FFinite gyro-radius multidimensional electron hole equilibria
I H Hutchinson a) Plasma Science and Fusion CenterMassachusetts Institute of TechnologyCambridge, MA 02139, USA
Finite electron gyro-radius influences on the trapping and charge density distribution of electron holes oflimited transverse extent are calculated analytically and explored by numerical orbit integration in low tomoderate magnetic fields. Parallel trapping is shown to depend upon the gyro-averaged potential energyand to give rise to gyro-averaged charge deficit. Both types of average are expressible as convolutions withperpendicular Gaussians of width equal to the thermal gyro-radius. Orbit-following confirms these phenomenabut also confirms for the first time in self-consistent potential profiles the importance of gyro-bounce-resonancedetrapping and consequent velocity diffusion on stochastic orbits. The averaging strongly reduces the trappedelectron deficit that can be sustained by any potential profile whose transverse width is comparable to thegyro-radius r g . It effectively prevents equilibrium widths smaller than ∼ r g for times longer than a quarterparallel-bounce-period. Avoiding gyro-bounce resonance detrapping is even more restrictive, except for verysmall potential amplitudes, but it takes multiple bounce-periods to act. Quantitative criteria are given forboth types of orbit loss. I. INTRODUCTION
Solitary potential peaks, of extent a few Debye-lengthsparallel to the ambient magnetic field, are frequently ob-served by satellites in space plasmas . They are usu-ally interpreted as being electron holes: a type of non-linear Bernstein, Greene, Kruskal [BGK] Vlasov-Poissonequilibrium in which a deficit of electrons on trappedorbits causes the positive charge density that sustainsthem. Such structures are frequently observed to form innon-linear one-dimensional particle simulations of unsta-ble electron distribution functions of types such as two-stream or bump-on-tail . One-dimensional dynami-cal analysis is, however, not sufficient to describe nat-urally occurring holes fully, because their transverse di-mensions are limited. The observational evidence, backedup by simulations in a variety of contexts, is that whileholes are generally oblate, in other words more extendedin the perpendicular than in the parallel (to B ) direction,their aspect ratio can be as low as L ⊥ /L (cid:107) ∼
1. Moreoverone-dimensional holes have been shown analytically and computationally to be unstable to perturbationsof finite transverse wavelength, which cause them quicklyto break up into multidimensional structures unless themagnetic field ( B ) is very strong. Multi-satelite mis-sions (e.g. Cluster and Magnetosphere Multiscale )are now beginning to document the transverse potentialstructure of electron holes in space.The present work culminates a series of theoreticalstudies addressing for multidimensional electron holeequilibria the important effects of finite transverse size.Beyond the one-dimensional BGK treatment that hasdominated past analysis, the phenomena that need to beunderstood and accounted for can be listed as (1) trans-verse electric field divergence, (2) supposed gyrokinetic a) Electronic mail: [email protected] modification of Poisson’s equation, (3) orbit detrappingby gyro-bounce resonance, and (4) gyro-averaging of thepotential and density deficit.The first (transverse divergence) has been theoreticallyinvestigated for a long time , but predominantly sup-posing that the potential shape is separable of the form φ z ( z ) φ r ( r ), where z is the parallel and r the transversecoordinate. In a recent paper in this series (whichshould be consulted for a review of prior multidimen-sional equilibrium studies) it was shown that electronholes that are solitary can never be exactly separable,and shown how to construct fully self-consistent poten-tial electron holes synthetically. Transverse divergence ispresent even in the limit of small gyro-radius, and so thisphenomenon has been treated in the context of purelyone-dimensional electron motion.Effect (2) (gyrokinetic polarization modification ofPoisson’s equation) was hypothesized as an explanationof statistical observations that electron hole transversescale (oblateness) increases with gyro-radius, or inversemagnetic field strength . And it has often been invokedsince . However, it is based on a misunderstandingof gyrokinetic theory, as has been explained and demon-strated in recent work in this series . The anisotropicshielding phenomenon envisaged does not occur and socannot explain electron hole aspect-ratio trends.Phenomena of type (3), gyro-bounce resonance, lead toislands in trapped phase-space where the gyro-frequencyis an even harmonic of the parallel bounce frequency ofelectrons. Orbits involving overlapped islands becomestochastic. They start trapped with negative parallelenergy, but energy is resonantly transferred from per-pendicular to parallel velocity causing them to becomeuntrapped after some moderate number of bounces. Cor-responding effects cause initially untrapped orbits to be-come trapped. Although this is a finite gyro-radius effect,it can be treated in a small gyro-radius approximationthat accounts only for parallel motion. An analytic treat-ment then proves to be feasible and agrees extremely well a r X i v : . [ phy s i c s . p l a s m - ph ] J a n with full numerical orbit integration in a potential whoselocal radial potential gradient is specified. The analysis,presented in a paper in this series , shows that stochas-tic orbits, giving rise to effective velocity-space diffusion,begin (and are always present to some degree) at thephase-space boundary between trapped and untrappedorbits (zero parallel energy). Increasing transverse elec-tric field or, more importantly, decreasing magnetic fieldcauses the energy range over which orbits are stochasticto extend downward into the bulk of the trapped orbits.Depletion of trapped phase-space sets in rapidly whenlow harmonic resonances are present. They are, whenΩ / ( ω p √ ψ ) (cid:46) ω p the plasma frequency, and ψ the peak electron holepotential in units of electron temperature T e /e ). ThusΩ / ( ω p √ ψ ) (cid:38) not small relative to the perpendicular scale length.It is discussed analytically in section II, and validated bynumerical orbit integration in section III, which includesexploration of resonant detrapping over the entire radialprofile of axisymmetric holes whose velocity distributionfunctions are constrained by the effective diffusivity ofdetrapping regions. In section IV the consequences ofdetrapping for the feasible shape and parameters of elec-tron holes are discussedThe treatment is, for specificity, of axisymmetric po-tential structures, independent of the angle θ of a cylin-drical r, z, θ coordinate system. Fig. 1 illustrates anexample. However, most of the gyro-averaging effectsare immediately generalizable to non-axisymmetric po-tentials, and the drift orbits are always normal to thepotential gradient. Throughout this discussion, ions areconsidered an immobile neutralizing background chargedensity and excluded from the analysis. This is justifiedwhen the electron hole moves (in the parallel direction) atspeeds much exceeding the ion thermal or sound speeds,because ions’ much greater mass gives them much slowerresponse than electrons. Electron holes can travel at anyspeed from a few times the ion sound speed up to theelectron thermal speed. Therefore for the vast major-ity of this speed range ignoring ion response is a goodapproximation. II. GYRO-AVERAGING ANALYTICS
Phenomena (1) and (3) can be treated using a puredrift kinetic treatment, taking the electrons to be suf-ficiently localized in transverse position not to requirefinite gyro-radius to be accounted for in trapping andbounce motion. When, however, the gyro-radius is notnegligible compared with the scale-length of transverse
FIG. 1. Example electron hole shape, showing contours ofpotential φ ( r, z ). Only a quadrant of the structure is shown,it being reflectionally symmetric about the r and z axes. variation, the effects of varying potential and parallelelectric field around the gyro-orbit must be considered.This gyro-averaging (rather than in the mistaken sup-position (2)) is where the sorts of techniques familiar ingyrokinetics become influential.There are actually two conceptually different ways toexpress the “center” of an approximately helical gyro or-bit of a particle in magnetic and electric fields. One isto define it in the stationary reference frame as R = x + v × ˆ z / Ω, where x and v are the instantaneous po-sition and velocity of a particle, and Ω = qB/m is the(cyclotron- or) gyro-frequency in the magnetic field (heretaken uniform) B = B ˆ z . This definition is commonlyused in gyrokinetics derivations. The other, which wasassociated with the original development of orbit drifttreatments (see e.g. Northrop (1961) ), and is there-fore historically the first definition, is to regard it as theposition of the particle averaged over the gyration, inthe frame moving with the mean drift velocity. This first definition was originally, and will be here calledthe “guiding-center”. If the (reference frame) perpen-dicular drift velocity is v d then the particle velocity inthe drifting frame is v − v d and the guiding-center is at X = x + ( v − v d ) × ˆ z / Ω = R − v d × ˆ z / Ω about which themotion of the particle is approximately a centered circle.These two definitions therefore differ by a distance v d / Ωnormal to the direction of v d . With uniform B , the driftis v d = E × B /B ; so X − R = E ⊥ /B Ω. This differenceis the integrated polarization drift, or simply the “polar-ization”, that would arise by increasing the electric fieldfrom zero to E ⊥ . Because the reference frame definition“gyro-center” ( R ) is actually not at the center of the cir-cle which the orbit follows in the drift frame, when anorbit average is carried out relative to R , then an extraterm representing the polarization X − R must be in-cluded. It accounts for the gyro-radius distance from R to the particle varying with phase angle around the orbit.It is more convenient for our current purposes to adoptthe guiding-center ( X ) as our reference position, becausethe gyro-radius (guiding-center to particle) is then (to rel-evant order) independent of gyro-angle; and polarizationdrift is irrelevant. Naturally a thorough mathematicaltreatment (which is not the present purpose) will get thesame results from either perspective. We will hencefor-ward denote the guiding-center as x c and the gyro-radius relative to it as r g .Now consider a finite gyro-radius particle orbit thatmoves along the magnetic field. We are most interestedin particles whose total energy, W = mv / qφ , ispositive because when it is negative, this exactly con-served energy guarantees trapping. To give sufficientsustaining charge deficit, electron holes require exten-sive trapping of orbits with positive W . Whether ornot such an electron is trapped depends on whether theparallel electric field attracting it to z = 0 is sufficient,when integrated over the orbit, to reduce its parallel mo-mentum to zero and hence cause it to bounce. In theabsence of gyration, this condition would be deducedfrom m ˙ v (cid:107) = qE (cid:107) , in the integrated form [ mv (cid:107) /
2] = (cid:82) m ddt ( v (cid:107) ) dt = − q (cid:82) dφdz dz = − [ qφ ] (parallel energy con-servation) and requires W (cid:107) ≡ mv (cid:107) / qφ <
0. Clearly,in the presence of gyration, we must regard the poten-tial (difference) as involving instead the potential gradi-ent averaged over the orbit , including the variation aris-ing from gyro-motion when φ varies in the perpendicu-lar coordinate. This is a (finite-duration) gyro-averagein which end-effects introduce some possible variationwith initial gyro-angle (e.g. at z = 0: the plane of re-flectional symmetry of the hole and maximum potentialalong lines of constant r ). But provided sufficiently manygyro-periods elapse in moving to distant z where φ → z = 0 and z = ∞ to be given by thegyro-averaged potential at z = 0:¯ φ ( x c ⊥ , ≡ (cid:90) π φ ( x c + r g cos ξ, y c + r g sin ξ, dξ π . (1)This potential gyro-average about the guiding-center isapproximately what determines particle trapping. Anorbit is (provisionally) trapped if − q ¯ φ > mv (cid:107) / z = 0.A second gyro-average is required to derive the particledensity (which is what determines the charge density inPoisson’s equation) from the guiding-center density. Atany position x , contributions to the particle density arisefrom all guiding-center positions x c on a circle a perpen-dicular distance r g away. Consequently if the distribu-tion function of guiding-centers is f c ( x c , v ⊥ , v (cid:107) ), and wewrite the vector r g = (cos ξ, sin ξ, r g , the corresponding distribution of particles is f ( x , v ⊥ , v (cid:107) ) = (cid:90) π f c ( x c , v ⊥ , v (cid:107) ) dξ π = (cid:90) π f c ( x − r g , v ⊥ , v (cid:107) ) dξ π . (2)[If there is an electric field gradient ∇ ⊥ E ⊥ , and hencedrift gradient ∇ ⊥ v d , then if r g is taken constant, v ⊥ varies with gyroangle ξ in f . This higher order effectcan be ignored for present purposes.] A. Gyro-averaged potential
In standard gyrokinetic theory, the evaluation of thegyro-averages usually relies upon a Fourier transform ofthe perpendicular potential variation. When φ ( x ) = φ ( k x )e ik x x , its gyro-average is proportional to a Besselfunction¯ φ = φ ( k x ) (cid:90) e ik x r g cos ξ dξ π = J ( k x r g ) φ ( k x ) . (3)In gyrokinetic theory, there is a perturbation in theguiding-center distribution function ( f c ) that is propor-tional to the gyro-averaged potential. This perturbationarises from the integration along orbits of the Vlasovequation (from a distant/past equilibrium). To find theparticle distribution function, needed for current- andcharge-density (Poisson’s equation in the electrostaticlimit) then requires a second gyro-average, eq. (2), whichintroduces another J ( k x r g ) factor. The double gyro-average and integration over a Maxwellian v ⊥ distribu-tion then gives rise to a modified Bessel function formproportional to e − r gt I ( r gt ), where r gt is the thermal gy-roradius v t / Ω.For trapped orbits in an electron hole equilibrium,the distribution function cannot feasibly be derived froma distant/past thermal equilibrium. A trapped parti-cle deficit (relative to reference distribution) arises byhighly complicated non-linear processes leading up tothe hole formation, and is then “frozen” on trapped or-bits. It is therefore not appropriate to suppose that onecan calculate the trapped particle distribution function.It must be specified in a form constrained by plausi-bility and non-negativity, and be expressed as a func-tion of the constants of motion: total energy and (ap-proximately) magnetic moment; so that it is consistentwith an eventually steady Vlasov hole equilibrium. Inany case, for an electron hole the detailed transversevelocity dependence of the distribution is not a criti-cal part of our interest. It is mathematically conve-nient to adopt a separable Maxwellian v ⊥ form, so that f c = f c (cid:107) ( v (cid:107) ) . e − ( v ⊥ /v t ) / / πv t (where v t = (cid:112) T /m ).The plausibility of this separable form requires us tosuppose that there is not a strong cross-coupling be-tween parallel and perpendicular velocity that compro-mises separation. Since the gyro-averaging process meansthat parallel trapping does depend to some extent on v ⊥ as well as v (cid:107) , the separable form will be poorly justifiedunless the phase-space density deficit ˜ f c is small near thetrapped-passing boundary for the majority of the v ⊥ dis-tribution. Fortunately ˜ f c being negligible near W (cid:107) = 0 isone of the plausibility constraints we must enforce any-way, to account for orbit stochasticity there.With this caveat we can perform the integration over v ⊥ and gyro-angle ξ simultaneously as follows, for a singleguiding-center position.¯ φ ( x c ) = (cid:90) φ ( x c + r g )e − ( v ⊥ /v t ) / v ⊥ dv ⊥ dξ πv t = (cid:90) φ ( x c + r g )e − ( r g /r gt ) / d r g πr gt . (4)This shows that the simultaneous gyro-averaging andperpendicular velocity integration simply convolves thequantity of interest (potential in this case) with a two-dimensional (unit area) Gaussian transverse profile, ofwidth r gt ≡ v t / Ω. Thus, the effective trapped particleconfining potential is modified by finite gyroradius Gaus-sian smoothing in the transverse direction. This smooth-ing will have the effect of decreasing the height of theeffective potential peaks (regions of negative d φ/dr )and increasing the height of potential troughs (regionsof positive d φ/dr ). Trapping is reduced near the ori-gin and enhanced in the wings, for a domed potentialdistribution. B. Gyro-averaged density deficit
The hole potential is sustained by a deficit ˜ n in trappedelectron density near z = 0, which gives rise to positivecharge density there. With a similar caveat about separa-bility of the transverse velocity distribution function, thegyro-averaging of the trapped phase-space-density deficit˜ f in the back-transformation from ˜ f c gives rise to a den-sity deficit smoothed in transverse position in the sameway as the potential. That is,˜ n = (cid:90) ˜ n c ( x − r g )e − ( r g /r gt ) / d r g πr gt , (5)where ˜ n c ( x c ) is the deficit in the guiding-center density( ˜ f c integrated over velocity).Suppose we compare a situation of finite r g with ahigh magnetic field case: such that r g (cid:39) n ( r, φ ) and ˜ n c ( r, ¯ φ )were taken to be the same function [˜ n ( r, φ ) = ˜ n c ( r, φ )],except with different argument ( φ versus ¯ φ ), then for anygiven potential shape ψ ( r ), the particle density deficit ˜ n at finite gyro-radius would differ from ˜ n by two Gaussianconvolutions of this form. One represents the effect ofgyro-averaging of the potential, and the other the trans-formation from guiding-center density to particle density.Such ˜ n would not then be consistent with Poisson’s equa-tion. Instead, to keep the ψ ( r ) fixed, ˜ n c ( r, φ ) would have to be more peaked in r by two “deconvolutions” than˜ n ( r, φ ) so as to give ˜ n = ˜ n . Alternatively, if one insiststhat ˜ n and ˜ n c must be the same function, then the radialpotential variation ψ ( r ) of the two cases cannot be thesame but must have ψ ( r ) smoothed by two Gaussian con-volutions relative to ψ ( r ). Two Gaussian convolutionsof the same width r gt are equivalent to one convolutionof width √ r gt . III. VALIDATION BY DIRECT ORBIT INTEGRATION
Since our discussion has alerted us to the fact thata gyro-averaged treatment is only an approximation inrespect of trapping, it is worth exploring how good anapproximation it is and what phenomena compromise itsaccuracy. This has been done by performing full (6-D)orbit numerical integrations using an updated version ofthe code described in reference . The main upgradesconsist of ability to use any axisymmetric hole poten-tial profile provided from a different code in the form ofan input file, and the calculation and plotting of vari-ous gyro-averaged quantities. In the code and this andthe following section we work in normalized units : lengthin Debye-lengths λ D , time in inverse plasma frequencies1 /ω p and energies in T e /e , for which the electron chargeis then -1 and velocity units are v t = (cid:112) T e /m . The con-vention throughout is that the value of potential φ atposition z = 0 is written ψ [= φ ( r, ψ at r = 0 is ψ which is the potential at the origin.Thus ¯ ψ is the gyro-average of potential at z = 0, but ¯ φ isthe average over the orbit at the instantaeous position: φ ( r, z ).An example of an orbit is shown in Fig. 2. The pro-jection of the orbit on to the transverse plane shows thedrift in the axisymmetric θ direction of the circular orbitin this situation where the gyro-radius is a moderate frac-tion of the guiding-center radius and transverse potentialscale-length. Simultaneously the orbit is bouncing in theparallel direction, as shown by the three-dimensional per-spective. The shape of the potential in which this particleis moving is given by the contours of Fig. 1.In Fig. 3 is shown the variation of some other parame-ters of the orbit. The top panel shows the instantaneousradial position r ( z ) (red) and its single-gyro-period run-ning gyro-average ¯ r (green). The dashed line is the ini-tial radial position of the orbit’s guiding-center (a chosenparameter r c for this orbit). [Because there are manybounces of the orbit, individual lines are hard to discernexcept at large magnification.] Also shown are the initialparticle position r (square) and the positions of the nom-inal extrema of the orbit based on the initial gyro-radius,namely r c ± r g (triangle and circle). For this examplethe gyro-averaged radius ¯ r has very small excursions. Onaverage ¯ r slightly exceeds r c . The initial particle radiusexceeds even ¯ r , but is controlled by the initial gyro-phase(a chosen parameter), which in this case is that the initialvelocity is directed along the guiding-center radius from FIG. 2. Example of a trapped orbit (a) projected in twodimensions showing the axisymmetric direction drift, and (b)in three dimensions showing the parallel bounce motion.FIG. 3. Orbit evolution, showing the particle radius (red)and gyro-averaged radius (green) as a function of parallel po-sition z (top panel), and the parallel energy W (cid:107) (red) and itsgyro-average ¯ W (cid:107) (green) together with the potential at theguiding-center ( r c ) and minimum radius r min (second andthird panels). See text for detailed explanation. FIG. 4. Orbit evolution for a more shallowly trapped orbitthan Fig. 3, but otherwise the same parameters. the origin; so r = r c + r g .The second panel shows the instantaneous and run-ning gyro-averaged value of the parallel energy W (cid:107) = v (cid:107) / − φ ( r, z ). Because of the gyration up and downthe substantial radial potential gradient, the perpendic-ular kinetic energy W ⊥ = v ⊥ / W (cid:107) = W − W ⊥ varies by anequal and opposite amount, since W is exactly conserved.(In the code W is monitored and confirms accuracy to afraction level of ∼ − .) Even though W (cid:107) can instan-taneously exceed zero, the orbit remains trapped for inexcess of 200 parallel bounces. The gyro-averaged ¯ W (cid:107) shows some limited variation with z and gyro-angle, butremains below zero; so the orbit remains trapped. Thelowest panel plots, in addition to W (cid:107) , the potential en-ergy at r c and r min . One can observe that the z -extremaof the orbit are approximately where ¯ W (cid:107) = − φ ( r c ) and W (cid:107) min = − φ ( r min ) corresponding to zero parallel kineticenergy.Fig. 4 shows for comparison an orbit in exactly thesame potential (whose peak is ψ = 0 .
16) and mag-netic field strength (Ω = 0 . W = 1), but with a starting parallel energy that is closerto the trapped-passing boundary ( W (cid:107) = − . ψ ( r c )instead of − . ψ ( r c )). The orbit starts trapped, butonly just. Consequently its parallel excursions extendfarther out in | z | . Focusing (at high resolution) on thegreen line in the second panel ¯ W (cid:107) , one can see that thefirst bounce position (at positive z ) is at | z | (cid:39) . W (cid:107) (cid:39) − . × − . However each subsequent bounceoccurs at an increasing value of | z | because ¯ W (cid:107) increasestoward zero, until after 10 bounces it becomes positiveand the orbit escapes the hole, moving out to large posi-tive z . In the inner regions of the hole, ¯ W (cid:107) -variation withgyro-angle is very significant, but at the ends near thebounce the excursions become small, because the radialpotential gradient is small there, and it is naturally the FIG. 5. Poincar´e plot of the parallel energy and gyro-angleat passages through z = 0 for the potential form of Fig. 1.(a) for the magnetic field strength Ω = 0 . . value at the ends that determines whether the particlebounces or not.In Fig. 5(a) is shown a Poincar´e plot of the parallel en-ergy, W (cid:107) ( r c ), evaluated taking the potential to be thatat position r c (which is a proxy for the gyro-averaged po-tential ignoring ∂ φ/∂r requiring no averaging), againstthe gyro-angle for each passage of an orbit through z = 0.A sequence of starting W (cid:107) orbits is shown in a corre-sponding sequence of colors. The orbit of Fig. 3 cor-responds to W (cid:107) ( r c ) /ψ ( r c ) (cid:39) − .
5. At that energyall the orbits are permanently trapped and no obviousphase-space islands are observed. The ratio of the gyro-frequency Ω to twice the bounce frequency 2 ω b there is approximately 3.5, so it is not close to a bounce-cyclotronresonance ( (cid:96)ω b = Ω with (cid:96) an even integer). The nearestresonance is (cid:96) = 8 at W (cid:107) ( r c ) /ψ ( r c ) (cid:39) − .
35 where anisland is obvious. By contrast, the orbit of Fig. 4 cor-responds to a starting energy W (cid:107) ( r c ) /ψ ( r c ) (cid:39) − . (cid:96) = 12 and 14 reso-nances. Actually the progressive gyro-averaged energyincrease observed in Fig. 4 does not require the orbitliterally to be stochastic in order to become detrapped.But nevertheless detrapping like this is observed to takeplace mostly in proximity to parameters that have over-lapped islands and stochastic orbits. Regions of detrap-ping (which are also, by time reversal, regions of trap-ping) and stochasticity experience strong effective energydiffusion which will limit any gradients in f (cid:107) and hencesuppress ˜ f . In this case, a distribution in which ˜ f (cid:39) W (cid:107) ( r c ) /ψ ( r c ) (cid:39) − .
25 would be plausible, butnot one that called for substantially non-zero ˜ f closer to W (cid:107) = 0.Fig. 5(b) shows what happens if the magnetic fieldstrength is lowered but with all else the same. Thegyro-resonance energy is changed so that the (cid:96) = 4 is-land centered on W (cid:107) ( r c ) /ψ ( r c ) (cid:39) − . W (cid:107) ( r c ) /ψ ( r c ) > − .
26 are lost immediately withouteven a single bounce, because the gyro-averaged poten-tial happens to rise above zero in the first quarter of thebounce period for all such orbits with the standard start-ing gyro-phase. At different starting gyro-phase one canfind higher energy orbits with a few (up to 10) bouncesprior to loss, but the overall lost region (whether afterzero or a few bounces) of phase-space is essentially thesame. High enough magnetic field avoids such strongresonances.Further trends with guiding-center position are illus-trated in Fig. 6, where different guiding-center position( r c ) cases are shown for the same hole parameters as Fig.5(b). Fig. 6(a) is for a larger guiding-center radial po-sition r c = 8 in the exponentially decaying radial wingof the hole where the peak potential ψ is substantiallysmaller. It has considerably less of its phase-space de-trapped than Fig. 5(b). Fig. 6(b) shows instead a casewhere r c < r g so that the gyro-orbit itself encircles theorigin ( r = 0). The resulting excursion of the gyro-phase ξ (which is defined as the angle between the perpendicu-lar velocity and the particle radius vector from the origin)is now less than 2 π . It also, though, shows a region ofdetrapping that is comparable to Fig. 6(a) and shallowerthan 5(b). Thus the intermediate radii where r c lies inthe steep radial gradient of the potential, exemplified by5(b), are the most susceptible to stochastic detrapping. FIG. 6. Poincar´e plots at different guiding-center radii, andthe lower magnetic field.
The extreme limit of the origin-encircling orbit type iswhen r c = 0, and the orbit becomes a circle centeredon the origin. There are then no resonant effects and nodetrapping because the orbit phase never changes.What Figs. 4 and 5 validate is that combining the gyro-averaged potential energy ( − ¯ φ ) plus the parallel kineticenergy v (cid:107) / W (cid:107) whose sign rather accu-rately determines whether or not an orbit bounces; andthat, although this quantity has significant excursions asthe orbit moves through the low- | z | strong potential re-gions, those do not generally lead to detrapping unlessthere is systematic or stochastic enhancement of the ¯ W (cid:107) at the bounce positions due to bounce-gyro resonance.This finding is summarized by results for a large num-ber of orbits in Fig. 7. It shows as a function of guiding- FIG. 7. Highest permanently trapped orbit initial parallelenergy ¯ W (cid:107) (colors) divided by gyro-averaged potential ¯ ψ , as afunction of guiding-center radius for different magnetic fieldsindicated by the length of the thermal gyro-radius and thevalue of Ω /ψ / . Also radial profile of ψ and ¯ ψ (black). Holeorigin potential: (a) ψ = 0 .
36, (b) ψ = 0 . center radial position, by colored lines the maximumvalue of the ratio of initial parallel energy ¯ W to gyro-averaged potential ¯ ψ (both at z = 0) for which the orbitis permanently trapped. Each plot shows cases for fivedifferent magnetic field strengths. That strength is indi-cated by the length of the thermal gyro-radius horizontalbars at the bottom left. It is also given by the labels con-sisting of the value of Ω /ψ / , which is a proxy for Ω /ω b at the hole origin and organizes the results. The radialprofile of the peak potential ψ ( r ) and the correspondinggyro-averaged potential ¯ ψ (to which ¯ W is normalized inthe colored lines) is also plotted in black with the scaleon the right. The different line styles correspond to thedifferent gyro-radii. Plots have potential at the origin (a) ψ = 0 .
36 and (b) ψ = 0 .
04, differing by a factor of nine.Yet generally, within the uncertainty implied by the fluc-tuations of the curves, they tell the same story. At highmagnetic field, the ratio ¯ W (cid:107) / ¯ ψ is close to zero, indicat-ing that the gyro-averaged potential determines paralleltrapping over the entire radial profile for all gyro-radiiand ψ : a key result. This limit is approached whenΩ /ψ / (cid:38)
2. For lower magnetic fields, detrapping bygyro-bounce resonance begins at significantly lower ¯ W (cid:107) ,strongest in the central regions of largest E ⊥ ( − dψ/dr ),but eventually deep detrapping occurs across most of thehole radius. Smaller amplitude ( ψ ) holes allow largergyro-radius to be accommodated before deep detrappingsets in: Fig. 7(b). FIG. 8. Highest permanently trapped orbit initial parallelenergy ¯ W (cid:107) divided by gyro-averaged potential ¯ ψ , as in Fig. 7except with a more peaked radial profile extending to muchlarger radius. Fig. 8 shows a much wider but more peaked radialprofile case. Detrapping is now strongest relatively closeto the potential peak, which is where the radial field isstrongest. A threshold below which gyro-bounce reso-nance causes major detrapping is still present. It is quan- titatively at somewhat lower ratio Ω / √ ψ ∼
1, in partbecause of the E r reduction.The ¯ ψ curves of Figs. 7 and 8 illustrate how gyro-averaging is changing the effective potential, lowering itnear r = 0, and raising it in the positive curvature wings. IV. THE CONSEQUENCES OF GYRO-AVERAGINGAND RESONANCE DETRAPPING
The key consequence of gyro-averaging is that trappeddensity deficits become less effective at sustaining thehole potential. That reduced effectiveness requiresdeeper deficits in order to sustain the same potential,leading eventually to a violation of non-negativity of thedistribution function. It is easiest and most relevant tothink of this in terms of the convolution spreading of thepotential profile given ˜ n ( r ).A Gaussian radial potential profile ψ ( r ) = ψ e − ( r/a ) / provides the simplest illustration of what gyro-averagingdoes. When convolved with the Gaussian of width √ r gt it yields a modified finite-gyro-radius Gaussian profile¯ ψ ( r ) of greater width (cid:113) a + 2 r gt . Of course the totalvolume (cid:82) ∞ ψ ( r )2 πrdr is conserved during convolution,consequently the height of the smoothed ¯ ψ is smaller bythe ratio ¯ ψ /ψ = (1 + 2 r gt /a ) − . This factor gives astrong reduction of the potential height at the origin once2 r gt exceeds a . The same spreading occurs for any shapeof profile ψ ( r ), when its width is considered to be givenby the variance (cid:104) r (cid:105) = (cid:82) ∞ r ψ ( r )2 πrdr/ (cid:82) ∞ ψ ( r )2 πrdr .For a Gaussian of width a , (cid:104) r (cid:105) = 2 a . For a general pro-file, when it is convolved with a Gaussian of width √ r gt ,its variance increases as (cid:104) ¯ r (cid:105) = (cid:104) r (cid:105) +4 r gt . The differenceis that for a non-Gaussian initial profile its relative shapechanges (toward Gaussian) when convolved, and so theratio of central heights ¯ ψ /ψ is not exactly the same,even though the ratio of the average heights is the same( (cid:82) ¯ ψ πrdr/ (cid:104) ¯ r (cid:105) ) / ( (cid:82) ψ πrdr/ (cid:104) r (cid:105) ) = (1+2 r gt /a ) − . Weshall take the average height suppression to represent themain effect. This amounts to supposing (correctly) thatthe inner regions of the hole are the most important.The relative importance of the two main multidimen-sional electron hole modifications, gyro-averaging andtransverse electric field divergence, can be estimated bywriting the field divergence ∇ φ ∼ − ( a − + L − (cid:107) ) φ . Thepotential is thus suppressed by divergence in a hole oftransverse dimension ∼ a relative to a one-dimensionalhole with the same charge density by a factor F − ⊥ ∼ (1 + L (cid:107) /a ) − where L (cid:107) is a parallel scale length ∼ λ D (1in normalized units). The gyro-average potential spread-ing factor (1 + 2 r gt /a ) − thus becomes more impor-tant than the effects of transverse electric field divergencewhen r g (cid:38) λ D that is ω p (cid:38) Ω or Ω (cid:46) ∼ ψ / , a hole depth ψ much smaller than 1 (times T e ) will be more strongly sup-pressed by gyro-averaging than by transverse divergence,near the gyro-bounce threshold. This point is made morequantitative in the following. A. Relationship between electron distribution deficit andpotential
Poisson’s equation relates the potential and the elec-tron distribution (with the ions as an immobile unitybackground charge density). Integrating it along the par-allel direction requires that0 = 12 (cid:18) ∂φ∂z (cid:19) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) z =0 = (cid:90) ∞ ( n ( φ ) − − ∇ ⊥ φ ) ∂φ∂z dz ≈ − ˜ V + (1 − (cid:104)∇ ⊥ (cid:105) ) ψ / , (6)where we have written − ˜ V ≡ (cid:82) ˜ ndφ , (cid:104)∇ ⊥ (cid:105) ≡ (cid:82) ∇ ⊥ φdφ/ψ , and n ( φ ) = n f + ˜ n is the sum of a refer-ence distribution that is independent of parallel velocityin the trapped region and ˜ n is the deficit of the trappedelectron density with respect to it. The final equalityof eq. (6) approximates n f (cid:39) φ and approximatesthe potential form as separable. ∇ ⊥ φ/φ is then a func-tion only of r and equal to (cid:104)∇ ⊥ (cid:105) which can be takenout of the z -integral; for non-separable potential it mustremain in integral form. Since ˜ n = (cid:82) √ φ ˜ f dv , the classi-cal potential ˜ V scales like ∼ − ˜ f ψ / , and eq. (6) means − ˜ f ∼ (1 − (cid:104)∇ ⊥ (cid:105) ) ψ / . This scaling is universal.In the “power deficit model”, worked out in detailelsewhere , the trapped parallel distribution deficit ˜ f is presumed to be of the specific form˜ f ( W (cid:107) ) = ˜ f ψ (cid:18) W j − W (cid:107) W j + φ (cid:19) α (7)varying between zero at parallel energy W (cid:107) = v (cid:107) / − φ = W j and ˜ f ψ at parallel energy W (cid:107) = − ψ (the bottom ofthe potential well). Fractional powers of negative quan-tities are by convention taken to be zero, so there is zerodeficit for energies W (cid:107) > W j , i.e. near the trapped-passing boundary W (cid:107) = 0. In the absence of gyro-averaging, the condition (6) on the classical potential V = (cid:82) ρ dφdz dz for the power deficit model has analyticcoefficients and becomes: − ˜ f ψ ψ / (1 + W j /ψ ) / Gα + 3 / − F ⊥ V f ( ψ ) (cid:39) F ⊥ ψ / . (8)Here G = (cid:112) π/ α +1) / Γ( α +3 /
2) is a constant depend-ing only on α , expressible in terms of the Gamma func-tion, V f ( φ ) is the classical potential of the flat-trappedscreening density n f (cid:39) φ minus the backgrounduniform ion density, and F ⊥ is the local correction fac-tor F ⊥ ( r ) = (1 − (cid:104)∇ ⊥ (cid:105) ) that accounts for perpendicu-lar electric field divergence in Poisson’s equation. Now F ⊥ and V f ( φ ) (cid:39) − φ / . But the left hand side term ˜ f ψ ( ψ + W j ) / represents the trapped electron guiding-center distribu-tion deficit integrated with respect to ¯ φ , and must ac-count for gyro-averaging. So in the presence of finitegyro radius it must be taken as ˜ f c ¯ ψ ( ¯ ψ + ¯ W j ) / , and onaverage ¯ ψ/ψ (cid:39) (1 + 4 r gt / (cid:104) r (cid:105) ) − . We will suppose that¯ W j / ¯ ψ = W j /ψ is independent of gyro-averaging. If in-stead ¯ W j = W j were kept constant as ψ was averaged, theconstraints about to be discussed would be made moresevere.To compensate for the reduction in trapped energyrange caused by ¯ φ averaging reduction, if we wished topreserve ψ ( r ) while increasing r gt , the condition (8) thenrequires a greater trapped guiding-center deficit˜ f c ¯ ψ = ˜ f ψ (1 + 4 r gt / (cid:104) r (cid:105) ) / . (9)The resulting cubic dependence of | ˜ f c ¯ ψ | on r gt , if r gt / (cid:104) r (cid:105) is allowed to increase significantly beyond unity by low-ering the magnetic field strength, will soon cause thetrapped electron deficit to exceed the allowed maximumvalue (0.399) permitted by non-negativity of the to-tal guiding-center distribution function. Non-negativitytherefore constrains (cid:104) r (cid:105) not to be very much smallerthan r gt . An alternative way to accommodate gyro-averaging increase might be to decrease ψ , thereby re-ducing the required | ˜ f c ¯ ψ | . However, if the magnitudeof | ˜ f c ¯ ψ | is fixed by non-negativity, then eq. (9) requires ψ ∼ (1+4 r gt / (cid:104) r (cid:105) ) − , an even stronger (sixth power) de-pendence on gyro-radius, which would rapidly force thehole potential to become negligible. We can concludethat electron hole equilibria exist in the presence of gyro-averaging only if increase of gyro-radius is accompaniedby increase of transverse potential extent approximatelykeeping pace with gyro-radius . Or expressed more briefly:4 r gt (cid:46) (cid:104) r (cid:105) . Although we have demonstrated this effectfor a particular model of the parallel distribution, thescalings, if not the precise coefficients, of all the phe-nomena are virtually independent of the model or thetransverse profiles.It should be remarked that the empirical scaling ofFranz et al, L ⊥ /L (cid:107) ∼ (cid:112) ω p / Ω) , is in the presentterminology (cid:104) r (cid:105) = L (cid:107) (1 + 1 / Ω ) = L (cid:107) (1 + r gt ), and tak-ing L (cid:107) = 2 is indistinguishable (within their significantobservational uncertainty) from (cid:104) r (cid:105) = 4 r gt . B. Allowable electron hole parameter space
Let us write equation (8) at the origin (the most de-manding place) with the maximum allowable − ˜ f c ¯ ψ =1 / √ π to avoid a negative distribution function. Sincewe are focussing on r = 0, take the transverse potentialshape there to be approximately Gaussian ∝ e − ( r/a ) / (cid:39) − ( r/a ) / a = (cid:104) r (cid:105) / / √ (cid:104)∇ ⊥ (cid:105) = − /a , so the divergence and gyro-averaging factors are0 FIG. 9. The limits of allowable peak potential permitted bynon-negativity of f (colors) and avoidance of gyro-bounce res-onant detrapping (black). Non-negativity depends on the ra-dial width a = ( −(cid:104)∇ ⊥ (cid:105) / − / . It also varies dependingon the parallel-peaking coefficient α ; curves are highest for awaterbag deficit α = 0. The range is indicated by the verti-cal bar. The approximate uncertainty range of the resonanceloss is indicated similarly. The parameters must lie belowboth relevant lines. (1 + 2 /a )[1 + 2 / (Ω a )] / . Then ψ / = (1 + W j /ψ ) / (1 + 2 /a )[1 + 2 / (Ω a )] / G (cid:112) π/ α + 3 / . (10)Now at the maximum permissible value of (1+ W j /ψ ), thecoefficient (1 + W j /ψ ) / G/ ( a + 3 /
2) is approximately2.4 at α = 0 and falls to about 1.2 at α = 1 becauseof the z -peaking of the profile. If we adopt the α = 0value (the most forgiving) and write 2 . / (cid:112) π/ (cid:39) ψ (cid:39) /a ) − [1 + 2 / (Ω a ) ] − . (11)This gives the maximum of ψ versus Ω for some chosen a . Lines below which ψ must lie are shown for a range ofwidths a in Fig. 9 For α = 1 the lines shift downward by afactor ∼ a =1. Permissible equilibria lie below the lines. Obviouslypoints for which − ˜ f ψ is actually less than 1 / √ π , so thetotal trapped distribution is greater than zero, lie belowthe line by an additional factor, which is − ˜ f ψ π .The other major constraint on equilibrium is the de-trapping of electrons by gyro-bounce resonance. Basedon the prior studies of exponential potential gradients , ψ must lie below ψ ∼ Ω / ψ ∼ Ω . These limit lines give the typical range of res-onance loss thresholds, shown by two black straight linesin Fig. 9. The strong detrapping effects take place notat r = 0 but in the steep region of the potential profile;even so, they prevent a steady equilibrium shape having ψ above the threshole, and actually tend to shrink thetransverse width, making the losses worse and tending tocollapse the hole entirely. V. DISCUSSION
The theory described here has addressed the threemain effects of finite transverse extent on electron holeequilibria (1) transverse electric field divergence, (3) or-bit detrapping by gyro-bounce resonance, and (4) gyro-averaging effects on the potential and density deficit.[Supposed modification of the shielding, enumerated (2)in the introduction, does not occur.] It has explored asampling of different self-consistent transverse shapes.In addition to the plausibility constraint that the dis-tribution deficit ˜ f should be zero at and immediatelybelow the trapped-passing boundary, the two criticalphysics constraints of non-negativity and avoidance ofresonant detrapping have been applied to obtain boundsrelating the peak potential ψ , magnetic field strength Ω(equivalent to the inverse of the thermal gyro-radius r gt ),and the transverse extent a (= (cid:112) (cid:104) r (cid:105) / magnitude , not direction, of its gradi-ent. Since in the wings Debye shielding gives rise to |∇ ⊥ φ/φ | (cid:39) E ⊥ is weaker than its dependence on Ω / √ ψ until E ⊥ be-comes very small. Overall, since Fig. 9 is independentof hole geometry apart from the parameter a , whose in-verse equals the square root of the normalized Laplacian( (cid:112) ∇ ⊥ φ/φ ) at the origin, it may be expected to applyto non-axisymmetric electron holes to approximately thesame degree that it applies to axisymmetric ones.It should be recognized that while non-negativity andPoisson’s equation are instantaneous requirements, res-onant detrapping takes a significant time to depletethe trapped deficit: typically some moderate number of1bounce-times, and certainly at least ∼ the gyro-period.Therefore, if a multidimensional electron hole is formedapproximately in a bounce-time or faster, for exampleby a highly nonlinear bump-on tail instability or somesubsequent process such as transverse break-up of an ini-tially one-dimensional hole, then it takes a longer timefor the resonant detrapping to become important. Itmight then be possible to observe multidimensional elec-tron holes shortly after their formation that violate theresonant detrapping criterion. Indeed, avoiding all res-onance loss is such a severe requirement that it pre-vents the long term sustainment of multidimensional elec-tron holes of any significant potential amplitude (e.g. ψ > − T e /e ) below magnetic field strength corre-sponding to Ω /ω p ∼ .
1. The present results thus suggestthat any solitary potential structures observed in spaceplasmas where Ω /ω p (cid:46) . f is not a functionof energy. It is also much less likely that such an ob-ject would be observed, because of its short life. If thenthe resonance loss constraint (black curve) was violatedfor some moderate time duration, but the non-negativityconstraint was not, and became the determining factor,then the simplified theoretical scaling (cid:104) r (cid:105) (cid:38) r gt wouldbe expected. Equality in this equation is indistinguish-able from the empirical scaling of Franz et al . Never-theless the experimental proxy they used for aspect ratio( E (cid:107) /E ⊥ ) has been shown to be an extremely uncertainmeasure of L ⊥ /L (cid:107) ; so one should not make too much ofthis agreement with observations. A great deal remainsto be done to establish observationally what the struc-ture of multidimensional holes in space actually is. Onecan look forward to a critical comparison of the theorypresented here with future observations. Acknowledgements and Supporting Material
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