PParticle Trapping in Axisymmetric Electron Holes
I H HutchinsonPlasma Science and Fusion Centerand Department of Nuclear Science and Engineering,Massachusetts Institute of Technology, Cambridge, MA 02139, USA.
Abstract
Electron orbits are calculated in solitary two-dimensional axisymmetric electro-static potential structures, typical of plasma electron holes, in order to establish theconditions for the particles to remain trapped. Analytic calculations of the evolutionof the parallel energy caused by the perturbing radial electric field are shown to agreewell with full numerical orbit integration Poincar´e plots. The predominant mechanismof detrapping is resonance between the gyro frequency in the parallel magnetic fieldand harmonics of the parallel bounce frequency. A region of phase-space adjacentto the trapped-passing boundary in parallel energy is generally stochastic because ofisland overlap of different harmonics, but except for very strong radial electric field per-turbation, more deeply trapped orbits have well-defined islands and are permanentlyconfined. A simple universal quantitative algorithm is given, and its results plotted as afunction of magnetic field strength and hole radial scale-length, determining the phasespace volume available to sustain the electron hole by depression of the permanentlytrapped distribution function.
Electron holes are steady solitary electrostatic positive potential structures that sustainthemselves by an electron density deficit arising from depressed phase-space density ontrapped orbits [1–4]. They are frequently observed in one-dimensional non-linear simula-tions of plasma kinetic instabilities [4–8], and in observations of space plasmas [9–21]. Theone-dimensional theory of these hole equilibria is well established, being a type of BGKmode [22]. However, in multiple dimensions, both the equilibrium and stability of theseself-sustaining structures is far less well understood. Satellite observations show that elec-tron holes are generally three-dimensional [23–26], oblate structures, more extended in thedirection perpendicular to the ambient magnetic field, than parallel, but by an amount thatvaries with plasma and hole parameters. Also, analysis and simulation have shown thatinitially-one-dimensional holes are subject to instabilities [8, 27–37] that break them up inthe transverse dimension, forming multidimensional remnants.1 a r X i v : . [ phy s i c s . s p ace - ph ] A p r significant magnetic field is known theoretically to be necessary for the existence ofmultidimensional electron hole equilibria in non-pathological background electron distribu-tions [38–40]. When the field is strong enough that the gyro-radius ( ρ ) is very small, theequilibrium becomes locally one-dimensional [41, 42], with minor corrections to Poisson’sequation to account for any transverse electric field divergence, but eventually negligibleinfluence on the particle orbits. At the other extreme, the magnetic field cannot be so weakas to make the gyro-radius large compared with the hole’s transverse dimensions, otherwiseit provides little transverse confinement. But there is a big parameter range between thesetwo limits, in which virtually no theory beyond order of magnitude heuristics has been com-pleted. A high proportion of observed electron holes have equilibrium parameters lying inthis unexplained region.This article presents a first step to carry out rigorous analysis of multidimensional electronhole equilibria. It adopts a model potential that is axisymmetric (independent of the angle θ in a cylindrical coordinate system) which is a representative subset of three-dimensionalholes. The electron orbits in this equilibrium are analysed and calculated numerically, todiscover which regions of phase space are permanently trapped; and, in contrast, the regionsthat initially possess small enough parallel kinetic energy to be trapped by the parallel electricfield, but evolve soon to become untrapped, by the transfer of energy from perpendiculargyration.The present work does not solve the (still unsolved) full problem of finding a self-consistent equilibrium in which only the velocity distribution function on the permanentlytrapped orbits is allowed to differ from the background distribution. But it does give lim-its on what fully self consistent solutions can exist, and indicates what their distributionfunctions might look like. We consider the orbits of electrons in a potential φ ( r, z ) that is axisymmetric about thecoordinate z , i.e. independent of the angle θ . This is a 2D problem, meaning there is justone ignorable coordinate θ . A 2D cartesian geometry in which one cartesian coordinate isignorable would give essentially the same result, and can be considered to be the limit inwhich the radius r is large. There is good reason to suppose that a fully 3D situation, pos-sessing for example elliptical equipotentials in the transverse plane, will also approximatelyconform to our results if it is reasonably represented by a drift kinetic formulation, but norigorous discussion of that case is given. Although the calculation will remain as general aspossible, we shall have in mind equipotentials that are oblate: varying faster in the parallel( z ) direction than in the transverse ( r ) direction. And where it is convenient we make theassumption that the potential is approximately separable, so that φ (cid:39) φ r ( r ) φ z ( z ). In anycase the shapes φ r ( r ) and φ z ( z ) have single positive peaks where their arguments are zero,and φ z is a function only of | z | (i.e. it is reflectionally symmetric). The negatively chargedelectrons are therefore attracted to the origin, and some are trapped. A uniform magneticfield in the z direction is also present.We work in units where time is measured in inverse plasma frequencies ( ω p = (cid:112) n e e /m e (cid:15) ),length in Debye lengths ( λ De = (cid:112) (cid:15) T e /e n e ), and energies (and potential) in electron tem-2a) (b)Figure 1: Example of a trapped electron orbit in a model electron-hole potential φ = ψ exp([ r − r ] /L ⊥ ) sech ( z/ x = r , y = 0 , z = 0 , v x = 1 , v y = 0 , v z = 1.Viewed (a) in the transverse x, y plane, and (b) in three-dimensions showing the bounc-ing parallel to the magnetic field ( z ) direction. (Parameters: ψ = 1, Ω = 0 . r = 10,1 /L ⊥ = 0 . T e ). Thus if primes denote dimensioned parameters, and unprimed the normalizedquantities, t = ω (cid:48) p t (cid:48) , x = x (cid:48) /λ (cid:48) De , and energy W = W (cid:48) /eT e . The parameters T e and n e arethe temperature and density of the unperturbed electron distribution far from the hole. Inthese units the electron mass is eliminated from the equations, and the electron charge is q e = −
1, so the total energy of an electron can be written W = v − φ , and we shall referto the parallel energy as W (cid:107) = v (cid:107) − φ and perpendicular as W ⊥ = v ⊥ = ( v r + v θ ). Themagnetic field strength is represented by the (normalized) cyclotron frequency Ω(= Ω (cid:48) /ω (cid:48) p ).The equation of an electron orbit is then d v dt = ∇ φ − v × Ω ˆ z . (1)In the 1D case where φ depends only on z , there are two constants of the motion W and W ⊥ . The perpendicular motion is then entirely decoupled from the parallel and canbe ignored. However, when φ varies with r , and a transverse electric field E r exists, the W ⊥ (magnetic moment) invariance is broken, and the only strict invariant in addition to W is the canonical angular momentum about the z -axis: p θ = r ( v θ − Ω r/ p θ conservation is mostly to restrict the range of variation of the orbit’s radius to whatin the probe literature are called “magnetic bottles”. At radii greater than approximatelythe gyro-radius ρ = v ⊥ / Ω, conservation of p θ contributes little to parallel particle dynamics,serving mostly to localize the orbit in radial position, within approximately one gyro-radius.Fig. 1 illustrates the kind of orbit that results.For an electron hole to sustain itself, it requires a substantial fraction of the particle orbitsto be trapped. These orbits can then permanently possess a phase-space density ( f ) lessthan those of untrapped orbits. Because in a collisionless plasma f is constant along orbits,3he untrapped orbits have phase-space density corresponding to their distribution function atinfinity; whereas the permanently trapped orbits have f determined by initial conditions: thehole formation processes etc. The key question concerning the existence of a steady solitaryelectron hole equilibrium is whether there are enough trapped orbits to provide a negativeelectron density perturbation that can sustain the potential structure self-consistently.It is known that in the absence of a magnetic field, 2D and 3D electron hole equilibria donot exist (for non-pathological external velocity distributions), because the trapped phasespace is then only orbits which have W <
0, and in multiple dimensions this is insufficient[38–40]. It is also known from particle in cell simulation and drift-orbit analysis that thereexist axisymmetric 2D equilibria when the magnetic field is strong enough that the gyro-radius ( ρ ) is negligibly small. Essentially this existence arises because of the orbit invarianceof W (cid:107) in the limit of small ρ and consequently larger trapped phase-space volume requiringonly W (cid:107) <
0. For the intermediate case, where ρ is finite, yet transverse φ -variation exists,the challenge is this. Given that energy can be exchanged during the orbit between W (cid:107) and W ⊥ , can one quantify whether and to what extent the amount exchanged is limited, and anorbit remains trapped in the z -direction ( W (cid:107) <
0) even if it has so large a W ⊥ that W > φ ) is small in the sense that E r /φ (cid:28) /ρ ,which may also be written L ⊥ (cid:29) ρ , where L ⊥ ≡ φ/E r = φ/ | ∂φ/∂r | is the transverse lengthscale of potential variation. It is this electric field and its time dependence, arising fromthe orbit’s motion in the non-uniform configuration, that is responsible for the transfer ofenergy between perpendicular and parallel. Starting from the drift limit (which is essentially ρ/L ⊥ → z under the influenceof the parallel electric field, and simultaneously rotates azimuthally in θ under the (time-varying) influence of E r × B /B . Trapped orbits (our main focus) bounce in z and experiencean effectively periodic E r as a consequence. The mean value of E r over a period determinesthe average azimuthal rotation. The varying component of E r is the perturbation responsiblefor the transfer between perpendicular and parallel energy. To first order, the fractionaltransfer of energy in a bounce period is small. Then during a single period, z ( t ) can beapproximated as being given by parallel motion with fixed W (cid:107) , which is simply the 1D orbitproblem. Moreover E r ( t ) = E r ( r, z ( t )) = − ∂φ ( r, z ( t )) /∂r can be approximated as being atfixed radius r (again provided ρ is small enough relative to L ⊥ ). The instanteous energytransfer rate is simply the rate of doing work on the electron by E r , namely dW (cid:107) dt = − E r ( t ) v r ( t ) . (2)The important velocity component in this equation arises from the gyro motion of the elec-tron, v r = v ⊥ cos(Ω t ). When this equation is integrated over many bounces and gyro-periods,large excursions in W (cid:107) will occur if there is a resonance between the gyro frequency and aharmonic of the bounce frequency ω b . These are the orbits that are liable to lead to detrap-ping, because the energy transfer is consistently unidirectional (between W ⊥ and W (cid:107) ) overmany bounce periods. If resonant orbits do not lead directly to detrapping, by raising W (cid:107) W (cid:107) versusrelative phase angle (to be explained more fully in a moment). The result then is that theorbits remain trapped.Since there are multiple resonances arising from the harmonics of ω b , the orbits canbecome stochastic and the islands broken up. Very generally, stochasticity begins in Hamil-tonian systems approximately when there is overlap between the separatrices of adjacentislands [44, 45]. Indeed, this principle is often called the Chirikov criterion in recognition ofits discoverer who studied resonances between gyro motion and bounces along the magneticfield in magnetic traps [46]: a close analog of our current concern. If an orbit is stochastic,it is generally not permanently trapped, and in principle cannot contribute to hole sus-tainment. The quantitative determination of the circumstances of trapped orbits thereforereduces, in this perturbative approach, to the determination of which islands (in which partsof W space) overlap, and which do not. We perform this calculation localized in r , andso obtain a measure of the trapped phase space volume which would eventually have to beintegrated over the transverse domain. When discussing resonant perturbation islands in a Hamiltonian system, one generally re-quires an angle-like coordinate that amounts to the phase difference between the Hamiltonianorbit and the perturbation. In the magnetized electron hole with (presumed) uniform Ω, thephase difference we require is between the gyro motion (of v ⊥ and hence phase of v r ) and aperturbing electric field which we will take as the Fourier component E n at some harmonic n of the slowly varying bounce frequency: ω n = nω b . The Fourier component has a fixedphase with respect to the z motion, which we will take as zero when z = 0. But because ω b isvarying, the bounce phase has a variable rate of change with respect to the gyrophase, whosephase we have taken as zero when v r = v ⊥ . We shall write the phase difference betweenbounce and gyromotion as ξ , so that dξdt = ω n − Ω , (3)and seek the locus of orbit motion in the plane ξ, W (cid:107) . Orbits will then have dW (cid:107) dt = − E n v ⊥ cos( ω n t ) cos(Ω t )= − E n v ⊥ [cos( ω n − Ω) t + cos( ω n + Ω) t ] (cid:39) − E n v ⊥ cos ξ, (4)and we have dropped the term cos( ω n + Ω) t , because it is a fast oscillation, compared withthe presumed slow evolution of ξ = ( ω − Ω) t . We shall mention it later.For an electron bouncing in a finite parallel potential energy well, the frequency ω b (andhence ω n ) varies with bounce amplitude, and hence with W (cid:107) . Let us suppose for initialillustrative purposes that the variation can be approximated linearly as dξdt = ω n − Ω = dω n dW (cid:107) ( W (cid:107) − W (cid:107) R ) = dω n dW (cid:107) ∆ W (cid:107) , (5)5here W (cid:107) R is the value of W (cid:107) at which exact resonance occurs ( ω n = Ω), and that we cantake E n and dω n dW (cid:107) to be independent of W (cid:107) . Then eq. (2) becomes dω n dW (cid:107) ∆ W (cid:107) dW (cid:107) dξ = dω n dW (cid:107) d ∆ W (cid:107) dξ = − E n v ⊥
12 cos ξ. (6)This expression can be integrated as dω n dW (cid:107) ∆ W (cid:107) = − E n v ⊥ sin ξ + C. (7)This is the island locus, and different values of the integration constant, C , give rise todifferent trajectories, effectively different starting W (cid:107) s. The island’s separatrix correspondsto C = E n v ⊥ . The x-point (if dW (cid:107) dω b is negative) is at ξ = − π/ ξ = π/
2) is then | ∆ W (cid:107) | = (cid:115) E n v ⊥ (cid:12)(cid:12)(cid:12)(cid:12) dW (cid:107) dω n (cid:12)(cid:12)(cid:12)(cid:12) . (8)The island center is at ξ = π/ C = − E n v ⊥ dW (cid:107) dω n . Fig. 2 illustrates two possible trajectoryFigure 2: Illustration of relatively narrow islands in energy when dω n /dW (cid:107) and E n v ⊥ canbe taken as constant, for two different magnetic field strengths and hence different resonantparallel energies w (cid:107) R = − W (cid:107) /ψ . ( E r = 0 . ψ = 1, n = 2.)islands for different cyclotron frequencies and hence resonant energies. They are plotted ascontours of the constant C . The perturbation E n , and hence island separatrix width is smallenough that the present approximations are good. (The islands are actually obtained by amore elaborate calculation that accounts for the variation of | dω n /dW (cid:107) | and | E n | with W (cid:107) and v ⊥ with W ⊥ , to be explained later, but are small enough that the present illustrativeapproximations are excellent.) 6 The electric field harmonics
We must now obtain expressions for ω b and E n as a function of W (cid:107) for a model electron hole.These depend upon the z -profile of the potential, which will be taken as φ ( r, z ) = ψ ( r )sech ( z/ . (9)This z -dependence is what is obtained for 1-D shallow holes whose trapped distribution isa Maxwellian of negative temperature [47]. More importantly, it falls off at large distances ∝ exp( − z ), which is required for essentially any 1-D Debye shielded potential (at small holevelocity) that does not have infinite velocity distribution derivative at W (cid:107) = 0 [4]. Thereforethe results we obtain from this model potential will apply to shallow trapped orbits for a widerange of acceptable potential profiles (which is not the case for Gaussian shaped potentials,often used.) We approximate the orbit, for the purpose of determining the z -motion, asoccurring at fixed r (because of p θ conservation), so it is effectively a 1-D problem in space. It has been shown recently [37] that for shallow-trapping ( − W (cid:107) (cid:28) ψ ) the (1D) bouncefrequency is ω b (cid:39) (cid:112) − W (cid:107) / W (cid:107) + ψ (cid:28) ψ ) havebounce frequency in the approxiately parabolic bottom of the potential energy well ω b = √ ψ/
2. This expression is exact for the sech profile chosen, but the potential shape at thepeak can (unlike the hole wings) be different, so this is a choice of a particular shape of hole.It has been found by numerical orbit integration as shown in Fig. 3 that an interpolation ofthe universal form ω b / (cid:112) ψ = (cid:2) ( − W (cid:107) /ψ ) − / − / (cid:3) − / √ , (10)represents the dependence over the entire trapped energy range extremely well, within ap-proximately the thickness of the line. The inverse of this expression is − W (cid:107) /ψ = [(2 ω b /ψ ) − / + 1 − / ] − . (11)The shallow W (cid:107) → ω b = (cid:112) − W (cid:107) / ω b / (cid:112) ψ = ( − W (cid:107) /ψ ) / / , (12)the dot-dash line with constant slope of 1 / E r . First, observe thatfor a mirror symmetric potential such as eq. (9) the period of the variation of E r = − ∂φ/∂r with z at constant r is actually π/ω b , and so only even harmonics nω b are non-zero. Theharmonics n > E r ( t )from a pure sinusoid.Let us introduce convenient energy parameter notation involving positive values normal-ized to ψ , and (for future use) cyclotron frequency to √ ψ as w (cid:107) ≡ − W (cid:107) /ψ, w ≡ W/ψ, b ≡ Ω / (cid:112) ψ ; (13)7igure 3: The energy dependence of bounce frequency ω b for trapped 1-D motion in apotential energy well − ψ sech ( z/ w (cid:107) running from 0 to 1, and the orbits that can become untrappedhave w >
0. For a given magnetic field value b , and harmonic number n , the resonancecondition is nω b = b , which gives a resonant parallel energy w (cid:107) R = [(2 b /n ) − / + 1 − / ] − (14)corresponding to eq. (11). For shallowly-trapped orbits, the E r ( t ) has the form of a train of relatively narrow impulsesof width ∼ τ t and period π/ω b , which peak briefly as the orbit passes rapidly through z (cid:39) z , where the parallel electric fieldis very small; and this dwell duration determines the period [37]. The total impulse in asingle passage can be written A = (cid:82) E r dt , and its duration is approximately the potentialwidth divided by the peak speed [ τ t (cid:39) / (cid:112) ψ + w (cid:107) ).] In the limit w (cid:107) → (cid:82) E r ( z ) dt = (cid:82) E r sech ( z/ dz/v (cid:107) ( z ) can be performed exactly and yields A = 8 E r / √ ψ .The Fourier decomposition of E r ( t ) then gives the following Fourier mode amplitudes E n ,for even n when τ t (cid:46) π/ω n (i.e. (cid:112) − W (cid:107) /ψ = √ w (cid:107) (cid:46) π/ n ): E n = A ω b π (cid:39) E r π (cid:113) − W (cid:107) /ψ = E r π √ w (cid:107) . (15)We will refer to this as the impulse limit. 8 .3 High Bounce Harmonics An alternative perspective of the impulse limit is to note that each impulse gives an energychange δw (cid:107) = − Av r = − Av ⊥ cos ξ , every δt = π/ω b . If δξ and δw (cid:107) during a single passagethrough z = 0 are small, we may approximate the effect as an average energy rate of change dw (cid:107) dt = δw (cid:107) δt = − Av ⊥ ω b π cos ξ, (16)in agreement with eqs. (15), and (4).However, if Ω (cid:29) ω b , so that only high harmonics of ω b are resonant, the continuumlimit is inappropriate. Moreover, it will always be the case that Ω (cid:29) ω b near the trappingboundary, w (cid:107) →
0, because ω b → τ t (which does not itself become sig-nificantly longer as w (cid:107) → ξ ) at which each succeeding impulse occursbecomes effectively random relative to the previous impulse. So, rather than a systematiccontinuous flight in the ( ξ, w (cid:107) ) space, the evolution consists of steps of virtually randomamplitude δw (cid:107) cosine-distributed between ± Av ⊥ . This represents an effective diffusion in w (cid:107) with a diffusion coefficient ∼ ( Av ⊥ ) ω b /π . The diffusion connects the trapped orbit re-gion w (cid:107) < w (cid:107) > w (cid:107) = 0, with the result that the distribution function in this region has only limited gradient | df /dw (cid:107) | , and a value approximately equal to the external distribution f ∞ at v (cid:107) = 0 (in theframe of reference of the hole). Orbits that are deeply trapped, having − W (cid:107) /ψ a significant fraction of unity, are not ac-curately described by the impulse approximation of the previous section. Instead of beingstrongly anharmonic, the φ ( z ) is approximately parabolic for them, and their orbit’s z -position varies approximately sinusoidally in time. In the limit w (cid:107) →
1, only the lowestFourier mode, n = 2 is important and the higher harmonics become negligible. Moreover, E r variation depends on the orbit excursion size; so even for the lowest relevant harmonic ω n = 2 ω b , the electric field Fourier amplitude E n = E can become small.The Taylor expansion of the potential φ ( z ) = ψ sech ( z/
4) about z = 0 is φ ( z ) (cid:39) ψ [1 − z / z /
768 + O ( z )], which leads to the sinusoidal bounce frequency ω b = √ ψ/ z and higher terms are dropped. In the presumed separable radial electric field, similarly, E r ( z ) (cid:39) E r (1 − z / w (cid:107) the amplitude z max of the z -oscillation satisfies z max / − w (cid:107) ), and E r ( t ) = E r [1 − z max sin ( ω b t ) /
8] = E r [(1 − z max /
16) + z max cos(2 ω b t ) /
16] then yields E = z max E r /
16 = E r (1 − w (cid:107) ) / . (17)This dependence on (1 − w (cid:107) ) replaces the √− w (cid:107) dependence of eq. (15).9igure 4: Comparison between numerically integrated Fourier coefficients for a sech ( z/ E n expression It is helpful to have an approximate analytic interpolation for the Fourier harmonics E n that spans the entire range 0 < w (cid:107) <
1. Observe that 1 − w (cid:107) = (1 − √ w (cid:107) )(1 + √ w (cid:107) );so an alternative expression to eq. (17), which is equally valid in the limit w (cid:107) →
1, is E = E r (1 − √ w (cid:107) ). Realize also that the higher harmonics, n = 4 , , . . . , arise fromcorrespondingly higher order terms in the Taylor expansion of φ ( z ), and that therefore,as w (cid:107) → E n will become proportional to correspondingly higher powers: (1 − √ w (cid:107) ) n/ .Consider then the following proposed interpolation between the two limits of w (cid:107) : E n = E r (cid:34) n/ − √ w (cid:107) ) n/ + π √ w (cid:107) (cid:35) − . (18)The first term predominates as w (cid:107) →
1, and the second as w (cid:107) →
0. In their respective limits,these two terms give the correct values for E , in agreement with eqs. 15 and 17. In the w (cid:107) → − √ w (cid:107) . Their numerator n/ Solving the w (cid:107) tra jectories Using the approximate expression (12) for ω b giving ω n = ( n/ (cid:112) ψw (cid:107) , the energy trajectoryequation (4) ignoring the fast ω n + Ω term becomes dw (cid:107) dt = ( ω n − Ω) dw (cid:107) dξ = n √ ψ √ w (cid:107) − √ w (cid:107) R ) dw (cid:107) dξ = 12 ( E n /ψ ) v ⊥ cos ξ, (19)where w (cid:107) R = 4Ω /n ψ is the resonant parallel energy at which ω n = Ω. Substituting theinterpolation for E n from eq. (18), and v ⊥ = (cid:112) w + w (cid:107) ) ψ , it can be written n √ w (cid:107) − √ w (cid:107) R √ √ w + w (cid:107) (cid:34) n/ − √ w (cid:107) ) n/ + π √ w (cid:107) (cid:35) dw (cid:107) dξ = ( E r /ψ ) cos ξ. (20)This equation can be integrated analytically in terms of elementary functions to obtain F n ( w (cid:107) , w, w (cid:107) R ) − ( E r /ψ ) sin ξ = const., (21)where for each n = 2 , , , . . . , F n is a fairly complicated algebraic expression detailed inthe appendix. For chosen total energy, magnetic field strength, and perturbing field (i.e. w , w (cid:107) R , and E r /ψ ) the trajectories can most easily be plotted as contours of the left handside expression, F n − ( E r /ψ ) sin ξ , in the plane ( ξ, w (cid:107) ). In these calculations it improvesaccuracy to use the more accurate equation (14) for w (cid:107) R in terms of b , in F n ; and we adoptthis practice forthwith, ignoring the minor inconsistency.In Fig. 5 are shown examples of the energy trajectories for w = 1, E r /ψ = 0 . ψ = 1,and three values of the magnetic field strength, and hence of the resonance energy w (cid:107) R forthe lowest harmonic n = 2. Unlike the previous illustration in Fig. 2 the perturbing field isstronger and we can see that the trajectories near the top or bottom of the potential energywell (i.e. near to − w (cid:107) = 0 or -1) are compressed asymmetrically at those limits because of theform of F n . For an energy away from those limits, the contours are approximately symmetricabout the resonant energy. If the magnetic field strength is big enough that Ω /ψ >
1, thenthis n = 2 resonance does not exist.Fig. 6(a), instead shows trajectories for fixed magnetic field, and hence fixed n = 2resonance frequency, ( b = Ω / √ ψ = √ . E r /ψ = 0 . n = 2 , , , , . . . . The resonance energy is w (cid:107) R = [(2 b /n ) − / + 1 − / ] − . The higherharmonics bunch together near the top of the potential energy well, corresponding to lowbounce frequency. And in fact the n = 6 and n = 8, islands overlap: indicating that thisregion of energy has stochastic orbits and so the orbits there are not permanently trapped.Lower in the well, no overlap occurs with the n = 2 island; so orbits there are permanentlytrapped. 11igure 5: Example energy trajectories for different magnetic field strengths, all for the lowestharmonic n = 2. (a) (b)Figure 6: (a) Example analytic energy trajectories for different harmonics ( n ), and fixedmagnetic field strength. (b) Poincar´e plot of the corresponding numerically integrated orbit.12 .2 Numerical orbits: Poincar´e Plots In order to verify the analytic calculation and to show what happens when its applicableparameter limits are exceeded, it is helpful to perform a numerical integration of the trappedorbits. The full (non-relativistic) equations of motion for the model potential have beenimplemented in cylindrical coordinates using a 4th order Runge-Kutta numerical scheme withtimestep chosen short enough that the (known) conservation of W and p θ are reproducedfor long orbits to no worse than 10 times machine precision. This is observed to requireΩ .dt (cid:46) .
05. Fig. 1 is an example of an orbit so calculated.Poincar´e plots of the energy trajectories for such orbits are obtained by collecting valuesof W (cid:107) and the phase of v r (i.e. atan2 ( v θ , v r )) at successive instants when the orbit passesthrough z = 0 (at which the orbit phase is zero or π and the phase of E n is zero for all even n ). The phase difference, ξ , thus equals the phase of v r . We place a point at each of thecorresponding positions in ξ, W (cid:107) space. We also, for convenience, start all orbits at z = 0and with v θ = 0, v r positive: ξ = 0. We abandon as escaped any orbits that acquire positive W (cid:107) or pass beyond | z | = 20. A technical subtlety is that it is most appropriate to use for W (cid:107) = v z / − φ , not the value of φ at the orbit, but rather the value of φ at the gyrocenter ,which gives significantly smaller oscillatory excursions of W (cid:107) . It therefore more effectivelysuppresses the ω n + Ω term and expresses the approximate magnetic moment conservation.Fig. 6(b) shows an example of a Poincar´e plot, alongside its analytic energy trajectories6(a). The agreement is excellent. Orbits are initialized at equally spaced W (cid:107) values. Ofcourse, they cannot trace island contours well inside their separatrices where the island doesnot extend past ξ = 0. The position and W (cid:107) -width of the n = 2 island agree quantitativelyvery well between (a) and (b). And the n = 4 and n = 6 islands are also readily seen at theirexpected positions. Between the islands, the Poincar´e points trace the open contours. Abovethe position of the n = 6 island ( W (cid:107) /ψ ≥ − .
05) and near its x-point the plot shows ratherincoherent scatter of the points. Orbits above this energy are stochastic, and terminateafter some tens of bounces by leaving the domain. Again, this agrees well with the analyticobservation of overlap between n = 6 and 8, but not between n = 4 and 6 islands.Fig. 7, by comparison, shows what happens if the amplitude of the perturbing transversefield is increased by a factor of 8, other parameters unchanged. The n = 4 and 6 islandsnow overlap strongly, and the entire region W (cid:107) /ψ (cid:38) − . ξ become visible. For example the chain of 3 islands at W (cid:107) /ψ (cid:39) − .
45, orof two islands at W (cid:107) /ψ (cid:39) − .
34. These additional chains arise from nonlinearity, and are notrepresented in the analytic linearized approximation. Still, the overall extent of the n = 2island is quite well captured by the analytics, which predict that it should remain intact, asit does. If the perturbing E r /ψ is increased to 0.1, then overlap and stochasticization of eventhe n = 2 island begins, as illustrated by Fig. 8. Soon beyond it, by E r /ψ = 0 .
13, essentiallythe whole of the phase space becomes stochastic.Fig. 9 shows what happens for a lower magnetic field, Ω / √ ψ = 0 .
6. In this case, a field E r /ψ = 0 .
04 is sufficient make the n = 2 island stochastic, but when that happens, therestill remain some permanently trapped orbits at energies sufficiently below the resonancevalue ( ∼ . ).In contrast, as shown in Fig. 10, increasing the magnetic field to Ω / √ ψ = 1 .
8, removes13a) (b)Figure 7: (a) Analytic energy trajectories for different harmonics ( n ), and fixed magneticfield strength and (b) Poincar´e plot of the corresponding numerically integrated orbit, for astronger transverse electric field. (a) (b)Figure 8: (a) Analytic energy trajectories for different harmonics ( n ), and fixed magneticfield strength and (b) Poincar´e plot of the corresponding numerically integrated orbit, foran extremely strong transverse electric field. 14a) (b)(c) (d)Figure 9: (a),(c) Analytic energy trajectories fixed magnetic field strength and (b),(d)Poincar´e plots of the corresponding numerically integrated orbits, for a lower magnetic field,and two perturbation amplitudes. 15a) (b)(a) (b)Figure 10: Analytic energy trajectories (a,c), and corresponding Poincar´e plots (b,d), forΩ / √ ψ = 1 /
8, and E r /ψ = 0 .
04 (a,b) or 0.2 (c,d).16he n = 2 resonance; and because the higher resonances are weaker, the orbits can sustainhigher E r /ψ before becoming stochastic. This stabilizing effect is enhanced by the resultingreduction in gyroradius ρ . In the previous section we have shown that the island overlap criterion successfully predictswhich trajectories are stochastic (and hence become untrapped) and which are permanentlytrapped. We therefore rely on this success and formulate an analytic condition for particlesat different locations in phase space to be permanently trapped. We will take those parallelenergies W (cid:107) to be trapped which lie below the bottom of the lowest overlapped island and allothers to be subject to detrapping. This criterion describes within typically 10% in W (cid:107) whathas been observed in the example cases we have shown.The function F n , for fixed w and w (cid:107) R , is stationary at resonance ( √ w (cid:107) = √ w (cid:107) R ), and itsderivative in the vicinity of the resonance can be taken from eq. (20) as ∂F n ∂ √ w (cid:107) = 2 √ w (cid:107) ∂F n ∂w (cid:107) (22)= n ( √ w (cid:107) − √ w (cid:107) R ) √ √ w + w (cid:107) (cid:34) √ w (cid:107) n/ − √ w (cid:107) ) n/ + π (cid:35) . Consequently the width of the island separatrix, which occurs at ξ = π/
2, is determined bythe √ w (cid:107) value for which F n ( w (cid:107) ) − F n ( w (cid:107) = w (cid:107) R ) (cid:39) ( √ w (cid:107) − √ w (cid:107) R ) ∂ F n ∂ √ w (cid:107) is equal to E r /ψ .Therefore, regarding the second derivative as constant (adopting just the second-order termin a Taylor expansion of F n ) we can express the island (half-) width as δ n ≡ √ w (cid:107) − √ w (cid:107) R (23) (cid:39) (cid:34) E r ψ √ n (cid:112) w + w (cid:107) R (cid:35) / (cid:34) √ w (cid:107) R n/ − √ w (cid:107) R ) n/ + π (cid:35) − / . We write w + w (cid:107) R = w ⊥ ( √ w ⊥ = v ⊥ / √ ψ ) and recognize that together the parameters n , E ro v ⊥ /ψ / , and w (cid:107) R determine δ n as follows. Analytic Algorithm
Eq. (14) w (cid:107) Rn = [( n / b ) / + 1 − / ] − enables us to find theenergy of the upper and lower island limits of island n as √ w (cid:107) Rn ± δ n = (24) √ w (cid:107) Rn ± (cid:20) E r v ⊥ ψ √ ψ n (cid:21) / (cid:34) √ w (cid:107) Rn n/ − √ w (cid:107) Rn ) n/ + π (cid:35) − / . Overlap occurs between the n and n +2 harmonic islands when √ w (cid:107) Rn − δ n < √ w (cid:107) Rn +2 + δ n +2 .Beginning at the lowest value of n for which a resonance exists (requiring w (cid:107) Rn <
1) determinefrom evaluation of √ w (cid:107) Rn − δ n and √ w (cid:107) Rn +2 + δ n +2 whether it overlaps with the n + 2 island.17a) (b)Figure 11: (a) Contours of the energy boundary W (cid:107) t between trapped and detrapped orbitsas a function of perturbation strength E r v ⊥ /ψ / and magnetic field Ω / √ ψ . (b) The energyboundary W (cid:107) t between trapped and detrapped orbits from eq. (25) compared with the lowestdetrapped orbits found from numerical orbit integration.If so, then it is the lowest energy overlapped island ; if not, increment n by 2 and repeat untiloverlap is found. The resulting n is the harmonic whose island’s lower energy limit is sought,which is W (cid:107) t /ψ = − w (cid:107) t = − ( √ w (cid:107) Rn + δ n ) . (25)Energies below this approximate bound are predicted trapped, energies above have stochasticorbits and are detrapped.Figure 11(a) shows the universal contours that result. Where W (cid:107) t /ψ is close to zero (lightregions), very few orbits are detrapped; while where W (cid:107) t /ψ is close to -1 (dark regions) almostall orbits are detrapped. Discontinuities in W (cid:107) t occur where n changes: it starts at 2 at thebottom (right, below Ω / √ ψ (cid:39)
1) and increments through 4,6,. . . as one moves to largerΩ / √ ψ . To avoid almost complete detrapping for Ω / √ ψ (cid:46)
1, extremely weak perturbationis required. In contrast, for Ω / √ ψ (cid:38) W (cid:107) t as a function of b . These lines are each accompanied bypoints, each of which comes from full numerical orbit integration. A point gives the loweststarting energy that escapes during the first 200 bounces (which might take as many as amillion time-steps). We observe that there is rather good agreement (even in respect of thediscontinuities) at moderate magnetic field Ω / √ ψ (cid:46) ξ = 0, or becauseof limited orbit integration length. It is known theoretically [45] that there can sometimes be18xtremely slow trajectory diffusion coefficients even in stochastic regions. This warns us thatit is an over-simplification to suppose that stochastic orbits can have negligible depressionof phase-space density f ( v ) from the marginally untrapped value. In any case, agreement inthe loss boundary deduced from analytic and numerical treatments is within 0.1 in W (cid:107) t /ψ .A more approximate form of the island widths can be obtained by substituting the moreapproximate frequency fit w (cid:107) R = 4Ω /n ψ = 4 b /n , and approximating (1 − √ w (cid:107) R ) n/ =(1 − b/n ) n/ (cid:39) e − b . Then we find δ n is approximately proportional to 1 / √ n and can bewritten δ n (cid:39) (cid:20) E r ψ √ w ⊥ n (cid:21) / (cid:104) b e b + π (cid:105) − / (26)With reference to this approximation, the behavior can readily be understood as follows.Island overlap leading to stochastic trajectories occurs if δ n is too large; that is if E r v ⊥ istoo large, or b (= Ω / √ ψ ) is too small, or n is too large. The last of these cases (high n atmodest E r and b ) predicts that there is in principle always a stochastic region at very small w (cid:107) , where the bounce frequency is correspondingly small and the resonant bounce harmonicnumber large, regardless of the exact E r and b values. Consequently, a steady electronhole of limited transverse extent will always have a stochastic transition between trappedand passing orbits that in practice smooths out any steep f -gradients at the separatrix.Our numerical orbit integration confirms this prediction (at the computing cost of very longintegrations and fine energy resolution) but not with particularly accurate agreement on W (cid:107) t ,as already mentioned.When b (= Ω / √ ψ ) is large, the term e b makes δ n small, regardless of E r , and suppressesoverlap. This effect can be considered to arise because when the gyro-period is small com-pared with the central transit time ( τ t ∝ / √ ψ , the duration of the impulse), the Fouriertransform of a single impulse has become exponentially small at the cyclotron frequency. Thesuppression applies at essentially all w (cid:107) up to 1, because the impulse width is a rather weak(slowly increasing) function of w (cid:107) . Only the exponentially-large- n orbits at exponentially-small- w (cid:107) will then be stochastic. And the region of stochasticity is limited to very small w (cid:107) .High enough magnetic field thus justifies the drift orbit treatment, and eventually imposesno minimum L ⊥ requirement for a long-lived hole to exist.The opposite case b (cid:28) r only if the transverse length scale remains greater than the gyro-radius E r /ψ = 1 /L ⊥ (cid:46) /ρ = Ω /v ⊥ = b √ ψ/v ⊥ so E r v ⊥ /ψ / (cid:46) Ω / √ ψ . The valid region of Fig. 11(a) is thereforeabove the diagonal straight line E r v ⊥ /ψ / = Ω / √ ψ drawn in purple. And in Fig. 11(b), thelines are drawn only in the valid region. In the invalid region one can expect the permanenttrapping to be poor, and this is confirmed by the points.A perhaps more intuitive way to portray typical results is as in Fig. 12, where are shownexamples of boundaries between trapped and untrapped orbits in velocity-space (based onthe island overlap calculation). The important regions of this domain extend to thermalvelocities ( v ⊥ (cid:38)
1, not just v ⊥ (cid:38) √ ψ ). We need orbits to be permanently trapped for mostof the range of possible v ⊥ to allow the depression of f ( v (cid:107) , v ⊥ ) to contribute sufficient positivecharge to sustain the hole. For smaller ψ the effective perturbation strength ∝ v ⊥ / √ ψL ⊥ becomes stronger for given L ⊥ , which makes orbits more easily detrapped. However, theeffects of varying resonance condition as ψ changes are very strong; so the boundaries do19a) (b)Figure 12: The boundary in velocity-space measured at z = 0 between trapped and un-trapped orbits at different ψ -values: (a) for low magnetic field Ω = 0 .
5, (b) for higherΩ = 1 . ψ . When the n = 2 resonance is avoided, as in Fig. 12(b),the boundary lies at fairly high velocity near W (cid:107) = 0. That leads us to expect qualitativelythat a distribution f ( v (cid:107) ) that is approxiately flat above v (cid:107) t , in the stochastic region, canstill sustain an electron hole with these parameters.In all cases, increasing L ⊥ and making the hole more oblate, i.e. closer to one-dimensionalreduces the detrapped phase-space area. But unless Ω / √ ψ (cid:38)
2, holes of large transversedimension are unstable to transverse perturbations that grow in a few hundred plasmaperiods and break up the holes into shorter transverse lengths, causing them to collapse.So there is a competition between the requirements of equilibrium and stability.
It has been shown that parallel energies of deeply trapped orbits in axisymmetric elec-tron holes have limited excursions in parallel energy, provided the transverse electric fieldperturbation is weak enough. There is a parallel energy threshold which is a function of per-turbation strength and magnetic field, above which the parallel energy trajectory becomesstochastic, and is no longer limited in extent, instead becoming detrapped. Such orbits can-not therefore contribute to the electron deficit needed to sustain the hole. The stochasticityarises when trajectory islands overlap, as has been confirmed by numerical orbit integration.The parallel energy threshold for detrapping has been quantitatively evaluated using the
Analytic Algorithm as a universal function of the hole parameters. Magnetic fields strongenough that Ω / √ ψ (cid:38) / √ ψ (cid:46) Appendix: Mathematical Function Details
The integrated expressions for F n are as follows F n = n √ g n + nπ √ g , (27)with g = (cid:90) ( √ w (cid:107) − √ w (cid:107) R ) dw (cid:107) √ w + w (cid:107) √ w (cid:107) ; (28) g n = (cid:90) ( √ w (cid:107) − √ w (cid:107) R ) dw (cid:107) √ w + w (cid:107) (1 − √ w (cid:107) ) n/ . (29)The first function is easy: g = 2[ √ w + w (cid:107) − w √ w (cid:107) R ln( √ w + w (cid:107) + √ w (cid:107) )]. To evaluate g n ,define the integrals I m ( a, x ) = (cid:90) dx √ a + x (1 − x ) m , (30) J m ( a, x ) = (cid:90) xdx √ a + x (1 − x ) m ; (31)then, since x = ( x − x + x , it is easy to show that g n = 2[(1 − √ w (cid:107) R ) J n/ ( w, √ w (cid:107) ) − J n/ − ( w, √ w (cid:107) )] . (32)The J m and I m are related by J m ( a, x ) = (cid:90) ( x −
1) + 1 √ a + x (1 − x ) m dx = I m − I m − . (33)Also J m can be integrated by parts as J m = √ a + x (1 − x ) m − m (cid:90) (1 − x ) − − x ) + 1 + a √ a + x (1 − x ) m +1 dx = √ a + x (1 − x ) m − mI m − + 2 mI m − m (1 + a ) I m +1 . (34)21liminating J m between (33) and (34), and gathering terms we obtain the following recursionrelation: I m +1 = (cid:34) √ a + x (1 − x ) m + (2 m − I m − ( m − I m − (cid:35) m ( a + 1) . (35)Given J = √ a + x , and the initial values of the recursion: I = ln( √ a + x + x ), and I = [ln( √ a + 1 √ a + x + a + x ) − ln(1 − x )] / √ a + x we can efficiently obtain by iteration I m and J m for m as high as required. This iterative scheme has been implemented andverified, and is used to give the island plots in this paper. Acknowledgements
I am grateful for useful discussions about transverse structure of electron holes with IvanVasko. The codes that were used to do the calculations and create the figures in this articleare publically available at doi:10.5281/zenodo.3746740 . This work was partially fundedby NASA grant NNX16AG82G.
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