JOURNAL OF GEOPHYSICAL RESEARCH, VOL. ???, XXXX, DOI:10.1029/,

Magnetotail Origins of Auroral Alfvenic Power

B. Zhang,1 W. Lotko,1 O. Brambles,1 P. Damiano,1,2 M. Wiltberger,3 and J. Lyon4

Abstract.The generation of Alfvenic Poynting flux in the central plasma sheet and its polar dis-

tribution at low altitude are studied using three dimensional global simulations of thesolar wind-magnetosphere-ionosphere interaction. A 24-hour event simulation (4-5 Feb2004) driven by solar wind and interplanetary magnetic field data reproduces the globalmorphology of Alfvenic Poynting flux measured by the Polar satellite, including its dawn-dusk asymmetry. Controlled simulations show that the dawn-dusk asymmetry is regu-lated by the spatial variation in ionospheric conductance. The asymmetry disappears whenthe conductance is taken to be spatially uniform. The simulated Alfvenic Poynting fluxis generated in the magnetotail by time-variable, fast flows emerging from nightside re-connection. The simulated fast flows are more intense in the premidnight sector as ob-served; this asymmetry also disappears when the ionospheric conductance is spatially uni-form. Analysis of the wave propagation in the plasmasheet source region, near xGSM ≈−15 RE, shows that as the fast flow brakes, a portion of its kinetic energy is transformedinto the electromagnetic energy of intermediate and fast magnetohydrodynamic waves.The wave power is dominantly compressional in the source region and becomes increas-ingly Alfvenic as it propagates along magnetic field lines towards the ionosphere.

1. Introduction

Nondispersive Alfven waves can propagate with low lossacross the vast reaches of geospace. Of particular interest istheir capacity to transmit electromagnetic power generatedby solar wind and magnetospheric dynamos to low-altitudeabsorption layers where the wave energy is converted to par-ticle beams and heat. Alfven waves are magnetically guidedto a high degree so their low-altitude amplitude distributionsreflect the spatial structure of their generators.

The global morphology of Alfven wave power flowing tolow altitudes has been derived from electric and magneticfields measured on the polar orbiting FAST [Chaston et al.,2003] and Polar [Keiling et al., 2003] satellites. The FASTsatellite sampled Alfvenic Poynting fluxes at altitudes of2000 to 4200 km where much of the Alfven wave energyis converted to accelerated particles. With apogee at 9 RE,the Polar satellite was able to sample the amplitude dis-tribution of Alfvenic Poynting fluxes above the absorptionlayers traversed by FAST.

In deriving the Alfvenic Poynting flux from data acquiredon an orbiting platform, the electric and magnetic fieldsmust be filtered in a frequency band that captures the intrin-sic time variation of the waves as well as satellite Dopplershifted spatial structure of longer period Alfven waves. Thedata reduction should also minimize standing wave compo-nents resulting from wave reflection at the ionosphere. Thebandpass filter appropriate for the altitude of the Polar mea-surements resolves waves with periods in the range 6−180 s

1Thayer School of Engineering, Dartmouth College,Hanover, NH 03755 USA.

2Now at Princeton Plasma Physics Lab, MS29 C-Site,Princeton, NJ 08543 USA.

3High Altitude Observatory, National Center forAtmospheric Research, Boulder, CO 83103 USA.

4Department of Physics and Astronomy, DartmouthCollege, Hanover, NH 03755 USA.

Copyright 2012 by the American Geophysical Union.0148-0227/12/$9.00

[Keiling et al., 2000; Wygant et al., 2000]. In this passband,Alfven wave activity is observed with the highest intensitiesin the nightside auroral zone and dayside cusp region. Aone-year average of the Alfvenic Poynting flux recorded bythe Polar satellite exhibits higher intensity fluxes in the pre-midnight auroral zone relative to the postmidnight region[Keiling et al., 2003]. This significant dawn-dusk asymme-try in Alfvenic Poynting flux is evidently a robust feature ofthe magnetosphere-ionosphere (MI) system. It appears instatistical patterns derived from 4 years of FAST measure-ments of both Alfven wave energy flux and associated field-aligned electron energy flux [Chaston et al., 2003] and inten-year averages of precipitating “broadband” electron en-ergy flux derived from DMSP satellite measurements [Newellet al., 2009]. The dawn-dusk asymmetry in wave power flow-ing into the low-altitude auroral region presumably reflectsasymmetries in the plasmasheet dynamo processes generat-ing the Alfven wave power, but the causes of this asymmetryhave not been studied and remain unknown.

Angelopoulos et al. [2002] investigated the generation ofauroral Alfven wave pulses in the plasmasheet during a sub-storm event. The pulses were recorded contemporaneouslyby the Polar satellite at altitudes of ∼ 5 RE and the Geo-tail satellite at altitudes of ∼ 18 RE in the central plasmasheet. They identify earthward-directed, bursty bulk flows(BBFs) in the plasmasheet as the primary energy source forthe Alfven waves in this event [Angelopoulos et al., 1992].BBFs are fast convective flows (several hundreds of kilome-ters per second) typically of 10 minutes duration or longerwith individual peaks of the order of 1 minute [Angelopou-los et al., 1992]. In the Angelopoulos et al. [2002] study,the electromagnetic energy carried by the Alfven waves wasfound to be a small portion of the kinetic energy in theBBFs. The wave power measured in the Alfven wave fre-quency band at the lower altitude of Polar was also an orderof magnitude smaller than the magnetically mapped powermeasured in the same band at Geotail. This finding, com-bined with an estimate for the wave impedance at Geotailthat exceeded the local Alfven speed, led Angelopoulos etal. to conclude: i) the observed waves are kinetic Alfvenwaves, and ii) the waves encounter substantial electron Lan-dau damping in their transit between the Geotail and Polarsatellites. If Landau damping is responsible for the wave

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X - 2 ZHANG ET AL.: MAGNETOTAIL ORIGINS OF AURORAL ALFVENIC POWER

attenuation between satellites, one would expect to see ac-celerated electrons on the same field lines as the Alfven wavePoynting fluxes. That fast streams of field-aligned electronsdid not accompany the reduction in Alfvenic power betweenGeotail and Polar was attributed to the low temporal resolu-tion of the electron measurements on Polar and to temporalaliasing of the electron measurements on Geotail.

Earthward flow bursts in the magnetotail are also closelyassociated with auroral brightenings in the high-latitude up-per atmosphere [Fairfield et al., 1999; Zesta et al., 2000].The morphology of auroral brightenings called polewardboundary intensifications [Nishimura et al., 2010] is sim-ilar to that of Alfvenic power flowing into the nightsideionosphere. The electron precipitation that causes auroralbrightenings typically carries upward field-aligned currents,and it is well known that changes in field-aligned currentare communicated by Alfven waves [e.g., Lysak , 1990]. Al-though auroral intensifications and magnetotail flow burstsare most intense during substorm events [Baumjohann etal., 1990; Angelopoulos et al., 1992], they also occur duringquiet times and during intervals of steady magnetosphericconvection [Sergeev et al., 1996]. The most plausible cause ofplasmasheet flow bursts is spatially and temporally variablenightside reconnection [Sergeev et al., 1992; Angelopouloset al., 1996; Nagai et al., 1998]. Flow bursts are deflectedand decelerated by the strong outward pressure force nearthe inner edge of the plasmasheet [Shiokawa et al., 1997].The braking is accompanied by oscillatory compressions inthe flow and Alfven wave activity known as Pi 2 pulsations[Shiokawa et al., 1998; Kepko et al., 2001]. As the flowburst brakes, its kinetic energy is transformed into electro-magnetic energy through the generation of fast magnetohy-drodynamic waves and shear Alfven waves that carry field-aligned currents to the ionosphere [Volwerk et al., 2007].

The distribution of BBF events in the plasmasheet clus-ter in the 2100-0100 MLT sector with a centroid near 2300MLT [McPherron et al., 2011]. Given the statistical distri-bution of BBFs, the event study by Angelopoulos et al. [2002]connecting an Alfven wave burst at Polar satellite altitudeswith BBFs in the plasmasheet and the propensity of brakingBBFs to generate Alfven wave and associated auroral activ-ity, it seems plausible that the dawn-dusk asymmetry in thedistribution of Alfven wave power reported by Keiling et al.[2003] is connected with the dawn-dusk asymmetry in theplasmasheet distribution of BBFs. But why are both mostprevalent in the premidnight sector?

We know that the ionosphere and its spatial distributionof conductivity also exhibit significant dawn-dusk asymme-try. We propose the hypothesis here that the ionosphereregulates the distributions of plasmasheet flow bursts andauroral Alfvenic power. It is difficult to test this hypothesiswith the sparse measurements available in geospace. In or-der to conduct such a study, one would need events with es-sentially the same solar wind and IMF driving conditions fordifferent ionospheric states, especially different ionosphericconductances. Satellites recording Alfven wave activity andBBFs would have to be approximately on the same mag-netic field line connecting the plasmasheet region and theionosphere for these special conditions. These requirementsmake it very difficult observationally to determine the effectsof the ionosphere on magnetotail dynamics. Consequently,we will use simulations to test the proposed hypothesis.

Global models of the solar wind-magnetosphere-ionosphereinteraction can be used to follow the temporal evolutionof fields and plasmas in the magnetosphere driven by solarwind conditions derived from actual upstream satellite data.Controlled numerical experiments can be designed to isolatethe effects of magnetosphere-ionosphere coupling on the sys-tem dynamics, including the type of asymmetries describedabove. In this paper, we use the three-dimensional Lyon-Fedder-Mobarry (LFM) global simulation model [Lyon etal., 2004] to examine the generation of Alfvenic power in

the central plasma sheet and to test the hypothesis that theionosphere regulates the dawn-dusk asymmetry observed inauroral Alfvenic power.

Two types of simulation studies are conducted. The firstis a 24-hour event simulation with upstream conditions de-rived from the NASA OMNI dataset. This simulation ex-hibits geomagnetic activity with average levels (as measuredby the Kp index) similar to those in the statistical studyof Keiling et al. [2003]. The results provide a statisticalsample of simulated Alfven wave activity that can be com-pared directly with the Polar satellite statistical distribu-tions. The second simulation study involves controlled ex-periments that eliminate all asymmetries and variability insolar wind driving so as to isolate the effects of differentionospheric states on the simulated distributions of auroralAlfven wave activity and their magnetotail generators.

The next sections are organized as follows. 2: Com-pare global distributions of observed and simulated AlfvenicPoynting flux. 3: Evaluate effects of ionospheric conduc-tance based on controlled simulations. 4: Examine genera-tion of Alfvenic Poynting flux in the magnetotail. 5: Ana-lyze propagation of Alfvenic power from the plasmasheet tolow altitude. 6: Summarize results and conclusions.

2. Morphology of Alfvenic Poynting Fluxes

The global morphology of Alfvenic Poynting flux was an-alyzed by Keiling et al. [2003] for the complete set of 470Polar satellite orbits in 1997 when the satellite’s high incli-nation over the northern polar cap was between 25,000 kmand 38,000 km above the ground. The field-aligned Poynt-ing flux was calculated from measured, perturbation electric(δE) and magnetic (δB) fields as

S|| =1

µ0δE× δB · B

B(1)

where µ0 is the permeability of free space, and B is the meanvector magnetic field calculated from a 180-s running aver-age of the locally measured magnetic field. Perturbation δEand δB were calculated by subtracting a 180-s running av-erage of each measured field from 6-s averages of each field.To reduce the effect of standing waves (e.g., superposition ofincident and reflected waves resulting from interaction withthe ionosphere), S|| was averaged over 30-s intervals. Theresulting Poynting flux values were then projected to a ref-erence ionospheric altitude of 100 km by mapping S||/B =constant along IGRF model field lines. The full databaseof mapped S|| values was then separated into upgoing anddowngoing fluxes with each sorted into 2◦ MLAT × 0.75MLT (magnetic latitude × magnetic local time) bins.

The average distribution of downgoing S|| reported byKeiling et al. [2003] is reproduced in Figure 1a. The re-sults show that Alfven-wave electromagnetic power statisti-cally flows into the dayside cusp region and nightside auroralzone with a significant dawn-dusk asymmetry. The averagePoynting fluxes exhibit two main regions of enhanced inten-sity. The dayside enhancement occurs between 1300 - 1500MLT and 74◦ - 80◦ MLAT. The nightside enhancement oc-curs in the premidnight auroral region between 2100 - 0100MLT and 66◦ - 76◦ MLAT.

The average Kp index for all of 1997 was 2-. However,the average Kp for the intervals when the high-latitude datarepresented in Figure 1a were acquired may be different fromthe annual average. It is impractical to do the exact one yearsimulation of the global magnetosphere and compare to theobservations. However, we can choose a sample event of theyear 1997 which has some statistical similarity. To compareAlfvenic Poynting fluxes derived from LFM simulations with

ZHANG ET AL.: MAGNETOTAIL ORIGINS OF AURORAL ALFVENIC POWER X - 3

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Figure 1. Global morphology of downward AlfvenicPoynting flux from a) a one-year average derived fromPolar spacecraft measurements [Keiling et al., 2003], andb) a 24-hour average derived from the LFM global sim-ulation running from 1200 UT, 04-Feb-2004 to 1200 UT,05-Feb-2004. The white curves in both figures show theFeldstein [1967] statistical auroral oval.

the Polar results, a 24-hour simulation was performed for in-terplanetary plasma and magnetic field conditions given inthe NASA OMNI 1-minute combined dataset for the interval4 Feb 2004, 1200 UT to 5 Feb 2004, 1200 UT. The intervalwas selected for its moderate activity level with Kp rangingfrom 3- to 3+ and a mean of 3. Although the average Kp in-dex for year 1997 was 2-, active time periods with higher Kpvalue definitely contributed to the observations. Therefore,the Kp 3 event was selected instead of Kp 2-. A descriptionof the simulation method, grid and boundary conditions isincluded in the Appendix.

The simulated Alfvenic Poynting flux was calculated from(1) using the electric and magnetic fields at grid cells nearthe inner boundary of the LFM simulation (at approxi-mately 2.8 RE geocentric and 0.8 RE above the sphericallow-altitude simulation boundary). The procedure for eval-uating the mapped S|| in the simulation is the same as inthe Polar study with minor exceptions: 1) Rather than com-puting 6-s average fields as in the Polar study, the values ofsimulated E and B are written to a data file every 5 s.These dropfile values are used to calculate the perturbation

and 180-s, running-average mean fields (comparatively littlepower resides at short periods in the simulations and Polardata so the choice of 5 s or 6 s sampling is relatively in-consequential); 2) S|| is averaged over one-minute intervalsrather than 30-s intervals to remove standing waves; a sim-ulation dataset of 24×60 one-minute average samples of S||is accumulated at each grid cell on the r = 2.8 RE fiducialsurface; 3) the 24×60 samples at each cell are then averagedto give a 24-hour average S|| at each grid cell on the r = 2.8RE fiducial surface; 4) the resulting average downward S|| ismapped to 100-km altitude along dipole rather than IGRFfield lines taking S||/B = constant as in the Polar study. Theaverage magnetic field at altitudes below 2.8 RE geocentricis dipolar in the simulation to a good approximation; and5) the mapped, average values are then interpolated into 2◦

MLAT × 0.75 MLT ionospheric bins to produce a polar plotlike the one in Figure 1a. The mapped simulation grid is ac-tually about 50% denser than the 2◦ MLAT × 0.75 MLTionospheric bins.

The resulting distribution of average downward AlfvenicPoynting flux from the 24-hour simulation is shown in Figure1b. The LFM simulation captures the prominent postnoonand premidnight enhancements evident in the morphologyof the Polar study with comparable intensities in both stud-ies. That the intensities are similar is somewhat surprisingbecause the Polar measurements resolve intense dispersiveAlfven waves that are subgrid in the simulations. Further-more, the physics of dispersive Alfven waves is not includedin the one-fluid MHD model of the simulations. Noteworthydifferences in the Polar and simulation results include:

1. The simulated S|| on average is more intense thanthe Polar average values. This difference may be a due tothe higher average geomagnetic activity during the 24-hourLFM run (mean Kp = 3) relative to that of the one-yearinterval of the Polar study (mean Kp = 2−). The simulatedintensities might be even higher if the simulations were ableto resolve dispersive Alfven waves.

2. The Polar results are more symmetric within ±0.75◦

MLT of midnight than the simulation results. We havefound that simulated substorms generate more symmet-ric distributions of Alfvenic Poynting fluxes near midnight,while steady magnetospheric convection states (SMCs) ex-hibit the average asymmetry shown in Figure 1b (to be dis-cussed in more detail below). The difference between Polarand simulation results near midnight would be expected ifthe data used in the one-year Polar study contains a higherpercentage of substorms than the 24-hour simulation study.

3. The premidnight enhancement in Polar average fluxesoccurs at lower latitude than that of LFM average fluxes.They peak near 69◦ MLAT and mainly lie within the Feld-stein statistical auroral oval (white dotted curves) while thepremidnight LFM fluxes peak near 75◦ MLAT with most ofthe distribution lying poleward of the Feldstein oval. Thislarge offset may be due to inaccuracies in the IGRF mappingused by Keiling et al. [2003] to map the Polar fluxes fromrelatively high altitude where the field may deviate signifi-cantly from the IGRF, especially during the active periodsproducing the most intense Alfven-wave fluxes. Two otherobservational studies differ in detail with the morphology re-ported by Keiling et al. [2003]. The statistical distributionof median peak values of dispersive Alfven-wave Poyntingfluxes derived from low-altitude FAST measurements peaksnear 73◦ MLAT [Chaston et al., 2003]. The statistical dis-tribution of the energy flux of “broadband” electron pre-cipitation in the 10-year DMSP particle data set exhibits abroad premidnight peak from 71◦−77◦ MLAT for “high so-lar wind driving conditions” and a narrower and less intensepeak near 72◦ MLAT for “low solar wind driving” [Newellet al., 2009]. Broadband electron precipitation is attributedto electron acceleration by dispersive Alfven waves and its

X - 4 ZHANG ET AL.: MAGNETOTAIL ORIGINS OF AURORAL ALFVENIC POWER

MLT-MLAT distribution may be considered as a proxy forAlfven wave energy flux [Chaston et al., 2003]. These otherresults appear to be consistent with the higher latitude dis-tribution of the intense Alfvenic Poynting fluxes in the sim-ulation. Thus it is not clear at this juncture whether thePolar or LFM-simulated distributions are statistically morerepresentative of the phenomena.

4. The Polar distribution exhibits flankside spots ofAlfvenic activity at 1800-1845 MLT and 0345-0430 MLTthat are absent in the simulated distribution. These en-hancements in the Polar average fluxes may be a featureof the moderate storms that occurred in 1997 sample usedby Keiling et al. [2003]. We have found in simulations thatas the IMF Bz becomes more negative, nightside Alfvenicactivity tends to spread toward the dawn and dusk merid-ians. The 24-hour simulated interval represented in Figure1b contained no storms.

However one views these differences, the degree of similar-ity in the observed and simulated distribution is actually re-markable and suggests that LFM simulations can be used tounderstand the origins of the observed Poynting fluxes. Tothis end, controlled numerical experiments were conductedto isolate cause and effect of features in the simulations thatresemble the observations.

3. Cause of Dawn-Dusk Asymmetry

A controlled 5-hour simulation was performed for steadyinterplanetary conditions with Bx = By = 0, Bz = −5 nT,n = 5/cm3, T = 10 eV, vx = −400 km/s and vy = vz = 0 atthe upstream boundary of the LFM simulation domain atxGSE = 30 RE. No substorm events are observed during thelast three hours of the simulation, and the magnetosphericconvection becomes quasi-steady but still variable. It resem-bles the steady magnetospheric convection event simulatedby Pulkkinen et al. [2010] using the LFM model, with char-acteristics similar to those shown in their Figures 2 and 3.

Before discussing the average properties of the AlfvenicPoynting fluxes in the idealized simulation, we note thatthe simulation exhibits considerable time variability, despitethe imposed steady solar wind and IMF conditions [Claude-pierre et al., 2008]. Most of the variability occurs in themagnetotail and along the magnetopause boundary. Themagnetopause boundary variations are most likely due toKelvin-Helmholtz waves which develop large vortical struc-tures along the magnetotail flanks. As these vortices evolveand expand, they interact with and modulate nightside re-connection, which causes, among other effects, variability insunward-directed reconnection flows resembling bursty bulkflows [Angelopoulos et al., 1992]. As discussed in the nextsection, this nightside variability generates bursty Alfvenicfluctuations which propagate along field lines to low alti-tudes. Figure 2 shows four snapshots in time of downwardflowing Poynting fluxes calculated from (1) and mapped to100-km altitude from the inner simulation boundary duringthe last hour of the idealized simulation. Thus the averagepatterns of downward Alfvenic Poynting flux exhibited inFigures 1b and 3a (discussed next) are actually ensemblesof temporally and spatially varying Alfven-wave packets orbursts.

Figure 3a shows the distribution of 1-hour averaged,downward Poynting fluxes for the controlled simulation plot-ted in the same format as Figure 1b and calculated using thesame procedure, except that the 1-hour average of S|| makesit unnecessary to compute 1-minute averages of S|| to removestanding waves. The most intense Alfvenic Poynting fluxesflow into the nightside auroral zone between 2100 − 2400MLT and 74◦ − 82◦ MLAT.

In contrast with the simulation results in Figure 1b, lit-tle downward Poynting flux flows into the cusp region inthe controlled simulation. The relative absence of dayside

Alfvenic activity is a consequence of the steady upstreamdriving conditions in the controlled simulation, which min-imally disturb the subsolar magnetopause and produce lit-tle Alfven-wave activity there. Since the subsolar mag-netopause is magnetically connected to the low-altitudecusp, Alfven-wave activity is also weak in the low-altitudecusp. Alternatively, magnetopause buffeting by the time-variable upstream conditions in the 24-hour simulation gen-erates considerable Alfvenic activity near the subsolar mag-netopause and in the low-altitude cusp in Figure 1b.

The simulated event (Figure 1b) and controlled simula-tion (Figure 3a) exhibit similar spatial distributions of aver-age downward flowing Alfvenic Poynting flux in the premid-night auroral region. This result was initially surprising tous because the steady interplanetary conditions are symmet-ric in the idealized simulation. Statistical results from thePolar study (Figure 1a) also show a similar dawn-dusk asym-metry. The question therefore arises: What causes dawn-dusk asymmetry in the distribution of average downwardAlfvenic Poynting flux on the nightside?

The asymmetry in simulated Alfvenic power is due to adawn-dusk asymmetry in the ionospheric conductance. Inthe LFM model, Pedersen and Hall conductances are calcu-lated using the empirical relations derived by Robinson etal. [1987]. The energy flux and average energy of the pre-cipitating electrons required to evaluate the Robinson et al.conductances are inferred from MHD variables near the low-altitude, spherical simulation boundary (see Appendix andWiltberger et al. [2009]). Enhancements in the ionosphericconductance are observed in regions of enhanced electronprecipitation, especially in regions of upward field-alignedcurrent density [Robinson et al., 1985; Ohtani et al., 2009].These same enhancements occur in the LFM simulations,with diffuse precipitation producing enhanced conductanceat lower latitudes and monoenergetic precipitation produc-ing enhanced conductance most prominently where intenseduskside region-1 currents flow out of the ionosphere (cf.Figure 6 of Wiltberger et al. [2009]).

To verify the effect of the asymmetry in conductance onnightside Alfvenic power, a simulation with constant iono-spheric conductance was performed using the same con-trolled solar wind and IMF conditions. Figure 3b shows

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Figure 2. Four snapshots of instantaneous AlfvenicPoynting flux derived from the controlled LFM simula-tion and mapped to the reference altitude of 100 km.

ZHANG ET AL.: MAGNETOTAIL ORIGINS OF AURORAL ALFVENIC POWER X - 5

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Figure 3. (Right) Distributions of 1-hour averaged downward Alfvenic Poynting flux mapped to the100-km reference altitude from the controlled simulation with a) electron precipitation-enhanced iono-spheric conductances and b) 5 mho constant ionospheric conductances. (Left) Corresponding nightsideequatorial plasmasheet fields with flow vectors (black), Bz = 0 contour (white) and magnetic field linesusing average B-field (blue) overlaid on log10 S|| in color. The spherical inset shows the distribution oflog10 S|| on a geocentric sphere at 3.6 RE, produced by upward mapping of S||/B = constant along dipolefield lines from the ionosphere altitude of 100 km.

the distribution of average downward Alfvenic Poynting fluxderived from the test simulation in which the Pedersen andHall conductances were taken to be constant throughout thesimulation ionosphere with ΣP = 5 S,ΣH = 0 S. The simu-lation result shows that the dawn-dusk asymmetry in down-ward Alfvenic Poynting flux is eliminated by introducingconstant conductance in the ionosphere. To determine howthe asymmetry in ionospheric conductance regulates the dis-tribution of Alfvenic power, we now examine the generationof Alfvenic power in the magnetotail.

4. Generation of S|| in the Magnetotail

A synoptic map showing the near equatorial distributionof 1-hour averaged, earthward-flowing S|| in color, evalu-ated on the z = 0.25 RE plane, is shown in Figure 3c forthe controlled simulation corresponding to Figure 3a. MHDvelocity vectors (vx and vy components in black) and the Bz

= 0 contour (white line), both evaluated in the equatorial(z = 0) plane, are overlaid on the S|| distributions. Thespherical inset shows the S|| distribution on a geocentricsphere at 3.6 RE, produced by mapping S||/B = constantfrom 100-km altitude in Figure 3a upward along dipole fieldlines to the 3.6 RE sphere. In the synoptic plot, 6-sec aver-ages of E and B derived from 1-sec samples are used in thecalculation rather than 5-sec samples as was done in con-

structing Figure 1b, but as noted above, this choice makeslittle difference in the end result; a 180-s running average ofE and B is subtracted to give the perturbation fields, δEand δB, required to evaluate S|| in equation (1), for boththe ionospherically mapped distribution of S|| and its planedistribution at z = 0.25 RE. Finally, select magnetic fieldlines, calculated from the 1-hour average B-fields at eachsimulation grid cell, are traced between the z = 0.25 RE

plane and the r = 3.6 RE sphere to illustrate connectivityof S|| between the regions of interest. Note that the log-scale color bar of the spherical inset reveals small-amplitudefeatures in the S|| distribution that are less evident whenplotted using a linear-scale color bar as in Figure 1b or 3a.

As mentioned above, the most intense Alfvenic Poynt-ing fluxes in the controlled simulation are generated alongthe magnetopause flanks by Kelvin-Helmholtz surface wavesand in the magnetotail (dominantly premidnight in thissimulation) by time-variable, braking fast flows emergingfrom nightside reconnection [Angelopoulos et al., 1992]. TheAlfvenic Poynting fluxes reaching low altitude from the mag-netotail sources are more intense than those from the flank-side sources due to the lensing effect of converging mag-netic flux tubes. The flux-tube convergence and the ef-fective lensing power is greater between the magnetotailand the ionosphere than between the magnetopause bound-ary region and the ionosphere. Let subscripts I, MT and

X - 6 ZHANG ET AL.: MAGNETOTAIL ORIGINS OF AURORAL ALFVENIC POWER

MP, denote ionosphere, equatorial magnetotail and equato-rial magnetopause regions, respectively. Along sample fieldlines in Figure 3c, we find BI/BMT = 4320 and BI/BMP

= 675. Thus for Poynting fluxes of the same intensity inthe magnetotail and near the magnetopause, conservation offield-aligned electromagnetic energy flow (S||/B = constant)and the greater convergence of magnetotail flux tubes yieldsfluxes at the base of magnetotail flux tubes that are 6.4×larger than those at the base of magnetopause boundarylayer flux tubes. Some of the difference might also be dueto diversion of a greater portion of the magnetopause-regionfield-aligned Poynting flux into thermal or kinetic MHD en-ergy flux or perpendicular Poynting flux as it flows towardthe ionosphere (not investigated).

For comparison, Figure 3d shows the equatorial-regionsynoptic plot corresponding to the simulation with a 5 S,constant Pedersen conductance ionosphere (Figure 3b). Thedawn-dusk asymmetry evident in Figure 3c is absent in Fig-ure 3d. The enhancements and/or gradients in ionosphericconductance produced by electron precipitation evidentlymodify the distributions and intensities of the field-alignedcurrents and the Poynting fluxes flowing between the iono-sphere and the magnetotail.

The time variability in plasma sheet convection is respon-sible for generating the time-variable Poynting fluxes shownin Figure 3. The dawn-dusk asymmetry in Poynting flux iscontrolled by the dawn-dusk asymmetry in simulated plasmasheet convection. Statistical averages of plasma sheet con-vection derived from satellite data exhibit a similar dawn-dusk asymmetry [Nakamura et al., 1991; Raj et al., 2002;Guild et al., 2008; Juusola et al., 2011; McPherron et al.,2011]. The simulations show that the asymmetry is ulti-mately controlled by the MI interaction. The closure offield-aligned current flowing between the two regions regu-lates the flow channelization across the plasma sheet anddetermines the extent of earthward penetration and spa-tial variation of the fast flow. Thus plasma sheet dynamoand ionospheric load characteristics and the electromagneticpower flows between them are also controlled by the MI in-teraction [Haerendel et al., 2009].

A more extensive analysis of the physical mechanisms ofthe MI coupling processes manifested in Figure 3 will bereported separately. It is sufficient to note here that theprocesses that control the distribution and intensity of theduskside fast flow channel also control the distribution andintensity of the Alfven-wave power flowing into the iono-sphere. The comparison between Figure 3c and 3d clearlyreveals the importance of electron precipitation and its ef-fects on ionospheric conductance in regulating the dawn-dusk asymmetry of plasma sheet convection and earthward-flowing MHD wave power.

5. Propagation from the Plasmasheet

To evaluate more definitively the mode of propagation,we show in Figure 4 simulation diagnostics for the controlledsimulation. The positions of two stationary point probes areshown in Figure 4a, together with the instantaneous mag-netic field lines connecting the probes and the ionosphere.Probe A is located in the premidnight plasma sheet regionat (XGSM, YGSM, ZGSM) = (−14.4, 4.5, 0.25) RE. Probe Bis located in the inner magnetosphere at (−5.4, 3.5, 5.2) RE.The polar Poynting flux distribution at r = 2.06 RE nearthe low-altitude boundary of the simulation (surface C) isalso shown at time 04:28 ST.

The time-bursty Poynting flux recorded at simulationprobes A and B is shown in Figure 4b. The Alfven wavepacket carrying the Poynting flux takes about 145 s to prop-agate from probe A to probe B (that the packet at probe B

propagates as a shear Alfven wave is demonstrated below).As shown in Figure 4a, in this time the magnetic flux tube,along which the wave pulse propagates, convects earthwardfrom the location of probe B to the location shown in Figure4a at 04:27:05 ST. The effective velocity of the shear Alfvenwave packet therefore includes two components: one due towave propagation along the ambient magnetic field and onedue to convection across the magnetic field [Mallinckrodtand Carlson, 1978]. The smaller pulse amplitude (≈ 1/6)recorded in the Poynting flux at probe B relative to that atprobe A is consistent with electromagnetic power being car-ried mainly in the fast MHD mode at probe A and mainly inthe shear Alfven mode at probe B, with only a small fractionof the power at probe A going into the shear Alfven mode.The fast mode propagation is essentially isotropic, so thewave power decreases as r−2 with distance from a fast modesource. The fast mode power at probe B would be a smallfraction of that at probe A if fast mode power is generatedby time variable reconnection at the approximate locationof the Bz = 0 contour in Figure 4a.

In the LFM simulation, the phase speed of propagationof an Alfven wave is limited by the Boris [1970] correction,which includes the effect of the perpendicular displacementcurrent. The resulting linear dispersion relation gives thefollowing values for δE⊥/δB⊥ for propagating shear Alfvenand fast modes:

δE⊥

δB⊥=

ω

kP= vp

{1 Alfven wave

1/ cos θ Fast mode(2)

with vP ≈ vA/√

1 + v2A/c2. The above expression for the

fast mode is valid when either c2s/v2P ≪ 1 or v2A/c

2 ≫ 1;cs and vA are the sound and Alfven speeds, respectively; cis the speed of light in vacuum. To satisfy numerical con-straints in the simulation (principally the Courant condi-tion on the time step), c is set to 1100 km/s in the Boriscorrection. At probe (A, B), the simulation results givevA/c ≈ (10, 100). Thus vP ≈ c at both probes in the LFMsimulation, and the condition for (1) to be valid for the fastmode is well satisfied. The parallel phase speed of a shearAlfven wave propagating between probes A and B, and theratio δE⊥/δB⊥ for a propagating Alfven wave, are essen-tially constant in the simulation with value c.

Figure 4c shows |δE⊥(f)|/|δB⊥(f)| versus frequency, nor-malized to vP , at probe A in the duskside plasma sheet(| · | indicates modulus of the complex amplitude). The fre-quency dependent, complex electric and magnetic perturba-tions are derived from Fourier transforms of the recordedamplitudes for δE⊥(t) and δB⊥(t) in a 5-minute intervalsurrounding the time 04:24:40 ST when the Poynting fluxburst in Figure 4b was recorded at probe A. The ratio|δE⊥(f)|/|δB⊥(f)| in Figure 4c is greater than the effec-tive Alfven speed (1) in the MHD simulation, suggestingthat the Poynting flux burst recorded at probe A is carriedby the MHD fast mode. The color band in Figure 4c showsthe expected range of the phase speed for a propagating fastmode wave in the simulation when 60◦ < θ < 80◦.

The simulated spectrum of |δE⊥(f)|/|δB⊥(f)| shown inFigure 4d is derived from Fourier transforms of δE⊥(t)and δB⊥(t) in a 5-minute interval surrounding the time04:27:05 ST when the Poynting flux burst in Figure 4b wasrecorded at probe B. The spectrum consists of two compo-nents. Above 50 mHz, it asymptotes to the constant value,vP ≈ c. This region of the spectrum is due primarily tothe propagating Alfven wave pulse evident in Figure 4b.The spectrum below 50 mHz increases with frequency. Thistype of variation in |δE⊥(f)|/|δB⊥(f)| arises from a super-position of downward and upward propagating waves, withthe upward propagating component produced by reflectionfrom the ionosphere [Knudsen et al., 1992; Nagatsuma etal., 1996]. Reflection occurs at the low-altitude simulation

ZHANG ET AL.: MAGNETOTAIL ORIGINS OF AURORAL ALFVENIC POWER X - 7

f, Hz

c

04:21:40 − 04:26:40 ST

10:40 - 11:00 UT

Geotail

LFM Probe A

70°60°

plasma "ow vector

mW

/m2

4

0

A

B

C B

Z = 0 contour

04:24:40 ST(−14.4, 4.5, 0.25) R

E

04:27:05 ST(−5.4, 3.5, 5.2) R

E

04:28:00 STr = 2.06 R

E

S|| at C

x

y

z

04:28:00 ST

a

04:10 04:20 04:30 04:40

0

15

30

ST

mW

/m2

S|| at A

6xS|| at B

b

f, Hz

10 2

10 1

10 0

10 -1

10 -2

10 2

10 1

10 0

10 -1

10 -2

10 010 -110 -210 -3 10 010 -110 -210 -3

Polar

LFM Probe B

d

1/µ0Σ

Pv

A (Polar)

1/µ0Σ

Pv

P

04:24 - 04:29 ST

11:00 - 11:20 UT

12 MLT

15

18

21

24

03

06

09

80°70°

60°

60°

80°

3 S

10 S

13 Nov 1996

13 Nov 1996

VP =

vA , Geotail/Polar

≈ { √1 + vA2/c2

vA c , LFM

δE ⊥/δB

⊥v

P

δE ⊥/δB

⊥v

P

Figure 4. a) The positions of two stationary probes A and B in the magnetosphere together with theinstantaneous magnetic field lines connecting the probes and the ionosphere, average plasma flow vectorin the equatorial plane and Bz = 0 contour. The polar plot shows the instantaneous distribution of S||at the ionospheric altitude of 100 km at 04:28:00 ST (simulation time). The time indicated at probesA and B shows the simulation time of maximum S|| recorded at the corresponding probe position. Thesphere C shows the distribution of S|| on a geocentric sphere at r = 2.06 RE, produced by upward dipolemapping of S||/B = constant from the ionosphere altitude. b) The Alfvenic Poynting flux S|| recordedby probes A (blue) and B (green) as a function of time and mapped to the reference altitude of 100 kmassuming S||/B = constant. c) The blue curve shows |δE⊥(f)|/|δB⊥(f)| versus frequency, normalizedto the phase velocity vP , at probe A from the simulation. The black curve shows the same quantityobserved on 13 Nov 1996 by the Geotail satellite at (-18,2,-3) RE. The color band shows the expectedrange of the phase speed vP of a propagating fast mode wave for 60◦ < θ < 80◦. d) The green curveshows |δE⊥(f)|/|δB⊥(f)| versus frequency at probe B from the simulation. The black curve shows thesame quantity observed on 13 Nov 1996 by the Polar satellite at (-4,5,-3) RE. The color band shows theexpected range for |δE⊥(f)|/|δB⊥(f)| = 1/µ0ΣP when 3 S < ΣP < 10 S.

boundary at r = 2 RE where the ionospheric boundary con-dition is imposed. This standing wave component of thespectrum is evident in Figure 4b as the quasi-periodic wave-form preceding the Alfven wave pulse at probe B. The stand-ing wave spectrum of |δE⊥(f)|/|δB⊥(f)| should asymptoteto 1/ΣP as f → 0 Hz, and the spectrum does shows someindication of bending toward an asymptote at the lowest re-solved frequency of 3.3 mHz. If the asymptote is 1/µ0ΣP ,the effective conductance at the ionospheric base of the fluxtube must be greater than 10 S in the simulation.

MHD waves observed during a substorm on 13 Nov1996 by the Polar satellite at (XGSM, YGSM, ZGSM) ≈(−4, 5, 3) RE and about 20 minutes earlier in the magneto-tail by the Geotail satellite at (−18, 2,−3) RE [Angelopou-los et al., 2002] have been overlaid on the simulation curvesin Figure 4c,d. The solar wind and IMF conditions for thisevent are similar to those of the simulation but with variable

and slightly weaker IMF Bz between −2 and −4 nT. Kp = 2during the event. The Polar and Geotail satellites were inthe same general regions, but not precisely at the same lo-cations, of probes A and B in the simulation. The satellitedata are superposed in the figures to illustrate the range ofmeasured values for actual data. It is somewhat surprisingthat results from Polar and the LFM simulation in Figure 4dare so similar, especially given the use of the Boris correctionin the simulation. The similarity may be fortuitous. A phys-ical explanation implies that the Alfven wave impedance, atboth the Polar satellite and simulation Probe B, is domi-nated by standing waves at the lowest frequencies, and thatthe eigenmode structure is similar in both cases. Some por-tion of the low-frequency spectrum recorded at Geotail (Fig-ure 4c) may include propagating Alfven waves, particularlybelow 30 mHz where δE⊥/δB⊥ < vA. The observed ratioδE⊥/δB⊥ would be nearly equal to vA at these lower fre-quencies if a 2.5% concentration of O+ were present in the

X - 8 ZHANG ET AL.: MAGNETOTAIL ORIGINS OF AURORAL ALFVENIC POWER

plasmasheet at the time of the Geotail observations. Thesimulation results suggest that at frequencies above 30− 40mHz Geotail may be seeing fast mode waves, although ki-netic Alfven waves may also be present as suggested by An-gelopoulos et al. [2002].

The simulation results are based on the equations of idealone-fluid MHD and do not include the two-fluid, dispersiveeffects that produce kinetic Alfven waves or kinetic effectsthat introduce Landau damping of kinetic Alfven waves.Both effects may be operative in the Geotail and Polarfields reported by Angelopoulos et al. [2002]. Nevertheless,some basic features of the observed Alfven wave event arecaptured in the simulation including: 1) generation of theAlfven wave packet by a braking earthward flow burst; ii)similar variation in wave impedance versus frequency in theplasmasheet and at Polar satellite altitude; and iii) compa-rable ratio of magnetically mapped pulse amplitude in theplasmasheet to that at Polar altitude – factor of 6 in the sim-ulation, factor of 10 in the event – even though the absoluteintensity of the observed pulses is 6 - 40 x larger than thatof the simulated pulses. The difference in absolute intensityis not surprising because smaller earthward flowing energyfluxes are expected during the simulated SMC conditionsrelative to the 13 Nov 1996 substorm conditions.

The simulation results suggest a plausible alternative tothe interpretation that the observed Poynting flux burstis carried mainly by kinetic Alfven waves [Angelopoulos etal., 2002]. In the simulation, the reduction in magneti-cally mapped amplitude between Geotail and Polar alti-tudes is not due to Landau damping of the Alfven wavebut to the fact that the signal is dominantly a fast com-pressional mode in the plasmasheet, although intermedi-ate mode, shear Alfven power is also present. The wavepower at Geotail diminishes in its transit to Polar due tothe more isotropic propagation of the fast mode, with itsPoynting flux decreasing essentially as the inverse square ofdistance from the fast mode source. In contrast, the mag-netic guidance of the Alfven wave causes its Poytning fluxto increase between Geotail and Polar in proportion to theratio of the magnetic field intensity at the two measure-ment points along the field line. It is not surprising thata braking flow burst would generate substantial power inthe compressional Alfven mode. The fact that the waveimpedance in the plasmasheet exceeds the Alfven speed forthe wave pulse reported by Angelopoulos et al. [2002], andthat density and temperature perturbations appear duringthe interval of field-aligned Poynting flux bursts observedon Geotail (Figures 7 and 10 of Angelopoulos et al.), arealso consistent with the Poynting flux burst observed in theplasmasheet being carried mainly by fast mode waves. Theapparent absence of elevated fluxes of field-aligned electronsat Polar and Geotail during the event is also consistent withthe power at Geotail being dominantly fast mode.

6. Conclusions

The physics of the generation and propagation of bursty,nightside Alfvenic Poynting fluxes has been analyzed us-ing global MHD simulation of the magnetosphere-ionospheresystem. The power of both shear Alfven and fast MHDwaves is generated by braking bursty bulk flows stimu-lated by localized, time-variable magnetic reconnection inthe magnetotail. Simulation results show that as the fastmode and shear Alfven waves propagate earthward fromtheir plasmasheet region of origin, the power in the com-pressional signal diminishes rapidly due to its essentiallyisotropic propagation, leaving a dominantly Alfvenic Poynt-ing flux at lower altitudes on auroral field lines.

When the LFM global simulation is driven by actual so-lar wind conditions, it reproduces with reasonable fidelitythe observed statistical morphology and amplitude distri-bution of auroral and cusp-region Alfvenic Poynting fluxes

reported by Keiling et al. [2003]. The cause of the observedenhancement in Alfvenic Poynting flux in the premidnightauroral sector relative to that in the postmidnight sectoris due in part to asymmetries in the spatial distributionof ionospheric conductance. Enhancements in electron pre-cipitation and ionospheric conductance in the dusk sectorinduce higher fluxes of Alfvenic power in the premidnightsector. Controlled simulations show that when the iono-spheric conductance is set to a constant, uniform value, thepre/postmidnight asymmetry in the distribution of earth-ward flowing Alfvenic power disappears.

The controlled simulations also show that the ionosphericconductance regulates the distribution of plasmasheet flowbursts during intervals of steady magnetospheric convection,with fast flows being more intense in the premidnight plas-masheet. To the extent that the global simulations ad-equately model the relevant underlying physics, we maytherefore conclude that the distribution of ionospheric con-ductance has a great influence on magnetotail dynamics andstructure, of which the generation of auroral Alfvenic poweris just one manifestation.

Despite the successes of the one-fluid simulations in cap-turing key features of auroral Alfven dynamics, they do notinclude Alfven wave dispersion and wave-particle interac-tions. We know that both effects can be important in thepropagation and absorption of Alfven wave power, particu-larly in boundary layers [Hasegawa et al., 1976] and at lowaltitudes where conversion of Alfvenic power to the precipi-tating electron power causes the Alfvenic aurora [Chaston etal., 2003]. Nonideal effects may also have other importantconsequences for energy conversion in the nightside recon-nection exhaust flows that produce plasmasheet flow bursts.

Appendix

LFM Global Simulation

The LFM global simulation uses a finite-volume methodon a nonorthogonal grid to solve one-fluid, ideal MHD equa-tions for evolution of mass, momentum and energy for thesolar wind and magnetospheric plasmas. The solar wind-magnetosphere-ionosphere interaction is simulated with:

1. boundary conditions on the solar wind and interplan-etary magnetic field at the sunward boundary of the com-putational domain;

2. a point dipole to represent Earth’s magnetic field;

3. ionospheric electrodynamics, including a proxy forelectron precipitation, at the inner (low-altitude) boundary.

The simulation domain extends from XSM = 30 RE sun-ward to XSM = −300 RE anti-sunward, with a 100 RE

radial span in the YSM−ZSM plane. The variable cell size ofthe computational grid provides higher resolution near thebow shock, the magnetopause, in the magneotail and nearthe low-altitude boundary, with lower resolution far fromthe earth in the solar wind and at the outer boundaries.The inner boundary is a sphere at a geocentric radial dis-tance of 2 RE where the electric drift velocity derived fromelectrostatic coupling to a 2-D ionosphere is imposed as aboundary condition on the fluid velocity [Lyon et al., 2004].

M-I Coupling Model

The basic equation of the M-I coupling module combinesOhm’s law with current continuity and the electrostatic ap-proximation to obtain the following elliptic equation for theionospheric electric potential Φi, given the field-aligned cur-rent density J||i at the top of the ionospheric conductinglayer and the height-integrated conductance tensor Σ:

∇ ·Σ · ∇Φi = J||i cosα. (A1)

ZHANG ET AL.: MAGNETOTAIL ORIGINS OF AURORAL ALFVENIC POWER X - 9

The equation is solved on a two-dimensional spherical sur-face located at 1.02 RE geocentric [Merkin and Lyon, 2010].The dip factor is cosα = b·r where b is a unit vector pointingalong the dipole magnetic field at the top of the conduct-ing layer and r is the radial unit vector in a spherical polarcoordinate system with θ measured from the north pole. Inaddition to the limitations implied by the electrostatic ap-proximation [Lotko, 2004], (A1) assumes the length scale forvariation in θ satisfies Lθ ≫ h |tanα| where h ∼ 100 km isthe effective height of the ionosphere.

J||i is derived from the field-aligned current in the mag-netosphere at the top surface (2.38 RE) of the bound-ary cell, mapped to the top of the ionospheric conductinglayer (1.02 RE geocentric) along dipole field lines assumingJ||/B = constant.

The solution Φi is then mapped along dipole field lines tothe low-altitude magnetospheric boundary, where the elec-tric field, calculated as E = −∇Φi, serves as part of thelow-altitude boundary condition for the MHD equations.

The conductance tensor in (A1) may be represented inspherical polar coordinates as [Goodman, 1995]

Σ = θθΣP − θϕΣH cosα+ ϕθΣH

cosα+ ϕϕΣP . (A2)

ΣP and ΣH are the height-integrated Pedersen and Hall con-ductances. Empirical equations derived by Robinson et al.[1987] are used to modify the spatiotemporal distribution ofthe auroral contribution to ionospheric conductance, giventhe average energy flux Fε and energy ε of electron precip-itation derived from the LFM precipitation model. Theseempirical relations are

ΣPAuroral =40ε

16 + ε2F 0.5ε , (A3)

ΣHAuroral = 0.45ε0.85ΣP . (A4)

The conductance also includes a contribution from solarEUV ionization, which varies with solar zenith angle andthe observed solar radio flux at 10.7 cm wavelength. Theempirical models for ΣP,HEUV are discussed in Wiltberger etal. [2004]. The total conductance is

ΣP,H =√

Σ2P,HAuroral

+Σ2P,HEUV

. (A5)

Electron Precipitation Model

Precipitating electrons are assumed to be isotropic andMaxwellian in the magnetospheric source region, with al-lowance for acceleration along the magnetic field by a quasi-static potential drop V if the their thermal loss-cone flux F0

is insufficient to carry the MHD field-aligned current. Adi-abatic kinetic theory gives the ionospheric number flux ofelectrons precipitating in the loss cone [Knight , 1973; Frid-man and Lemaire, 1980]:

FN = F0

RM − (RM − 1) e−eV/Te(RM−1), V > 0

eeV/Te , V < 0

(A6)

RM = 8 is the mirror ratio between the ionosphere and theelectron source region (assumed to be located in the magne-tosphere at the cell center of the low-altitude boundary ofthe MHD computation), Te is the electron thermal energyin the source region, and V is the parallel potential drop.The thermal electron number flux in the loss cone withoutparallel acceleration (V = 0) is

F0 = Ne

(Te

2πme

)1/2

. (A7)

Ne is the electron density in the loss-cone.The electron variables Ne and Te are chosen to be pro-

portional to the dynamically computed MHD density andtemperature in each cell center at the low-altitude compu-tational boundary:

Te = aTMHD, Ne = bNMHD. (A8)

The empirical proportionality constants, a = 1.72 andb = 0.023, have been previously optimized for agreementbetween simulated and observed precipitation [Fedder et al.,1995; Slinker et al., 1999]. Parameter b effectively specifiesthe degree of loss cone filling in the source region.

The field-aligned potential drop in (A6) has the form

V = η∣∣J||i

∣∣ with η =T 0.5e

NeH

(J||i

)[kV/(µA/m2)]. (A9)

The requirement that V > 0 for field-aligned currents flow-ing out of the ionosphere (upward) and V < 0 for downwardcurrents is included in the step function

H(J||

)=

{η↑, J||i upward

η↓, J||i downward(A10)

where η↑ = −5 η↓ = 11.25 [Fedder et al., 1995]. η has unitsof kV/(µA/m2) when Te is given in keV and Ne in cm−3.Its density and temperature dependence are derived in theappendix of Wiltberger et al. [2009].

Using (A6)-(A10), the average energy ε and energy fluxFε needed to evaluate (A3) and (A4) are determined by

ε = Te + eV and Fε = εFN . (A11)

Acknowledgments. The research was supported by NASAunder Heliophysics Theory Program Grant NNX11AO59Gand Geospace Supporting Research and Technology GrantNNX11AJ10G, by the NSF National Space Weather ProgramGrant AGS-1023346, and by the Center for Integrated SpaceWeather Modeling funded by the NSF Science and TechnologyCenters program under cooperative agreement ATM-0120950.Computing resources for the research were provided by the Na-tional Center for Atmospheric Research under CISL Projects36761008 and 3761009. B. Zhang and W. Lotko would like toacknowledge the hospitality and support of the UCAR AdvancedStudy Program during summers 2009, 2011 when portions of theresearch were completed. The National Center for AtmosphericResearch is sponsored by the National Science Foundation.

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B. Zhang, 14 Engineering Drive, Thayer School of En-gineering, Dartmouth College, Hanover, NH 03755 USA.(bzhang@dartmouth.edu)