VOL.
14, NO. 2
REVIEWS
OF GEOPHYSICS
AND
SPACE PHYSICS
MAY
1976
Magnetospheric Electric Fields and Their Variation with Geomagnetic Activity
MARGARET
GALLAND
KIVEI•SON
Departmentof Geophysics
and Space Physicsand Institute of Geophysics
and Planetary Physics
Universityof California, Los Angeles,California 90024
Diverse and independentmeasurementsof the variation of particle convectionboundarieswith geomagnetic activity are usedto obtain relations betweenthe magnitude of a large scaleelectric field and the
Kp index. The relations are compared with measuredfields and with models and are found to be
consistentwith diverseobservationsand probably more reliable than resultspreviouslyobtained.
INTRODUCTION
The importance of large scaleelectricfields in the dynamics
of the magnetospherehas been emphasizedboth in broad
reviews fAxford, 1969; Vasyliunas, 1970a; Chappell, 1974;
Stern, 1975] and in more specializedstudies[Kavanaghet al.,
1968: Chen, 1970; Mozer, 1971;Rycroft, 1971; Volland, 1973;
Mozer and Lucht, 1974]. These studiesconcludethat satisfactory qualitative interpretations of varied aspects of
magnetosphericdynamicssuchas particle injectionand acceleration, low-energy plasma convection, and the motion of
whistler ducts emergefrom analysisof the responseto simple
analytic representationsof large scale electric fields. It is apparent, however, that more detailed understanding of the effectsof electricfieldson particle behaviorrequiresfield models
of greater complexity, models which provide for localized
spatial effects [for example, Wolf, 1970; Mcllwain, 1974], and
modelswhich, though still static, vary in an empirically establishedway with the level of geomagneticactivity [Vasyliunas,
1968;Rycroft, 1971; Chenand Grebowsky,1974;Maynard and
Chen, 1975]. The present paper addressesthe latter problem,
obtaining a relation between the magnitude of a uniform
cross-magnetosphere
electric field and the Kp index which is
compatible with diverse and independent measurementsof
particle boundariesin different local time regionsof the magnetosphereand therefore probably more reliable than results
previously obtained.
In this paper the comparisonswith both theory and observa-
tion stresslocaltimesotherthandusk.Thisemphasis
is based
on the argumentthat the quasi-steadystate convectionmodel
alone is not likely to describephenomenanear the stagnation
point where the convectionelectricfield is counteractedby the
corotarion
electricfield.The largescaleele•:tricfieldsin this
region may be small with respectto localized and time-varying
electric fields. The action of localized and time-varying fields
may prevent the plasma from reaching equilibrium in the
combined convection and corotation fields in the 'bulge' region near dusk.
Correlations between various interplanetary and magnetosphericparametersand the Kp index have been noted in the
past, and though the correlationsare crude, they help illuminate the underlyingphysicalprocesses.For example,Snyderet
al. [1963] and Vasyliunas[1968] have related Kp to the solar
wind velocity, Wilcox et al. [1967] and Schattenand Wilcox
[1967] have related it to the magnitude of the interplanetary
magnetic field, numerousworkers have expressedthe distance
to the plasmapausein terms of this parameter [Carpenter,
1967; Binsack, 1967; Taylor et al., 1968; Bezrukikh, 1970;
.....•pyright ¸ 1976by the AmericanGeophysicalUnion.
Chappellet al., 1970a;Rycroft and Thomas, 1970], and, more
recently, Mauk and Mcllwain [1974] and Freeman [1974] have
expressed the local time at which substorm-injected lowenergy particles are observedby a geostationarysatellite in
terms of Kp. Analogouscorrelationsfor low-altitude phenomena include the dependenceof the auroral zone boundarieson
Kp and closely related indices [Feldstein and Starkov, 1967;
Feldstein,1972, 1974] and the dependenceof ionosphericelectric field magnitudeson Kp [Mozer and Lucht, 1974]. By inference from relations discussedabove, Vasyliunas [1968] and
Mendillo and Papagiannis[1971] have inferred relations between the solar wind velocity and the magnetosphericelectric
field, and Freeman [1974] has inferred the distanceto the inner
edgeof the plasma sheetat midnight as a function of Kp. In the
next section,the Kp dependences
of convectionboundariesin
the equatorial plane are used to obtain empirical relations
betweenKp and the magnitude of a uniform dawn-dusk electric field, E. The resultsobtained are compared with independent measurementsof the total potential drop acrossthe polar
cap [Heppner, 1972] and with low-altitude properties.Results
based on a uniform cross-magnetospherefield are compared
with thosebasedon more sophisticatedmodels[Volland, 1973;
Mcllwain, 1974; Stern, 1975; Maynard and Chen, 1975], and
differencesare not large for small Kp. Observations of the
plasmapauseduring a geomagnetic storm [Hoffman et al.,
1975] show that the relation obtained is consistentwith observations even at very large Kp. In the final section, a relation
betweensolar wind velocity and Kp is given, and propertiesof
the inner edgeof the plasmasheetat midnight are evaluated.
E VERSUS
Kp,•ROMEQUATORIAL
CONVECTION
BOUNDARIES
In the presenceof a dawn-duskelectric field the drift orbits
of low-energychargedparticleswhich complt/telyencirclethe
earth lie insidea boundarycalledthe Alfv0n layer [Schieldet
al., 1969]. (Particle orbits and Alfv0n layers in a uniform
dawn-dusk field have been discussed by Alfvdn and
Fiilthammar [1963], Kavanagh et al. [1968], Schield [1969],
Chen [1970], and Kivelson and Southwood[1975].) Nishida
[1966] and Brice [1967] recognizedthat this outermostclosed
drift orbit (or, equivalently, the last closed equipotential)
would, in a static situation, correspond to the plasmapause.
Experimentally, the plasmapausemust be defined as the outermost sustainedclosedequipotential contour [Carpenterand
Park, 1973; Chen and Grebowsky,1974]. The requirement that
a field configurationbe sustainedarisesfor somewhatdifferent
reasons when the field increases than when it decreases. In the
case of a field increase,filled flux tubes on newly open orbits
continue to drift around the earth until they reach the late
189
190
KIvELsON: MAGNETOsPHERIc ELEcTRIc FIELDs AND GEoMAGNETIc ACTIVITY
afternoon quadrant, where they drift out toward the magnetopause. As it takes • 1 day for plasma initially in the evening
sector to move away from the Alfv6n layer, the new
equilibrium plasmapauseposition is achieved only after •1
day. For this reason, in some parts of this paper the average
Kp for the previous24 hoursis usedto characterizethe magnetic conditions.On the other hand, freshly injectedparticles
on orbits open to the tail appear at the new Alfv•n boundary
in the evening-midnightquadrant within hours of the increase
in the convectionfield; hence,in the interpretationsbasedon
night side injection boundaries,the Kp index for the time of
observationis used.In the caseof a decreasing
field,flux tubes
consideredrepresentativeof the relations which follow from
this type of measurement.The functional form of (4) is partially a consequenceof the use of a linear fit to the plasmapause observations.A different form follows from the empirical relation for Loo, the distanceto the plasmapauseat
night
Loo = 5.64 - (0.78 + O.12)(Kp)•/•'
Kp < 6
(5)
which Rycroft and Thomas[1970] reportedas an improvement
over the linear fit to their data. As their measurements were
made at night, we set 4 = 0 in (2) to obtain
E = 0.71/[1 - O.14(Kp)•/•']•' kV/Re
on newly closedorbits refill completelyonly after 5-8 days
(6)
[Park, 1970]. For this reason the Alfv6n boundary based on
the 24-hour Kp may not correspondto the plasmapauseduring
intervals of decreasingactivity.
Inward displacementof the plasmapausewith increasing
magneticactivity [Carpenter, 1967;Rycroft and Thomas,1970]
is consistentwith the existenceof a Kp-related dawn-dusk
electric field acrossthe magnetosphere,as noted by Rycroft
[1974]. For the earth's dipole magneticfield the positionof the
Alfv6n layer in a uniform dawn-dusk electric field is found
from the equations
Equation (6) givesE = 0.71 and 1.44 kV/Re for Kp = 0 and
5-, respectively, as noted by Rycroft [1974], but the
dependencediffers slightly from the linear one he mentions.
Equations(4) and (6) are plotted in Figure 1. Also plotted in
Figure 1 are two points relating E and Kp obtained by Rycroft
[1971] from considerationsof the local time asymmetryof the
plasmapause.
A relation between E and Kp which agreeswell with the
resultsof the plasmapauseanalysis(equation (4)) can be obtained from an analogoustreatment of the observationsof the
local time of injection of substorm-associatedparticles at
-eEReL(½ ) sin 0 + C/L(½ ) = eEReLs + C/Ls (la)
geostationaryorbit. Mauk and Mc!lwain [1974] and Freeman
eEReL, = C/L,
(lb)
[1974] found there is an approximatelylinear relation between
Kp and the local time, primarily betweendusk and midnight,
[Kivelsonand Southwood,1975], where L(½) is the radig] disat which freshly injected particlesare encountered.Their obtance
(inearthradii)totheAifv6nlayerintheequatorial'plane
servationsare shown in Figures 2a and 2b. Low-energy parat azimuth 0 (0 measured eastward from midnight), C =
ticles, injected in the premidnight region by a crosse•2BoRe
•' -• 90 keV with • = 2•r/day, and Bo the equatorial
magnetosphereelectricfield associatedwith a substorm,must
field strength at 1 Re; L, is the distanceto the flow stagnation
be on drift orbits open to the geomagnetictail [Kivelsonand
point on the dusk meridian, and E is the magnitude of the
Southwood,1975]; that is, they must be outside their Alfv6n
dawn-duskelectricfield in the equatorial plane (in kV Re-•).
layer L(½), specifiedby (2). This boundary is plotted in Figure
Equations(la) and (lb) can be manipulatedto showthat L(½)
and E are related by
L•'(O)eERe = C/[1 - (1 + sin ½)x/2]/sin0}•'
1.8
(2)
The azimuthal variation (or local time dependence)of the
distance to the boundary given by (2) has been shown by
Rycroft [1971] to agree well with observationsfrom 2100 LT
through dawn to noon.
Equation (2) can be used in conjunction with one of the
empirical relationsbetweenKp and the distanceto the plasmapause to yield a relation between E and Kp. For example,
Carpenter [1967], using whistlers to determine the distanceto
the plasmapausenear dawn (Lo = L(90ø)), found a linear
relation betweenLo and the highest3-hour Kp index in the 24
hours precedinghis measurement.Using the fit to this observation [Carpenterand Park, 1973]
Lo -• 5.7 - 0.47Kp
Kp < 6
I
I
I
I
I
1.4
1.2
(3)
.2-
with (2), we find
E = 0.46/(1 - 0.082Kp)•' kV/Re
(4)
Kp
This relation is relatively insensitiveto the local time assigned
1. Magnitude of a uniform dawn-dusk electric field versus
to (3), whichis b•ised
on measurements
in themidnight-dawnKp.Fig.
The dashed line representsdata inferred from Carpenter's [1967]
sector. For example, if (3) is taken to representL(45ø), the
field of (4) increasesby 10%.
As the empirical relations between the plasmapausedistance and Kp obtained by Binsack [1967], Bezrukikh [1970],
Gringauz [1969], Chappell et al. [1970a], and Rycroft and
Thomas [1970] are quite similar to that of Carpenter [1967]
(seethe discussionin the review of Rycroft [1975]), (4) can be
measurementsof equatorial distanceto the dawn plasmapause(equation (4)). The solid line is data inferred from Mauk and Mcllwain's
[1974] measurementsof the low-energy particle injection boundary
(equation (11)). The dotted and dashed line representsdata inferred
from Rycroft and Thomas's[1970] measurementsof distanceto night
side plasmapause(equation (6)). The two points were found by Rycroft [1971] from studiesof the local time dependenceof the plasmapause.
KIVELSON:
MAGNETOSPHERIC
ELECTRIC
FIELDSANDGEOMAGNETIC
ACTIVITY
I[
I
I
I
I
I
0200
I
191
0000
\
'
i
2200
,
/
'
0400
2000
0600
1800
0800
1600
x •Kp >2
ß AKp< 2
I000
1200
1400
Fig. 3. Zero energy Alfv6n layers in the equatorial plane for different magnitudesof a uniform dawn-duskelectricfield. The boundaries are labelled with the values of E in kV RE-:, and the orbit of a
Fig. 2a. The local time of low-energyparticle injection at geosta- geostationarysatelliteis shownas a circle. In the eveningquadrantthe
intersection of the satellite with the Alfv6n layer is seen to occur at
tionary orbit by Mauk and Mcllwain [1974], (8).
later local times for smaller values of E.
3 for a number
of different
values of E. Also shown is the
approximateorbit of a geostationarysatellite;no corrections [Mauk and Mcllwain, 1974] or
have been made for dipole tilt. From the figure it is apparent
LT = 27.6 - 1.6Kp
Kp < 8(9)
that in the late evening quadrant the satellite crossesthe
boundary at local times which are earlier the greater the [Freeman, 1974] to obtain a relation betweenE and Kp.
strength of the electric field. From (2), with L(½) = 6.6, we.
As the naturally occurringtime variable in the model is sin
obtain
½, and the time variable in terms of which E is most simply
expressed
is s = Icos [(½ - 90ø)/211(seeequation(7)), we
E= 2.1{[1
-- (1q-sin
4•):/2]/sin
4•]}
2 kV/RE (7) could obtain an algebraicrelation betweenE and Kp if Kp
E = 2.1/{ 1 q- 2:/2 Icos[(4>-- 90ø)/211}" kV/RE
were correlated with s instead of L T. Accordingly, we have
foundthe leastsquarefit to s asa functionof Kp from the data
of Mauk and Mcllwain. Having verified that our data set
as the relation between the local time (½) at which the first
freshlyinjectedparticlesshould appear at geostationaryorbit
closelyreproducestheir fit (LT = 25.3 - 1.42Kp,standard
and the strengthof the dawn-duskelectricfield. Equation (7)
deviation: 1.58), we obtained a fit for s versusKp
can be usedin conjunctionwith eitherof the relationsbetween
s = 0.825 - 0.148Kp
(10)
local time of encounter and Kp:
LT = ½/15 ø -- 25.5 - 1.5gp
gp • 5+
The standarddeviationfor this fit showsit to be comparable to
(8)
(DEGREES)
0
I
45
i
i
I
.•
6
•.!•
'
•
90
I
i
I
I
I
I
I
I
i
Ii
I I i ..-
Kp
=2-FOR
PREP/OUR
$-HOUR
INTEl?
VA
L7
X.x•
2-
HIGHESTKpV
ON
DAYS
WITH
ø''•
I-
•
135
I
ø
'•
eeeee
• t I ! I ! ! I I I • • • • • 05t 06• 0708
v I 09I I0J
14 15 16 17 18 19 20 21 22 2:5 O0 01 02 0304
LOCAL
TIME
(HOURS)
Fig. 25. The local time oœlow-energyparticleinjectionat geostationaryorbit by Freeman[1974],(9).
192
KIVELSON:MAGNETOSPHERIC
ELECTRICFIELDS AND GEOMAGNETICACTIVITY
the fit for LT versusKp. From (7) and (10), we then find (see TABLE 1. Total Dawn-Dusk Potential Drop in Kilovolts Predicted
Acrossa 40-RE MagnetosphereversusKo
Figure 1)
E - 0.44/[(1 - 0.097Kp)•'1 kV/Re
(11)
Analogoustreatment of Freeman'sdata is not usefulunless
his two pointsat Kp > 6 are excluded,becausethe theoretical
model cannot predict the local time of injectionif the electric
field is large enough to move the stagnationpoint inside of
geostationaryorbit. For the observationsat Kp < 6 the fit
s = 0.954 - 0.141Kp
EquationNo.
0
2
4
6
4
6
11
13
15
19
28
18
16
32
27
44
27
23
74
42
54
47,
36
116
74
64
101
66
160
(12)
COMPARISON
yieldsa resultsimilarto that of (4) and (11), i.e.,
WITH
IONOSPHERIC
MEASUREMENTS
The position of the auroral oval is known to shift
equatorwardwith increaseof geomagneticactivity [Feldstein,
1974].Equatorwarddisplacementof the ionosphericboundary
The electricfieldsof (11) and (13), deducedfrom observations may be interpreted as the signatureof inward convectionof
of the equatorial convectionboundary identifiedat each local the plasmasheetin the equatorial plane, for the inner edgeof
time as the minimum distance to which newly injected low- the plasma sheet has been found to coincidewith the equaenergyparticlespenetrateduringa substorm,shouldin a steady torwardedgeof the auroral oval [Vasyliunas,1970b;Lossen,
state convection model correspondto the field (4) inferred 1974]. The convectionmodel can then be invoked to provide
from the location of the equatorial'plasmapauseidentified an estimateof the equatorial electricfield basedon the auroral
(13)
E = 0.39/[(1 - 0.086Kp)•'] kV/Re
as the radial distanceat whichthe gradientof coldplasma
zon.e
density increasesrapidly with radial distance. Despite the
problems discussedin the introduction of identifying the
plasmapausein a time-varying situation and despitethe fact
The linear fit to the Kp dependenceof the magneticlatitude
(XM) of the equatorwardboundary of the auroral oval shown
by Feldsteinand Starkov [1967] satisfies
that the two boundaries
are measured in different
local time
sectors,the three equationsare in good agreementover a large
range of Kp. For large values of Kp, one might expect the
relation found from the plasmapauseposition to be more
dependablethan (11) and (13) becausethe convectionboundary, which limits. the inward injection observed by a
geostationary satellite, lies increasinglyclose to the 'bulge'
region of the plasmapause(seeFigure 3) for increasinglylarge
E, and the 'bulge' region is known to be highly variable in
structure [Chappellet al., 1970b;Rycroft, 1975]. For E >• 1.5
kV/Rs, small departuresfrom the assumedsymmetry would
produce large changesin the local time at which injectionwas
first observed.Inspectionof Figure 2a suggests
that the linear
relation betweenLT and Kp may not be significantfor LT <
1800 or for Kp > 4. Figure 2b, on the other hand, showsthat
the only two measurementsat L T < 1800 (Kp = 6+ and 8-)
are consistent
with
the fit obtained
from
measurements
at
later local times. We shall return to this point in the next
sectionwhere it is suggestedthat, despitethe argumentspresented here, the data for large Kp are somewhat better approximated by (11) than by (4). The electric fields obtained
from (4), (6), and (11) agree within a factor of 2 over the
entire range 0 < Kp < 5. For Kp < 4.5, the electricfieldsof
(6) and (11) are bounded by the values previouslygiven by
Rycroft [1971] and by Rycrofi and Thomas [ 1970].
COMPARISON WITH
LoW-ALTITUDE
measurements.
MEASUREMENTS
The values of the electric fields found in (4), (6), (11) and
(13) can be comparedwith Heppner's[1972] measurementsof
polar cap electric fields. Heppner found that the integrated
potential drop acrossthe polar cap ranged from 20 to 100 kV,
with typical valuesfor moderateKp, near 3, falling in the range
40-70 kV. For comparison, the potential drops across an
assumed40-Rs magnetospherepredicted from the equations
mentioned are tabulated for severalvaluesof Kp in Table 1.
The predictions of (11) appear most closelyconsistentwith
Heppner'smeasurements.Equation (6) givesvaluestoo large
for small Kp and too small for large Kp.
XM = 65.2ø - 1.04øKp
(14)
in the vicinity of local midnight. To map this to the equator,
we note that in the Olson-Pfitzer model, the ATS field line
(L = 6.6) at quiet time maps to approximately65.2ø magnetic
latitude [Olsonand Pfitzer, 1974]. An estimateof the equatorial L value LA correspondingto the midnight auroral boundary is obtained by normalizing the dipole mapping so that it
correspondsto the Olson-Pfitzer model when Kp = 0. It follows that
LA = 1.16/[cos•' (65.2ø - 1.04øKp)]
(15)
If this boundary is also the inner edgeof the plasma sheet,
the strength of the electric field can be determined from the
convectionmodel. In this model, near the midnight meridian,
electron convection boundaries lie at greater radial distances
for more energeticparticles [SchieM,1969]. The inner edgeof
the plasma sheet should thus correspondto the convection
boundary evaluated for the low-energy end of the range of
electronenergiesfound in the plasma sheet(100 eV to 10 keV)
[Vasyliunas,1970b]. For these 100-eV electronsthe correction
to the zero energysolution is negligible[Kivelsonand Southwood, 1975], so (2) may be usedto give
E -- 22.5
2 -- 16.7cos
4(65.2
ø-- 1.04
øKp)--kV
LA
R•
(16)
and this is plotted in Figure 4. For small values of Kp the
correspondencewith the resultsobtained directly from equatorial observations(equation(11)) is close.For largevaluesof
Kp the mappingof field linesto the equatoris not realistic,and
the discrepanciesare not surprising.
Direct
measurements
of the electric field between
0100 and
0600 LT in the auroral ionospherehave shownthat thoughthe
magnitudeof the field increaseswith Kp, local turbulentvariations of the field dominate the weak effect [Mozer and Lucht,
1974]. The ambiguitiesinvolved in mapping electricfieldsbetween the equator and the auroral zone (discussedby Mozer
KIVELSON:MAGNETOSPHERIC
ELECTRICFIELDSAND GEOMAGNETICACTIVITY
TABLE 2.
193
Explorer 45 PlasmapauseObservations,
August 2-4, 1972
2.0
Orbit
LT
L••
E
E4
E•
(Kp)
818
819
820
821
822
823
824
825
826
827
828
13.84
13.02
12.04
12.04
11.31
10.19
ßßß
10.42
9.82
10.88
12.64
4.6
4.2
3.5
3.1
2.8
2.3
<2.1
2.2
2.2
2.6
3.8
1.4
1.5
1.8
2.4
3.0
3.5
>4.2
3.7
3.7
3.0
1.7
0.56
0.78
1.1
1.4
2.2
2.5
3.9
3.0
2.5
1.9
1.5
0.56
0.83
1.3
1.8
3.4
4.3
8.8
5.5
4.3
2.8
2.0
1.2
2.8
4.3
5.2
6.6
7.0
8.0
7.4
7.0
6.2
5.5
1.8
1.6
Observationswere by Hoffman et al. [1975]; Lvv is measureddistance to plasmapauseat the local time indicated, E is inferred from
(2), (Kp) is the average 3-hour Kp for the previous 24 hours, E4
and E• are obtained from (Kp) and (4) and (11), respectively.
polated to the encounter time and found a worse fit if they
usedKp for 2 hours postencounter,whereasKp of the 2-hour
preencounter time enhanced the scatter somewhat near dusk
0
i
2
3
4
5
6
but improved it slightly in the postmidnight region. These
featurescan be interpreted in terms of the assumedmodel. If E
Fig. 4. Convection electricfield versusKœ.Solid curve, empirical increaseswhen Kp increases,convection begins immediately;
thus freshlyacceleratedparticlesappear at the Alfv6n layer in
equation (11) œrominjection boundaries by lidauk and Mcllwgin
[1974]. Dashed curve œromthe equatorward boundary of the auroral the evening quadrant with delays of less than 2 hours, as
oval [Feldsteinand Starkov, 1967] mapped to the equatorial plane observedby Mauk and Mcllwain [1974] and Freeman [1974].
using (16).
On the other hand, adjustment of the plasmapauseto an
enhanced electric field can take a full day, for the plasma
which lies between the old and new Alfv6n layers must drift
and Lucht [1974]) limit the value of detailedintercomparisons,
around to the day side before the newly establishedopen drift
but typical valuesof quiet time auroral zone fieldsmappedto
paths diverge from the initial closed orbits. Only after a full
the equator exceed the values obtained by other means by
day will all flux tubes have convectedto the day side, and thus
factorsof 5-10. This overestimateprobably reflectsthe importhe new boundaries will be established.At the dawn plasmatant contribution
to auroral zone measurements of localized
pause the new equilibrium should occur with a delay of more
turbulent fields which are superimposedon the large scale
than 12 hours, which accountsfor Carpenter's result. Plasma
convectionfield [Mozer and Manka, 1971; Mozer et al., 1974].
convectingto the day side, having been presentin the critical
STORM TIME COMPARISONS
Observationsof plasmapauseerosion near local noon during the geomagnetic storm period, August 4-6, 1972, have
been describedby Hoffman et al. [1975]. These observations
provide a usefultest of the empiricalexpressionof this paper,
both becausethey apply to the predictionsat very large Kp,
and becausethey were made at local times excluded in the
measurementsusedto establishthe relation. Table 2 givesthe
observedparametersof the plasmapause,the convectionfield
strength required to place the plasmapauseat the observed
location (equation (2)), the mean 3-hour Kp in the previous24
hours, and the field strength calculated for this average Kp
from (4) and (11). The empirical relations and the values
inferred from observationsare plotted in Figure 5. On one
orbit the plasmapauselay insideperigee,so only a lower limit
for the field is indicated in the table and on the figure. Once
again, the agreementwith both (4) and (11) is good, even at
very disturbed times. Equation (6), obtained from the results
of Rycroft and Thomas[1970], is also plotted in Figure 5 and,
as in the previouscomparisonwith polar cap potential drops,
appearsto underestimatethe field for high valuesof Kp.
One point should be made regardingthe valuesof Kp used
in the above analysis.It has beennoted that Carpenter[1967]
associatedthe dawn plasmapausewith the largestKp observed
in the previous24 hoursbecausethis improvedthe correlation.
__M_auk
and Mcllwain [1974], on the other hand, usedKp inter-
/0
I
i
i
i
i
I
i
6
lower
ound
Fig. 5. Magnitude of a uniform dawn-duskelectricfield versusKp
for very disturbed times. The solid curve represents(11), the dashed
curve (4), and the dotted and dashed curve (6) [Rycroft and Thomas,
1970]. The dots are obtained from Explorer 45 observations of plasmapause erosion during a magnetic storm [Hoffman et al., 1975].
194
KIVELSON: MAGNETOSPHERICELECTRIC FIELDS AND GEOMAGNI:TIC ACTIVITY
flow region for most of a day, should be bestcharacterizedby
some long-time average of the geomagneticconditions, and
hencewe have usedthe stated24-hour averageKp.
2.0
1.8
RELATION
OF OTHER
MODELS
OF THE ELECTRIC FIELD
16
Analytic models of the convection electric field somewhat
more complex than the uniform dawn-dusk field have been
developed in recent years [Volland, 1973; Stern, 1975], and if
the Kp-dependent particle boundaries were interpreted in
terms of thesemodels,the empiricalrelationsbetweenKp and
field strength would be somewhat different. As a more complex model field varies in space,its magnitudemust be characterized by some averagevalue suchas the total potential drop
acrossthe 30-RE width of the magnetospherein the dawn-dusk
///
1.4
1.2
1.0
/
iiii
.8
.6
meridian.
A simplified version of Volland's [1973] field, attributed to
Stern, is used by Maynard and Chen [1975]. This VollandStern-Maynard-Chen (VSMC) field is symmetric about the
dawn-duskmeridian, and its amplitudeincreaseswith Kp; the
Alfvfin layer for Kp -- 3.7 is illustrated in Figure 6. Also shown
is the Alfvfin boundary for the uniform field correspondingto
Kp - 3.7 in (11) and a portion of the orbit of a geostationary
satellite. This diagram indicates that the field strengthsobtained from the local time versusKp fits of (8) and (9) would
be only slightly modified by use of the VSMC model, and this
point is confirmed by Figure 7, on which the filled circleswere
evaluated from the VSMC model by a procedureanalogousto
that used to derive (11). Numbers in parenthesesshow the
percent deviation of the electric field values from the curve
(equation (11)) and are seen to be of the order of 30%. The
large separation near dawn of the Alfvfin boundaries of the
uniform and VSMC models seen in Figure 6 suggeststhat
discrepanciesbetweenthe modelsmay increasepostmidnight.
For example, in the VSMC model the dawn plasmapause
never movesinsideL -- 4, clearly at variancewith Carpenter's
[1967] observations.Consequently,the model does not produce self-consistentrelations between field strength and Kp
when usedto interpret observationsin differenttime intervals.
For these reasonswe have preferred to use the uniform field
model, which seemsreasonablyconsistentwith most observations except very near the dusk meridian.
Other field modelswhich have'beengiven by Volland[1973]
and Mcllwain [1974] are illustrated in Figures8 and 9, and on
.4
i
i
1
Kp
Fig. 7. Convection electric field versus Kp. Solid curve, (11).
Filled circlesinferred from A TS 5 injection boundariesof Mauk and
Mcllwain [1974] and the Volland-Stern-Maynard-Chen electric field
model describedin the text. The diamond indicatesthe averageelectric
field across30 RE in the dawn-dusk meridian for the Volland [1973]
quiet time (Kp = 0-1) model; the squareindicatesthe same quantity
for the Mcllwain [1974] EH3 model. Numbers in parenthesesare the
percentdisagreementof the fields based on the two models. Dashed
curve is the fit of Chen and Grebowsky[1974] given in (17),
Figure 7 two points are included showing the average field
strength of the Mcllwain model and of the Volland 'quiet'
model, both basedon data for a Kp rangefrom 0-1. The points
fall closeto the curve representing(11). Volland's 'disturbed'
field, said to apply for Kp in the range 2-4, correspondsto an
averageelectricfield of 2.2 kV/RE, which is much larger than
the values given by (11). Note in Figure 8 that Volland's
'disturbed' field placesthe plasmapausenear dawn at L - 3.3,
i.e., at a distancenormally expectedfor Kp - 5. The suggestion therefore is that the 'disturbed' field may be better
assignedto the range 4-6 for Kp, which would bring its average magnitude into the range of the predictionsof (11).
A linear relation between E and Kp used by Chen and
Grebowsky[1974]
0000
E = 0.795(1 + •Kp)
eostationary
Satellite
i
0600
//
'
kV/Re
(17)
also plotted in Figure 7, seems to overestimate the field
strengthfor all Kp.
Roederer and Hones [1974] have developed a model timedependentelectric field to describethe injection of particlesat
geostationaryorbit during substorms.In their model the potential drop across 20 Rs in the dawn-dusk meridian plane
increaseslinearly for 10 min from 0 to 60 k V, then decreases
linearly to 0 in 120 min. The averageelectric field acrossthe
dawn-dusk meridian during the 130-min interval containing
the model substorm is 1.5 kV/RE, while for a full 3-hour
interval spanning an isolated model substorm the average
electricfield is 1.1 kV/RE. For E -- 1.1 kV/Rs, (11) givesKp =
Fig. 6. Zero energyAlfvfin layersand a portion of geostationary 4-, a value which is frequently regarded as 'moderately dis-
orbit. The dashed curve is based on a uniform
dawn-dusk
electric field
of 1.2 kV RE-1, which correspondsto Kp = 3.7 in (11). The solidcurve
is based on the Volland-Stern-Maynard-Chenfield describedin the
text and is evaluatedfor Kp -- 3.7, which impliesan averagefield of 39
kV RE-1 acrossa 30-RE magnetosphere.
turbed.'
The Roedeter-Hones
isolated substorm electric field is
therefore in reasonableagreementwith (11).
Kuznetsov[1972] has obtained a linear relation betweenE
and Kp by examination of morning-eveningasymmetriesof
KIVELSON;
MAGNETOSPHERIC
ELECTRIC
FIELDSANDGEOMAGNETIC
ACTIVITY
6h
195
e•
Fig. 8. Volland's[1973] equatorialelectricfield equipotentials,includingthe corotationfield. On the left is the quiet
time (Kp = 0-1) model. On the right is the disturbed(Kp = 2-4) model which, as describedin the text, may be more
correctly associatedwith Kp -- 4-6.
the flux of > 35-keV outer belt electrons. The innermost L shell
on which asymmetry is detected(L•<) is found to move earthward with increasingKp. Attributing the asymmetry to the
presenceof newly acceleratedplasma sheet electronson the
morning side, Kuznetsov interprets L•< as the boundary between open and closeddrift orbits for energeticelectrons.The
argumentsare similar to those usedto obtain (11), but as the
boundaryfor 35-keV electronslieswell outsidethe boundaries
for low-energyelectrons,his analysisis more sensitiveto departures from the dipole magneticfield used in both casesto
interpret the data. Furthermore, the accelerationof particles
by the strongly time-dependentmagnetic fields which frequently occur near midnight during substormswill systematically increasethe electric fields inferred from the morningevening asymmetries.Nonetheless,the relation obtained by
Kuznetsov [1972]
E = 0.87(1 + 0.71Kp) kV/RE
APPLICATIONS
From the injection boundariesdescribedby (9), Freeman
[1974] inferred the radial distanceat midnight to the inner
boundaryof convection(inner edgeof the plasmasheet)by
assuming that the particles forming the ATS 1 injection
boundary included only particlesof magneticmoment # and
greater. His results,basedon the electricfield of Vasyliunas
[1968], given here as (19), indicated that electrons of a wide
range of magneticmomentscould penetrateto geostationary
orbit at midnightfor moderatelydisturbedgeomagneticconditions.
We have reexaminedthe midnight plasma sheetboundary,
assumingfrom the injection spectrausedby Mauk and Mcllwain [1974] that the injectedparticleshave magneticmoments
of >0 and that consequentlythe plasma sheetat midnight is
composed of particles convected in as far as their Alfv6n
(18)
differsfrom (11) only by a factor of <3.2 in the rangeKp from
0600
0to8.
Finally, a comparisonwith the pioneer relation betweenE
and Kp [Vasyliunas, 1968] should be made. Vasyliunas obtained his expression
ELECTRIC
MODELS E3H, M2
FIELD
POTENTIALS
E - (2.$5 kV/RE)/(1 - 0.1gp) 2
(19)
by relating the distance to the dusk stagnation point in a
uniform field model to the electric field, but for the distance to
the stagnationpoint, he usedBinsack's[1967] measurements
20kV
of the distanceto the plasmapause
averagedover all local
times (L = 6 - 0.6Kp). The asymmetryof the Alfv6n layers
illustrated in Figure 3 shows that this procedure seriously
underestimatesthe distanceto the stagnationpoint. Indeed,
Binsack'sexpressi9nis identicalwith the Carpenterand Park
IOkV
[1973]
expression
forthedistance
tothedawnplasmapause
to
onesignificant
figure.
Ontheotherhand,intheuniform
field
model the ratio of the distancesto the plasmapauseat dawn
and dusk is 0.4. As E varies as the inverse square of the
OkV
distanceto the stagnationpoint, (19) shouldbe modifiedby
multiplying it by (,0.4)2, and it would then be approximately
the same as (4). (This argument has been given in on
unpublished
document:
A. J. Chen,personal
communicotion,i
1800 LOCAL TIME
ß
1975•.
•
'':•.•
Fig. 9.
Mcltwain's [1974] E3H model ctcctricfield for Kp = 0-1.
196
KIVELSON'
MAGNETOSPHERIC
ELECTRIC
FIELDS
AND
GEOMAGNETIC
ACTIVITY
0.4
boundaries.We have usedthe approximations
for finite •t
Ay(Re)
• (V•w/1000)(1
-- V•/1000)
2 (23)
electrons
givenby Kivelson
andSouthwood
[1975]andhave
included
corotation
fields.At midnight
for electron•
of 90ø whichfor V•w= 500gives•3 Rs forthewidthof themerging
pitchangleandsufficiently
large•, thedistance
totheAlfv6n region.
boundaryLM satisfies
Kp = 0 LIMIT
0.47B0•
21
Theempirical
relation
between
E andKp(equation
(11)) has
eERE
= L3• -• L3•2
(20)a nonzerolimitforKp-• O,andif thislimitisnotspurious,
it is
interesting,
foritsuggests
thepresence
ofanelectric
wherethe particlekineticenergyis •tBo/L
• in keY.(The physically
fieldin theabsence
of anyothersignificant
geomagnetic
activcoefficients
differslightly
when• isverysmall.)ValuesofLM
ity.
HoweVer,
the
position
of
the
plasmapause,
which
we
have
forsmall• anda range
ofKparegiveninTable3,fromwhich usedto determinethe field,cannotbe observed
at timesof
it is clearthatonlyparticles
withextremely
smallmagnetic
quietconditions
asit becomes
exceedingly
ill demoments
canreachgeostationary
orbitat midnight
forKp -- prolonged
fined
[Chappell
et
al.,
1970a].
If
particle
injections
occur
for
Kp
2. Thedifference
between
ourconclusion
andFreeman's
isdue
= 0, wearemorelikelyto conclude
thattheKpindexisnota
to the muchsmallerelectricfieldswe have used.
of thelevelofactivity
thanto believe
thatwe
An estimate
of the dependence
of the dawnsdusk
electric goodmeasure
fieldonthesolarwindvelocity(Vs,o),
analogous
to thosegiven
have learnedsomething
aboutconvection
fields.Unfortu-
nately,
present
dataareinsufficient
to testthesignificance
of
byVasyliunas
[1968]
andMendillo
andPapagiannis
[1971
],can thelimitingvalueof theconvection
electricfield.
befoundfrom(4) andtherelation
between
Vs,o
andKpobtained
bySnyder
etal. [1963],
which,
if expressed
interms
of
SUMMARY
the average3-hourKp in the previous
24 hours,is
The empirical
expression
(equation
(11)) E (kV/Rs) -0.44/(1- 0.097Kp)
• accords
wellwithobservations
of the
(21) dependence
onKpoftheaverage
positions
ofdiverse
particle
approximately
V•okm/s = 68Kp+ 330
boundaries
in themagnetosphere
andwithmeasured
polarcap
electricfieldsfor 0 < Kp •< 8. This relation,whichis
From(19)and(11)it follows
that
approximately
thesame
asthose
givenin (4) and(13),has
0.2 kV R•--1
beenfavored
primarily
because
it agrees
somewhat
better
with
E= [(1-- V.w
km/s)/1000]
2
(22)polarcapfieldmeasurements
andbecause
it predicts
storm
timelocationsof the plasmapause
quitewell.
Thevalidity,
evenina statistical
sense,
of(22)isrestricted
to
Naturally,
theuniform
fieldmodelmustbeused
withcare.
therange
of solarwindvelocities
whichSnyder
et al. usedto For example,
the position
of the Alfv6nboundary
in the
establish
(21).In theirwork,330km/s( V•,o( 750km/s,and electric
fieldgivenby(11)isnotexpected
to correspond
to a
theaverage
val,ue
wasV,,o- 504km/s.Obviously
then,one clearly
defined
gradient
in theplasma
density
during
intervals
shouldconclude
neitherthata convection
electricfieldexistsin ofdecreasing
magnetic
activity
whenfluxtubes,
emptybecause
theabsence
l•)•f
a solarwindnorthattheelectric
fieldbecomestheylieoutside
a previous
Alfv6n
layer,
havenotyetregained
infiniteat V•,o- 1000km/s. It is worthnotingthat (22) equilibrium
withtheionosphere.
Similarly,
because
of the
absence
of a sharpgradientat the plasmapause
at low Kp
corresponds
quiteclosely
to a one-pa,rlameter
quadratic
form
(E -- 3.2(Vs•o/l•00)
•',kV/Rs),
selected
togiveagreement
forthe
average
solarwindvelocity
previously
cited,anddiverges
considerably
• froman analogously
chosen
linearform(E = 1.6
[Chappell
etal., 1970a],
theposition
oftheboundary
depends
sensitively
on the criterion
usedto defineit (e.g.,density
greater
than10cm-sor 100cm-S),
andthisdependence
onan
(V•w/1000)).
Thisobservation
supports
th•[heoretical
con, arbitrary
criterion
canpresent
problems
foranalysis
of quiet
clusion
ofMendillo
and
Papagiannis
[1971
] that
tl-ie
convection
time properties.
electricf•eldvariesmQyenearlyqhadra•teal15
' thanlinearly
Although
a relation
between
theconvection
fieldandKpcan
bevalidatbestonlyina statistical
sense,
it should
beuseful
for
From(22),onecanestimate
thesizeof thedayside'merg- calculating
theexpected
driftpathsofparticles
duringmagneting'region
required
to accouht
fortheconvection
field,again ically
active
times.
Equation
(11) may,forexample,
beapplied
limiting'applications
to330km/s_(V•o'• •50km/s.
Fora to furtheranalysis
of theredistribution
of low-energy
plasma
solarwindmagnetic
fieldof •5 'y,w:itha component
of the during
geomagnetically
active
timesof thetypeinitiated
by
orderof 3 'y parallelto theearth'sdipole,thewidthof the ChenandGrebowsky
[1974,andreferences
therein].
In addimerging
region(Ay)across
thedaysidescales
as
tion,thebehavior
of particles
of nonzero
energy,
whichare
alsoaffected
byconvection
electric
fields,
canbestudied
by
using
theestimated
fields.
Thevariation
inthefieldamplitude
mayalsoprovide
anestimate
of thepower
in
TABLE3. Dista•n9es
in EarthRadiito theAlfv6nLayers
onthe asKpchanges
Midnight
Meri•ittn
for Electrons
of Magnetic
Moment
# for
low-frequency
electric
fieldvariations
which
maybeofusein
with the solar wind ve16city.
different Kp
studies
of radialdiffusion
[Fiiltharnrnar,
1968].However,
asKp
#, keV/'•
0
1
2
3
4
5
0
0.05
7.5
8.7
6.7
8.0
5.7
7.1
5.0
6.5
4.3
5.9
3.8
5.3
0.1
9.6
8.8
7.9
7.3
6.6
6.1
0.16
10.3
9.6
8.6
7.9
7.3
6.7
is itselfaveraged
overbothtimeandspace,
it canat best
characterize
onlylargescale
activity
andtime-averaged
behavior,Accordingly,
theelectric
fieldsassociated
withKpin the
results
of thispapershould
beapplied
to theanalysis
oftimeaveragedlargescalebehavioronly.
Acknowledgments.
Stimulating
discussions
withC. T. Russell
are
gratefully
acknowledged.
Thisworkwassupported
byNSFthrough
KIVELSON: MAGNETOSPHERICELECTRICFIELDS AND GEOMAGNETICACTIVITY
grant DES 74-23464.This article is Institute of Geophysicsand Planetary Physicspublication 1478-82.
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