On deformation of electron holes in phase space
eepl draft
On deformation of electron holes in phase space
R. A. Treumann, , C. H. Jaroschek and R. Pottelette Department of Geophysics, Munich University, Theresienstr. 41, D-80333 Munich, Germany Department of Physics and Astronomy, Dartmouth College, Hanover, NH 03755 Department of Earth and Planetary Sciences, Tokyo University, Tokyo, Japan CETP/CNRS St. Maur des Foss´es, Cedex, France
PACS – Auroral phenomena
PACS – Wave-particle interactions
PACS – Radiation processes
Abstract. - This Letter shows that for particularly shaped background particle distributionsmomentum exchange between phase space holes and the distribution causes acceleration of theholes along the magnetic field. In the particular case of a non-symmetric ring distribution (ringwith loss cone) this acceleration is nonuniform in phase space being weaker at larger perpendicularvelocities thus causing deformation of the hole in phase space.
Introduction. –
In configuration space, phase spaceholes appear as localised intense electrostatic fields E (cid:107) h = −∇ (cid:107) φ h with broadband spectral signature parallel to theambient magnetic field. In velocity space they form nar-row regions of lacking particles of one signature kept alivefor a limited time by the electrostatic field. Ion holes arelocal deficiencies of ions while electron holes are local defi-ciencies of electrons. Thus the former correspond to weaknegative, the latter to positive space charges Q i,e . In thisLetter we deal with electron holes which can be excited bybeam or current instabilities parallel to the ambient mag-netic field B , like the two-stream instability which worksfor electron drifts v d > v e , larger than the electron ther-mal velocity v e [3, 4]. At lower drifts this instability isreplaced by a modified version [14] which is a form of themodified two-stream instability [13, 16, 17, 24]. Their the-ory has been given by Bernstein, Greene and Kruskal [2],Schamel [30–32], Dupree [8, 9] and Turikov [38]. Simula-tions by Newman et al. [22, 23], Muschietti et al. [20, 21]and others have shown that electron holes are the natu-ral nonlinear state of these instabilities, being Debye scaleentities along B in configuration space, and of short exten-sion in the parallel velocity component v (cid:107) in velocity space.They contain a dilute component of trapped electrons ofdensity N t of low energy mv t / ≤ | eφ h | . In configurationspace they are oblate in the direction perpendicular to themagnetic field (pancakes). Their behaviour in v ⊥ has notyet been investigated in detail. It is, however, reasonableto assume that the holes are either gyro-limited, being of transverse spatial extension up to the thermal gyrora-dius ∆ h ⊥ ∼ r c = v e /ω c or inertia limited ∆ h ⊥ ∼ c/ω p .Their life time is determined by the stability of the holeswith respect to the generation of whistlers, trapped par-ticle instabilities, particle trapping, heating and diffusionand the corresponding generation of dissipation (see, e.g.,Newman et al. [23]). One might believe that these micro-scopic entities are of minor importance for the behaviourof the plasma. However, in collisionless plasmas they forman important dynamical source of dissipation. They heatand accelerate electrons, cause beam cooling, and are sus-pected to provide a substantial part of the dissipation thatis needed in collisionless shocks and in reconnection. Incollisionless shocks they might contribute to the emissionof radiation causing the badly understood type II bursts.Some time ago we proposed [25] that phase space holescontribute to electron cyclotron maser emission [35] gen-erating auroral kilometric radiation in the upward currentsource region where the holes have been identified [26]subsequently, forming what we called ‘elementary radi-ation sources’. For this to work the holes must becomedeformed in phase space in order to attain a perpendicu-lar phase space gradient ∂F ( v (cid:107) , v ⊥ ) /∂v ⊥ on the electrondistribution function, which is required by the cyclotronmaser mechanism [35]. A qualitative discussion of how thiscan be achieved has rcently been provided [36]. Momen-tum exchange between the background electron distribu-tion and the hole has been made responsible for defor-mation of the phase space shape of the hole, with thep-1 a r X i v : . [ phy s i c s . s p ace - ph ] O c t . A. Treumann, C. H. Jaroschek, and R. Pottelettedynamics of the hole depending sensitively on the shapeof the background electron distribution. In this Letter wepresent a more quantitative mechanism which is developedfor electron holes. However, in a similar way it should alsowork for ion holes in the presence of, say, ion conics, whichhave been found in multitude under auroral conditions. Mechanism. –
Under auroral upward current con-ditions the bulk distribution is kind of a non-symmetric(downgoing) ring distribution with loss cone (due to thepresence of the absorbing ionosphere) as shown in Fig-ure 1. We assume that some appropriate instability gener-ates an electron hole that propagates along the magneticfield B at velocity v h (cid:28) V R much less than the nomi-nal electron ring velocity V R . The question, in which waythe hole is generated is of secondary importance for thepurpose of this Letter. The qualitative discussion of [36]referred to the two-stream instability [3] as generator ofthe hole. Under the dilute plasma conditions in the up-ward current auroral region where ω ce /ω pe ∼ − γ B ∼ ω B ∼ ( m e / m i ) ∼ . ω pe isless than the global electron cyclotron maser growth rate γ ecm ∼ (10 − − − ) ω ce (cf., e.g. [27–29, 40]). However,hole generation is not affected by the global maser insta-bility which has only a minor effect on the bulk electrondistribution not causing a substantial energy loss for theelectrons. Quasilinear flattening of the distribution whichpartially fills in the inner part of the ring (shown as weakbackground in Fig. 1) just where the hole is located inphase space is mainly caused by VLF turbulence generatedunder the same conditions [15, 35]. However, at fixed v ⊥ the electron drift velocity v d might not exceed the electronthermal speed v e , in which case the drift cyclotron insta-bility (MTSI) takes over. Its growth rate is of the order ofthe lower hybrid frequency γ mts ∼ ω lh ∼ ω pi which in thedilute plasma is also small. Hence, hole formation is a pro-cess comparable to or slower than the emission of radiationby the global maser instability. In the auroral kilometricradiation the latter might thus provide the backgroundradiation level while the steep gradients produced in holedeformation [36] generate the narrow intense short liveddrifting emission bands that have been observed.The hole-related localised deficiency of electrons on theelectron background in phase space is centred at the in-stantaneous velocity v h ( t ) of the hole (Fig. 1). In completeanalogy to solid state physics it represents a localised pos-itive charge Q h in the electron fluid that is attached to thehole. The hole moves with velocity v h ( t ) on the electronfluid. In the presence of an electric field E (cid:107) this charge Q h will become accelerated, with its collisionless dynam-ics being described by the equation of motiond v h ( t )d t (cid:12)(cid:12)(cid:12)(cid:12) v ⊥ = Q h M h E (cid:107) . (1)Since the hole can move only parallel to the ambient mag-netic field, this is a one-dimensional equation of motion losscone V T V T V T V ||downwardelectronsupwardions Vh V R V R|| V R||hole ringdistribution Δ v h Fig. 1: Schematic of a ring distribution with loss cone. In thisrepresentation the ring is given a varying phase space densitywith increasing pitch angle ending at a large empty loss cone.Also shown is an electron hole at location of parallel velocity v h and for a range of perpendicular velocities v ⊥ . The holeis assumed to be a straight line initially at constant v h beinglocated between the hot electron ring distribution and the coldion distribution. The latter propagates into opposite directionto the electrons at much lower speed. The hole is slow againstthe electrons while its velocity might be comparable to theion speed. Also shown are the components of the nominal ringvelocity V R a line of constant v ⊥ used in the calculations. that holds at every fixed perpendicular velocity v ⊥ , as in-dicated on the left hand side. The interaction of the hole ispurely electrostatic, and the electric field is external to thehole given by the conditions in the plasma. These are de-scribed by the presence of the downward electrons whichmove with respect to the ambient ions a shown in the fixedlaboratory reference frame in Figure 1. Moreover, the ionand electron beams are distributed about homogeneouslyover an area that is large against the extension of the holein any direction. This implies that the electric field can beexpressed through the current flowing in the plasma as ∂E (cid:107) ∂t = eN(cid:15) v d (cid:12)(cid:12)(cid:12)(cid:12) v ⊥ , where v d = (cid:104) v e (cid:107) (cid:105) − (cid:104) v i (cid:107) (cid:105) . (2)Here N is the plasma density. The drift speed is the differ-ence between the average bulk speeds of the electrons andions. In order to find an equation for the parallel velocityof the hole, we take the time derivative of Eq. (1), remem-bering that the hole charge Q h ( v h ) itself is a function ofthe hole speed. Defining q = Q h /e and m = M h /m e thisyields d v h d t − d ln q ( v h )d v h (cid:18) d v h d t (cid:19) = ω pe v d (cid:20) q ( v h ) m (cid:21) . (3)In deriving this expression we used Eqs. (1-2) in order toeliminate the electric field, and we have suppressed thep-2hase space hole deformationindex v ⊥ understanding that Eq. (3) holds for every con-stant perpendicular velocity. To be able to proceed wemust determine the charge and mass of the hole. The to-tal mass of the hole is given by M h = m e N t V h , with N t the number density of electrons that are trapped in thespatial volume V h of the hole. This number is assumed tobe constant during the evolution of the hole, a simplifi-cation which neglects any possible exchange of electronsbetween the hole and its environment which determinesits life time. Hence over the entire evolution of the hole m = const does not change. This is not so for the chargeof the hole Q h = e ( N | v h − N t ) V h . Any acceleration of thehole in real space corresponds to a displacement in veloc-ity space. Hence Q h depends on the fraction in ambientnumber density of ambient electrons of different velocityseen by the hole when it moves in velocity space. We needto know only the ratio q/m = ( N | v h − N t ) /N t , (4)and, hence, the volume V h of the hole cancels out leavingus with the problem to calculate N | v h . In order to dothis we assume, for simplicity, that the background ringdistribution is a rotated Maxwellian F ( v (cid:107) , v ⊥ ) = Nπ v e exp (cid:26) − ( v − V R ) v e (cid:27) . (5)Here V R = const is a constant radius in velocity space,and v = ( v (cid:107) , v ⊥ ) is the velocity vector. (In global electron-cyclotron maser theory and simulations as, e.g., in [29], thethermal width of the distribution is not important and a δ -ring distribution is used.) At constant v ⊥ the variationis in the parallel component ( v (cid:107) − V R (cid:107) ) in the argument ofthe exponential. The fractional number density of the holeis obtained by integration just over the velocity volume ofthe hole. Since this is very small we assume that the hole isof rectangular shape of width ∆ v h = (cid:112) φ h /m e in parallelvelocity. Then N | v h = 2 π ∆ v h F | v h v ⊥ d v ⊥ . (6)The indication v h on F means that in F the parallel ve-locity v (cid:107) → v h is to be replaced by the hole speed. Itis then easy to show that the coefficient of the secondterm on the left in Eq. (3) is d ln q/ d v h = − ( v h − V R (cid:107) ) /v e ,which is linear in v h . This fact would enable us to rewriteEq. (3) in terms of the variable x = v h − V R (cid:107) . However,it is more convenient to introduce dimensionless variables τ = ω pe t, u = v h /v e , U R = V R (cid:107) /v e , U d = v d /v e , n = N/N t and rewrite Eq. (3) as u (cid:48)(cid:48) − ( U R − u )( u (cid:48) ) = ( U d − u ) (cid:110) nC e − ( U R − u ) − (cid:111) , (7)where (cid:48) = d / d t , and C for fixed v ⊥ is a constant fac-tor that is determined from the background distribution F . The right-hand side of this equation is linearly pro-portional to the average parallel beam velocity U d which is the driving force. U d = 0 corresponds to a symmetricring distribution with no loss-cone. The rigth-hand sidevanishes for U d = u , in which case the equation is solvedtrivially (expressed in decaying error functions in u ), andit can be shown that an initially finite velocity hole willcome to rest. Generally one may suspect that asymptoti-cally u → U d . Hence, for a non-symmetric ring distributionthat, for instance, involves a loss cone, the section of thehole corresponding to v ⊥ = const is accelerated parallel to B until settling near U d . This causes deformation of thehole in phase space and is described by the solutions ofEq. (7) for the most interesting case U d (cid:29) u .The hole is a slowly moving entity of velocity u (cid:28) U R .Observation of the displacement of the fine structure inthe auroral kilometric radiation suggests that holes moveat velocity v h (cid:46)
100 km/s [25] compared to electron beamvelocities of V R (cid:107) ∼ km/s. Hence, in the coefficient ofthe second term on the left in Eq. (7) u can be neglected.Moreover, the expression in front of U d − u on the right-hand side becomes a constant A ≡ nC exp( − U R / U d (cid:29) u , which holds for a non-symmetricelectron distribution like that shown in fig.1, the solutionof Eq. (7) can be found by multiplication with u (cid:48) and inte-grating once with respect to d τ . This yields the followingexpression for u (cid:48) ( τ ) = (cid:112) AU d /U R tan (cid:104)(cid:112) AU R U d ( τ f − τ ) (cid:105) , (8)with τ = τ f the final time when the hole comes to rest and u (cid:48) = 0. The restriction on the argument of the tangentfunction implies the following restriction on τ f : τ f (cid:46) π (cid:16) (cid:112) AU R U d (cid:17) − . (9)The larger the electron drift velocity U d , the faster the holetends to reach its final velocity. In other words, the fasterthe electron beam moves, the less time it takes for the de-formation of the hole in phase space. This is what has beenexpected from the very beginning. Eq. (8) also shows thatthe acceleration of the hole is a monotonically decreasingfunction of time τ until the hole arrives at its final state.The velocity u ( τ ) of the hole is obtained by integratingEq. (8) with respect to time. In the initial state for times τ short against τ f a solution is found by expanding theintegrand. Restricting to the first two terms yields u (˜ τ ) ∼ ˜ τU R (cid:26) tan ˜ τ f −
12 ˜ τ cos ˜ τ f (cid:27) , (10)where ˜ τ = τ √ AU R U d . Initially the hole velocity increaseslinearly with time, and the acceleration is proportional to √ U d , the root of the electron drift velocity and vanishes for U d → u is of the same orderas U R , Eq. (7) cannot be solved analytically. Since thiscase is of lesser interest and would exceed the limits ofp-3. A. Treumann, C. H. Jaroschek, and R. Pottelettea Letter, we leave its investigation for a later publication(Jaroschek et al., to be submitted). Instead we proceed toinvestigate the variation of the hole velocity with the shapeof the electron distribution. This is closely related to thevariation with v ⊥ and the drift velocity U d . To this end westudy the effect of A = nC exp( − U R / − C . The constant C is a differential in v ⊥ C = 2∆ v h √ πv e exp (cid:20) − ( v ⊥ − V R ⊥ ) v e (cid:21) v ⊥ d v ⊥ v e . (11)At a phase space section parallel to v (cid:107) all particles in thedistribution have the same v ⊥ = V R ⊥ , and the exponentialfactor in C is unity. We may define the electron drift speedand electron temperature at constant v ⊥ as the first andsecond moments of the electron distribution function atconstant v ⊥ , respectively. The local drift velocity enteringinto the expressions for the evolution of u ( τ ) is the parallelcomponent of the first moment of the electron distributionfunction at constant v ⊥ . For a half ring distribution with V R (cid:107) > U d | + v ⊥ = 2 (cid:18) V R v e − v ⊥ v e (cid:19) v ⊥ d v ⊥ v e , (12)an expression that decreases with increasing v ⊥ . The sameexpression holds for the part of the ring in Fig. 1 that hasno expression at negative velocities in the loss cone on theleft of Fig. 1. The decrease in U d with v ⊥ is crucial for theevolution of the parallel velocity u ( τ ) of the hole in phasespace in the presence of a half ring. Since the hole veloc-ity and acceleration of the hole in parallel direction areproportional to U d the decrease implies that the largestvariation in u is for small v ⊥ . The velocity space deforma-tion decreases with increasing perpendicular velocity v ⊥ .Hence, the hole attains the largest velocity at small v ⊥ thus becoming bent in velocity space, as was suggestedearlier [36] from qualitative considerations.If the loss cone is not empty, the effective drift velocity U d = U + d − U − d the hole experiences is reduced by thecontribution of the ring at negative velocities U d | − v ⊥ = 2 α (cid:18) V R v e − v ⊥ v e (cid:19) v ⊥ d v ⊥ v e , (13)where α is the fractional density of the half-ring distribu-tion with V R (cid:107) <
0, while the inclusion of the upward iondistribution at low v ⊥ would increase the effective driftspeed for the small values of v ⊥ to which the ions extendin velocity space (see Fig. 1).The case U d < u is not of vital interest. In this case u →
0, and the hole will be about at rest. Such cases mayrefer to standing narrow band emissions in the auroralkilometric radiation with the loss-cone about filled.
Discussion. –
The above considerations show that anelectron phase space hole experiences a particular dynam-ics in phase space that depends on the shape of the dis-tribution function. This Letter was restricted to the mere investigation of the possibility of deformation of an ini-tially straight (in velocity space) phase space hole (BGKmode) by momentum exchange with the ambient electrondistribution. We argued that momentum exchange causesdifferential acceleration of the hole different for different v ⊥ .The evolution of phase space holes has so far been con-sidered in theory and simulations only for Maxwellian dis-tributions in which case their phase space dynamics issimple (see, e.g., [23, 33]). Observations in space, for in-stance under conditions in the auroral kilometric radiationsource and also in collisionless shocks, suggest that phasespace holes evolve when the phase space particle distribu-tions deviate strongly from Maxwellian shape [6,10–12]. Inparticular, the combination of an electrostatic field alongthe magnetic field and a magnetic mirror geometry trans-form a beam distribution into a ring distribution [5] withloss cone (colloquially called a horseshoe). Such distribu-tions are the rule in the aurora and are also expected insuper-critical quasi-perpendicular collisionless shocks [37].In their presence a hole should undergo deformation inphase space of the kind described in this Letter. Holes havealso been predicted in relation to reconnection in collision-less current sheets [7] where their signatures might havebeen observed in situ in space [34, 39].The phase space deformation might not be of over-whelming importance for the dynamics of the plasma eventhough its importance for dissipation has not yet beeninvestigated. However, as suggested in [36] it should beof crucial importance for involving electron holes intothe emission of electromagnetic radiation by the electron-cyclotron maser mechanism acting in the auroral regionsof planetary magnetospheres and in a variety of astrophys-ical objects (cf., e.g., [1]). This requires the generation ofsharp positive phase space gradients ∂F ( v (cid:107) | , v ⊥ ) /∂v ⊥ > ω pe . This mechanism isviable, for instance, at collisionless shocks and might beresponsible for the so-called backbone radiation in type IIradio bursts observed in the solar corona and interplane-tary space.However, for electron holes to become directly involvedinto the electron cyclotron maser mechanism, generationof perpendicular gradients is absolutely necessary. This isprovided by the phase space deformation mechanism pro-posed in this Letter, which bends the hole in phase spaceand transforms its natural parallel phase space gradient ∂F ( v (cid:107) , v ⊥ ) /∂v (cid:107) (cid:54) = 0 into a perpendicular gradient. Since ∂F ( v (cid:107) , v ⊥ ) /∂v (cid:107) (cid:54) = 0 is steep and increases when the holeis accelerated into the bulk of the distribution [36], thenew ∂F ( v (cid:107) | , v ⊥ ) /∂v ⊥ is also very steep and will readilyp-4hase space hole deformationcontribute to maser emission. Its frequency maps the lo-cal electron cyclotron frequency or its lower harmonics.We are not going here into the details of the radiationprocess as this has been described in the literature forthe global electron distribution (see e.g. [18, 19, 27]). Sincethe hole possesses two boundaries, one of them causes apositive, the other a negative phase space gradient. It hasbeen argued [36] that, theoretically, these gradients corre-spond to emission and absorption separated by the wholewidth. Current instrumental resolution and possibly nat-ural line broadenings do not allow for a discrimination ofthese emissions and absorptions, however. Whistler sauceremissions from holes might be another indication of evo-lution of positive perpendicular phase space gradients.The emission frequency changes when the hole is dis-placed in space along the magnetic field. Because of thesteepness of the gradient, the emission is very narrowband. From its spectral displacement the electron holespeed can be determined while from its spectral widthproperties of the hole can be inferred [36]. In the auroralregion the hole speed is found to be of the order of (cid:46) v ⊥ phase space sections.We also note that ion holes behave in a similar way ex-periencing bending in phase space if the ion distributiondiffers from a Maxwellian. This is the case in regions whereion conics are generated in the presence of field alignedelectric potentials and mirror geometries for upward go-ing ions. The evolving steep perpendicular velocity spacegradient may excite ion cyclotron waves from ion holes. ∗ ∗ ∗ This research is part of a Visiting Scientist Programmeat ISSI, Bern. CHJ acknowledges a JSPS Fellowship ofthe Japanese Society for the Promotion of Science. CHJand RT thank M. Hoshino for hospitality, support anddiscussions. This research has also benefitted from a Gay-Lussac-Humboldt award of the French Government.
REFERENCES[1]
Begelman M J, Ergun R E and
Rees M J , Astrophys.J. (2005) 51.[2]
Bernstein I B, Greene J M and
Kruskal M D , Phys.Rev. (1957) 546.[3]
Buneman O , Phys. Rev. Lett. (1958) 8, doi: 10.1103/PhysRevLett.1.8.[4] Buneman O , Phys. Rev. (1959) 503, doi: 10.1103/PhysRev.115.503. [5]
Chiu Y T and
Schulz M , J. Geophys. Res. (1978) 629.[6] Delory G T et al. , Geophys. Res. Lett. (1998)2069,doi: 10.1029/98GL00705.[7] Drake J F, Swisdak M, Cattell C, Shay M A, Rog-ers B N and
Zeiler A , Science (2003) 873, doi:10.1126/science.1080333 .[8]
Dupree T H , Phys. Fluids (1982) 277.[9] Dupree T H , Phys. Fluids (1983) 2460.[10] Ergun R E et al. , Astrophys. J. (2000) 456.[11]
Ergun R E et al. , Phys. Rev. Lett. (2001) 045003.[12] Ergun R E et al. , Phys. Plasmas (2002) 3695.[13] Gladd N T , Plasma Phys. (1976) 27.[14] Kindel J M and
Kennel C F , J. Geophys. Res. (1971)3055.[15] LaBelle J and
Treumann R A , Space Sci. Rev. (2002) 295.[16]
Lampe M, McBride J B, Orens J H and
SudanR N , Phys. Lett. A (1971) 129, doi: 10.1016/0375-9601(71)90583-4.[17] Lampe M, Manheimer W M, McBride J B, Orens JH, Papadopoulos K, Shanny R and
Sudan R N , Phys.Fluids (1972) 662.[18] Louarn P, Roux A, de Feraudy H, Le Qu´eau D and
Andr´e M , J. Geophys. Res. (1990) 5983.[19] Louarn P and
Le Qu´eau D , Planet. Space Sci. (1996)199.[20] Muschietti L, Ergun R E, Roth I and
Carlson C W , Geophys. Res. Lett. (1999a) 1093.[21] Muschietti L, Roth I, Ergun R E and
Carlson C W , Nonlin. Process. Geophys. (1999b) 211.[22] Newman D L, Goldman M V, Ergun R E and
Man-geney A , Phys. Rev. Lett. (2001) 255001.[23] Newman D L, Goldman M V and
Ergun R E , Phys.Plasmas (2002) 2337.[24] Ott E, McBride J B, Orens J H and
Boris J P , Phys.Rev. Lett. (1972) 88, doi: 10.1103/PhysRevLett.28.88.[25] Pottelette R, Treumann R A and
Berthomier M , J. Geophys. Res. (2001) 8465.[26]
Pottelette R and
Treumann R A , Geophys. Res. Lett. (2005) L12104, doi: 10.1029/2005GL022547.[27] Pritchett P L , J. Geophys. Res. (1984) 8957.[28] Pritchett P L , J. Geophys. Res. (1986) 10673.[29] Pritchett P L, Strangeway R J, Ergun R E and
Carlson C W , J. Geophys. Res. (2002) A1437.[30]
Schamel H , Plasma Phys. (1972) 905.[31] Schamel H , J. Plasma Phys. (1975) 139.[32] Schamel H , Phys. Rep. (1986) 161.[33]
Singh N , Geophys. Res. Lett. (2000) 927.[34] Sundkvist D, Retin´o A, Vaivads A and
Bale S D , Phys. Rev. Lett. (2007) 025004, doi: 10.1103/Phys-RevLett.99.025004.[35] Treumann R A , Rev. Astron. Astrophys. (2006) 229.[36] Treumann R A, Jaroschek C H and
Pottelette R , physics.space-ph arXiv (2007) 0712.0185v1.[37] Treumann R A and
Jaroschek C H , astro-ph arXiv (2008) 0805.2181v1.[38] Turikov V A , Phys. Scr. (1984) 73.[39] Vaivads A et al. , Phys. Rev. Lett. (2004) 105001,doi: 10.1103/PhysRevLett.93.105001.[40] Yoon P H and
Weatherwax A T , Geophys. Res. Lett. (1998) 4461.(1998) 4461.