Diffusion and Radiation in Magnetized Collisionless Plasmas with High-Frequency Small-Scale Turbulence
aa r X i v : . [ phy s i c s . p l a s m - ph ] D ec Diffusion and Radiation in Magnetized Collisionless Plasmas with High-Frequency Small-ScaleTurbulence
Brett D. Keenan ∗ and Mikhail V. Medvedev Department of Physics and Astronomy, University of Kansas, Lawrence, KS 66045
Magnetized high-energy-density plasmas can often have strong electromagnetic fluctuations whose correla-tion scale is smaller than the electron Larmor radius. Radiation from the electrons in such plasmas, whichmarkedly differs from both synchrotron and cyclotron radiation, and their energy and pitch-angle diffusion aretightly related. In this paper, we present a comprehensive theoretical and numerical study of the particles’transport in both cold, “small-scale” Langmuir and Whistler-mode turbulence and its relation to the spectra ofradiation simultaneously produced by these particles. We emphasize that this relation is a superb diagnostic toolof laboratory, astrophysical, interplanetary, and solar plasmas with a mean magnetic field and strong small-scaleturbulence.
I. INTRODUCTION
High-energy density plasma environments are generally thesites of turbulent, high-amplitude (i.e., larger or comparable topreexisting ambient magnetic fields) electromagnetic fluctua-tions, which often exist at scales below a Larmor scales. Suchturbulence is a common feature of astrophysical and spaceplasmas, e.g., at high-Mach-number collisionless shocks andin reconnection regions in weakly magnetized plasmas [1–7]. Additionally, turbulent magnetic fields existing on “sub-Larmor-scales” play a critical role in laboratory plasmas; es-pecially in high-intensity laser plasmas, as is observed in ex-periments at the National Ignition Facility (NIF), OmegaEP,Hercules, Trident, and others [8–11].Small-scale electromagnetic turbulence can be of variousorigin and thus have rather different properties. “Weibel-like”turbulence [12–14] may occur in non-magnetized plasmas,i.e., plasmas possessing no ambient (mean) magnetic field. Incontrast, several turbulence-producing electromagnetic insta-bilities require a preexisting magnetic field, e.g., the Whistler-mode, mirror-mode, fire-hose, Bell’s-type instability and oth-ers [15–23].If the electromagnetic fields are substantially small-scaleand statistically random, which is usually the case of turbu-lence because of the random phases of fluctuations, the pathsof the particles diffusively diverge. If the turbulence is sub-Larmor-scale (for the electrons) then the radiation simultane-ously produced by the electrons is neither cyclotron nor syn-chrotron (for non-relativistic or relativistic particles, respec-tively) but, instead, carries information about the spectrum ofturbulent fluctuations.In our previous works, we found the relation between thetransport of ultrarelativistic [24] and non-relativistic/trans-relativistic [25] particles in isotropic three-dimensional small-scale (mean-free) magnetic turbulence and the radiation spec-tra simultaneously produced by these particles. In Ref. [24],we found that the radiation spectrum, in the ultrarelativistic(small deflection angle) regime, agrees with the small-anglejitter radiation prediction [26–30]. Furthermore, we demon-strated that the pitch-angle diffusion coefficient is directly re- ∗ [email protected] lated to, and can readily be deduced from, the spectra of theemitted radiation. These results were then generalized to non-relativistic and trans-relativistic velocities in Ref. [25].Our previous studies strictly considered a “Weibel-like”magnetic turbulence. This means that we treated the electro-magnetic turbulence as static, i.e., with zero real frequencyand no mean field. In this study, we will extend our model toinclude sub-Larmor-scale electromagnetic turbulence in plas-mas with ambient magnetic fields. The instabilities, in thiscase, are usually driven with non-zero real frequency, andthus, they induce random electric fields. For this reason, oneshould not only consider stochastic transport via magneticpitch-angle diffusion, but transport via electric-field-inducedenergy and pitch-angle diffusion as well. Additionally, wewill show that the energy diffusion coefficient is proportionalto the (sub-Larmor-scale) magnetic, pitch-angle diffusion co-efficient. The exploitation of the inter-relation between thetransport and radiative properties of these plasmas should pro-vide a powerful diagnostic tool for examination of small-scaleturbulence in magnetized plasmas.We will, furthermore, consider the transport of, and ra-diation production by, relativistic electrons moving through“small-scale” Langmuir turbulence – which is purely electricturbulence.Moreover, we omit the resonant wave-particle interactions,which support the underlying electromagnetic turbulence,from our analysis and consider non-resonant particles only –as the resonant ones constitute a nearly infinitesimal test par-ticle population.We will, principally, focus on realizations of Whistler-modeturbulence, because Whistler waves are regularly seen in avery wide variety of magnetized enviroments. Given certainconditions, the (temperature anisotropy) Weibel instability –in pre-magnetized plasmas – may evolve into a Whistler-modeinstability [31]; thus, for example, Whistler-modes may spon-taneously appear in environments where Weibel-like instabil-ities may take hold.Many examples of Whistler waves in space and astrophys-ical plasmas exist. Whistler waves near collsionless shocks inthe solar system, in particular, have been observed in situ fordecades. These wave-modes have, addtionally, been stronglyassociated with interplanetary shocks – appearing both in theupstream and downstream regions [32–34]. The solar windturbulence, as well, appears to host Whistler-modes [35, 36].The rest of the paper is organized as follows. Section IIpresents the analytic theory. Sections III and IV describe thenumerical techniques employed and the obtained simulationresults. Section V presents as special case of jitter radiationfrom a thermal distribution of electrons. Finally, Section VI isthe conclusions. All equations appear in cgs units. II. ANALYTIC THEORY
Consider a test particle (electron) moving through a non-uniform, random magnetic field with velocity, v . Assume thatthe magnetic field has the mean value, h B i , where h·i is anappropriately chosen average over space and, possibly, time.Consequently, we write the total random magnetic field as: B ( x , t ) = B + δ B ( x , t ) , (1)where B ≡ h B i is the mean field and δ B ( x , t ) is themean-free “fluctuation” field, that is h δ B i = 0 but δB ≡h δB i / = 0 .Next, the motion of an electron in a random magnetic fieldis, in general, very complicated. It is the spatial scale of inho-mogeneity, i.e., the correlation length of the field fluctuation,that fundamentally determines the dynamics. These mag-netic fluctuations are deemed sub-Larmor-scale (or “small-scale”) when the electron’s fluctuation Larmor radius, r L ≡ γβm e c /eδB (where β = v/c is the dimensionless particlevelocity, m e is the electron mass, c is the speed of light, e isthe electric charge, and γ is the electron’s Lorentz factor) isgreater than, or comparable to, the correlation length of thefield, λ B , i.e., r L & λ B . We introduce the “gyro-number”,which fully characterizes the small-scale regime [25] as fol-lows: ρ ≡ r L λ − B . (2)Notice that we are considering only the fluctuation componentof the magnetic field, δ B . This is because the motion canbe separated into two components: the regular gyro-motionabout the mean magnetic field, and the random deflections dueto the small-scale random component. In the discussion tofollow, we will presuppose that ρ ≫ .Next, because the fluctuation Lorentz force on the electronis random, the electron velocity and acceleration vectors varystochastically, leading to a random (diffusive) trajectory. Ad-ditionally, the magnetic Lorentz force acts only upon the com-ponent of velocity transverse to the local magnetic field, lead-ing only to energy-conserving (i.e., β = constant ) deflec-tions. Only an electric field can change the particle energy.When this electric field is random, transport via energy diffu-sion may occur – we will explore this later.Ignoring, for the moment, the presence of any electricfields: the electron motion has two limiting regimes – depend-ing upon the relative strength of the magnetic fluctuations withrespect to the mean field. These are a “straight line” trajectorywith random (transverse) deflections (i.e., δ B ≫ B ), and aslightly “perturbed” helical motion about the mean magnetic field (i.e., δ B ≪ B ). In the latter case, we will ignore theregular component of the motion. Doing so allows us to con-sider only the transport in mean-free, small-scale, magneticturbulence, which we have explored previously.This picture is, of course, only correct if any presentelectric fields are ignored. Electric turbulence, likewise, caninduce transport via pitch-angle diffusion – as we will showlater. However, the contribution to the total transport due toelectric fields in small-scale Whistler turbulence, specifically,is negligible. A. Transport via Magnetic Pitch-angle Diffusion
The pitch-angle diffusion coefficient in mean-free, sub-Larmor-scale, magnetic turbulence is a known function of sta-tistical parameters. It may be obtained by considering thatthe electron’s pitch-angle experiences only a slight deflection, δα B , over a single magnetic correlation length. Consequently,the ratio of the change in the electron’s transverse momen-tum, ∆ p t , to its initial momentum, p , is δα B ≈ ∆ p t /p ∼ e ( δB/c ) λ B /γm e v , since ∆ p t ∼ F L τ B – where F L =( e/c ) v × δ B is the transverse Lorentz force and τ B ∼ λ B /v is the time to transit λ B . The subsequent deflection will bein a random direction, because the field is uncorrelated overthe scales greater than λ B . As for any diffusive process, themean squared pitch-angle grows linearly with time. Thus, thediffusion coefficient appears as [24, 25]: D αα ≡ h α i t = λ B γ c h β B i / h Ω δB i , (3)where α is the electron deflection angle (pitch-angle) with re-spect to the electron’s initial direction of motion, h β B i / isan appropriate ensemble-average over the (transverse) elec-tron velocities, and Ω δB ≡ eδ B m e c . (4)In general, the pitch-angle diffusion will be path-dependent,owing to the dependence on the magnetic correlation length, λ B . To properly treat the correlation length, we must intro-duce the two-point autocorrelation tensor of the magnetic fluc-tuations [25], R ij ( r , t ) ≡ h δB i ( x , τ ) δB j ( x + r , τ + t ) i x ,τ , (5)with the path and time dependent correlation length tensor de-fined as: λ ijB (ˆ r , t ) ≡ ˆ ∞ R ij ( r , t ) R ij (0 ,
0) d r. (6)To evaluate this expression, we must consider the physics in-volved. In magnetic deflections, only the component of themagnetic field transverse to the particle velocity is involvedin the acceleration. Thus, for magnetic fields, we only con-sider fields transverse to the direction of motion. In contrast,electric fields will have a “longitudinal” and “transverse” cor-relation length. The former is important for energy diffusion –whereas, the latter governs pitch-angle diffusion, since trans-verse deflections do no work.Evaluation of Eq. (6) can be very difficult, in a general case.If we make some simplifying assumptions about the magneticturbulence, however, we may evaluate Eq. (6) exactly. If thetransit time of a particle over a correlation length is shorterthan the field variability time-scale, then we can treat the mag-netic field as static. Additionally, assuming statistical homo-geneity and isotropy permits us to use a simple expression forthe correlation tensor.The pitch-angle diffusion coefficient, under these simplify-ing assumptions, has been derived previously [25]. We repeatthose results here. The magnetic correlation length assumesthe form [25]: λ B = 3 π ´ ∞ k | δ B k | d k ´ ∞ k | δ B k | d k , (7)where | δB k | is the spectral distribution of the fluctuationmagnetic field in Fourier “ k -space”. Thus, the pitch-anglediffusion coefficient, for sub-Larmor-scale electrons movingthrough (isotropic/homogeneous) magnetic turbulence, is: D αα = 3 π r ´ ∞ k | δ B k | d k ´ ∞ k | δ B k | d k h Ω δB i γ cβ , (8)where we have assumed a mono-energetic distribution of elec-trons with velocity, β .Since the magnetic turbulence is assumed to be statisticallyhomogeneous and isotropic, the deflection angle, α , may bechosen with respect to an arbitrary axis. The component ofthe velocity parallel to B is unaffected by this mean field.Consequently, without loss of generality, we can define α asthe conventional pitch-angle – i.e., the angle of the velocityvector with respect to the mean (ambient) magnetic field. B. Pitch-angle Diffusion in Small-Scale Electric Fields
The derivation for pitch-angle diffusion in general small-scale electric turbulence follows in a similar fashion. Supposean electron test particle is moving, with speed v , through anexternal random electric field. This may be an electrostaticfield (i.e., Langmuir-like turbulence), or – as in the more gen-eral case – it may be the electric component of electromag-netic turbulence (e.g. Whistler-mode turbulence). We will as-sume that the electric field fluctuates very slowly – such thatthe particle dynamics, on relevant time-scales, are largely un-affected by the field’s time-variability. Furthermore, we willignore any present magnetic fields – for the moment.For “small-scale” turbulence, the principal time-scalewhich governs particle transport is the time to transit a singleelectric field correlation length, λ tE – where the “ t ” superscriptindicates that the correlation length is specified along the pathwith a “transverse” component of the electric field (which wedid with the magnetic field). If the (pitch-angle) transit time, τ tE ∼ λ tE /v , is much less than the field-variability time-scale, Ω − , then we may treat the electric field as approximatelytime-independent.To proceed, it will be instructive to first discuss the radia-tion produced by an electron moving through an external ran-dom field. First, regardless of the acceleration mechanism,the radiation of an ultrarelativistic electron will be beamedalong a narrow cone with opening angle, ∆ θ ∼ /γ . In a ran-dom electromagnetic field, the acceleration occurs principallyalong the extent of a correlation length. Since the electron ismoving ultrarelativistically, it will undergo a slight deflection, δα E , as it traverses a correlation length. If δα E ≪ ∆ θ , thenthe electron will move approximately rectilinearly, undergo-ing only slight random deflections along its path; the radiationwill then be beamed along the extent of the electron’s rela-tively fixed direction of motion. Consequently, an observer onaxis would see a signal for the entire trajectory of the elec-tron. Furthermore, the radiation spectrum will be wholly de-termined by the statistical properties of the underlying accel-eration mechanism [37]. When the acceleration mechanismis a random (static) magnetic field, the electron emits radia-tion in the small-angle jitter regime [24–30]. The radiationproduced by ultrarelativistic electrons moving through elec-trostatic turbulence, in this small deflection angle regime, isnearly identical – which has lead to its designation as a sub-class of small-angle jitter radiation [38].We have previously shown that these random deflectionsinitiate pitch-angle diffusion in sub-Larmor-scale magneticturbulence, and that this diffusion coefficient is intimately re-lated to the radiation spectrum [24, 25]. We expect that anelectric field analog of this diffusion exists for the (small-angle) jitter regime in small-scale electric turbulence. Here,we consider an electric field as “small-scale”, with respect tothe test electrons, if: Ω − ≫ τ tE , (9a) ∆ θ ≫ δα E . (9b)Since the electron is moving ultrarelativistically, the compo-nent of its acceleration transverse to its direction of motionwill be far larger than the longitudinal component. Thus, itsmotion occupies the small deflection angle regime – which isthe reason its radiation spectrum resembles the jitter spectrum.Additionally, transverse accelerations leave the particle’s ki-netic energy fixed. For this reason, we will assume a constant v . Next, since the deflections are small, δα E ∼ ∆ p t /p – aspreviously noted for magnetic deflections. Since ∆ p t /τ tE ∼ eE t , where E t is the component of the electric field per-pendicular (transverse) to the electron’s direction of motion, ∆ p t /p ∼ eE t /γm e v ; thus: δα E ∼ eE t γm e v τ tE . (10)Consequently, the electric diffusion coefficient must be: D elec. αα ∼ δα E /τ tE ∼ e E t γ m e v (cid:18) λ tE v (cid:19) . (11)Finally, the exact numerical coefficients depend upon the sta-tistical properties of the turbulent fluctuations. Given statisti-cally isotropic and homogeneous turbulence, h E t i = h E i .Thus, the diffusion coefficient follows as: D elec. αα = 23 λ tE γ cβ h Ω E i , (12)where: Ω E ≡ eE/m e c. (13)When a magnetic field is introduced, the (small-scale) pitch-angle diffusion coefficient will be the sum of the magnetic andelectric components – i.e. Eq. (12) and Eq. (3).As we mentioned previously, and will demonstrate later, theelectric pitch-angle diffusion is negligibly small compared tothe magnetic equivalent in small-scale Whistler-mode turbu-lence. For this reason, the electric contribution to the radiationproduction is, also, insignificant. Nonetheless, the electricfield will still uniquely affect the particle motion via energydiffusion. C. Energy Diffusion in Small-Scale Electric Turbulence
All electromagnetic turbulence results from instabilities,dynamo-action, etc. with some finite growth rate. So longas the growth (or dissipation) time-scale is much greater thanthe correlation length transit time, we can ignore the time-dependence of the magnetic field in our model.In contrast to Weibel magnetic fields in (initially) unmagne-tized plasmas, however, MHD/kinetic instabilities (which re-quire an ambient magnetic field) may grow random fields withnon-negligible real frequency, Ω r . That is to say, these mag-netic fields will possess oscillating wave-modes, whose time-dependence may not be completely ignored. The Faraday-induced electric fields, E , may influence the particle motionon relevant time-scales, e.g., the gyro-period time-scale in theregular (ambient) magnetic field.These random electric fields may induce transport via en-ergy diffusion. Although diffusive energy transport in elec-tromagnetic turbulence has long been a topic of investiga-tion [39], energy diffusion in strictly sub-Larmor-scale elec-tromagnetic fields has yet to be – to the best of our knowledge– explored. This topic has proved to be richly complicated, sowe have limited ourselves to a particularly simple regime.Furthermore, we emphasis that the “energy” diffusion co-efficient – rather than the “velocity-space” analog – is a moreuseful quantity for our purposes. Although it possesses a num-ber of favorable properties, its prominent feature is that it isdirectly proportional to the electric field’s correlation length.This feature is not present in the “velocity-space” coefficient,however.Next, we must consider the time-scales involved. There aretwo such characteristic time-scales: the “acceleration” time, τ lE and the electric field “auto-correlation” time, τ ac . The lat-ter time-scale characterizes the temporal inhomogeneity of theelectric field. Diffusive (energy) transport may arise not only from spatial stochasticity in the electric field but temporal ran-domness as well.The former quantity, τ lE , characterizes the spatial stochas-ticity. This is the time required to transit an electric field cor-relation length, λ lE – with the “ l ” superscript indicating the“longitudinal” transit time; i.e. the time required to transversea “longitudinal” electric correlation length, λ lE , which is alongthe direction of motion. Assuming that a λ τ lE ≪ v E , where a λ is the acceleration over λ lE , and v E is the component of theelectron velocity parallel to the electric field, the transit timeis: τ lE ∼ λ lE v E . (14)While transiting a single correlation length, the electron issubject to a nearly uniform electric field. These “accelera-tions” are uncorrelated on a spatial-scale dictated by the elec-tric field correlation length.The diffusion regime we will explore will consider the “spa-tial” diffusion to be the dominant process, i.e., τ lE ≪ τ ac . (15)Furthermore, to ensure that the energy change is random onthe time-scale of consideration, we require that: τ lE ≪ t. (16)Next, an equation for the electron energy, W e , may be ob-tained directly from the Lorentz Force Equation of Motion. Itis: dW e dt = e ( v · E ) . (17)Since the electron energy changes over the characteristic time-scale, τ lE , we may write: ∆ W λ τ lE ∼ ev E E. (18)If the random process is, indeed, diffusive: D W W ≡ h W e i t . (19)Thus: D W W ∼ (∆ W λ ) τ lE ∼ e v E E λ lE , (20)where we have used Eq. (14). With the usual assumptionsof statistical homogeneity/isotropy and an initially mono-energetic distribution of electrons, we may write the energydiffusion coefficient, thusly: D W W = r e h E i vλ lE . (21)This result may be contrasted with the “temporal”, i.e. res-onant , energy diffusion coefficient. The physics of this typeof diffusion may be understood by considering the, so called,“Quasilinear” energy diffusion coefficient. As before, we willconsider only small corrections to the electron’s initial veloc-ity – hence, we will assume the zero-order trajectory: r ( t ) = v t + r , (22)where r is the electron’s initial position. Let us suppose thatthe electric field assumes a simple sinusoidal profile, i.e. E ( x , t ) = E cos ( k · x − Ω t ) . (23)Thus, using Eqs. (17) and (23), we have: dW e dt = e ( v · E ) cos ( k · v t + k · r − Ω t ) . (24)Integrating Eq. (24), averaging over all possible initial posi-tions, and squaring the result, gives the energy variance: h ∆ W e i = (cid:20) e ( v · E )(Ω − k · v ) (cid:21) sin (cid:20) (Ω − k · v ) t (cid:21) . (25)Finally, with Ω t ≫ , we may employ the relation [39]:sin (cid:20) (Ω − k · v ) t (cid:21) ∼ πδ (Ω − k · v ) , (26)Thus the (Quasilinear) diffusion coefficient is: D res. W W ≡ h ∆ W e i t ∼ π (cid:20) e ( v · E )(Ω − k · v ) (cid:21) δ (Ω − k · v ) . (27)In general, turbulence will contain a spectrum of waves;hence, an integration of Eq. (27) over | E k , Ω | is required toproduce the complete diffusion equation.Nevertheless, much can be gathered by examining the func-tional form of this simplified expression. For example, owingto the dependence on the quantity, δ (Ω − k · v ) , only par-ticles that are in resonance with the wave participate in thediffusive process.Moreover, since Ω t ≫ , this “temporal” diffusion processoccurs on a much greater time-scale than τ lE (when the electricfield is small-scale). For this reason, the non-resonant energydiffusion coefficient – Eq. (20) – is much greater than the res-onant equivalent – at least, for the “small-scale” population ofelectrons.As an important side note, the“Quasilinear” diffusion equa-tion derived here applies for non-magnetized plasmas. Whenan ambient magnetic field, B , is present, the “resonance”condition generalizes to [39]: Ω − k k v k = n Ω ce /γ, (28)where Ω ce ≡ eB /m e c is the non-relativistic gyro-frequency,the “parallel” direction is along the ambient (mean) mag-netic field, and n is an integer. Electrons moving throughelectromagnetic turbulence are not “magnetized”, in the for-mal sense, with respect to the “small-scale” fluctuation fields. Hence, the small-scale fields do not contribute to the higher-order (magnetic) resonances – such as the Cherenkov reso-nance at n = 1 . Thus, with regard to the “small-scale” sub-population of electrons, we may disregard resonant diffusionin general.Finally, to evaluate the (non-resonant) energy diffusion co-efficient – Eq. (21) – we need an expression for the electricfield, h E i , and its “longitudinal” correlation length, λ lE . Tothis end, we must relate the electric field to the underlyingmagnetic turbulence that produces it, i.e., we need to specifythe wave turbulence dispersion relation, Ω r ( k ) .In general, this may be done via the dielectric tensor, ↔ ǫ k , Ω .Using Amp ` ere’s law, and the definition of the dielectric tensor,we write [41]: k × δ B k , Ω = − Ω c ↔ ǫ k , Ω · E k , Ω . (29)Suppressing the time-dependent in the field amplitudes, i.e.ignoring wave growth/damping, the electric spectral distribu-tion may be expressed as: | E k | = (cid:12)(cid:12)(cid:12) ↔ ǫ − k , Ω · ˆ b t k (cid:12)(cid:12)(cid:12) n | δ B k | , (30)where ˆ b t k is the unit vector in the direction of k × δ B k , Ω , and n ≡ kc/ Ω is the index of refraction.Next, using Eq. (30) and Parseval’s theorem, h E i be-comes: h E i = ´ (cid:12)(cid:12)(cid:12) ↔ ǫ − k , Ω · ˆ b t k (cid:12)(cid:12)(cid:12) n | δ B k | d k ´ | δ B k | d k h δB i . (31)Finally, the general expression which relates the (electric)energy diffusion and (magnetic) pitch-angle diffusion coeffi-cients follows from Eqs. (31), (21), and (3). It is: D W W = √ W e β ´ (cid:12)(cid:12)(cid:12) ↔ ǫ − k , Ω · ˆ b t k (cid:12)(cid:12)(cid:12) n | δ B k | d k ´ | δ B k | d k (cid:18) λ lE λ B (cid:19) D αα , (32)where W e ≡ γm e c is the electron’s total energy, and wehave assumed statistical isotropy/homogeneity to produce thenumerical prefactor.Eq. (32), despite its apparent complication, offers a fairlysimple interpretation when the dielectric tensor assumes ascalar value, ǫ . Recalling that √ ǫ = n , so that ǫ − = 1 /n ,Eq. (32) simplifies to: D W W = √ W e β h β ph i dist. (cid:18) λ lE λ B (cid:19) D αα , (33)where h β ph i dist. is the distributional average, over the mag-netic spectrum, of the normalized wave phase velocity, β ph ≡ Ω /kc . Thus, D W W ∝ (cid:0) m e v h v ph i dist. (cid:1) D αα , (34)which is what we would expect, given the general relation be-tween the “velocity space” diffusion coefficient, D vv , and thepitch-angle diffusion coefficient; i.e. D vv ∼ v ph D αα [42]. D. Particle Transport in Magnetized Plasmas with ElectricFluctuations
As mentioned previously, the combined effect of electricand magnetic fields can lead to fairly complicated particledynamics. Particle drifts , for example, involving both theelectric and magnetic fields, should be considered. Here, wepresent two realizations of the drift phenomenon. We will,subsequently, argue that these effects are of negligible impor-tance for diffusion in small-scale fields.In Section II A, we argued that sub-Larmor-scale magneticfluctuations result in trajectories that occupy the small deflec-tion angle regime. For this reason, the “guiding center approx-imation”, that underlies the drift theory, breaks down. Conse-quently, the notions of curvature drift and
Grad-B drift loseall meaning in this regime.Nonetheless, a magnetized plasma contains a large-scalemagnetic field – which is, by construction, “super-Larmor-scale”. Hence, drifts that involve the electric field and theambient (mean) field are, in principle, important to consider.The first of these that we will explore is the, so called,
Ecross B drift . We will, once more, assume a sinusoidal elec-tric field. In this case, however, we assume that an ambientmagnetic field, B , is present. Furthermore, we suppress thetime-dependence; hence: E ( x ) = E cos ( kx )ˆ x, (35)where the x -direction is along k . Assuming non-relativisticvelocities, the y -component of the electron, in the ambientmagnetic field, will have the solution [40]: d v y dt = − Ω ce v y − Ω ce cE B cos [ kx + kr L sin (Ω ce t )] (36)where x is the initial position, and r L = m e βc /eB isthe (ambient) Larmor radius. This solution presupposes thatthe electric field will only perturb the electron orbit about theambient field. Hence, our substitution of the zero th -order so-lution.Next, we average Eq. (36) over a gyro-period. Thus, h v y i + cE B h cos [ kx + kr L sin (Ω ce t )] i = 0 , (37)since h dv y /dt i = 0 – i.e. the drift velocity is constant.Next, assuming that kr L ≪ , we may write the solutionfor h v y i as [40]: h v y i /c = − E ( x ) B (cid:18) − k r L (cid:19) . (38)Finally, recognizing that, in the general case, i k → ∇ , wewrite the solution for an arbitrary electric field as [40]: v E × B = c (cid:18) r L ∇ (cid:19) E × B B , (39)where v E × B is the drift velocity. The second term, i.e. thatwhich involves the Laplacian operator, is a correction known as a finite-Larmor-radius effect . When kr L ≫ , the Larmorradius is much larger than the field wavelength. In this case,the particle is acted upon, by the electric field, on a time-scalemuch shorter than the gyroperiod. Consequently, the drift ap-proximation is not appropriate for “small-scale” fields, sincethe perturbation is implicitly assumed to act on a time-scaleof many gyroperiods.A similar drift phenomenon occurs when we consider thetime-dependence of the electric field. Assuming that Ω ≪ Ω ce , the particle will drift with velocity [40]: v p = ± c Ω ce B d E dt , (40)The quantity, v p , is known as the polarization drift velocity.Similarly, the small-scale processes – by construction – occuron time-scales much shorter than Ω − . Hence, the polariza-tion drift time-scale will be far greater than either τ lE or τ tE .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 tur-bulence. E. Energy Diffusion in Small-Scale Whistler Turbulence
Next, to evaluate Eq. (21), we consider a concrete exam-ple of electromagnetic turbulence in a magnetized plasma.Whistler-mode turbulence in a “cold” plasma admits a sim-ple dispersion relation [43]: Ω r ( k ) = Ω ce k c k c + ω pe cos( θ k ) , (41)where ω pe ≫ Ω ce is the electron plasma frequency, and θ k ∈ (0 , π/ , is the angle between the wave-vector, k , andthe ambient magnetic field, B . We will assume a (nearly)steady-state, so that the instability is non-linearly saturated,that is the instability growth rate, Ω i , is much less than all rel-evant frequency-scales, and thus is negligible. This treatmentassumes that the turbulence is “linear”, i.e. δB ≪ B . Wewill further assume that: γv Ω ce > λ B , (42)where Ω ce /γ is the relativistic gyro-frequency.Eq. (42) implies that ρ ≫ , since δB ≪ B – thus, the testelectrons are sub-Larmor-scale with respect to the fluctuationmagnetic field, δ B .It is worth mentioning that, formally, the cold plasma ap-proximation requires that kv/ Ω ce ≪ [44]. This conditionwould imply that the electron population is “super-Larmor-scale” with respect to the magnetic field, since λ B ∼ k − B ,where k B is the wave-number of the dominant wave-mode.For this reason, our model implicitly presupposes the exis-tence of a cold population of super-Larmor-scale electronswhich support the Whistler-modes. Consequently, our testparticles will be comprised of a “hot”, albeit smaller, popula-tion of sub-Larmor-scale electrons. This situation may be ap-proximately realized by the “Super-halo” electron population[45, 46], as it propagates through the “colder” solar wind tur-bulence, which appears to include small-scale Kinetic-Alfvénand Whistler-wave modes [47].An examination of Eq. (41) reveals that Ω r ( k ) ≪ Ω ce inthe regime where kc ≪ ω pe . Restricting ourselves to thisregime motivates the introduction of a new parameter, whichwe call the “skin-number”. It is: χ ≡ d e λ − B , (43)where d e ≡ c/ω pe , is the electron skin-depth. Thus, theregime of interest is characterized by χ ≪ .It is noteworthy that, in principle, the test electron veloci-ties may be large enough so that Ω ce /γ ∼ Ω r . By restrict-ing the electron velocities to the mildly relativistic regime, wemay safely presuppose that the field-variability time, Ω − r , issufficiently greater than the time to transit a magnetic corre-lation length, thus permitting the static field treatment for themagnetic field and avoiding the wave-particle resonance treat-ment.Next, in the χ ≪ regime, the electric field perpendicularto B is much greater than the component parallel to the ambi-ent magnetic field; i.e. E ⊥ ≫ E k [43]. Furthermore, it can beshown that in the frame moving along the direction of B withvelocity equal to the parallel phase velocity, v k ph ≡ Ω r /k k , theperpendicular electric field is approximately zero [43]. Con-sequently, this allows us, via Lorentz transformation of theelectromagnetic fields, to relate the magnetic spectral distri-bution to the electric distribution. It is, thusly: | E k | ≈ | E ⊥ k | ≈ β ph | δ B ⊥ k | , (44)where ⊥ refers to the spectrum perpendicular to the meanmagnetic field, and β ph ≡ v k ph c = Ω r ( k ) k k c . (45)Given isotropic/homogeneous magnetic turbulence: | δ B ⊥ k | = | δ B k | cos ( θ k ) . (46)This relation then allows us to express h E i in terms of themagnetic field as: h E i = 23 ´ β ph | δ B k | d k ´ | δ B k | d k h δB i . (47)Next, the electric field correlation length may be obtainedfrom the electric field correlation tensor. For isotropic tur-bulence, one may write the general expression for the Fourierimage of the electric field two-point auto-correlation tensoras: Φ ij ( k ) = | E t k | ( δ ij − k i k j k ) + | E l k | k i k j k . (48) Isotropy is an approximation here, given the polar asymme-try indicated by Eq. (46). Using Maxwell’s Equations, wemay relate the longitudinal, | E l k | and transverse, | E t k | dis-tributions to | δ B k | (where “longitudinal” and “transverse”are with respect to the wave-vector, not the electron velocity).To wit: ( | E t k | = Ω r k c | δ B k | | E l k | = | E k | − | E t k | (49)In the χ ≪ regime, we may substitute Eq. (44) to expressthe tensor completely in terms of the magnetic spectrum. Thetrace of the correlation tensor is then given by: T r h ↔ Φ ( k ) i = 2 β ph | δ B k | cos ( θ k ) . (50)While integrating Eq. (50) along a selected path, we only con-sider the component of the electric field parallel to the trajec-tory, owing to the dot product with velocity in Eq. (17). Thisallows us to draw an analogy to the “monopolar” (magnetic)correlation length considered in Ref. [25] – permitting us towrite the expression immediately as: λ lE ≡ λ T rE ( x ˆ x ) = 3 π ´ ( v k ph ) k | δ B k | d k ´ ( v k ph ) k | δ B k | d k , (51)where the integration path was chosen to be along the x-axis.By comparing Eq. (7) to Eq. (51), we see that the electriccorrelation length differs from the magnetic correlation lengthonly by a factor of a few. For this reason, we may concludethat τ lE is less than τ ac ∼ Ω − r . Consequently, the energydiffusion will be dominated by the electric field’s “spatial”stochasticity, as per Eq. (15).Additionally, χ ≪ and Ω ce ≪ ω pe demand that v k ph ≪ c .This implies that h δB i ≫ h E i . Consequently, the pitch-angle diffusion will be dominated by the magnetic deflections,and thus we may neglect the contribution due to the electricfield.Finally, given Eq. (47), the energy diffusion coefficient maybe related directly to the (magnetic) pitch-angle diffusion co-efficient via the relation: D W W = 2 √ W e β ´ ( β k ph ) | δ B k | d k ´ | δ B k | d k λ lE λ B D αα . (52)Eqs. (52) and (8) will be confirmed, given isotropic small-scale Whistler turbulence, via first-principle numerical simu-lation in Section IV. F. Radiation Production in Magnetized Plasmas withSub-Larmor-Scale Magnetic Fluctuations
As mentioned previously, radiation production by electronsmoving through (mean-free) sub-Larmor-scale magnetic tur-bulence has been explored thoroughly by a number of authors.The ultrarelativistic regime, specifically, is characterized by asingle parameter, the ratio of the deflection angle, δα B (over asingle magnetic correlation length) to the relativistic beamingangle, ∆ θ ∼ /γ . The ratio [24, 26, 28]: δα B ∆ θ ∼ eδBm e c λ B ≡ δ j , (53)is known as the jitter parameter. If δ j ≪ , which implies that ρ ≫ , then a distant observer on the line-of-sight will seethe radiation along, virtually, the entire trajectory of the par-ticle. This radiation is known as small-angle jitter radiation[26–28]. Jitter radiation is distinctly not synchrotron radia-tion. The jitter radiation spectrum is wholly determined by δ j and the magnetic spectral distribution. Consider an isotropicpower-law magnetic spectrum for a time-independent field,such as: (cid:26) | B k | = Ck − µ , k min ≤ k ≤ k max | B k | = 0 , otherwise (54)where the magnetic spectral index, µ is a real number, and C is a normalization. It has been shown [27–30] that monoener-getic ultrarelativistic electrons in this prescribed sub-Larmor-scale turbulence produce a flat angle-averaged spectrum be-low the spectral break and a power-law spectrum above thebreak, that is: P ( ω ) ∝ ω , if ω < ω j ,ω − µ +2 , if ω j < ω < ω b , , if ω b < ω, (55)where the spectral break is ω j = γ k min c, (56)which is called the jitter frequency. Similarly, the high-frequency break is ω b = γ k max c. (57)Recently, we have generalized the small-scale jitter regime tonon-relativistic and mildly relativistic velocities [25]. Non-relativistic jitter radiation, or “pseudo-cyclotron” radiation,differs markedly from both synchrotron and cyclotron radi-ation. 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. ρ ≫ .Given a monoenergetic distribution of electrons, and the spec-tral distribution indicated by Eq. (54), the pseudo-cyclotronspectrum has a slightly more complicated structure than ultra-relativistic jitter radiation. It appears as [25]: P ( ω ) ∝ A + Dω , if ω ≤ ω jn F ω − µ +2 + Gω + K, if ≤ ω bn , if ω > ω bn , (58)where µ = 2 and A , D , F , G , and K are functions of spec-tral/particle parameters (e.g. µ , k min , and β ). The break fre-quencies generalize to non-relativistic velocities as expected;namely: ω jn = k min βc, (59) and the break frequency indicated by the smallest spatialscale, i.e. the maximum wave-number, becomes: ω bn = k max βc. (60)Finally, a series of Lorentz transformations allow the general-ization of these results to all velocities [25].The introduction of a mean magnetic field will compli-cate this picture. The topic of radiation production by ultra-relativistic electrons in magnetized plasmas with small-scalemagnetic fluctuations was originally considered in Ref. [48].In the case of strictly sub-Larmor-scale magnetic turbulence,with a mean field, the spectrum will simply be the sum of asynchrotron/cyclotron component (corresponding to the meanmagnetic field) and the jitter contribution from the small-scalefluctuations, i.e. P ( ω ) = P jitter ( ω ) + P synch ( ω ) . (61)Since a plasma is a dielectric medium, dispersion may affectthe form of the radiation spectrum. The effect is mostly neg-ligible in the ultrarelativistic limit, but dispersion may be re-quired for a complete description of the mildly relativistic andnon-relativistic regimes – in real plasmas. Nonetheless, thedispersion-corrected spectrum has already been consideredfor small-angle jitter radiation, Ref. [25], and synchrotron ra-diation [49]. For this reason, we will ignore plasma dispersionin this study.When the electric field is stronger, or comparable, to themagnetic field, its contribution must be included. As shownby Ref. [38], the radiation from ultrarelativistic particles inthe “small-scale” regime resembles jitter radiation. At non-relativistic velocities, however, the deflection angle may befairly large – since the parallel acceleration on the electroncannot be neglected in this regime. Consequently, the radia-tion – in the non-relativistic case – may fall outside the small-angle jitter prescription.Fortunately, since h E i ≪ h δB i for small-scale Whistlerturbulence, we can completely ignore this electric contribu-tion.In Section IV, we will confirm Eq. (61) (via ab initio numerical simulation) in the case of small-scale (isotropic)Whistler turbulence. III. NUMERICAL MODEL
In Section II, we made a number of theoretical predictionsconcerning the transport and radiation properties of magne-tized plasmas with small-scale turbulent electromagnetic fluc-tuations. Additionally, we considered a concrete realizationof this in the form of a cold, magnetized plasma embedded insub-Larmor-scale Whistler turbulence. Here we describe thefirst-principle numerical simulations we employed to test ourpredictions.As stated previously, we assumed the existence of a back-ground of cold plasma which supports Whistler-mode turbu-lence. We then inject a smaller population of hot electrons(test particles) that are sub-Larmor-scale with respect to thesepreset Whistler magnetic fields. First, we consider the numer-ical generation of the Whistler magnetic and electric fields.Our principal assumption, in generating electromagneticturbulence, is that these stochastic electromagnetic fields arethe linear superposition of a large number of wave-modes withrandomized propagation direction and relative phase. Giventhis assumption, we can construct the turbulent fields directlyfrom the plasma waves which are characteristic of the under-lying instability.In general, the properties of these electromagnetic wave-modes, and their dispersion relation, are derived from theplasma dielectric tensor – the determinant, of which, providesa system of characteristic equations. Given the“cold” plasmaapproximation, these equations admit the dispersion relationspecified by Eq. (41) – valid in the frequency range [44]: Ω ci ≪ Ω r ≪ Ω ce (62)where Ω ci ≡ eB /m i c is the ion cyclotron frequency and m i is an ion mass. The inequality is understood to hold for all ionspecies. The equations, additionally, specify the polarizationof the wave-modes. Given obliquely (with respect to the ambi-ent magnetic field) propagating whistler waves, the magneticcomponent will be right-circularly polarized with the follow-ing relations among its components [43, 44] δB x = − θ k ) δB z = i cos( θ k ) δB y , (63)where B is along the z -direction, and the wave-vector is inthe x - z plane. Because the magnetic field is divergenceless, k ⊥ δ B . Given these conditions, the magnetic field will rotateabout the direction of the wave-vector – which, in the χ ≪ regime, will have a period much greater than all other relevanttime-scales. Since the phase is randomized for each wave-mode, this indicates that the magnetic field is approximatelylinearly polarized with a random polarization axis.Next, the electric field is (generally) elliptically polarized.It obeys the following relations [43, 44] : (cid:26) E x /E y = − i Θ E z /E x = Θ , (64)where Θ ≡ k c sin( θ k ) cos( θ k ) ω pe + k c sin ( θ k )Θ ≡ Ω r ω pe + (Ω r − Ω ce ) k c Ω r ω pe Ω ce . (65)These equations suggest that the electric field parallel to theambient magnetic field may be expressed in terms of the mag-netic fluctuations via the relation [43]: | E z k | = Ω r Ω ce kc | B k | tan( θ k ) . (66)Then, specifying a magnetic spectral distribution, e.g. Eq.(54), allows the complete description of each wave-mode. We then add a large number of these waves (given random relativephases and “k-vectors”) to simulate Whistler turbulence.Next, we describe the numerical solution of the equationof motion for our test electrons. Obviously, the test particlesdo not interact with each other, nor do they induce any fields.Additionally, any radiative energy losses are neglected. Anindividual electron’s motion is, consequently, determined onlyby the Lorentz force equation given by: d β dt = − γ [ Ω E + β × Ω B − β ( β · Ω E )] , (67)where Ω E ≡ e E /m e c and Ω B ≡ Ω ce ˆ z + Ω δB .Eq. (67) was solved via a fixed step 4 th -order Runge-Kutta-Nyström method, or a (symplectic) 2 nd -order Boris method.In our test runs, we found little variation between these twomethods – barring numerical instability due to using an insuf-ficiently small step-size in time. This is likely because oursimulation time is limited by actual computational time, andthus, we were unable to realize the slow accumulation of er-rors in the total energy characteristic of non-symplectic nu-merical integrators.With all the particle positions, velocities, and accelerationscalculated, the numerical radiation spectrum was obtained di-rectly from the Liénard-Wiechert potentials. The radiationspectrum (which is the radiative spectral energy, dW per unitfrequency, dω , and per unit solid-angle, dη ) seen by a distantobserver is given by [37, 50]: d Wdω dη = e π c (cid:12)(cid:12)(cid:12)(cid:12) ˆ ∞−∞ A κ ( t ) e iωt d t (cid:12)(cid:12)(cid:12)(cid:12) , (68)where A κ ( t ) ≡ ˆ n × [(ˆ n − β ) × ˙ β ](1 − ˆ n · β ) e − i κ · r ( t ) . (69)In this equation, r ( t ) is the particle’s position at the retardedtime t , κ ≡ ˆ n ω/c is the wave vector which points along ˆ n from r ( t ) to the observer and ˙ β ≡ d β / d t . Since the observeris assumed to be distant, ˆ n is approximated as fixed in time tothe origin of the coordinate system.Next, the total radiation spectrum is obtained by “sum-ming” over the spectra of the individual particles. For themoment, we will only consider mean-free, small-scale mag-netic turbulence in the following discussion.Given an isotropically distributed (in velocity-space) en-semble of electrons, the “summed” spectrum will be equiv-alent to the angle-averaged, i.e. dW/dω , spectrum for a singleelectron. There are two, usually equivalent, methods for doingthis “summation”. First, one may add the spectra coherentlyby summing over each particle’s A κ , and then performing asingle integration via Eq. (68). This method is more physical.Alternatively, we may add the spectra incoherently (i.e., by in-tegrating each particle’s A κ separately, and then summing theresults of each integration). As discussed in Ref. [51], bothmethods will result in the same spectra, since the wave phasesare uncorrelated. However, an incoherent sum will produce aspectrum that is less noisy (for a given number of simulation0particles) than the coherently summed spectrum. For this rea-son, we employ the incoherent approach in this study – as wehave done previously.In contrast to our previous studies, Refs. [24, 25], the non-vanishing mean magnetic field introduces a previously non-existent anisotropy; the “summed” spectrum will, as a result,depend upon the location of the observer. However, if themagnetic turbulence is statistically homogeneous/isotropic,then the synchrotron/cyclotron (mean field) component of thespectrum, alone, will possess this dependence. Since theangle-averaged synchrotron spectrum is a known function, wemay simply add it to the jitter spectrum, obtained via the“summation” method above. Lastly, the contribution due tothe electric field may be neglected, since h E i ≪ h δB i .Finally, the electron pitch-angle (with respect to the z -axis)and kinetic energy, W e ≡ ( γ − m e c , were calculated ateach time-step. Using the definitions, Eqs. (3) and (19), weobtained the pitch-angle and energy diffusion coefficients di-rectly from the simulation data. IV. NUMERICAL RESULTS
In Section II, we made a number of theoretical predictionsconcerning the transport and radiation properties of magne-tized plasmas with small-scale turbulent electromagnetic fluc-tuations – in particular, sub-Larmor-scale Whistler-modes.Additionally, we anticipated that an inter-connection betweenthe transport and radiative properties of electrons movingthrough small-scale Whistler turbulence exists, as it does forstrictly, “Weibel-like”, mean-field-free turbulence [24, 25].
A. Whistler Turbulence
First of all, we explore the particle transport by testing ourpredictions concerning the energy and pitch-angle diffusioncoefficients in small-scale Whistler turbulence. The diffusioncoefficients depend on various parameters, cf. Eqs. (8) and(52), namely the particle’s velocity, β , the magnetic fluctua-tion field strength, h Ω δB i , and the field correlation scale, λ B .To start, we must confirm the fundamental assumption ofdiffusion. As stated previously, a diffusive process requiresthat both h ∆ W e i and h α i increase linearly in time – atleast, on some characteristic time-scale of the system. With δB/B ≪ , the “gyro-period” T g ≡ πγ Ω ce = 2 π γm e ceB , (70)is such a characteristic, “macro” time-scale. On a multi-ple gyro-period time-scale, the electron velocities will changevery slightly. Consequently, we may treat the magnitude ofthe electron velocity as approximately constant.To establish diffusion, mono-energetic electrons ( β =0 . ) were injected into Whistler turbulence with k min = 32 π (arbitrary simulation units), k max = 10 k min , h δB i / /B =0 . , Ω ce = 1 , ρ ≈ , χ ≈ . , and µ = 4 . The simula-tion time included several gyroperiods; T = 10 T g . Additional g A v e r a g e s qu a r e p it c h - a ng l e < α > Figure 1. (Color online) Average square pitch-angle vs. normal-ized time. Relevant parameters are β = 0 . , (number of sim-ulation particles) N p = 5000 , k min = 32 π , k max = 10 k kmin , h δB i / /B = 0 . , Ω ce = 1 , ρ ≈ , χ ≈ . , and µ = 4 . Thelinear nature of the curve (solid, “red”) confirms the diffusive natureof the pitch-angle transport. Here, the dashed (“blue”) line indicatesa line of best fit (simple linear regression) with Pearson correlationcoefficient: . .Figure 2. (Color online) Average square change in electron energy(in simulation units) vs. normalized time. Relevant parameters are β = 0 . , (number of simulation particles) N p = 5000 , k min = 64 π , k max = 10 k kmin , h δB i / /B = 0 . , Ω ce = 1 , ρ ≈ , χ ≈ . ,and µ = 4 . The linear nature of the curve (solid, “red”) confirms thediffusive nature of the energy transport. Here, the dashed (“blue”)line indicates a line of best fit (simple linear regression) with Pearsoncorrelation coefficient: . . simulation parameters include: the time-step ∆ t = 0 . (arbitrary units), and the number of Whistler wave-modes N m = 10000 . In Figure 1, we see that the average squarepitch-angle (as measured with respect to the z -axis, i.e. themean field direction) does, indeed, grow linearly with time.Figure 2 confirms that the electron energy undergoes a clas-sical diffusive process as well. With the existence of pitch-angle and energy diffusion established, we then proceeded tocompare the slope of h α i and h ∆ W e i vs time (the numerical1 β -7 -6 -5 -4 -3 P it c h - a ng l e d i ff u s i on c o e ff i c i e n t ( a r b . un it s ) numerical diffusion coeff. (squares)theoretical diffusion coeff. (circles) Figure 3. (Color online) Pitch-angle diffusion coefficient, D αα vsthe normalized electron velocity, β . Relevant simulation parametersinclude: N p = 5000 , k min = 32 π , k max = 10 k kmin , h δB i / /B =0 . , Ω ce = 1 , χ ≈ . , and µ = 4 . The (purple) empty “squares”indicate the D αα ’s obtained directly from simulation data (as theslope of h α i vs. time), while the (green) filled “circles" are theanalytical pitch-angle diffusion coefficients, given by Eq. (8). β -17 -16 -15 E n e r gy d i ff u s i on c o e ff i c i e n t ( a r b . un it s ) numerical diffusion coeff. (squares)theoretical diffusion coeff. (circles) Figure 4. (Color online) Energy diffusion coefficient, D WW vs thenormalized electron velocity, β . Relevant simulation parameters in-clude: N p = 5000 , k min = 32 π , k max = 10 k kmin , h δB i / /B =0 . , Ω ce = 1 , χ ≈ . , and µ = 4 . The (blue) empty “squares” in-dicate the D WW ’s obtained directly from simulation (as the slope of h ∆ W e i vs. time), while the (red) filled “circles" are the analyticalenergy diffusion coefficients, given by Eq. (52). pitch-angle and energy diffusion coefficients) to Eqs. (8) and(52). In Figure 3, the numerically obtained pitch-angle diffu-sion coefficients are compared to Eq. (8) for a range of possi-ble electron velocities. In each, the theoretical and numericalresults differ only by a small factor of O (1) . Next, in Figure4, we see decent agreement with Eq. (52) and the numericalenergy diffusion coefficients. Figures 3 and 4, furthermore,confirm that our theoretical diffusion coefficients are valid forall electron velocities – including relativistic speeds.
100 1000Inverse magnetic correlation length λ B-1 (arb. units)10 -6 -5 -4 P it c h - a ng l e d i ff u s i on c o e ff i c i e n t ( a r b . un it s ) numerical diffusion coeff. (squares)theoretical diffusion coeff. (circles) Figure 5. (Color online) Pitch-angle diffusion coefficient, D αα vs theinverse of magnetic field correlation scale, λ − B . Relevant simulationparameters include: γ = 3 , N p = 1000 , k min = 8 π , π , π , π ,and π , k max = 10 k kmin (for each k kmin ), h δB i / /B = 0 . , Ω ce = 1 , χ ≈ . , and µ = 4 . For each data point, the theoreticaland numerical results differ only by a small factor of O (1) .
100 1000Inverse magnetic correlation length λ B-1 (arb. units)10 -17 -16 -15 -14 -13 -12 E n e r gy d i ff u s i on c o e ff i c i e n t ( a r b . un it s ) numerical diffusion coeff. (squares)theoretical diffusion coeff. (circles) Figure 6. (Color online) Energy diffusion coefficient, D WW vs theinverse of magnetic field correlation scale, λ − B . Relevant simulationparameters include: γ = 3 , N p = 1000 , k min = 8 π , π , π , π ,and π , k max = 10 k kmin (for each k kmin ), h δB i / /B = 0 . , Ω ce = 1 , χ ≈ . , and µ = 4 . The theoretical and numericalresults differ only by a small factor of O (1) . Another important parameter which strongly influences thediffusive transport is the magnetic field correlation length. InFigure 5, the correlation length was varied by changing k min ,while keeping all other parameters fixed. Once more, we seeclose agreement with Eq. (8). Similarly, the numerical andtheoretical energy diffusion coefficients continue to show de-cent agreement – see Figure 6.Lastly, we consider the magnetic spectral index, µ – i.e.the power-law exponent in Eq. (54). With k min = 32 π and k max = 10 k min , we varied the magnetic spectral index, µ from − to . In Figure 7, we see that the numerical pitch-2 -4 -2 0 2 4 6 8 10Magnetic spectral index µ -6 -5 P it c h - a ng l e d i ff u s i on c o e ff i c i e n t ( a r b . un it s ) numerical diffusion coeff. (squares)theoretical diffusion coeff. (circles) Figure 7. (Color online) Pitch-angle diffusion coefficient, D αα vsthe magnetic spectral index, µ . Relevant parameters are N p = 2000 , k min = 32 π , k max = 10 k max , h δB i / /B = 0 . , Ω ce = 1 , and χ ≈ . . Notice that the numerical results have nearly the samefunctional dependence on µ as the analytical squares, as given byEq. 8. -2 0 2 4 6 8 10Magnetic spectral index µ -16 -15 -14 -13 -12 -11 E n e r gy d i ff u s i on c o e ff i c i e n t ( a r b . un it s ) numerical diffusion coeff. (squares)theoretical diffusion coeff. (circles) Figure 8. (Color online) Energy diffusion coefficient, D WW vs themagnetic spectral index, µ . Relevant parameters are N p = 2000 , k min = 32 π , k max = 10 k max , h δB i / /B = 0 . , Ω ce = 1 , and χ ≈ . . angle diffusion coefficient closely matches the analytical re-sult. Similarly close agreement was, once again, realized be-tween the energy diffusion coefficients; as may be seen in Fig-ure 8.Finally, we consider the radiation spectra. As discussedin Section III, the radiation spectra are expected to be thesummation of synchrotron (cyclotron) and jitter (psuedo-cyclotron) components. For an ultrarelativistic electron, theangle-averaged synchrotron radiation spectrum is the knownfunction [37, 50]: dWdω = √ e c γ ωω c ˆ ∞ ω/ω c K / ( x ) d x, (71) ω / ω jn -5 -4 -3 -2 -1 N o r m a li ze d S p ec t r a l e n e r gy d W / d ω Figure 9. (Color online) Radiation spectrum for a monoenergetic,isotropic distribution of γ = 5 ( χ ∼ ; ρ ≈ ; h δB i / /B =0 . ) electrons moving through small-scale Whistler turbulence. Thefrequency is normalized by ω jn = γ k min βc – the relativistic jitterfrequency. The solid (“red”) curve is from simulation data, whereasthe dashed (“blue”) curve is the analytic estimate. Clearly, the spec-trum is well represented by a superposition of synchrotron + jittercomponents. Note the lower-frequency synchrotron componentand a higher-frequency power-law component corresponding to thesmall-angle jitter radiation. where K j ( x ) is a modified Bessel function of the second-kind, and ω c = 3 / γ Ω ce is the critical synchrotron fre-quency. This result applies for an electron moving in the planetransverse to the ambient magnetic field, i.e. when α = π/ .Nonetheless, we find the expression fits the synchotron com-ponents fairly well; especially when γ is decently large.We illustrate two numerical spectra here, along with theircorresponding analytical estimates – for details concerningthe jitter component, see Ref. [25]. First, we considered a γ = 5 electron population for Figure 9. In this plot, the rele-vant parameters are: N p = 1000 , ∆ t = 0 . , k min = 2 π , k max = 20 π , h δB i / /B = 0 . , Ω ce = 0 . , µ = 4 , ρ ≈ , χ ∼ , and the total simulation time was T = 5 T g .We see that the synchrotron + jitter fit closely resembles thenumerical spectrum.Next, we explored the non-relativistic regime. In Figure10, we assumed a population of sub-Larmor-scale β = 0 . electrons. As expected, a peak in the spectrum may be ob-served near the cyclotron frequency Ω ce – confirming that thetotal spectrum is the hybrid of psuedo-cyclotron + cyclotronradiation. Additionally, to provide a point of comparision, wehave superimposed a simulation result for γ = 4 electrons. B. Langmuir Turbulence
In Section II B, we predicted the pitch-angle diffusion co-efficient for ultrarelativistic electrons moving in small-scaleelectric turbulence. Here, we will numerically confirm Eq.(12) – via our first-principle simulations. We will treat the3 ω / Ω ce -4 -2 N o r m a li ze d S p ec t r a l e n e r gy d W / d ω Figure 10. (Color online) Radiation spectrum for a monoenergetic,isotropic distribution of β = 0 . electrons ( χ ∼ . ; ρ ≈ ; h δB i / /B = 0 . ; Ω ce = 2 ; k min = 64 π ; k max = 10 k min ; µ = 5 ; T = 50 T g ); superimposed with a spectrum given a popu-lation of γ = 4 electrons ( χ ∼ ; ρ ≈ ; h δB i / /B = 0 . ; Ω ce = 0 . ; k min = π ; k max = 10 π ; µ = 4 ; T = 5 T g ). Thenormalization on the y -axis is arbitrary, whereas the x -axis is nor-malized to the β = 0 . population’s cyclotron frequency, i.e. Ω ce = 2 . The “thick” solid (“red”) curve is from simulation datafor the β = 0 . population, the dashed (“blue”) curve is the cor-responding analytic estimate for “pure” psuedo-cyclotron radiation,the “thin” solid line is the simulation data for the γ = 4 population,and the “dot-dashed” ( “black”) line is the γ = 4 analytic estimate.Notice, for the β = 0 . spectrum, that the spectrum peaks near thecyclotron frequency, Ω ce – hence we see the signature of cyclotronradiation. The additional harmonics, which are purely a relativisticeffect, are the signature of emerging synchrotron radiation. electric field as quasi-static, i.e. k × E k ≈ . To this end, weemploy a model identical to that used by Ref. [38] for the nu-merical generation of the electrostatic (Langmuir) turbulence.Essentially, a background of “cold” langmuir wave-modes areassumed to be present, with Ω r ∼ ω pe .It was assumed that the Langmuir oscillations are “cold”,i.e. possessing real frequency, Ω r ( k ) ≈ ω pe (where ω pe isthe electron plasma frequency). In this case, the parameterswhich characterize the radiation/transport regime are the jitterparameter [38]: δ j ≡ δα E ∆ θ ∼ eE ⊥ λ E m e c (72)and the “skin-number”: χ ≡ d e λ tE = cω pe λ tE . (73)Additionally, we considered an electric field with a spectraldistribution identical to Eq. (54) – with δ B k → E k . Fur-thermore, the simulation procedure was identical – with theexception that E k k , rather than peripendicular to the wave-vector.This form of turbulence may be realized in a number ofways. “Cold” Whistler turbulence with χ ≫ – i.e. the β L a ng m u i r P it c h - a ng l e d i ff . c o e ff . ( a r b . un it s ) numerical diffusion coeff. (squares)theoretical diffusion coeff. (circles) Figure 11. (Color online) Electric pitch-angle diffusion coefficient, D elec. αα vs the normalized electron velocity, β for small-scale Lang-muir turbulence. Relevant simulation parameters include: k min =8 π , k max = 10 k kmin , h Ω E i = 4 . , χ ≈ . , δ j ≈ . , and µ = 5 . The (purple) empty “squares” indicate the D lang. αα ’s obtaineddirectly from simulation data, while the (green) filled “circles" arethe analytical pitch-angle diffusion coefficients, given by Eq. (12).Notice that the small deflection approximation, which is the founda-tional assumption behind Eq. (12), holds well for velocities that aremildly relativistic ( γ ∼ ). opposite regime to that considered in the previous sections– has an the electric field which is approximately electro-static; i.e. resembling an anisotropic realization of Lang-muir turbulence (ignoring the magnetic field), with Ω r ( k ) ≈ Ω ce cos ( θ k ) . For strictly sub-Larmor-scale magnetic fields, thecorrelation length transit time is much shorter than the aver-age gyro-period – hence the electric field is effectively time-independent. Conceptually, the electric field may be compa-rable in strength to the magnetic field in this regime. Conse-quently, it may be necessary to include its contribution.Figure 11 shows the electric pitch-angle diffusion coeffi-cient as a function of particle velocity. In each scenario, monoenergetic electrons were injected into Langmuir turbu-lence with δ j ≈ . , χ ≈ . , k min = 8 π , k max = 10 k min ,and h Ω E i = 4 . (all simulation units are arbitrary). The elec-tron velocities vary for each run. We see that the numericalpitch-angle diffusion coefficient approaches the ultrarelativis-tic result as v → c . Furthermore, we see fairly close agree-ment, even in the mildly relativistic ( γ ∼ ) regime. The largediscrepancy seen from the most leftward data points may beattributed to the breakdown of the small deflection angle ap-proximation, which accompanies the existence of a compa-rable longitudinal acceleration. In Figure 12, we have plot-ted the corresponding numerical radiation spectra (the spec-tral energy per unit frequency) for electrons with v = 0 . c and γ = 2 . Details on the numerical implementation may befound in Refs. [24, 25]. We present the analytical solution forthe γ = 2 electron via the perturbation theory approach de-tailed in Ref. [25]. The resulting radiation spectrum is analo-gous to the (mildly) relativistic small-angle jitter spectrum ofan electron moving through sub-Larmor-scale magnetic tur-4 ω / ω jn N o r m a li ze d S p ec t r a l e n e r gy d W / d ω γ = 2 γ = 1 Figure 12. (Color online) Langmuir Radiation spectra for the γ = 2 and v = 0 . c electrons (see Figure 11 for details on the simulationparameters). The frequency is normalized by the characteristic jitterfrequency, i.e. ω jn ≡ γ k min βc . The lower (“red”) curve is fromsimulation data, and it corresponds to the v = 0 . c electron. Theupper (“blue”) curve is the simulation result for the γ = 2 electron,and the dashed curve is the analytic estimate. Clearly, the mildly rel-ativistic spectrum is well represented by the (Langmuir) jitter result. bulence, but it is morphologically distinct. This is because theelectrostatic field, owing to its curl-free presentation, has adifferent correlation tensor, Φ ij ( k ) , than the (divergenceless)magnetic equivalent. Thus, we require the substitution: Φ ij ( k ) ∝ (cid:16) δ ij − ˆ k i ˆ k j (cid:17) → ˆ k i ˆ k j . (74)The analytical solution, strictly, holds for the ultrarelativisticlimit. Nonetheless, as can be seen in Figure 12, the numericalsolution closely matches the analytic result for mildly rela-tivistic electrons with γ = 2 . This is consistent with the resultseen in Figure 11, which suggests the presence of the smalldeflection angle regime.In contrast, the third spectrum in Figure 12 differs markedlyfrom the analytic (jitter) prediction. This is the spectrum re-solved for a v = 0 . c , i.e. γ ≈ , electron. In accord withFigure 11, the deflection angle is large, thus the spectrum isoutside the small-angle jitter regime.It is noteworthy that the χ ≫ condition in Langmuir-like turbulence may not be physically realizable, since Lan-dau damping would likely eliminate wave-modes at sub-skin-depth spatial scales too quickly [38]. With the field variabil-ity time-scales of comparable order to the electric correlationlength transit time, it may be necessary to consider the rmselectric field as a function of time. Thus, a more realisticmodel may require a time-dependent pitch-angle diffusion co-efficient. V. THE JITTER/SYNCHROTRON SPECTRUM OF ATHERMAL DISTRIBUTION OF PARTICLES
In most cases, our sub-Larmor-scale electron distributionwill not be composed of mono-energetic electrons. Here, we consider the radiation spectrum one might expect froma Maxwell-Boltzmann (thermal) distribution of electrons insub-Larmor-scale magnetic fields.To obtain the jitter component of the spectrum, we mustaverage the single electron spectrum over an appropriate rela-tivistic Maxwell-Boltzmann distribution. We define the jitter emission coefficient , which is the total radiated power per fre-quency per volume, as thusly: (cid:18) dPdνdV (cid:19) jitt. = n e ´ P j ( ν, p ) e γ/ Θ d p ´ e γ/ Θ d p , (75)where Θ ≡ k B T e /m e c , k B is the Boltzmann constant, T e is the electron temperature, n e is the electron number density, ν = ω/ π , and P j ( ν, p ) ≡ πT dWdω ( p ) , (76)is the single electron (power) spectrum with kinetic momen-tum, p = γm e v , and at the observation time, T .Next, we require an expression for the angle-averaged ther-mal synchrotron emission coefficient. To that end, we employ: (cid:18) dPdωdV (cid:19) syn. = 2 / π / e n e ν / cK (1 / Θ) ξ / exp " − (cid:18) ξ (cid:19) / , (77)where ξ ≡ ω/ Ω ce Θ . This expression produces the correcttotal power, up to a factor of . , when Θ = 0 . [52]. With Θ = 0 . , the thermal Lorentz factor, γ T e = Θ + 1 = 1 . .Thus, this corresponds to the trans-relativistic regime.When the temperature approaches the ultrarelativistic limit,i.e., Θ ≫ , Eq. (77) gives a fairly accurate result, with acorrection factor of order unity ( ≈ . – see Ref. [52], fordetails).In figure 13, we have plotted the combined emission coeffi-cient for a scenario in which sub-Larmor-scale magnetic tur-bulence, with a spectrum identical to Eq. (54), is embeddedin an ambient magnetic field, B . We suppose the followingparameters: n e = 1 cm − , δB = B = 1 G , γ T e = 12 , k max = 50 k min , and k − min = d rel. e – where d rel. e ≡ c √ γ T e /ω pe isthe relativistic skin-depth. These parameters, other than Θ , arenot important for determining the overall shape of the spectra;thus, our selection is made only for instructional purposes.As may be readily seen in Figure 13, the jitter emissionspectrum – dashed (“blue”) line – dominates over the syn-chrotron component – three-dot-dashed (“red”) line – at lowfrequencies. This contrasts with the scenario depicted in Fig-ure 9, where the jitter portion dominates at high frequencies.Essentially, the ratio: ω jn /ω c , determines where the jitter com-ponent makes an appearance.Furthermore, the depicted jitter and synchrotron spectra arenearly identical to the mono-energetic equivalents. This is be-cause with γ T e = 12 – or, equivalently, Θ = 11 – the vastmajority of particles are moving ultrarelativistically. Hence,the thermal spread is very small.In contrast, with γ T e = 1 . , a considerable degree of ther-mal spread will be noticable on inspection. However, as5 Frequency (Hz)10 -26 -24 -22 -20 E m i ss i on C o e ff i c i e n t ( e r g s - H z - c m - ) Figure 13. (Color online) Emission coefficient vs. frequency fora thermal distribution of electrons moving through “magnetized”,sub-Larmor-scale magnetic turbulence. Relevant parameters: n e =1 cm − , δB = B = 1 G , γ T e = 12 , k max = 50 k min , and k − min = d rel. e – where d rel. e ≡ c √ γ T e /ω pe is the relativistic skin-depth.The jitter component – dashed (“blue”) line – overpowers the syn-chrotron portion – three-dot-dashed (“red”) line – at frequencies be-low ω bn ∼ γ T e k max v T e . This produces a distinctly non-synchrotronfeature, at low frequencies, in the total (summed) spectrum, solid(“purple”) line. we see in Figure 14 – where we consider an identical sce-nario, with Θ = 0 . – this spread does not obscure the trans-relativistic jitter (pseudo-cyclotron) feature; the jitter portionis still very clearly distinct from the thermal synchotron com-ponent.To summarize, the signature of jitter radiation — both inthe relativistic and trans-relativistic regimes – remains clearlyevident, even given a thermal distribution of electrons. Frequency (Hz)10 -30 -28 -26 -24 -22 -20 E m i ss i on C o e ff i c i e n t ( e r g s - H z - c m - ) Figure 14. (Color online). Emission coefficient vs. frequency fora thermal distribution of electrons moving through “magnetized”,sub-Larmor-scale magnetic turbulence. Relevant parameters: n e =1 cm − , δB = B = 1 G , γ T e = 1 . , k max = 50 k min , and k − min = d rel. e . Despite the presence of noticable thermal spread, thejitter component – dashed (“blue”) line – still overpowers the syn-chrotron portion – three-dot-dashed (“red”) line – at frequencies be-low ω bn ∼ γ T e k max v T e . The summed spectrum, solid (“purple”) line,remains distinctly non-synchrotron-like at low frequencies. VI. CONCLUSIONS
In this paper, we explored test particle transport (diffu-sion) and radiation production in magnetized plasmas withsmall-scale electromagnetic turbulence. In our previous works[24, 25], we showed that the pitch-angle diffusion coeffi-cient and the simultaneously produced radiation spectrum arewholly determined by the particle velocity and the statisti-cal/spectral properties of small-scale (mean-free) magneticturbulence. Here, we have generalized these results to the casewhen the magnetic field has a mean value.In fact, the pitch-angle diffusion coefficient, Eq. (8), re-mains unchanged by the addition of a mean field – so longas the pitch-angle, α assumes its conventional meaning, i.e. asthe angle between the electron velocity vector and the ambient(mean) magnetic field. Since magnetized plasmas character-ized by instability often include random electric fields, as isthe case for the Whistler turbulence considered here, we addi-tionally considered test particle energy diffusion. We showedthat the energy diffusion coefficient in small-scale Whistlerturbulence is directly proportional to the (magnetic) pitch-angle diffusion coefficient – see Eq. (52). Thus, it is alsointimately related to the field’s statistical properties. Conse-quently, transport via energy diffusion may provide, yet an-other, powerful diagnostic tool for the investigation of small-scale electromagnetic fluctuations in magnetized plasmas.Whistler turbulence, as conceived here, is dominated by themagnetic field. In constrast, the purely electrostatic Lang-muir turbulence is characterized by random electric fields. Weshowed that a generalization of the magnetic pitch-angle dif-fusion coefficient exists for the case of relativistic electrons6moving through small-scale electric turbulence. We, further,confirmed our analytic result via first-principle numerical sim-ulations of Langmuir turbulence.Next, we showed that the test particle radiation spectrum(which is predominately determined by the magnetic fieldin Whistler turbulence) is simply the summation of a small-scale, jitter/pseudo-cyclotron component and a regular, syn-chrotron/cyclotron component – see Eq. (61). We have, fur-ther, confirmed these theoretical results via first-principle nu-merical simulations.Additionally, we confirmed the result first shown in Ref.[38] that the spectrum of relativistic electrons in small-scaleLangmuir turbulence is a form of jitter radiation. We, fur-ther, expanded upon this result by resolving the spectrum fortrans-relativistic velocities – showing that the jitter prescrip-tion holds well even down to γ ∼ .Finally, we considered the radiation produced by aMaxwell-Boltzmann (thermal) distribution of electrons in amagnetized plasma with sub-Larmor-scale magnetic fluctua-tions. We demonstrated that the signature of the jitter compo-nent clearly remains when the fluctuation field is comparableto the ambient magnetic field – just as it did for the mono-energetic case considered previously.Our model implicitly considered a scenario whereby a tur-bulent magnetic field was generated in a cold, magnetized,background plasma. We then imagined the existence of a“hot” population of sub-Larmor-scale electrons that servedas our test particles. We suggested that the motion of high-energy, supra-thermal, “super-Halo” electrons through themagnetized solar wind is a promising candidate for the physi-cal realization of our model. Indeed, despite the fact that thispopulation only accounts for a small fraction of a percent ofthe solar wind, its high energy ( − keV ) makes it verysignificant [46, 53].Additionally, the super-Halo population is largely insen-stive to solar activity, and it is likely constantly present in theinterplanetary plasma [46] – thus, it is a relatively fixed sourceof high-energy particles. In fact, recent work has suggestedthat the super-Halo electrons may mediate Weibel-like insta-bilities in the solar wind plasma – facilitating the developmentof Kinetic-Alfvén wave (KAW) and/or Whistler-mode turbu-lence at sub-electron spatial scales [47].The nature of this wave turbulence, in the solar windplasma, is a matter of contention. Conficting accounts im-plicate either KAW or Whistler-modes (or both) [54, 55]. Anumber of reasons for this ambiguity have been given. Forexample, in situ measurements of these waves must be donein the spacecraft frame – which is usually moving at super-Alfvènic speeds with respect to the plasma [55]. Furthermore,the solar wind hosts a permanent source of turbulence; hence,many results implicating Whistler waves – via, for example,the observed power spectrum – may be the erroneous signa- ture of the, ever present, background turbulence [56].However, a more detailed analysis of the turbulent spec-trum may provide a means by which Whistlers and KAW maybe distinguished. In fact, the degree of anisotropy has beenfound to significantly differ between the two types of waveturbulence [55]. With regard to our model, the presence ofanisotropy will result in diffusion coefficients that differ per-pendicular and parallel to the anisotropy axis (which is typ-ically the direction of the ambient magnetic field), since thecorrelation lengths will depend upon the structure of the cor-relation tensor.Hence, we may imagine that the transport properties of“hot” electrons (e.g. sub-Larmor-scale, super-Halo electrons)may be different for Whistler-mode and KAW turbulence.The radiation spectrum would, additionally, distinguish theseforms of turbulence – as the anisotropy, which features intothe field correlation tensor, would alter the shape of the radia-tion spectrum in a characteristic way.Other cases where this work is of great interest include theupstream of collisionless shocks in astrophysical and inter-planetary systems. The “hot” population, in this case, wouldbe Cosmic Rays (CRs) – which are both non-relativistic andrelativistic in astrophysical systems. Relativistic CRs are ra-diatively efficient and radiation from them is observed in su-pernova remnant shocks (Tycho, Chandra, 1003, etc) pulsarwind nebulae, termination shocks, GRBs (internal and reverseshocks, if the ejecta is magnetized) and GRB remnants. Inthe latter case, the external shock may become weak and non-relativistic. Consequently, the ambient interstellar field maybecome significant, and Whistler-like instabilities may de-velop from an initial Weibel “seed”.Concerning Whistler turbulence and our energy diffusioncoefficient, our model’s principal limitation is the essential as-sumption of the “cold” plasma approximation. In many cases,thermal effects must be accounted for; i.e. the plasma “beta”is non-negligible. Nonetheless, under certain conditions, theunderlying plasma may be considered “cold”. As an example,the plasma outflow in ultrarelativistic “collisionless” shocksis beam-like, with very little dispersion; this permits a coldplasma treatment [57]. Therefore, since these shocks may bemediated in part by small-scale Whistler-modes, our roughestimates concerning the diffusive transport of electrons mayprovide some insight into the process of shock acceleration.To conclude, the obtained results reveal strong inter-relation of transport and radiative properties of plasmas tur-bulent at sub-Larmor scales – “magnetized”, i.e. possessing amean magnetic field, or otherwise. ACKNOWLEDGMENTS
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