\omega}_{pe}$ demand that $v_{ph}^{\Vert }\ll c$ . This implies that $\langle {\it\delta}B^{2}\rangle \gg \langle E^{2}\rangle$ . Consequently, the pitch-angle diffusion will be dominated by the magnetic deflections, and thus we may neglect the contribution due to the electric field.

Finally, given (2.35), the energy diffusion coefficient may be related directly to the (magnetic) pitch-angle diffusion coefficient via the relation:

(2.40) $$\begin{eqnarray}D_{WW}=\frac{2\sqrt{2}}{9}W_{e}^{2}{\it\beta}^{2}\frac{\displaystyle \int ({\it\beta}_{ph}^{\Vert })^{2}|{\it\delta}\boldsymbol{B}_{k}|^{2}\,\text{d}\boldsymbol{k}}{\displaystyle \int |{\it\delta}\boldsymbol{B}_{k}|^{2}\,\text{d}\boldsymbol{k}}\frac{{\it\lambda}_{E}^{l}}{{\it\lambda}_{B}}D_{{\it\alpha}{\it\alpha}}.\end{eqnarray}$$

Equations (2.40) and (2.8) will be confirmed, given isotropic small-scale Whistler turbulence, via first-principle numerical simulation in § 4.

2.5 Radiation production in magnetized plasmas with sub-Larmor-scale magnetic fluctuations

As mentioned previously, radiation production by electrons moving through magnetic turbulence has been explored thoroughly by a number of authors. Typically, radiation from particles in small-scale magnetic fields will markedly differ from synchrotron and cyclotron radiation. This is because the synchrotron/cyclotron prescription requires that the magnetic field be only weakly inhomogeneous on typical Larmor spatial scales – which may not hold on kinetic scales, as is the case with magnetic fields driven by Weibel-like instabilities.

For example, in the nonlinear evolution of the Weibel instability, a soliton magnetic wave may take hold. Electrons moving through this wave will emit radiation, which owing to the small-scale nature of the field, will not be of the synchrotron/cyclotron type (Schaefer-Rolffs, Lerche & Tautz Reference Schaefer-Rolffs, Lerche and Tautz2009; Tautz & Lerche Reference Tautz and Lerche2012).

Regardless of the nature of the magnetic field, the ultrarelativistic regime, specifically, is characterized by a single parameter, the ratio of the deflection angle, ${\it\delta}{\it\alpha}_{B}$ (over a single magnetic correlation length) to the relativistic beaming angle, ${\rm\Delta}{\it\theta}\sim 1/{\it\gamma}$ . The ratio (Medvedev Reference Medvedev2000; Medvedev et al. Reference Medvedev, Frederiksen, Haugbølle and Nordlund2011; Keenan & Medvedev Reference Keenan and Medvedev2013):

(2.41) $$\begin{eqnarray}\frac{{\it\delta}{\it\alpha}_{B}}{{\rm\Delta}{\it\theta}}\sim \frac{e{\it\delta}B}{m_{e}c^{2}}{\it\lambda}_{B}\equiv {\it\delta}_{j},\end{eqnarray}$$

is known as the jitter parameter. If ${\it\delta}_{j}\ll 1$ , which implies that ${\it\rho}\gg 1$ , then a distant observer in the line-of-sight will see the radiation along virtually the entire trajectory of the particle. This is why the radiation from a Weibel soliton wave is not synchrotron radiation. Similarly, if the magnetic field is turbulent on these small scales, the radiation will be further distinguished; it is known as small-angle jitter radiation (Medvedev Reference Medvedev2000, Reference Medvedev2006; Medvedev et al. Reference Medvedev, Frederiksen, Haugbølle and Nordlund2011). The jitter radiation spectrum is wholly determined by ${\it\delta}_{j}$ and the magnetic spectral distribution. Consider an isotropic power-law magnetic spectrum for a time-independent field, such as:

(2.42) $$\begin{eqnarray}|\boldsymbol{B}_{\boldsymbol{k}}|^{2}=\left\{\begin{array}{@{}ll@{}}Ck^{-{\it\mu}},\quad & \quad k_{min}\leqslant k\leqslant k_{max}\\ 0,\quad & \quad \text{otherwise},\end{array}\right.\end{eqnarray}$$

where the magnetic spectral index, ${\it\mu}$ , is a real number, and $C$ is a normalization. It has been shown Medvedev (Reference Medvedev2006), Reville & Kirk (Reference Reville and Kirk2010), Medvedev et al. (Reference Medvedev, Frederiksen, Haugbølle and Nordlund2011) and Teraki & Takahara (Reference Teraki and Takahara2011) that mono-energetic ultrarelativistic electrons in this prescribed sub-Larmor-scale turbulence produce a flat angle-averaged spectrum below the spectral break and a power-law spectrum above the break, that is:

(2.43) $$\begin{eqnarray}P({\it\omega})\propto \left\{\begin{array}{@{}ll@{}}{\it\omega}^{0}, & \text{if}~{\it\omega}<{\it\omega}_{j},\\ {\it\omega}^{-{\it\mu}+2}, & \text{if}~{\it\omega}_{j}<{\it\omega}<{\it\omega}_{b},\\ 0, & \text{if}~{\it\omega}_{b}<{\it\omega},\end{array}\right.\end{eqnarray}$$

where the spectral break is

(2.44) $$\begin{eqnarray}{\it\omega}_{j}={\it\gamma}^{2}k_{min}c,\end{eqnarray}$$

which is called the jitter frequency. Similarly, the high-frequency break is

(2.45) $$\begin{eqnarray}{\it\omega}_{b}={\it\gamma}^{2}k_{max}c.\end{eqnarray}$$

Recently, we have generalized the small-scale jitter regime to non-relativistic and mildly relativistic velocities (Keenan et al. Reference Keenan, Ford and Medvedev2015). Non-relativistic jitter radiation, or pseudo-cyclotron radiation, differs markedly from both synchrotron and cyclotron radiation. Since relativistic beaming is not realized in the non-relativistic regime, the jitter parameter loses its meaning here. Instead, the gyro number characterizes the regime, i.e.  ${\it\rho}\gg 1$ . Given a mono-energetic distribution of electrons, and the spectral distribution indicated by (2.42), the pseudo-cyclotron spectrum has a slightly more complicated structure than ultrarelativistic jitter radiation. It appears as Keenan et al. (Reference Keenan, Ford and Medvedev2015):

(2.46) $$\begin{eqnarray}P({\it\omega})\propto \left\{\begin{array}{@{}ll@{}}A+D{\it\omega}^{2}, & \text{if}~{\it\omega}\leqslant {\it\omega}_{jn}\\ F{\it\omega}^{-{\it\mu}+2}+G{\it\omega}^{2}+K, & \text{if}~\leqslant {\it\omega}_{bn}\\ 0, & \text{if}~{\it\omega}>{\it\omega}_{bn},\end{array}\right.\end{eqnarray}$$

where ${\it\mu}\neq 2$ and $A$ , $D$ , $F$ , $G$ and $K$ are functions of spectral/particle parameters (e.g.  ${\it\mu}$ , $k_{min}$ and ${\it\beta}$ ). The break frequencies generalize to non-relativistic velocities as expected, namely:

(2.47) $$\begin{eqnarray}{\it\omega}_{jn}=k_{min}{\it\beta}c,\end{eqnarray}$$

and the break frequency indicated by the smallest spatial scale, i.e. the maximum wavenumber, becomes:

(2.48) $$\begin{eqnarray}{\it\omega}_{bn}=k_{max}{\it\beta}c.\end{eqnarray}$$

Finally, a series of Lorentz transformations allow the generalization of these results to all velocities (Keenan et al. Reference Keenan, Ford and Medvedev2015).

The introduction of a mean magnetic field will complicate this picture. The topic of radiation production by ultrarelativistic electrons in magnetized plasmas with small-scale magnetic fluctuations was originally considered in Toptygin & Fleishman (Reference Toptygin and Fleishman1987). In the case of strictly sub-Larmor-scale magnetic turbulence, with a mean field, the spectrum will be the sum of a synchrotron/cyclotron component (corresponding to the mean magnetic field) and the jitter contribution from the small-scale fluctuations, i.e. 

(2.49) $$\begin{eqnarray}P({\it\omega})=P_{jitter}({\it\omega})+P_{synch}({\it\omega}).\end{eqnarray}$$

Since a plasma is a dielectric medium, dispersion may affect the form of the radiation spectrum. The effect is mostly negligible in the ultrarelativistic limit, but dispersion may be required for a complete description of the mildly relativistic and non-relativistic regimes – in real plasmas. Nonetheless, the dispersion-corrected spectrum has already been considered for small-angle jitter radiation, (Keenan et al. Reference Keenan, Ford and Medvedev2015) and synchrotron radiation (Zheleznyakov & Trakhtengerts Reference Zheleznyakov and Trakhtengerts1966). For this reason, we will ignore plasma dispersion in this study.

Furthermore, since $\langle E^{2}\rangle \ll \langle {\it\delta}B^{2}\rangle$ for small-scale Whistler turbulence, we can completely ignore this electric contribution.

In § 4, we will confirm (2.49) (via ab initio numerical simulation) in the case of small-scale (isotropic) Whistler turbulence.

4.1 Whistler turbulence

First of all, we explore the particle transport by testing our predictions concerning the energy and pitch-angle diffusion coefficients in small-scale Whistler turbulence. The diffusion coefficients depend on various parameters, cf. (2.8) and (2.40), namely the particle’s velocity, ${\it\beta}$ , the magnetic fluctuation field strength, $\langle {\it\Omega}_{{\it\delta}B}^{2}\rangle$ and the field correlation scale, ${\it\lambda}_{B}$ .

To start, we must confirm the fundamental assumption of diffusion. As stated previously, a diffusive process requires that both $\langle {\rm\Delta}W_{e}^{2}\rangle$ and $\langle {\it\alpha}^{2}\rangle$ increase linearly in time – at least, on some characteristic time scale of the system. With ${\it\delta}B/B_{0}\ll 1$ , the gyro period

(4.1) $$\begin{eqnarray}T_{g}\equiv \frac{2{\rm\pi}{\it\gamma}}{{\it\Omega}_{ce}}=2{\rm\pi}\frac{{\it\gamma}m_{e}c}{eB_{0}},\end{eqnarray}$$

is such a characteristic, ‘macroscopic’ time scale. On a multiple gyro period time scale, the electron velocities will change very slightly. Consequently, we may treat the magnitude of the electron velocity as approximately constant.

To establish diffusion, $5000$ mono-energetic electrons ( ${\it\beta}=0.25$ ) were injected into Whistler turbulence with $k_{min}=32{\rm\pi}$ (arbitrary simulation units), $k_{max}=10k_{min}$ , $\langle {\it\delta}B^{2}\rangle ^{1/2}/B_{0}=0.1$ , ${\it\Omega}_{ce}=1$ , ${\it\rho}\approx 400$ , ${\it\chi}\approx 0.04$ and ${\it\mu}=4$ . The simulation time included several gyro periods; $T=10T_{g}$ . Additional simulation parameters include: the time step ${\rm\Delta}t=0.00125$ (arbitrary units) and the number of Whistler wave modes $N_{m}=10\,000$ . In figure 1, we see that the average square pitch angle (as measured with respect to the $z$ -axis, i.e. the mean field direction) does indeed grow linearly with time. Figure 2 confirms that the electron energy undergoes a classical diffusive process as well. With the existence of pitch angle and energy diffusion established, we then proceeded to compare the slope of $\langle {\it\alpha}^{2}\rangle$ and $\langle {\rm\Delta}W_{e}^{2}\rangle$ versus time (the numerical pitch angle and energy diffusion coefficients) to (2.8) and (2.40). In figure 3, the numerically obtained pitch-angle diffusion coefficients are compared to (2.8) for a range of possible electron velocities. In each, the theoretical and numerical results differ only by a small factor of $O(1)$ . Next, in figure 4, we see decent agreement with (2.40) and the numerical energy diffusion coefficients. Figures 3 and 4, furthermore, confirm that our theoretical diffusion coefficients are valid for all electron velocities – including relativistic speeds.

Figure 1. Average square pitch angle versus normalized time. Relevant parameters are ${\it\beta}=0.25$ , (number of simulation particles) $N_{p}=5000$ , $k_{min}=32{\rm\pi}$ , $k_{max}=10k_{kmin}$ , $\langle {\it\delta}B^{2}\rangle ^{1/2}/B_{0}=0.1$ , ${\it\Omega}_{ce}=1$ , ${\it\rho}\approx 400$ , ${\it\chi}\approx 0.04$ and ${\it\mu}=4$ . The linear nature of the curve (solid, ‘red’) confirms the diffusive nature of the pitch-angle transport. Here, the dashed (‘blue’) line indicates a line of best fit (simple linear regression) with Pearson correlation coefficient: 0.9998.

Figure 2. Average square change in electron energy (in simulation units) versus normalized time. Relevant parameters are ${\it\beta}=0.25$ , (number of simulation particles) $N_{p}=5000$ , $k_{min}=64{\rm\pi}$ , $k_{max}=10k_{kmin}$ , $\langle {\it\delta}B^{2}\rangle ^{1/2}/B_{0}=0.1$ , ${\it\Omega}_{ce}=1$ , ${\it\rho}\approx 400$ , ${\it\chi}\approx 0.04$ and ${\it\mu}=4$ . The linear nature of the curve (solid, ‘red’) confirms the diffusive nature of the energy transport. Here, the dashed (‘blue’) line indicates a line of best fit (simple linear regression) with Pearson correlation coefficient: 0.9999.

Figure 3. Pitch-angle diffusion coefficient, $D_{{\it\alpha}{\it\alpha}}$ versus the normalized electron velocity, ${\it\beta}$ . Relevant simulation parameters include: $N_{p}=5000$ , $k_{min}=32{\rm\pi}$ , $k_{max}=10k_{kmin}$ , $\langle {\it\delta}B^{2}\rangle ^{1/2}/B_{0}=0.1$ , ${\it\Omega}_{ce}=1$ , ${\it\chi}\approx 0.02$ and ${\it\mu}=4$ . The (purple) empty ‘squares’ indicate the $D_{{\it\alpha}{\it\alpha}}$ ’s obtained directly from simulation data (as the slope of $\langle {\it\alpha}^{2}\rangle$ versus time), while the (green) filled ‘circles’ are the analytical pitch-angle diffusion coefficients, given by (2.8). Numerical error bars (red, lines centred in the ‘squares’) from the standard deviations on each $\langle {\it\alpha}^{2}\rangle$ also appear. Note: that the symbol sizes are much greater than the statistical/numerical error bars.

Figure 4. Energy diffusion coefficient, $D_{WW}$ versus the normalized electron velocity, ${\it\beta}$ . Relevant simulation parameters include: $N_{p}=5000$ , $k_{min}=32{\rm\pi}$ , $k_{max}=10k_{kmin}$ , $\langle {\it\delta}B^{2}\rangle ^{1/2}/B_{0}=0.1$ , ${\it\Omega}_{ce}=1$ , ${\it\chi}\approx 0.02$ and ${\it\mu}=4$ . The (blue) empty ‘squares’ indicate the $D_{WW}$ ’s obtained directly from simulation (as the slope of $\langle {\rm\Delta}W_{e}^{2}\rangle$ versus time), while the (red) filled ‘circles’ are the analytical energy diffusion coefficients, given by (2.40). Numerical error bars (green, lines centred in the ‘squares’) from the standard deviations on each $\langle {\it\alpha}^{2}\rangle$ also appear. Note: that the symbol sizes are much greater than the statistical/numerical error bars.

Another important parameter which strongly influences the diffusive transport is the magnetic field correlation length. In figure 5, the correlation length was varied by changing $k_{min}$ , while keeping all other parameters fixed. Once more, we see close agreement with (2.8). Similarly, the numerical and theoretical energy diffusion coefficients continue to show reasonable agreement – see figure 6.

Figure 5. Pitch-angle diffusion coefficient, $D_{{\it\alpha}{\it\alpha}}$ versus the inverse of magnetic field correlation scale, ${\it\lambda}_{B}^{-1}$ . Relevant simulation parameters include: ${\it\gamma}=3$ , $N_{p}=1000$ , $k_{min}=8{\rm\pi}$ , $16{\rm\pi}$ , $32{\rm\pi}$ , $64{\rm\pi}$ and $128{\rm\pi}$ , $k_{max}=10k_{kmin}$ (for each $k_{kmin}$ ),  $\langle {\it\delta}B^{2}\rangle ^{1/2}/B_{0}=0.1$ , ${\it\Omega}_{ce}=1$ , ${\it\chi}\approx 0.02$ and ${\it\mu}=4$ . For each data point, the theoretical and numerical results differ only by a small factor of $O(1)$ .

Figure 6. Energy diffusion coefficient, $D_{WW}$ versus the inverse of magnetic field correlation scale, ${\it\lambda}_{B}^{-1}$ . Relevant simulation parameters include: ${\it\gamma}=3$ , $N_{p}=1000$ , $k_{min}=8{\rm\pi}$ , $16{\rm\pi}$ , $32{\rm\pi}$ , $64{\rm\pi}$ and $128{\rm\pi}$ , $k_{max}=10k_{kmin}$ (for each $k_{kmin}$ ),  $\langle {\it\delta}B^{2}\rangle ^{1/2}/B_{0}=0.1$ , ${\it\Omega}_{ce}=1$ , ${\it\chi}\approx 0.02$ and ${\it\mu}=4$ . The theoretical and numerical results differ only by a small factor of $O(1)$ .

We consider the magnetic spectral index, ${\it\mu}$ – i.e. the power-law exponent in (2.42). With $k_{min}=32{\rm\pi}$ and $k_{max}=10k_{min}$ , we varied the magnetic spectral index, ${\it\mu}$ from $-3$ to $9$ . In figure 7, we see that the numerical pitch-angle diffusion coefficient closely matches the analytical result. Similarly close agreement was, once again, realized between the energy diffusion coefficients; as may be seen in figure 8.

Figure 7. Pitch-angle diffusion coefficient, $D_{{\it\alpha}{\it\alpha}}$ versus the magnetic spectral index, ${\it\mu}$ . Relevant parameters are $N_{p}=2000$ , $k_{min}=32{\rm\pi}$ , $k_{max}=10k_{max}$ , $\langle {\it\delta}B^{2}\rangle ^{1/2}/B_{0}=0.1$ , ${\it\Omega}_{ce}=1$ and ${\it\chi}\approx 0.05$ . Notice that the numerical results have nearly the same functional dependence on ${\it\mu}$ as the analytical squares, as given by (2.8).

Figure 8. Energy diffusion coefficient, $D_{WW}$ versus the magnetic spectral index, ${\it\mu}$ . Relevant parameters are $N_{p}=2000$ , $k_{min}=32{\rm\pi}$ , $k_{max}=10k_{max}$ , $\langle {\it\delta}B^{2}\rangle ^{1/2}/B_{0}=0.1$ , ${\it\Omega}_{ce}=1$ and ${\it\chi}\approx 0.05$ .

Finally, we consider the radiation spectra. As discussed in § 3, the radiation spectra are expected to be the summation of synchrotron (cyclotron) and jitter (pseudo-cyclotron) components. For an ultrarelativistic electron, the angle-averaged synchrotron radiation spectrum is the known function (Landau & Lifshitz Reference Landau and Lifshitz1975; Jackson Reference Jackson1999):

(4.2) $$\begin{eqnarray}\frac{\text{d}W}{\text{d}{\it\omega}}=\sqrt{3}\frac{e^{2}}{c}{\it\gamma}\frac{{\it\omega}}{{\it\omega}_{c}}\int _{{\it\omega}/{\it\omega}_{c}}^{\infty }\text{K}_{5/3}(x)\,\text{d}x,\end{eqnarray}$$

where $\text{K}_{j}(x)$ is a modified Bessel function of the second kind, and ${\it\omega}_{c}=3/2{\it\gamma}^{2}{\it\Omega}_{ce}$ is the critical synchrotron frequency. This result applies for an electron moving in the plane transverse to the ambient magnetic field, i.e. when ${\it\alpha}={\rm\pi}/2$ . Nonetheless, we find the expression fits the synchrotron components fairly well; especially when ${\it\gamma}$ is reasonably large.

We illustrate two numerical spectra here, along with their corresponding analytical estimates – for details concerning the jitter component, see Keenan et al. (Reference Keenan, Ford and Medvedev2015). First, we considered a ${\it\gamma}=5$ electron population for figure 9. In this plot, the relevant parameters are: $N_{p}=1000$ , ${\rm\Delta}t=0.00125$ , $k_{min}=2{\rm\pi}$ , $k_{max}=20{\rm\pi}$ , $\langle {\it\delta}B^{2}\rangle ^{1/2}/B_{0}=0.1$ , ${\it\Omega}_{ce}=0.512$ , ${\it\mu}=4$ , ${\it\rho}\approx 928$ , ${\it\chi}\sim 1$ and the total simulation time was $T=5T_{g}$ . We see that the synchrotron $+$ jitter fit closely resembles the numerical spectrum.

Figure 9. Radiation spectrum for a mono-energetic, isotropic distribution of ${\it\gamma}=5$ ( ${\it\chi}\sim 1$ ; ${\it\rho}\approx 928$ ; $\langle {\it\delta}B^{2}\rangle ^{1/2}/B_{0}=0.1$ ) electrons moving through small-scale Whistler turbulence. The frequency is normalized by ${\it\omega}_{jn}={\it\gamma}^{2}k_{min}{\it\beta}c$ – the relativistic jitter frequency. The solid (‘red’) curve is from simulation data, whereas the dashed (‘blue’) curve is the analytic estimate. Clearly, the spectrum is well represented by a superposition of synchrotron $+$ jitter components. Note the lower-frequency synchrotron component and a higher-frequency power-law component corresponding to the small-angle jitter radiation.

Next, we explored the non-relativistic regime. In figure 10, we assumed a population of sub-Larmor-scale ${\it\beta}=0.125$ electrons. As expected, a peak in the spectrum may be observed near the cyclotron frequency ${\it\Omega}_{ce}$ – confirming that the total spectrum is the hybrid of pseudo-cyclotron $+$ cyclotron radiation. Additionally, to provide a point of comparison, we have superimposed a simulation result for ${\it\gamma}=4$ electrons.

Figure 10. Radiation spectrum for a mono-energetic, isotropic distribution of ${\it\beta}=0.125$ electrons ( ${\it\chi}\sim 0.04$ ; ${\it\rho}\approx 160$ ; $\langle {\it\delta}B^{2}\rangle ^{1/2}/B_{0}=0.2$ ; ${\it\Omega}_{ce}=2$ ; $k_{min}=64{\rm\pi}$ ; $k_{max}=10k_{min}$ ; ${\it\mu}=5$ ; $T=50T_{g}$ ); superimposed with a spectrum given a population of ${\it\gamma}=4$ electrons ( ${\it\chi}\sim 1$ ; ${\it\rho}\approx 367$ ; $\langle {\it\delta}B^{2}\rangle ^{1/2}/B_{0}=0.1$ ; ${\it\Omega}_{ce}=0.512$ ; $k_{min}={\rm\pi}$ ; $k_{max}=10{\rm\pi}$ ; ${\it\mu}=4$ ; $T=5T_{g}$ ). The normalization on the $y$ -axis is arbitrary, whereas the $x$ -axis is normalized to the ${\it\beta}=0.125$ population’s cyclotron frequency, i.e.  ${\it\Omega}_{ce}=2$ . The ‘thick’ solid (‘red’) curve is from simulation data for the ${\it\beta}=0.125$ population, the dashed (‘blue’) curve is the corresponding analytic estimate for pure pseudo-cyclotron radiation, the ‘thin’ solid line is the simulation data for the ${\it\gamma}=4$ population and the ‘dot-dashed’ (‘black’) line is the ${\it\gamma}=4$ analytic estimate. Notice, for the ${\it\beta}=0.125$ spectrum, that the spectrum peaks near the cyclotron frequency, ${\it\Omega}_{ce}$ – hence we see the signature of cyclotron radiation. The additional harmonics, which are purely a relativistic effect, are the signature of emerging synchrotron radiation.