SSynthetic Multidimensional Plasma Electron Hole Equilibria
I H Hutchinson a) Plasma Science and Fusion CenterMassachusetts Institute of TechnologyCambridge, MA 02139, USA
Methods for constructing synthetic multidimensional electron hole equilibria without using particle simulationare investigated. Previous approaches have various limitations and approximations that make them unsuitablewithin the context of expected velocity diffusion near the trapped-passing boundary. An adjustable modelof the distribution function is introduced that avoids unphysical singularities there, and yet is sufficientlytractable analytically to enable prescription of the potential spatial profiles. It is shown why simple models ofthe charge density as being a function only of potential cannot give solitary multidimensional electron holes,in contradiction of prior suppositions. Fully self-consistent axisymmetric electron holes in the drift-kineticlimit of electron motion (negligible gyro-radius) are constructed and their properties relevant to observationalinterpretation and finite-gyro-radius theory are discussed.
I. INTRODUCTION
Active space plasma regions are observed to containlong-lived solitary positive potential peaks whose spatialextent is a few Debye lengths; they are mostly identi-fied as electron holes, having an electron charge deficiton trapped orbits . It has been known for a long timethat non-zero magnetic field is necessary for the sustain-ment of electron holes; if it is strong enough, the electronmotion and trapping becomes one-dimensional. One-dimensional dynamic treatments have predominated pasthole theory, where the gyro-radius r L is neglected relativeto the transverse scale length L ⊥ . Although the presenttheory continues to calculate using only parallel particledynamics, it takes account qualitatively of one recentlydiscovered important effect of the transverse electric fieldin a multidimensional electron hole, namely the reso-nant interaction of the trapped particle bounce motionwith the gyro-motion. This interaction essentially alwaysinduces a region of stochastic orbits near zero parallelenergy: the trapped-passing boundary of phase space.And the energy-depth of this stochastic layer increaseswith r L /L ⊥ . The anticipated result in the stochasticlayer is a large effective phase-space diffusion rate whichforces the distribution function (phase-space density) tobe approximately independent of energy in that region.It has also been shown recently , in contradiction ofa longstanding suggestion, that regardless of the rela-tive strength of the magnetic field, the screening of thetrapped electron deficit charge is isotropic: having ap-proximately Boltzmann dependence. This discredits onespeculation concerning how the transverse scale of elec-tron holes relates to magnetic field strength, and reem-phasizes the need to understand better multidimensionalelectron holes, especially since recent multi-satellite mea-surements are giving unprecedented information aboutmultidimensional holes in space plasmas.The purpose of the present work is to develop a ver- a) Electronic mail: [email protected]
FIG. 1. An oblate electron hole equilibrium, axisymmetricabout the magnetic field direction z , showing color contoursof the trapped electron density deficit ˜ n , and line contours ofthe self-consistent potential φ . satile model of self-consistent multidimensional electronhole equilibria that conform to the physical constraint ofhaving zero distribution slope at and for a controllabledepth inside the trapped-passing boundary. Because ofthe difficulties of taking fully into account finite gyro-radius dynamic effects, the model simply prescribes theparallel velocity distribution as a function of transverseposition r , and potential φ . This exactly represents theelectron dynamics only for the limit of high magneticfield (one-dimensional motion) where the gyro-radius issmall. However, even this first step has not previouslybeen achieved taking properly into account the full Pois-son equation and resulting non-separable form of its solu-tion for the potential. The present results provide phys-ically self-consistent potential profiles in which the con-sequences of transverse dynamics, and especially gyro-averaging, will be explored in a separate publication. Anillustrative equilibrium found by the present approach(explained fully later) is shown in Fig. 1.After a review of prior analysis of multidimensionalelectron hole equilibria (section 2), a model distributionfunction with shape controlled by adjustable parametersis introduced in section 3. It avoids the unphysical prop-erties of most previous choices, and from it explicit an-alytic expressions for the charge deficit and (for some a r X i v : . [ phy s i c s . p l a s m - ph ] J a n parameters) the one-dimensional potential profile can bederived. It is generalized to separable multidimensionalpotential forms. In section 4 is explained the subtlety ofhow to relax the potential iteratively to the non-separableform it always must take. It becomes clear that no soli-tary multidimensional equilibrium can in fact exist whenthe charge density is a function only of potential. It mustbe also an explicit function of transverse position. Iter-ative relaxation with effectively constant trapped chargedeficit embodies sufficient of the physical stabilizing dy-namics to produce a convergent scheme. And examplesof hole shapes are examined with particular emphasis onthe consequences for satellite observations.For convenience throughout this paper normalizedunits are used: Debye-length λ D for length, inverseplasma frequency 1 /ω pe for time, and background tem-perature T e ∞ for energy; with the result that velocityunits are (cid:112) T e ∞ /m e . Densities are normalized to the dis-tant background electron density n e ∞ . In steady state,the total energy W = v / − φ (normalized units) of anelectron is conserved, and neglecting collisions, the to-tal derivative of the distribution function df ( v ) /dt alongparticle orbits is zero. Ions are taken to be a uniformimmobile neutralizing background. II. TWO DIMENSIONAL HOLE EQUILIBRIA
Electron holes are a type of BGK equilibrium inwhich a deficit of the electron velocity distribution ontrapped orbits sustains the positive potential that trapsthem. For decades electron hole analysis was almost allone-dimensional, along the magnetic field and ignoredor highly simplified any transverse spatial dependence.This one-dimensional prior theory has been reviewed ex-tensively elsewhere and will not be discussed in de-tail here, but a brief review of the relatively few multi-dimensional electron hole analytic equilibrium studies isappropriate.Multidimensional electrostatic hole model studies werepublished for several years following 2000, motivatedby new observational data documenting finite transversehole extent in space plasmas. The model equilibria basedtheir particle dynamics on the one-dimensional Vlasovequation plus drift in a uniform magnetic field, but be-gan to account for transverse potential variation, gener-ally simplifying the problem to axisymmetric cases (in-dependent of θ in cylindrical coordinates, which is ourfocus here too).Chen and Parks specified the potential to have aseparable axisymmetric form and a constant value of ∇ ⊥ φ/φ , giving φ = φ (cid:107) ( z ) J ( l r/r s ) ( the Bessel func-tion J has its first zero at argument l ). The potentialis thus zero, but has finite ∇ ⊥ , at r = r s . This repre-sents a “waveguide” configuration, effectively equivalentto the (1979) analysis of Schamel , rather than an iso-lated hole far from boundaries. The convenience of theansatz is that the Laplacian in Poisson’s equation simply acquires from transverse divergence an extra term k ⊥ φ where k ⊥ = ( l /r s ) is independent of r . This term isreadily included into an expression for charge density inPoisson’s equation. With potential specified in this way,and an assumed Maxwellian untrapped distribution, theauthors solve the integral equation for the trapped par-allel distribution function f (cid:107) ( W (cid:107) ) (for parallel energy in-cluding potential W (cid:107) < f (cid:107) must be non-negative, which constrainsthe hole parallel length in a way that depends on theperpendicular scale r s .Later, Chen, Thouless, and Tang analyzed insteadGaussian potential shapes φ ∝ exp( − z /δ z − r /δ ⊥ ) (stillseparable but far from boundaries), showing that the re-quired trapped density as a function of potential, andtrapped distribution as a function of energy, can be de-rived in closed form for Maxwellian background distribu-tion. They also included the static response of a repelled(ion) Maxwellian species of different temperature.Muschietti et al. prescribed a separable potential φ ∝ [1 + η cosh( βz )] − exp[ − ( r/δ ⊥ ) ], which allows par-allel elongation of the hole by flattening its top via theparameter η , and prescribing its asymptotic scale-length1 /β at large z . Its perpendicular variation is Gaussian,which means that writing ∇ ⊥ φ ( z ) = k ⊥ φ , implies k ⊥ varies with r . By making a mathematically convenient(non-Maxwellian) choice of the external parallel distribu-tion, they found a closed analytic form for the trappeddistribution, satisfying the parallel integral equation self-consistently as a function of energy and radius r . Thiscalculation (like those of Chen et al) is carried out inthe drift kinetic approximation, assuming the ordering ω θ (cid:28) ω b (cid:28) Ω e where ω θ is the azimuthal drift frequencyof the gyrocenter about the axis, ω b is the parallel bouncefrequency, and Ω e the cyclotron frequency. It also ignoresthe distinction between guiding-center density and parti-cle density, effectively neglecting the gyro radius relativeto the perpendicular scale length. The work explores onlylimited transverse extent r/δ ⊥ ≤ . W (cid:107) is zero, f (cid:107) ( W (cid:107) ) − f (cid:107) (0) ∝ ( − W (cid:107) /ψ ) / , whichMuschietti et al illustrate. And they all use separablepotential form.Jovanovi´c et al. approached the problem instead bythe “differential equation” route, specifying the trappedparallel distribution function to be of the Schamel form( ∝ exp( βW (cid:107) /T )), which for small potential gives a den-sity difference from pure Debye shielding (i.e. n e /n =1 + φ ) proportional to − φ / , and no f (cid:107) -slope singu-larity. They supposed that the Poisson equation wasmodified by anisotropic dielectric shielding, becoming ef-fectively {∇ (cid:107) + [1 + ( ω p / Ω e ) ∇ ⊥ ] } φ = aφ − (4 b/ φ / .This supposed modification of Poisson’s equation is erro-neous when applied to electron holes, as has been shownelsewhere .Krasovsky, Matsumoto, and Omura show that in aplasma with (Debye) shielding length λ D , a sphericallysymmetric hole of Gaussian form φ = ψ exp( − R /λ )is self-consistent with a trapped parallel distribu-tion function deficit ˜ f (cid:107) = f (cid:107) trapped ( W (cid:107) ) − f (cid:107) (0) = − (2 | W (cid:107) | /π ) / [1 /λ D + (2 /λ )(1 + 2 ln(4 | W (cid:107) | /ψ )], gov-erned by the Vlasov equation. For mathematical con-venience, they invoke instead the presumption that thetrapped particle density (deficit), like the shielding den-sity, depends linearly on the potential, which is satis-fied by having ˜ f (cid:107) ∝ (cid:112) W j − W (cid:107) for W (cid:107) /W j > W (cid:107) /W j <
1, where W j = − φ j is some maxi-mum trapped energy ( W j < ∇ + k )( φ − const. ) = 0 except with the con-stant k = − dn/dφ different (in magnitude and sign) inthe two regions W (cid:107) /W j ≶
1. The potential in the innerregion can be expressed in terms of a sum of known har-monic solutions, by specifying the position in r and z ofthe potential contour at the join φ = φ j .In Krasovsky, Matsumoto, and Omura these au-thors venture beyond the pure drift approximation, not-ing that a potential of the additive form φ = ψ × [1 − ( z/L ) − ( r/R ) ] gives rise to integrable equations of mo-tion. Although they do not find an equilibrium, theynote that the resulting (2-D) Vlasov equation is satis-fied by f ( v ) being an arbitrary function of the energy-like constants of the motion, w (cid:107) = v z / z/L ) ψ , and w ⊥ = v r / r/R ) ψ + ( p θ /r + Br/ /
2, where p θ isthe conserved canonical angular momentum about theaxis of symmetry in the magnetic field B , and the totalenergy is actually W = w ⊥ + w (cid:107) − ψ . They concen-trate on the density at r = 0, z = 0, noting that or-bits that pass through the origin have p θ = 0, and aretrapped only if they never reach φ = 0, which is true if W (cid:107) + 1 / [1 + ( BR ) / ψ ] W ⊥ < ψ . This inequality definesthe interior of a trapped ellipse in velocity space at theorigin, and thereby limits the maximum possible chargedeficit contributable by trapped particles. To make thatexceed the passing particle density perturbation (whichat a minimum it must), when the field is weak B (cid:46) ψ (cid:28) B , they show requires ( r L /R ) √ ψ (cid:46)
1. Thus,a maximum thermal gyro radius r Le of order the hole’sradial scale length ( R ) times ψ − / is permitted. A sim-ilar criterion is derived by Krasovsky et al (2006).After 2006 I am aware of no published analytic assaultson the multi-dimensional hole equilibrium problem un-til Hutchinson showed that when transverse potentialgradients exist there is always a region of stochastic or-bits caused by bounce-cyclotron resonance near W (cid:107) = 0.The energy depth of the stochastic region is found as afunction of the peak potential ψ , E ⊥ , and the magneticfield strength B . It deepens rapidly when B/ √ ψ (cid:46) E ⊥ is significant, eventually extending to the fullhole depth. The presence of this stochastic region willprevent the formation of any strong energy gradients of f (cid:107) near W (cid:107) = 0, ruling out any distributions that donot have an approxmately flat region of f (cid:107) there. As has been noted before , this forces a requirement thatthe hole potential fall ∝ exp( − z/λ D ) at large z, but italso constrains φ ∝ exp( − r/λ D ) at large r . Thus, ex-act Gaussian potential forms, parallel or perpendicular,cannot be physical in the hole wings.These summaries motivate the key emphases of thepresent work: insisting upon physically plausible distri-bution dependence on energy, and abandoning the conve-nient but unphysical assumption that the potential formis separable. Observing these principles we constructtruly physical multidimensional electron hole equilibria,based for the first time on synthetic analysis rather thanparticle-in-cell simulation. III. POWER TRAPPED DISTRIBUTION DEFICITMODEL
Consider a solitary potential peak which in its ownframe of reference is time-independent. The distributionfunction satisfying the (parallel) Vlasov equation is con-stant on particle orbits. Since the potential is steady,energy is also constant on orbits, and in the drift ap-proximation in a uniform magnetic field the distributionis a function of parallel ( W (cid:107) = v (cid:107) / − φ ) and total ( W )energy, with those energies conserved.However, for orbits that are weakly trapped, conserva-tion of magnetic moment (and hence of W (cid:107) separatelyfrom W = W (cid:107) + W ⊥ ) begins to break down becauseof bounce-gyro resonance , and resulting velocity spacediffusion suppresses the difference ˜ f = f t − f ∞ (0) be-tween the trapped distribution and a flat distributionhaving the separatrix value f ∞ (0). We therefore adoptinitially the “differential equation” ( ˜ f specified) approachto constructing a self consistent equilibrium solution. Wesuppose, further, that there is a bounding trapped energy W j , lying between − ψ and 0, such that for parallel energy W (cid:107) > W j , ˜ f is zero. For the more deeply trapped regionwhere W (cid:107) < W j , ˜ f is taken proportional to the energydifference raised to a chosen power α : ˜ f ∝ ( W j − W (cid:107) ) α .This form is convenient because it is continuous andanalytically tractable, yet enables monotonic shapes of˜ f ( W (cid:107) ) to be represented from uniform waterbag ( α = 0),through rounded α ∼ / α = 1, and be-yond, peaked at the maximum trapping depth. It is herecalled the power deficit model. We take no account ofthe perpendicular velocity distribution and so f means f (cid:107) and we drop the parallel suffix for brevity henceforth.Fig. 2 illustrates some of the distribution function shapesthat can be prescribed by the power deficit ˜ f model. A. One-dimensional
If the overall magnitude of ˜ f is represented by its value˜ f φ at parallel energy − φ for some specific φ > − W j (i.e.in the non-zero region), then its value at all other energies FIG. 2. Illustration of shapes of the distribution function inthe trapped region of velocity space at potential ψ , for thepower deficit model. ( W (cid:107) < W j ) is˜ f ( W (cid:107) ) = (cid:18) W j − W (cid:107) φ + W j (cid:19) α ˜ f φ = C ( W j − W (cid:107) ) α , (1)where C ≡ ˜ f φ / ( φ + W j ) α is a constant (negative because˜ f is negative). We adopt for brevity hereafter the conven-tion that energy combinations taken to some real powergive zero if they are negative. The density perturbationproduced by ˜ f at potential φ is given by the integral˜ n ( φ ) = (cid:90) √ φ ˜ f dv = (cid:90) W j − φ ˜ f φ (cid:18) W j − W (cid:107) φ + W j (cid:19) α √ dW (cid:107) (cid:112) φ + W (cid:107) = 2 ˜ f φ ( φ + W j ) / √ (cid:90) (1 − ζ ) α dζ = 2 CG ( φ + W j ) α +1 / (2)[using the substitution ζ ≡ (cid:112) ( φ + W (cid:107) ) / ( φ + W j ), andwriting G for the ζ -integral (which is a known functionof α ) G ≡ √ (cid:82) (1 − ζ ) α dζ = (cid:112) π/ α +1) / Γ( α +3 / n t , pass-ing electrons n p , and passing plus a flat distribution oftrapped electrons n f . The charge-density ρ = 1 − n p − n t = 1 − n f − ˜ n is a function of φ , so Poisson’s equationin one-dimension d φ/dz = − ρ can be integrated onceto give12 (cid:18) dφdz (cid:19) = (cid:90) ( n f −
1) + ˜ ndφ ≡ − V f − ˜ V = − V total . (3)For our power deficit model ˜ f we have − ˜ V = 2 CG ( φ + W j ) α +3 / / ( α + 3 / f density for a Maxwellian at small φ is n f = 1+ φ , giving (the “classical potential”) − V f = φ /
2, and ˜ n can be evaluated at larger φ or forother distributions, e.g. shifted Maxwellians, numerically.In any case, the implicit solution for the potential formis then z ( φ ) = (cid:82) dφ/ √− V total . The crucial condition atthe hole peak, z = 0, where dφ/dz = 0, is then V total = 0.The peak potential ψ = φ (0) is thus the solution of − ˜ f ψ G ( ψ + W j ) / α + 3 / − CG ( ψ + W j ) α +3 / α + 3 /
2= ˜ V ( ψ ) = − V f ( ψ ) . (4)For the linear n f approximation V f = − φ /
2, this equa-tion gives˜ f ψ /ψ / = − ( α + 3 / G W j /ψ ) / , (5)which shows that ˜ f ψ ∝ √ ψ with a constant of propor-tionality that depends only on the ˜ f shape parameters α and W j /ψ . Also for the linear n f approximation CG = − ( α + 3 / ψ / − α (1 + W j /ψ ) / α ; (6)so ˜ n ( φ ) = − ( α + 3 / ψ ( φ + W j ) / α ( ψ + W j ) / α . (7)There does not appear to be a closed form solutionfor the z ( φ ) integral for general α . However, as was byKrasovsky et al , there is when one adopts the linear n f approximation and α = 1 /
2, because the resultinglinear dependence of the density on φ turns Poisson’sequation into the Helmholtz equation for φ > − W j andthe Modified Helmholtz equation for φ < − W j , whichmatch at the join. d φdz = φ + 2 CG ( φ + W j )= − k ( φ − CGW j /k ) ( − φ < W j )= φ ( − φ > W j ) . (8)The inner region solution is an offset cosine: φ − CGW j /k = ( ψ − CGW j /k ) cos( kz ) (9)where k = − (1 + 2 CG ), and the amplitude comes fromrequiring φ (0) = ψ . The outer region is an exponential φ = − W j exp( z j − z ) . (10)The join position is where cos( kz j ) = [ k +2 GC ] / [ k ψ/W j + 2 CG ] = W j / [ k ψ + 2 CGW j ]. For α = 1 / G is π/ (2 √ C = ˜ f ψ ( ψ + W j ) / = √ π W j /ψ ) . (11) FIG. 3. Dependence of the key coefficient on power α . We note that there are certain constraints on allowablevalues of the parameters of the power deficit model. Oneis that the distribution function cannot be negative, so | ˜ f | ≤ f ∞ (0) = 1 / √ π = 0 .
399 (unshifted Maxwellian);this limits the allowable maximum ψ to a value of orderunity. Another less obvious is that for α < / − W j /ψ for there toexist a solution to V ( ψ ) = 0, and when V f = − φ / − W j /ψ > / − α/
2. The minimum value 1 / f (i.e. α →
0) was noted long ago byDupree . Fig. 3 shows how the numerical coefficient4 G/ ( α + 3 /
2) varies with α . Taking into account theallowable minimum of − W j /ψ the maximum value of(1 + W j /ψ )4 G/ ( α + 3 /
2) does not rise as steeply for α < .
5. This combination is what determines the max-imum value of ψ / for a given ˜ f ψ through eq. 5, andhence how the non-negativity constraint varies with α .So far this is all one-dimensional analysis. B. Multidimensional Generalization: SeparablePotential-form
When there is transverse potential gradient, Poisson’sequation contains an additional transverse divergenceterm ∇ ⊥ φ that couples together Vlasov solutions on ad-jacent field-lines. If its value at a certain transverse posi-tion r is considered a known function of parallel position z , then the integration along z can still be carried outand the transverse divergence adds an extra term V ⊥ = (cid:82) ∇ ⊥ φ dφ , giving classical potential V total = ˜ V + V f + V ⊥ .In the separable case where ∇ ⊥ φ = φ/L , with L ( r )the transverse scale-length independent of z , the effec-tive density contribution is again proportional to φ , and V ⊥ = φ / L . Writing − V f ( ψ ) + V ⊥ ( ψ )] /ψ = F ⊥ ,equations (5), (6), and (7), coming from V total = 0, canbe generalized by multiplying the right hand sides by F ⊥ as ˜ f ψ /ψ / = − ( α + 3 / G W j /ψ ) / F ⊥ ; (12) CG = − ( α + 3 / ψ / − α (1 + W j /ψ ) / α F ⊥ ; (13)and ˜ n ( φ ) = − ( α + 3 / ψ ( φ + W j ) / α ( ψ + W j ) / α F ⊥ . (14)These expressions provide the trapped deficit distribution˜ f ( W (cid:107) , r ) required for a specified transverse potential vari-ation ψ ( r ) and join energy W j ( r ), if the potential wereseparable . However, adopting that distribution does notmake the potential separable . In fact φ is never exactlyseparable, because (at least) far from the hole (where˜ n is negligible) it becomes a Yukawa potential, whichis not separable. Therefore it must be emphasized thatsynthetic expressions (9) and (10) for potential propor-tional to cos( kz ) ( φ > − W j ) and exp( − z ) ( φ < − W j ), orother separable power deficit potential model approxima-tions, are not fully self-consistent equilibria. Fortunately,the approximation can be quite good, but demonstratingso requires us to find the exact self consistent potential φ ( r, z ) corresponding to this distribution using a numeri-cal solution of the Poisson system with the corresponding˜ n ( φ, r ). FIG. 4. Potential and density deficit z -profiles (at r = 0),illustrating different model parameter settings. Different types of parallel ( z ) profiles are illustrated inFig. 4. They are actually profiles at r = 0 of fully self-consistent multidimensional solutions, whose attainmentis the subject of the next section, but they are little differ-ent from the separable treatment of this section. Whenthe power is very small, α = 0 .
02, the ˜ f shape is es-sentially a waterbag (flat for W < − W j , and zero for W > W j ) giving a density deficit | ˜ n ( z ) | that is nearlyelliptical. This case gives the steepest onset of deficit,and has a ˜ f -slope singularity, but at W = W j , not theseparatrix. The potential has only moderate curvatureat z = 0.By contrast, when α = 1 the ˜ f ( W ) is linear, without aslope singularity anywhere, but naturally tends to peakmore near z = 0, giving charge and potential profiles thatare considerably narrower. The case α = 0 . | ˜ n | alinear function of φ ; and choosing a small − W j = 0 . − W j /ψ = 0 .
02) causes the profiles to extend to con-siderably larger z than the other examples, even thoughthe φ -curvature at z = 0 is almost the same as the wa-terbag case. A limitation of this power deficit model isthat it cannot represent the non-monotonic ˜ f ( W ) profilesrequired to give a flat region of φ ( z ) near z = 0. IV. RELAXATION TO TRUE NON-SEPARABLEEQUILIBRIUM
In addition to providing analytic self-consistent solu-tions when potential gradients are ignorable or separable,the model potential is also suitable for calculating numer-ically the full non-separable potential form for two andthree dimensional potential variation. Obviously, to pro-duce a hole of finite transverse dimension one must havetransverse (e.g. radial) variation of the peak potential ψ ,which will imply for the present model radial variationof ˜ f ψ and W j (and possibly α but here we shall suppose α is uniform).In principle it ought to be possible to construct self-consistent profiles for any specified distribution func-tion of the form ˜ f = ˜ f ( W (cid:107) , r ) which gives rise tocharge-density ρ ( φ, r ) = 1 − n p − n t = 1 − n f − ˜ n =1 − n ( φ, r ). When substituted into Poisson’s equation −∇ φ = ρ ( φ, r ), a well-posed problem arises with someboundary conditions. It is non-linear in general, ofcourse, so typically will require numerical solution, al-though in regions where ˜ n is zero, the Modified Helmholtzequation (Debye shielding) in multiple dimensions will beobtained when n f is linear in φ .Particle-in-cell (PIC) simulations have frequently ob-served multi-dimensional holes arising from initially uni-form unstable plasma distributions . And isolatedequilibria depending on non-uniform trapped distribu-tions have been constructed and demonstrated to per-sist for extended time durations using a PIC code bythe present author. But a PIC code uses more physicsthan just a knowledge of n ( φ, r ). It turns out that tofind a multidimensional potential structure that is soli-tary (having boundary conditions only at infinity), basedpurely on ρ ( φ ), is in fact not a well posed problem. Thework of Krasovsky, Matsumoto, and Omura using thetwo-region piecewise linear-density variation ρ ( φ ) (as dis-cussed above, with α = 1 / r ) purports to find a non-spherical equi-librium with a universal ρ ( φ ) dependence by prescrib-ing a non-spherical contour shape of the join potential φ j = − W j , solving the resulting Helmholtz equation for φ inside the contour as a sum of harmonics, and therebyprescribing the fixed ˜ n ( r, z ). However, their presumptionthat this solution can be extended without discontinuityto the external region appears to be incorrect. In any case— and this is the key problem of any multidimensionalsolitary equilibrium — a ρ ( φ ) non-spherical equilibriumprescribed by a φ j contour shape constraint is unstableto potential changes if the constraint is removed.A numerical iterative relaxation on a suitable multidi-mensional mesh of a potential profile in accordance with −∇ φ = ρ ( φ, r ) demonstrates this instability. Explo-rations of such relaxation approaches have shown thatschemes based upon iterations in which Poisson’s equa-tion is solved for given ρ followed by updating ρ in accor-dance with prescribed ρ ( φ, r ) do not generally convergeto electron holes. The potential either grows withoutlimit in the large- φ region, or collapses to zero giving anull ( φ = 0 everywhere) converged solution. And thisbehavior is independent of the degree of partial relax-ation, because it is not merely numerical. The instabilityis physically intuitive, since ρ is an increasing functionof φ in the positive charge region, and positive ρ gener-ally gives rise to positive φ in Poisson’s equation, whichamounts to positive feedback of a perturbation near thepotential peak.In one dimension, it is sometimes possible to stabilizerelaxation iterations so as to converge to the desired equi-librium by using an implicit scheme, but this requires theiteration to start close enough to the final potential pro-file, otherwise either collapse or stochastic fluctuationsresult. Normally of course one-dimensional equilibria areobtained for specified ρ ( φ ) not by relaxation but by directintegration using the “classical” potential V ( φ ) = (cid:82) ρdφ .Such an approach does not readily carry over to multi-dimensional problems, because the transverse field di-vergence contribution is known only when the potentialprofile is known. In multiple dimensions, even implicitrelaxation schemes experience instability.Physically steady, stable, solitary electron hole equi-libria nevertheless exist. How? The answer is that theiractual time-dependent electron dynamics is not repre-sented by prescribing ρ ( φ, r ). The most important dy-namic effect is probably that a time rate of change of thepotential violates the conservation of total energy alongparticle orbits, thereby dynamically changing the formof ρ ( φ, r ) in response to a non-steady perturbation. Asuccessful numerical relaxation scheme to find the equi-librium must represent some of that stabilizing physicaldynamics. One way to represent the dynamics approxi-mately is to regard the charge density as being specifiedas a function of the potential difference between the peakpotential ψ (at z = 0) and the potential elsewhere ( z (cid:54) = 0)so that ρ = ρ ( ψ − φ, r ). This mocks up the idea that ρ does not change in response to time-dependent evolutionof φ at only a fixed point, but rather has non-local dy-namic response contributions. Raising or lowering uni-formly the entire potential profile does not then change ρ ( z ).I have implemented schemes that take just the trappedparticle deficit ˜ f to be a fixed function of ψ − φ (and r ).The screening response, consisting of the passing parti-cles at positive energy and flat trapped distribution f f atnegative energy (which in total I call the flat-trap or ref-erence distribution) remains, as before, a fixed function of φ (approximately ∝ φ ). This amounts to supposing thatthe passing particle density plus the flat contribution inthe trapped region responds quickly to changes in poten-tial, while the trapped deficit does not. Heuristic justi-fication is that passing particles are rapidly exchangedout of the hole and the flat trap level f ∞ (0) does notdepend on φ . Screening is a stabilizing contribution, ap-proximately the Boltzmann response. The power modelof density deficit is used; so mathematically the densityderived from eq. (14) is˜ n ( φ ) = − ( α + 3 / ψ d ( φ − ψ + ψ d + W jd ) / α ( ψ d + W jd ) / α F ⊥ , (15)where ψ d , W jd , and F ⊥ (and α ) are fixed quantities ex-pressing the desired (subscript d ) ψ , W j , initialized be-fore the relaxation, and the F ⊥ value calculated fromthe ψ d profile V ⊥ d plus V f . The φ and ψ change withiteration, but by prescription ˜ n depends only on theirdifference.Two different algorithms for solving for the potentialupdate have been investigated, one using an ad hoc im-plicit advance along alternating directions. The other(much faster) simply solves Poisson’s equation as a Mod-ified Helmholtz equation in an rz -domain, given the dif-ference from pure linear screening (the non-Helmholtzpart of the charge density, ∆ ρ using the prior φ ) as asource: ∇ φ − φ/λ s = − ∆ ρ = ˜ n + n f − − φ/λ s , (16)where the screening length is taken as λ s = ( dn f /dφ ) − / at φ →
0. The Helmholtz equation (16) is solved ateach iteration using the cyclic reduction routine sepeli from the “Fishpack” library. Boundary conditions are dφ/dr = 0 at r = 0, dφ/dz = 0 at z = 0, d ln φ/dr = − /λ s − /r max at r = r max , and d ln φ/dz = − /λ s at z = z max . The distant boundaries do not matter much aslong as they are far enough away. These two codes usein addition a simple Pad´e approximation for the fullynonlinear screening (flat-trap) response n f of an exter-nal Maxwellian of arbitrary drift velocity (not just the n f − φ approximation). They thus accommodateany speed or depth of hole. The agreement between thetwo codes helps verify their coding. A. Global density functional yields only 1-D holes
One version of these relaxation calculations supposesthat the potential difference ψ − φ on which the electrondensity deficit ˜ n depends, is the quantity φ (0 , − φ ( r, z ).That is, the difference between the local potential andthe potential at the origin, which is where the globalpeak in potential lies. This may be called the “globalfunctional”. The other, called the “parallel functional”,is instead φ ( r, − φ ( r, z ); that is, the difference at fixedradial position between the local potential and the ridgeat z = 0 along the parallel coordinate z .A summary of the results of the global functionalmodel relaxation is simple. Electron hole equilibria arefound, but they are spherically symmetric, no matterwhat the initial starting state is. Even highly elongatedinitial potential shapes rapidly relax to spherically sym-metric equilibria that depend solely on the spherical ra-dial distance from the origin R = √ r + z . These aretherefore one-dimensional, not multidimensional, thoughthe one dimension is spherical radius not a cartesian co-ordinate. B. Radially varying density parallel functional gives 2-Dholes
In this section we examine the two-dimensional (ax-isymmetric) holes obtained when the density deficit takesthe form of the power model, eq. (15), in which ψ = φ ( r, n . The pro-file ψ d ( r ) is effectively a free choice but the results shownwill use the following parametrization whose monotonicshape is sufficiently versatile for present purposes: ψ d ( r ) = ψ d r t /λ s )cosh( r/λ s ) + (1 + Dr ) exp( r t /λ s ) ,W jd ( r ) = W j ψ d ( r ) /ψ d . (17)Here the adjustable (uniform) parameter r t , when posi-tive, is approximately the radius of a flattened potentialregion for r (cid:46) r t ; large negative r t removes all flatten-ing, giving a sech( r/λ s ) shape. Parameter D is approx-imately half an extra negative curvature − (1 /φ ) d φ/dr of the potential, modifying the top’s flatness if desired.The uniform screening length λ s is not considered tobe adjustable, but depends on the passing particle dis-tribution, notably its mean speed relative to the hole,or equivalently the hole speed. The form chosen foreq. (17) recognizes that at large r , where ˜ n is negli-gible, the transverse potential variation (at z = 0) isdominated by the screening length and asymptotes to1 / cosh( r/λ s ) → exp( − r/λ s ). The join energy is takento have a desired profile the same shape as ψ d , giving aconstant desired ratio W jd ( r ) /ψ d ( r ) = W j /ψ d , withvalues in the approximate range − .
05 to − . α values. Recall that, for W (cid:107) > W j , the trapped distribution function is indepen-dent of energy, equal to f (0), plausibly accommodating FIG. 5. Example prescribed desired radial profiles of peakpotential ψ and negative join energy − W j ( α = 1 / r t = 5, D = 0).FIG. 6. The final self-consistent hole solution based on theprescription of Fig. 5, showing logarithmically spaced con-tours of potential (lines) and the trapped-density deficit ˜ n (colors) that sustains it. rapid trapping and detrapping of particles on stochasticorbits with energies above W j . Fig. 5 shows an exampleof the prescribed radial profiles of peak potential ψ andnegative join energy − W j (for α = 1 / r t = 5, D = 0).The final fully self-consistent equilibrium found for thisexample (after 20 relaxation iterations) is shown in thecontour plot of Fig. 6. The chosen r t prescribes a holeconsiderably extended in the transverse dimension. Be-yond the region of non-zero electron density deficit ˜ n ,the potential contours gradually become less oblate. Thecontours there show outward decay of potential that isapproximately exponential φ ∂φ∂R ∼ const. ; the spacingof two adjacent contours is almost independent of anglein this rz -plane. At radial positions r (cid:38) r t the contourlines are nearly circular, concentric about r ∼ r t . Nearerthe origin, the potential contours are still oblate, but FIG. 7. The radial shapes of the solved equilibrium param-eters: potential ψ ( r ) /ψ (0) with (dashed) its desired value ψ d ( r ) /ψ d (0), and (dotted) their ratio, the peak distributionfunction deficit | ˜ f ψ | , the ratio of join energy to potential peak − W j ( r ) /ψ ( r ), and the potential logarithmic gradient E r /φ (at z = 0). it is evident that they do not follow lines of level den-sity deficit ˜ n . For this equilibrium, therefore, the deficitcharge density ˜ ρ = − ˜ n is not a function just of φ . Itis a function also of r . As has already been observedin the previous section, such explicit r -dependence is es-sential to obtain multidimensional (non-spherical) holes.That r -dependence is implicitly prescribed by ψ d ( r ) and W jd ( r ). All these qualitative comments are observed toapply to essentially the full range of possible equilibria.In Fig. 7 are shown profiles of several parametersalong the radial axis ( z = 0). The solution potential ψ ( r ) normalized to its peak at the origin ( ψ (0)) proves tohave a shape gratifyingly close to what was prescribed, ψ d ( r ) /ψ d (0) (the dashed line). A way of showing theabsolute agreement out to large radii is the dotted lineof ψ ( r ) /ψ d ( r ), which deviates significantly from unityonly at radii far from the hole, where the potential isalready small. The z = 0 value of the distribution func-tion deficit | ˜ f ψ | , which is fixed by its derivation prior torelaxation from ψ d and W jd , is slightly peaked off-axis(at r > ∇ ⊥ φ of the pre-scribed potential, which requires an enhanced deficit inregions of negative ψ -curvature. The deficit | ˜ f ψ | fallsquickly to zero at large radius where the potential profileis governed essentially by the screening effect of passingparticles. If a potential that did not have exponential ra-dial variation in this distant region had been prescribed, | ˜ f ψ | would not have become as quickly negligible as itdoes here. This exponential potential variation at dis-tant r is evident in the asymptotic approximately flat E r /φ at a value of approximately 1. It is not exactly 1 FIG. 8. Parallel profiles of potential shape for a range ofradial positions, showing the non-separable form of the solu-tion. Marked points show the analytic power deficit modelprescribed (‘+’ at φ d > − W j and ‘ × ’ at φ d < − W j ), in goodagreement at small radius. because, although λ s is effectively 1, the solution of theHelmholtz equation is φ ∝ r exp( − r/λ s ) with an extra1 /r factor contributing a correction of order λ s /r to theslope. The fractional join energy − W j /ψ shows variationwith radius that is the inverse of that of ψ ( r ) /ψ d ( r ) be-cause the iteration scheme holds the potential difference ψ − φ j = ψ d − φ jd = ψ d + W jd fixed.Fig. 8 shows at a range of r positions (0, 2, 4, . . . )the relative parallel shape φ ( z ) /ψ . For r (cid:46) r t this shapeis practically invariant. In this inner radial range thepotential is in fact separable to a good approximation.And, as shown by the point markers plotted over thelines, the agreement with the analytic solution for α =1 / C. Consequences for satellite observations
Since satellites encountering electron holes measure thecomponents of the electric field, rather than the potentialdirectly. It is helpful to derive from an equilibrium likeFig. 6 contours of the perpendicular and parallel electricfield. These are shown in Fig. 9(a). Generally holes movepredominantly in the parallel direction at a fraction of theelectron thermal speed, which is usually much faster thansatellites move. Therefore to a first approximation, thetransit of an electron hole past a satelite corresponds torelative motion of the satellite along a vertical (fixed- r )line through the profile of the hole. The anticipated time (a)(b)FIG. 9. (a) Contours of parallel and perpendicular electricfield corresponding to the equilibrium of Fig. 6. (b) Peakvalues of field along constant r and their ratio versus radialposition. dependence of electric field is the corresponding spatialdependence along such a line. As can easily be under-stood from the contours, such motion gives rise to bipolar E (cid:107) and simultaneously unipolar E ⊥ profiles. The occur-rence of such features is often used to identify electronhole events. One simple and frequently used measureof electron hole amplitude is the peak absolute value ofthe electric field. For E ⊥ this usually occurs at z = 0the point of closest approach; but for E (cid:107) it is always at | z | >
0. Fig. 9(b) therefore plots the peak values of | E ⊥ | and | E (cid:107) | along each line r = const as a function of r .The ratio of electric field peaks, E ⊥ /E (cid:107) , has often beenused in past satellite data analysis as a proxy for the as-pect ratio of an electron hole: that is, the ratio of thepotential’s scale lengths in the parallel and perpendicu-lar directions is supposed L ⊥ /L (cid:107) ∼ E (cid:107) /E ⊥ . Fig. 9 is animportant caution for this practice. The entire r -profileof this ratio comes from a single electron hole model equi-0librium, in which the actual aspect ratio L ⊥ /L (cid:107) of thedensity deficit is approximately 3 and the aspect ratioof the potential contours closer to 2. Nevertheless themodel shows that the observed ratio E ⊥ /E (cid:107) can, all de-pending on r , be anywhere between 0 and ∼
5. The mostprobable or mean value of the ratio to be observed, sup-posing that the nearest distance of approach is random,is determined by whatever selection criterion governs thedistant cut-off of selection of events as electron hole en-counters. If this cut-off is taken as sharp, correspondingto a specific radial distance (e.g. r = c ), or value of fieldcomponent relative to its peak (which would correspondto r = c ), then the probability distribution of detectedradii is p ( r ) = r/ (cid:82) c rdr = 2 r/c whose maximum is at r = c and whose mean is r = 2 c/
3. The correspondingprobability distribution of E ⊥ ( r ) /E (cid:107) ( r ) = A ( r ) is p ( A ) = p ( r ( A )) drdA = 1 c dr dA . (18)The most probable value is E ⊥ /E (cid:107) = A ( c ), and the meandepends on the shape of the ratio profile but if A ( r ) is ap-proximately parabolic it is E ⊥ /E (cid:107) ∼ A ( c/ c = 6,then for the profile shown we find the peak electric fieldratio is ∼ . ∼ .
18, while if c = 8 thesevalues become ∼ ∼ .
25. Thus, the most proba-ble field ratio gives naively an aspect ratio 4 to 6 timessmaller than the actual hole scale-length L ⊥ /L (cid:107) and themean field ratio gives roughly twice the actual. Thisis a disappointingly large interpretational uncertainty. Itreemphasizes the statistical importance of the hole detec-tion algorithm, and the great value of obtaining multiplesimultaneous satellite passages through an electron hole(such as can be obtained using MMS ) for estimat-ing the hole’s transverse extent without relying on the E ⊥ /E (cid:107) ratio. D. Different radial shapes
The earlier Fig. 1 shown in the introduction is a dif-ferent example that illustrates the versatility of the equi-librium prescription. The independent parameter adjust-ments make it more oblate but with a more peaked radialprofile ( r t = 10, D = 0 . α = 1 / ψ (0) = 0 . − W j = 0 . ψ ( r ), in Fig. 10(a), now has no flattop, but falls steadily out to a radius of approximately r = 12 under control of the more peaked ˜ f ψ profile, giv-ing a flatter E r profile. Then ˜ f ψ falls off rapidly intothe exponential region of ψ ( r ) where E r φ rises to unity.Notice that the maximum value of | ˜ f ψ | (cid:39) .
36 is closeto the maximum permissible (0.399), showing that thisequilibrium requires the deeply trapped orbits to be al-most completely depleted of electrons near the origin.The peak fields along fixed r , Fig. 10(b), show a more (a)(b)FIG. 10. Equilibrium solution parameter radial profiles (a),and profiles of electric field peak values (b), for the hole illus-trated in Fig. 1. extended region of flat E ⊥ /E (cid:107) before it rises in the ex-ponential region. Still, even the flat region E ⊥ /E (cid:107) (cid:39) . L (cid:107) /L ⊥ ∼ /
3. Despite the substantial ra-dial gradients, the parallel potential relative shapes (notshown) do not significantly differ until r (cid:38)
12. Theyagree with the analytic form in approximately the sameway as Fig. 8. Thus, they are approximately separableuntil a radius at which radial exponential decay sets in,like the previous equilibrium example.
V. CONCLUSIONS
A technique for constructing fully self-consistent mul-tidimensional electron hole equilibria in the drift limitof negligible gyro-radius has been described and illus-trated. It starts from a versatile phase-space distribu-1tion deficit form that satisfies the plausibility constraintof being zero at and immediately below the parallel en-ergy trapping threshold, and from a desired radial trans-verse potential variation. The analytic results of an ap-proximation of separability of the potential are derived.Dropping the separable approximation, which cannot ap-ply everywhere, requires an iterative relaxation to a fullyself-consistent (non-separable) equilibrium. Such relax-ation cannot be carried out under an assumption thatthe density deficit is a function only of potential, be-cause that leads to instability. An alternative scheme,embodying numerically some aspects of the electron dy-namic response, finds multidimensional equilibria. Butit also shows that no stable non-spherical solitary equi-librium exists without there being explict dependence ofdensity deficit on transverse position. The deviation ofthe full axisymmetric equilibria from the separable formproves to be remarkably small except in the profile wings.The resulting electric field component spatial dependen-cies, observable by satellites, are illustrated for differentoblate hole shapes. They show that the ratio of peakperpendicular and parallel electric fields E ⊥ /E (cid:107) is un-fortunately a very uncertain measure of the aspect ratio L (cid:107) /L ⊥ of the potential or charge distributions of elec-tron holes. This conclusion re-emphasizes the importanceof obtaining multiple simultaneous in-situ measurementssuch as can be obtained from multiple-satellite missions,in order to establish the true spatial structure of natu-rally occurring electron holes. Supporting Data Statement
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