method of characteristics in spherical geometry applied to a harang

Method of characteristics in spherical geometry appliedto a Harang-discontinuity situation

O. Amm

Technical University of Braunschweig, Institute of Geophysics and Meteorology, Mendelssohnstr. 3, D-38106 Braunschweig, Germany

Received: 7 April 1997 / Revised: 28 July 1997 /Accepted: 3 October 1997

Abstract. The method of characteristics for obtainingspatial distributions of ionospheric electrodynamic pa-rameters from ground-based spatial observations of theground magnetic disturbance and the ionosphericelectric ®eld is presented in spherical geometry. Themethod includes tools for separation of the externalmagnetic disturbance, its continuation to the iono-sphere, and calculation of ionospheric equivalent cur-rents. Based on these and the measured electric ®elddistribution, the ionospheric Hall conductance is calcu-lated as the primary output of the method. Byestimating the Hall- to-Pedersen conductance ratiodistribution, the remaining ionospheric electrodynamicparameters are inferred. The method does not assumer�~E � 0 to allow to study time-dependent situations.The application of this method to a Harang disconti-nuity (HD) situation on 27 October 1977, 17:39 UT,reveals the following: (1) The conductances at andnorth of the HD are clearly reduced as compared to theeastern electrojet region. (2) Plasma ¯ow across the HDis observed, but almost all horizontal current is divertedinto upward-¯owing ®eld-aligned currents (FACs)there. (3) The FACs connected to the Hall currentsform a latitudinally aligned sheet with a magnitudepeak between the electrically and magnetically de®nedHD, where break-up arcs are often observed. Theirmagnitude is larger than that of the more uniformlydistributed FACs connected to the Pedersen currents.They also cause the southward shift of the magneticallyde®ned HD with respect to the electrically de®ned one.(4) A tilt of the HD with respect to geomagneticlatitude as proposed by an earlier study on the sameevent, which used composite vector plot technique, andby statistical studies, is not observed in our single time-step analysis.

Key words. Ionosphere � Electric ®elds and currents �Instruments and techniques � Magnetospheric physics �Current systems

1 Introduction

During and after the International MagnetosphericStudy (IMS, 1976±1979), several methods have beendeveloped to obtain instantaneous spatial distributionsof the ionospheric electrodynamic parameters fromground-based measurements of the ground magneticdisturbance ®eld together with the ionospheric electric®eld. A detailed review of these e�orts is given byUntiedt and Baumjohann (1993). The two main ap-proaches have been the ``trial and error'' or three-dimensional modeling method and the method ofcharacteristics.

In the ``trial and error'' approach (e.g., Baumjohannet al., 1981; Opgenoorth et al., 1983), measurement-based models of the ground magnetic and ionosphericelectric ®eld distributions are used together with modelsof the unknown Hall and Pedersen conductances to varythe latter until the calculated ground magnetic distur-bance su�ciently ®ts the measured one. An advantageof this method is that it can be led through with virtuallyany data coverage. However, especially (though notonly) in case of sparse data, due to the nonuniquerelation between a certain ground-magnetic ®eld and theionospheric electrodynamic quantities that produce it,the range of possible models that ®t the observations toa speci®ed degree may be quite large, and no errorestimation can be provided for the ®nal results. Anotherdisadvantage of the ``trial and error'' method is that toreproduce the ground magnetic disturbance on a certainmodel area, the ionospheric quantities have to bemodeled on a considerably larger area. Since for thatlarger area often not enough data of a single time-stepare available, it becomes necessary to produce the input

Also at: Finnish Meteorological Institute, Geophysical Research,P.O. Box 503, FIN-00101 Helsinki, Finland Fax: +49 531 3915220; e-mail: [email protected]

Ann. Geophysicae 16, 413±424 (1998) Ó EGS ± Springer-Verlag 1998

data by the ``composite vector plot'' approach whichtranslates temporal changes of the observed quantitiesinto spatial distributions by assuming a stationarystructure moving with a certain velocity over the ®eldof view of the measurements (e.g., Kunkel et al., 1986).These assumptions limit the ability of the method toobtain instantaneous output distributions since thestationarity assumption may not hold, and for theuncertainties in the velocity estimation.

The method of characteristics (Inhester et al., 1992;Amm, 1995), on the contrary, is a forward method inwhich the ionospheric electric and ground magnetic ®eldmeasurements are used to solve a ®rst-order di�erentialequation for the Hall conductance ®rst. Using themeasured electric ®eld and a modeled Hall-to-Pedersenconductance ratio distribution, the remaining electro-dynamical quantities are then inferred from it. Thetypical range of the conductance ratio is relativelynarrow, between 1 and 2 for quiet periods (e.g.,Schlegel, 1988), increasing up to about 3 under dis-turbed conditions and about 5 for substorm situationsassociated with discrete aurora (e.g., Kirkwood et al.,1988; Olsson et al., 1996; Aikio and Kaila, 1996; Lesteret al., 1996; however, some extreme values up to 10directly inside an auroral break-up have been reportedby Olsson et al., 1996), and the e�ect of its modeling onthe ®nal results has been shown to be small by Amm(1995). Moreover, additional optical or riometer datamay be used for a rough estimate of this ratio. Usually,there are (mostly small) regions where the solution forthe Hall conductance is nonunique. However, theseregions are known, and an error estimation is available.No data are needed outside the actual area under study.Since divergence and curl of the input data have to beestimated, a reasonable data coverage inside that area isrequired.

Up to now, both methods have only been used inplanar geometry, mostly in the ``Kiruna coordinatesystem'' de®ned by KuÈ ppers et al. (1979) which is astereographic projection of the earth's surface to atangential plane that touches the earth near Kiruna,Sweden. Whereas neglecting the curvature of earth'ssurface may be justi®ed for small regions of interest,with the new possibility of obtaining electric ®eld dataover nearly half of the northern auroral zone with theSuperDARN radar (Greenwald et al., 1995), themethods should consequently be extended to sphericalgeometry.

An essential di�erence between the method of char-acteristics and the KRM (Kamide et al., 1981), AMIE(Richmond and Kamide, 1988), and IZMEM (Pap-itashvili et al., 1994) methods that are developed forspherical geometry, too, lies in their di�erent primaryinput and output quantities: the latter methods use theground magnetic ®eld (KRM), interplanetary magnetic®eld (IZMEM), or variable sets of measurements(AMIE) to deduce the ionospheric electric potential,and the conductance distributions are input to them.Since no large-scale spatial conductance measurementsare available, these distributions need to a large extentto be taken from statistical models. Whereas this does

no harm when studying ``typical'' or averaged situa-tions, it may restrict the ability of these methods tohandle special, instantaneous events. Spatial measure-ments of the ionospheric electric ®eld as needed as aninput to the method of characteristics are available fromcoherent-scatter radars. However, the application of themethod is limited to situations in which the radarsreceive enough backscatter to provide electric ®eld dataon a su�ciently large area.

In this paper, we will derive the method of charac-teristics for spherical geometry, including tools forextraction of the external part of the ground magneticdisturbance, its continuation to the ionosphere andcalculation of the ionospheric equivalent currents.

We will then apply the method, using magnetic ®elddata from the Scandinavian Magnetometer Array(SMA) (KuÈ ppers et al., 1979) and electric ®eld datafrom the STARE coherent-scatter radar (Nielsen, 1982),on a Harang-discontinuity situation on 27 October1977. We have selected this event for three di�erentreasons: (1) Densely spaced ground magnetic ®eld datafrom the SMA and STARE electric ®eld data with goodbackscatter directly over that array are available. (2)This event was studied earlier by Lampen (1985) [mainresults shown in Untiedt and Baumjohann (1993)], sothat we can compare our results using the method ofcharacteristics with those gained by the ``trial and error''method. (3) Finally, with the ®ndings of this study wetry to address some questions raised by Koskinen andPulkkinen (1995) in their review on the physics of theHarang discontinuity.

2 Theory

2.1 General

The method of characteristics requires as input distri-butions of the ground magnetic ®eld disturbance ~BG, theionospheric electric ®eld ~E, and the ratio of the Hall toPedersen conductance a. The ®rst task is to extract theexternal part of ~BG, then continue it to the ionosphericheight, and therefrom derive the ionospheric equivalentcurrents ~Jeq;Ion (Sect. 2.2). The ionosphere is treated asan in®nitely thin conducting layer at radiusRI � RE � 100 km, with RE being the earth's radius.Next, we have to prove that ~Jeq;Ion is equal to thedivergence-free part of the real horizontal ionosphericsheet currents, ~Jdf (Sect. 2.3). Then, the ``core equa-tions'' of the method are discussed in Sect. 2.4. Finally,in Sect. 2.5, we take a closer look at time-dependentsituations.

As in previous studies (e.g., Fukushima, 1976;Untiedt and Baumjohann, 1993), we assume that theearth's main magnetic ®eld is directed straight andperpendicular to the ionosphere. Moreover, we assumethat there is no deviation between the magnetic ®eldlines and the ¯ow direction of the ®eld-aligned currentsjk (FACs). Consequently, in spherical geometry theFACs ¯ow radially. Deviations of the real FAC ¯owfrom that direction are in our context only important

414 O. Amm: Method of characteristics in spherical geometry applied to a Harang-discontinuity situation

because of the ground magnetic ®eld that this deviationcauses (see Sect. 2.3), which has been shown to be smallin polar latitudes (cf. Richmond, 1974; Fukushima,1976; Tamao, 1986; Untiedt and Baumjohann, 1993,their Fig. 12; Amm, 1995). However, in mid- andespecially equatorial latitudes the e�ect of that deviationbecomes important, so that there we cannot use themethod described in this paper (see also Sect. 2.4). Adetailed discussion on the ground magnetic e�ect ofstraight but tilted FACs in planar geometry, includinganalytical expressions, and on the possible, but smallerrors in the output quantities of the method ofcharacteristics related to that can be found in Amm(1995).

2.2 Separation of the external magnetic disturbance ®eld,continuation to the ionosphere and calculation of iono-spheric equivalent currents: spherical cap harmonicanalysis

Decomposition of the magnetic ®eld disturbance ~BGinto its parts due to internal and external sources,and its height continuation by means of a sphericalharmonic analysis (SHA) go back to Gauû and isfully elaborated as early as in the classical book ofChapman and Bartels (1940). However, since thebasis functions of SHA extend over the whole earth,problems appear if the area of interest and ofmeasurements is con®ned to a part of the earth'ssurface only: the SHA coe�cients will then bepoorly de®ned, or ``virtual'' data points have to beadded. Moreover, since the maximum wavelengthresolution of an SHA analysis is kmin � �2pRE�=nmaxwhere nmax is the maximum degree used in theanalysis, for a desired resolution of kmin � 400 km,nmax � 100, i.e., a total number of 10201 coe�cientswould be required.

A way out of these problems is provided by sphericalcap harmonic analysis (SCHA) (Haines, 1985). Let usassume that a given data set can be covered by aspherical cap with midpoint �#p;up� (in geographicalcoordinates) and a half-angle #0 of the cap. The SCHAexpansion of the magnetic potential U in the current-freeregion between r � RE and r � RI in the sphericalcoordinate system with the midpoint of the cap as thenorthern pole is (Haines, 1985)

U�r; #;u� � RE

"XKi

k�0

Xk

m�0

RE

r

� �nk�m��1P m

nk�m��cos#�

� gm;ik cos�mu� � hm;i

k sin�mu�� XKe

k�0

Xk

m�0

rRE

� �nk�m�P m

nk�m��cos#�

� gm;ek cos�mu� � hm;e

k sin�mu�� #: �1�

The ®rst term of this equation gives the internal, thesecond sum the external part of the magnetic potential.Both of them can independently be calculated atdi�erent radii by inserting respective values for r. Thestructure of Eq. (1) is similar to the corresponding oneof SHA, but to yield appropriate basis functions on thecap, the SHA integral degree n has to be replaced by aSCHA nonintegral degree nk�m� where k � 0; . . . ;Ki forthe sum of internal coe�cients (index `i') andk � 0; . . . ;Ke for external coe�cients (index `e') is aninteger, with Ki and Ke determining the number ofinternal and external coe�cients to be taken intoaccount for the expansion of the potential. The nk�m�are determined by the boundary conditions for theassociated Legendre functions P m

nk�m��cos#� at # � #0dP m

nk�m��cos#0�d#

� 0 for k ÿ m even;

P mnk�m��cos#0� � 0 for k ÿ m odd;

�2�

i.e., for a given m and #0, those Legendre functionswhich ful®l Eq. (2) are searched with increasing nk�m�and are indexed by k. Accordingly, a de®nition ofLegendre functions is needed that does not rely on aninteger degree n (Hobson, 1931; Haines, 1985):

P mn �cos#� � Km

n � sinm #

� F mÿ n; n� m� 1; 1� m;1ÿ cos#

2

� �;

�3�where F �a; b; c; x� is the hypergeometric function and Km

nare normalization factors [in geophysics usually Schmidtnormalization, e.g., Chapman and Bartels (1940)].

One important fact to note is that nk�m� � k (seeTable 1 for an example with #0 � 20� and k � 0; . . . ; 8�.

Table 1. Spherical cap noninteger degrees nk�m� for cap half angle #0 � 20�

# km! 0 1 2 3 4 5 6 7 8

0 0.001 6.38 4.842 10.49 10.49 8.363 15.31 14.79 14.26 11.694 19.60 19.60 18.75 17.86 14.935 24.29 23.97 23.64 22.53 21.36 18.136 28.65 28.65 28.09 27.52 26.20 24.79 21.297 33.28 33.04 32.81 32.05 31.28 29.78 28.18 24.438 37.67 37.67 37.25 36.83 35.91 34.97 33.30 31.52 27.55

O. Amm: Method of characteristics in spherical geometry applied to a Harang-discontinuity situation 415

The minimum wavelength resolution of SCHA is similarto that already given for SHA, but with nmax substitutedby nk�m�max. For the example as given in Table 1, anSHA analysis with the same number of coe�cientswould have nmax � 8. As seen from the table, for SCHAnk�m�max � 37:67 holds. Hence, for our example we seethat with the same number of coe�cients, SCHA yieldsan about 4.7 times higher spatial resolution than SHA,in addition to the already-stated advantage that nounsampled data outside the spherical cap is needed.

The coe�cients gm;�

le

k and h

m;�

le

k are determined by

®tting ~BG � ÿrU to the given observations. Then, theexternal part of the magnetic disturbance is extractedand the height continuation is performed by taking onlythe second sum and setting r � RI in Eq. (1). Computerprograms for these purposes were given by Haines(1988).

Two notes of caution should be given: (1) As notedby Haines (1985), errors can occur at the boundary# � #0 during the upward continuation and spreadinward with increasing continuation height. Hainesattributed these errors to the missing independentcontrol of the second and higher derivatives at thatboundary. From our experience with the continuationfrom the earth's surface to the ionosphere, it is enoughto select #0 1±2� larger than demanded by the datapoints or the region of interest to prevent this e�ect fromin¯uencing that region. (2) Torta and de Santis (1996)found that SCHA may encounter numerical problems inthe process of ®eld separation when long wavelengthsare modeled over small caps (similar to problems thatoccur in Fourier transforms when wavelengths longerthan the transformation area are present). Their and ourown experiences agree in that these errors can becomecritical for the small and often quite homogeneousinternal part, but are not remarkable for the larger,usually more structured external part which is solely ofinterest for this study.

In planar geometry, the relation between the mag-netic ®eld disturbance immediately below or above aplane and the related equivalent currents in that plane islocal �~Jeq�~r� � 2=l0 �z�~B�~r��, where z is a unit vectorperpendicular to the plane and ~B the magnetic ®eld inthe immediate vicinity of the plane, if it is left in zdirection) since any segment of ~Jeq causes a horizontalmagnetic disturbance directly below and above itsposition only, and elsewhere in the plane a purelyvertical one. In spherical geometry, this relation be-comes nonlocal. Expressed in spherical (cap) harmonics,where Ue

k denotes the k-th harmonic of the externalmagnetic potential [i.e., the external part of Eq. (1)inside the k sum] immediately below the ionosphere, andWk the corresponding k-th harmonic of the ionosphericequivalent current function as de®ned by~Jeq;Ion � �rW,we get [cf. Richmond, 1974; Haines and Torta, 1994; thefactor 10=4p that appears in Chapman and Bartels(1940) and Kamide et al. (1976) is related to cgs units]

Wk � 1

l02ÿ 1

nÿ 1

� �Ue

k: �4�

Note that Eq. (4) can be inserted into Eq. (1) tocalculate W from the SCHA coe�cients in a one-stepprocedure. The constant factor 2=l0 is the same as inplanar geometry and represents the local part of therelation between W and Ue, whereas the second term inbrackets is responsible for the nonlocal part. The latterdisappears for n!1, corresponding to k! 0 when thecurvature of the sphere becomes negligible. To show thisnonlocality, Fig. 1 displays the horizontal magnetic ®eldof a constant sheet current of 1000 mA/m, ¯owingeastward with its center at constant latitude of 70� (thesheet current has 1� of width and extends over 180�longitude), as calculated by direct Biot-Savart integra-tion. To make the small vectors due to the nonlocal partof Eq. (4) visible, we deleted the 637-nT vectorsproduced by the local part at the center latitude of thecurrent ¯ow. In the case shown, the nonlocal vectorshave a magnitude of about 0.6% of the local ones at 1�latitudinal distance of the current, and decrease to 0.18%at 5� latitudinal distance.

In some studies, the three steps presented in thissection (i.e., extraction of the external part of themagnetic potential, its continuation to the ionosphere,and calculation of equivalent currents consideringspherical geometry) are completely skipped (e.g., LuÈ hrand Schlegel, 1994; Stauning, 1995), and the ionosphericequivalent currents are approximated as ~Jeq;Ion � 2=l0�~BG � r�, i.e., as equal to the total equivalent currents onthe ground. However, we feel that although dependingon the type and purpose of the study, considerable errorscan be introduced, especially if the upward continuationis skipped. Typically, the ionospheric equivalent cur-rents are not only larger than those on the ground, butmay also have a somewhat di�erent structure, since highwave numbers are ampli®ed in the process of upwardcontinuation [cf. Eq. (1), compare Sect. 3].

Fig. 1. Magnetic ®eld of an eastward-¯owing ionospheric sheetcurrent of 1000 mA/m at 70� latitude, immediately below theionosphere; the 637-nT magnetic ®eld vectors at the latitude ofcurrent ¯ow have been deleted to show the small ``nonlocal'' e�ectbetween current and horizontal magnetic ®eld on the sphere


2.3 Ionospheric equivalent currents and divergence-freepart of real ionospheric currents

As with any vector ®eld, we can decompose the realionospheric sheet current density~J into a curl-free and adivergence-free part uniquely:

~J �~Jcf �~Jdf : �5�For ~Jeq;Ion to be equal to ~Jdf it is thus required that ~Jcftogether with its accompanying FACs

jk � rh �~J � rh �~Jcf �6�causes no magnetic ®eld below the ionosphere (thesubscript h denotes the horizontal part of the divergencevector operator). This can be most easily proved bydecomposing ~J into spherical curl-free and divergence-free elementary current system (Amm, 1997; note thatthis can be done regardless of any considerations onconductances or the electric ®eld), and then proving thatthe curl-free elementary system together with its FACshas the desired property. The curl-free elementarysystem is

~Jcf ;el � I0;cf

4pRIcot�#0=2�e#0 ; �7�

where #0 � 0 is called the pole of the elementary system.Equation (7) is similar to a system that was attributed toPedersen currents in an earlier study by Fukushima(1976). There and in Amm (1997) proofs are given thatthe magnetic e�ect of the this system together with itsradially ¯owing FACs is zero below the ionosphere.Hence, we have

~Jeq;Ion �~Jdf �8�

2.4 Core equations of the method of characteristics

To summarize shortly our start equation, we have~Jeq;Ion � Fct.�~BG� from Sect. 2.2, and Eq. (5) togetherwith Eq. (8) as the link between ~Jeq;Ion and ~J fromSect. 2.3. Also noted in that section is Eq. (6) as therelation between ~J and its FACs that comes from thedivergence-freeness of the total current system. In itsgiven form, it is valid for the earth's main magnetic ®eldbeing perpendicular to the ionosphere, as assumed.Finally, we need Ohm's law which, under the sameassumption, has the form

J#Ju

� �� RP ÿRH

RH RP

� �� E#

Eu

� �: �9�

However, in the real case the preceding assumption isnot ful®lled and the tensor R in Eq. (9), includingpolarization e�ects, should read (e.g., Amm, 1996)

R �R0RP

CR0RH �ÿ cos e�

C0R0RH cos e

C RP � R2H sin2 e

C

!; �10�

where R0 is the conductance parallel to the magnetic®eld, e the angle between the magnetic ®eld lines and the

normal on the ionosphere, and C � R0 cos2 e� RP sin

2 e.The question arises, what errors are introduced for R bythe assumption of a radial main magnetic ®eld. Figure 2shows the tensor elements of Eq. (10) for a dipole mainmagnetic ®eld, with RP � 1 and RH � 2 (in relativeunits) which are also indicated in Fig. 2 by horizontaldotted lines. The deviation between the tensor elementsof Eqs. (9) and (10) is nearly unnoticeable for # � 30�,i.e., polar latitudes, still acceptable for # � 45�, but thenincreases dramatically toward the equator. Thus, inaddition to the considerations of Sect. 2.1, this isanother reason why the method presented cannot beused in low latitudes.

After the start equations are veri®ed, the remainingconsiderations are completely similar to those for planargeometry given by Inhester et al. (1992) and Amm(1995). This is because the remaining ``core equations''of the method of characteristics are local and can thusbe used in any geometry as long as the start equationshold (of course, the di�erential operators must beexpanded according to the respective geometry). Forcompleteness, we give a short summary of thoseequations and refer to the already cited for more detail.

We de®ne a vector ®eld

~V � ~E ÿ aÿ1~E � er �11�and two scalar ®elds

C � rh � ~V ; �12�D � ÿ r�~Jeq;Ion

� �r: �13�

By combining Eqs. (9) and (8) we obtain

1

RI

@RH

@#V# � 1

RI sin#

@RH

@uVu � Dÿ CRH : �14�

This ®rst-order partial di�erential equation can be splitup into many ®rst-order ordinary di�erential equations

Fig. 2. Components of conductance tensor R�, including polarizatione�ects, for a dipole main magnetic ®eld; in relative units, withRH � 2and RP � 1 (see horizontal dotted lines)


dRH �~r�`��d`

� C�~r�`��j~V �~r�`��j � RH �~r�`�� D�~r�`��

j~V �~r�`��j �15�

where ~r�`� are the characteristics of Eq. (14) with thegeometric path length ` de®ned by

d

d`~r�`� � 1

j~V �~r�`��jV#�~r�`��=RI

Vu�~r�`��=RI sinu

� �; �16�

i.e., they are everywhere tangential to ~V . Equation (15)can be solved straightforwardly to

RH �~r�`�� RH �~r0�eÿI�0;`� �Z `

0

D�~r�`0��j~V �~r�`0�j e

ÿI�`0;`�d`0 �17�

with

I�`0; `� �Z `

`0

C�~r�`00��j~V �~r�`00��j d`

00 �18�

The solution provided by Eq. (17) consists of a termdependent on the unknown initial value RH �~r0� (often,but not necessarily, given on a boundary point of theregion of interest) and another part solely determined bythe input quantities. The in¯uence of RH �~r0�, however,will decrease exponentially with (positive) I�0; `� andwill therefore usually be marginal for most part of acharacteristic. If I�0; `� is mainly negative along acharacteristic, the direction of integration is changedfrom �~V to ÿ~V direction, causing the sign of I toswitch, too. Since characteristics intersect the boundaryof the region under study twice (except for when theyreach a ``singular point'', see below), and the direction ofintegration is chosen as already described such that thecharacteristics spread up in that direction, boundaryvalues have to be given on at most 50% of the boundary,often much less (see Sect. 3). That part of the boundaryis called ``in¯uencing''.

An error estimation for regions where the solution ofRH is nonunique since the in¯uence of the unknownRH �~r0� persists, can be obtained by selecting an upperand lower physically reasonable RH �~r0� and integratingEq. (17) with both along the respective characteristic.

Two situations can occur for which RH �~r0� is known.First, there may be a direct pointwise conductancemeasurement available, or, second, an isolated pointwith ~E � 0 and rh �~E 6� 0 [called ``singular point'', seeAmm (1995)] is present at which RH � D=C is knownfrom the input directly. From the singular points,characteristics emerge in all directions, and they arethe only points where characteristics can meet (notintersect). In both cases, the points where RH �~r0� isknown will be selected as starting points for theintegration along the characteristic(s) passing throughit and make the solution of RH unique along them.

2.5 Time-dependent situations

It should be explicitly stated that the method ofcharacteristics does not require �r �~E�r � 0. Thus, itis able to handle nonstationary situations when largetime derivations of ~B cause a considerable amount of

�r �~E�r, as it may happen when large-magnitudeplasma waves are incident to the ionosphere (e.g.,Glaûmeier, 1988).

However, for the ®eld separation and upwardcontinuation in Sect. 2.2 we assumed that a magneticpotential exists, i.e., r�~B � 0 holds. This neglects thedisplacement currents caused by @~E=@t in nonstationarysituations. For purity, one could treat the upwardcontinuation in such a case as an inverse wavepropagation problem and use methods similar to seismicmigration to solve it. However, for several reasons wefeel that this e�ort would be inadequate and unneces-sary: (1) Between the ionosphere and the earth the wavestravel as electromagnetic waves with phase and groupvelocity c, and need only 0.3 ms to reach the earth'ssurface. The typical sampling interval of magnetometersis 10 s (that of radars mostly longer). Thus, according toNyquist's theorem the minimum cycle length of a waveresolvable is 20 s. These waves can therefore completeonly 1:5 � 10ÿ5 wave cycles. (2) On the spatial scale,magnetometer stations are typically not less than100 km apart, so the minimum wavelength they canresolve is 200 km. Thus, on a distance of 100 kmbetween the ionosphere and the earth's surface, only halfof a wavelength of the waves of interest can evolve. (1)and (2) together result in wave fronts that are quiteexactly planar in the range of interest. (3) Withapproximate values of ~E � 50 mV/m, ~B � 100 nT,@~E=@t � 10 mVmÿ1=10 s and @~B=@t � 10 nT/10 s, weget r�~B=j~Bj � 10ÿ5 r�~E=j~E|. Therefore neglectingr�~B can be justi®ed.

3 Data analysis on a Harang-discontinuity situation

In this section, we will apply the method of character-istics in spherical geometry as introduced in theprevious section to a Harang-discontinuity (HD)situation on 27 October 1977. This event is particularlysuitable to evaluate the results of our method since itwas studied earlier by Lampen (1985) by means of a``trial and error'' analysis. He composed observationsbetween 16:50 and 18:10 UT to a single distributionusing the ``composite vector plot technique'' [e.g.,Untiedt and Baumjohann (1993), where the mainresults of Lampen's work are also shown]. To avoiduncertainties that are inherent to this technique even ifmost carefully done, i.e., the need of a stationarityassumption and the velocity estimation of the sta-tionary structure, and to exploit the capabilities of ourmethod, which does not rely on the reproduction of theground magnetic ®eld and thus does not need toestimate the ionospheric currents on a much larger areathan that of actual interest, in this study we willperform a single time-step analysis. The need for suchan analysis was also emphasized by Koskinen andPulkkinen (1995) when they pointed out that the HDmay move rapidly in latitude even within a few minutes.We will discuss some similarities and di�erencesbetween the study of Lampen (1985) and ours furtheron in this section, and give a more general discussion of


our results in the context of previous works on the HDin the next.

We use magnetic ®eld data of the former Scandina-vian Magnetometer Array (SMA) (KuÈ ppers et al., 1979)and electric ®eld data of the STARE coherent-scatterradar (Nielsen, 1982). No all-sky camera data areavailable. The Kp index for the time under study is 5(15±18 UT), followed by 7 for 18±24 UT. The magne-tograms available suggest that our event takes placeduring the expansion phase of a substorm, and anaverage equatorward motion of the HD is expected. Forour single time-step analysis, we choose the time17:39 UT. At that time, the HD is located immediatelyabove the densest area of the SMA, and STAREreceives a good backscatter there. The latter fact is ofspecial importance, since a single time-step analysis ismainly restricted to the immediate ®eld of view of themeasurement devices. In our case, the STARE ®eld ofview is much smaller than that of the SMA. Figure 3shows the SMA ground magnetic, Fig. 4 the STAREelectric ®eld measurements. For the latter, we took onlyinto account vectors with at least 6 dB of backscatterintensity for both STARE radars to exclude possibleerratic vectors. The SMA data are shown in Fig. 3 as90� clockwise-rotated magnetic ®eld vectors (cf. discus-sion in Sect. 2.2). An eastward electrojet can be seenover Scandinavia between about 62� and 66� latitude,with an increasing northward component towardsnorth. Its maximum magnitude is about 250 nT at 64�latitude. Northward of that jet, the vectors turn topurely northern direction, indicative for the magneticHD, near 67:8� latitude, with magnetic ®eld distur-bances near 70 nT. Further northward, a westwardelectrojet with a large northward de¯ection is observedwith a slightly larger magnetic disturbance than at theHD reaching 100 nT.

The STARE radar provides good backscatter be-tween 67:6� and about 71� latitude (Fig. 4). The electricHD can be located quite precisely at 68:6� latitude,where vectors are present which point nearly directlywestward with about 25 mV/m of magnitude. Thevectors south of the HD, pointing northward with slightwestward deviation, have magnitudes of about 30 mV/m, whereas the vector population north of the HDbetween 69� and 70� latitude reaches up to 50 mV/mand points southward, with mostly small westwarddeviations, but also some eastward de¯ected vectors eastof 19� longitude.

We carry out our analysis in the area between 65�and 71� latitude and 15� and 24� longitude. Thisinvolves some southward extrapolation of the electric®eld in the eastward electrojet region, as was alsoapplied by Lampen (1985). An inverse distance methodwas used for this extrapolation. This approach isjusti®ed since the electric ®eld inside the eastwardelectrojet region has been reported as quite homoge-neous (e.g., Baumjohann et al., 1980; LuÈ hr et al., 1994).The ®nal electric ®eld distribution used as an input toour analysis is shown in Fig. 5b. Figure 5a shows theionospheric equivalent currents~Jeq;Ion as they result fromthe separation of the external part of the magneticdistrubance ®eld, upward continuation of that part tothe ionosphere and calculation of the equivalentcurrents as discussed in Sect. 2.2. In addition to thegeneral increase in magnitude, a comparison with Fig. 3indicates that the basic structure of the rotated groundmagnetic vectors is preserved, but slight changes can beseen, e.g., in the northwest of the analysis area where thenorthern de¯ection of the westward pointing vectors isreduced. As a ®nal input quantity, the modeled Hall-to-Pedersen conductance ratio a is displayed in Fig. 5c. A

Fig. 3. By 90� clockwise-related horizontal ground magneticdisturbance measurements of SMA and other ground magnetometersfor 27 October 1979, 17:39 UT. Latitudes and longitudes aregeographic

Fig. 4. STARE electric ®eld data for 27 October 1979, 17:39 UT,with Scandinavian coastline (data courtesy of E. Nielsen, Katlenburg-Lindau)


minimum of a of about 1.2 is modeled between theelectric and magnetic HD at about 68:2� latitude, withrespect to the results of Lampen (1985) and Kunkel et al.(1986). From there, a increases to the south up to 2 inthe eastward electrojet region. The smaller increase tothe north was modeled with respect to the smallerincrease in j~Jeq;Ionj in that direction. No longitudinalvariation is included. Note that this distribution is infact the only modeled input to our analysis, and none ofthe later conclusions relies on this special distribution. Itwould be equally possible just to set a uniform medianvalue of a (compare Amm, 1995), but we followed theprevious studies, which both used a similar distributionwith a minimum at and gradients perpendicular to theHD.

Two obvious di�erences to the study of Lampen(1985) are readily visible from the input distributions inFig. 5a, (as well as from the data in Figs. 3 and 4):whereas in the earlier study the HD is northwest-southeast aligned [as in the work of Kunkel et al. (1986)and also the statistical model of Heppner and Maynard(1987)], Figs. 4 and 5b show the HD to be almost alignedwith geographical latitude, with even a slight northwardtilt at the eastern side which results in an good alignmentwith geomagnetic latitude, too. Our observations agreewith STARE data observations of Koskinen andPulkkinen (1995, their Fig. 4), who found reasonablealignment of the HD with geomagnetic latitude for allsix events that they studied. Next, in our single time-stepobservations the electric HD is found about 0:8� (�0:2�)northward of the magnetic one as reported with similarvalues by Kamide and Vickrey (1983) and Kunkel et al.(1986), whereas in the composite vector plots of Lampen

(1985) the two almost coincide (cf. Untiedt andBaumjohann, 1993).

The output distributions of the method of character-istics are shown in Fig. 6. Figure 6a displays selectedcharacteristics as de®ned in Eq. (16). As in previousstudies in which this method was applied in planargeometry to an HD (Inhester et al., 1992; Amm, 1995), astrip of characteristics runs along the electric HD fromwhich characteristics spread out to the north and south.Points of in¯uencing boundary are marked as hatchedsquares. The in¯uencing boundary takes up only about25% of the whole boundary. The resultingRH is shown inFig. 6b. For that ®gure, in regions where an uncertaintyin RH persists due to the in¯uence of unknown boundaryvalues, we took the average of the upper and lowerconductance estimate (see Sect. 2.4). Such regions arepresent in the southwest and in the outer northeast. Thedi�erence between the two estimates has a maximumvalue of about 5 S there. In Fig. 6b, a strong north-southgradient of RH is seen, with very low RH values mostlybelow 2 S in the region north of the electric HD. Southof the magnetic HD, RH increases considerably to reachvalues between 10 and 20 S at the southern boundary ofour analysis region (i.e., still north of the center of theeastward jet, compare Fig. 3). The east-west gradient inRH in the eastward electrojet region is related to anopposite gradient in j~Ej (see Fig. 5b) and does not lead toa discontinuity in the resulting distribution of ~J , asindicated by Fig. 6c. It is instructive to compare ~J with~Jeq;Ion (Fig. 5a). The nearly symmetric distribution of~Jeq;Ion with respect to the magnetic HD turns out tocorrespond to a highly assymetric distribution of ~J withlarge northeastward pointing current vectors up to

Fig. 5a±c. Input data for the method of characteristics. a Ionosphericequivalent currents, as resulted from separation of the external part ofthe magnetic disturbance ®eld, its continuation to the ionosphericheight, and transformation to equivalent currents. b Gridded electric

®eld. c Modeled distribution of Hall-to-Pedersen conductance ratio[modeled with respect to results of Lampen (1985) and Kunkel et al.(1986)]


450 mA/m in the eastward electrojet region, but onlyvery weak west-to-southwest-directed current vectors inthe region north of the electric HD. Hence, the apparentwestward electrojet in ~Jeq;Ion is caused by the e�ect of asheet of large upward-¯owing FACs up to 2A=km2 asshown in Fig. 6d. That sheet appears to have its centerbetween the electric and the magnetic HD, but closerto the magnetic one. Upward FACs are visible innearly the whole analysis area, but decrease strongly

north of the electric HD. Slight downward FACs arealso seen in between, but they are clearly weaker thanthe upward ones. The general picture of the FACs isquite patchy, but this is in fact what has been detectedby satellite measurements above the HD, too (cf.Koskinen and Pulkkinen, 1995, and references therein).It would be interesting to compare our results with suchmeasurements, but unfortunately none are available forthis event.

Fig. 6a±f. Results of the method of characteristics. a Characteristics;squares mark points of in¯uencing boundary. b Hall conductance.c Ionospheric currents. d Total ®eld- aligned currents. e Field-aligned

currents connected with Pedersen currents. f Field-aligned currentsconnected with Hall currents. Crosses mark downward, squaresupward ®eld-aligned currents


However, the patchy structure at least partly resolveswhen looking at the FACs which are connected to thePedersen (Fig. 6e) and Hall (Fig. 6f) currents. Whereasthe former are distributed relatively homogeneouslysouth of the electric HD with magnitudes of 1A=km2 orless and disappear nearly completely north of it, thelatter exhibit a more patchy behavior and have theirhighest intensities up to 1:5A=km2 near the latitude ofthe magnetic HD.

It is not surprising that our result distributions arenot as even as those Lampen (1985) obtained with the``trial and error'' method, or as any results of thismethod, because in ``trial and error'' one starts to modelsmooth and symmetric conductance distributions ®rstand departs from them only if absolutely necessary. Incontrast, we start out with plain data distributionsexcept the one for a that has no decisive in¯uence at all.

Still, in a broad view our results agree with that ofLampen (1985) in that a sheet of upward FACs is presentalong theHD and that the conductances are clearly largerin the eastward electrojet region than in the HD andnorth of it. On the other hand, in Lampen's study thestrong decrease in RH is located considerably furthersouth of the HD, which results in two distinctly separatedsheets of upward FACs, one at the HD and one severaldegrees south of it, from which the latter is even thestronger. This feature is not supported by our results, norby those of Kunkel et al., (1986), in another HDsituation. In general, we feel that our output distributionsshow somewhat more detail than those provided by the``trial and error'' method, and that the amount ofassumptions and modeled input used is clearly reduced.Therefore, we feel encouraged to try to address somegeneral questions on the HD in the next section.

4 Discussion

Only few previous single-event studies of the HD areavailable that involve two-dimensional analysis of allionospheric electrodynamic parameters. The mostextensive are those by Kunkel et al. (1986), later re®nedby Inhester et al. (1992) and Amm (1995), and that ofLampen (1985), shown also in Untiedt and Baumjohann(1993), on the same event as studied in the presentpaper. We will concentrate on these studies in ourdiscussion. On the other hand, many works of di�erenttype (e.g., based on magnetometer or electric ®eldmeasurements only, or using optical or satellite-baseddata, as well as statistical studies) have been carried out.These are summarized in the review of Koskinen andPulkkinen (1995). We will use the questions raised inthat paper as a starting point of our discussion.

One question already mentioned in the previoussection is whether the HD is tilted against latitude innorthwest-southeast direction as indicated by thecomposite vector plot results of Lampen (1985) andKunkel et al. (1986), as well as by the statistical resultsof Heppner and Maynard (1987), or if it is mainlyaligned with geomagnetic latitudes, as it was proposedby Koskinen and Pulkkinen (1995) who studied STARE

single time-step data of a number of HD events. Ourdata set clearly supports the latter point of view. Wehave already discussed the possible shortcomings of thecomposite vector plot technique. Marklund (1993)pointed out that statistical pictures may be quitedi�erent from single-event results.

Another question is whether or not there is plasma¯ow across the HD. The STARE data of our event(Fig. 4) shows some nearly purely westward directedelectric ®eld vectors in the center of the electric HD,clearly indicating a southward plasma ¯ow across theHD.

However, in the context of currents, parts c and d ofFig. 6 indicate that, at least in the situation studied here,almost all of the eastward electrojet current, bothregarding the Hall and Pedersen parts is diverted intoFACs at or south of the electric HD. This agrees with theconclusions of Lampen (1985) and Kunkel et al. (1986),but is in contrast to the proposal raised by Kamide(1978) of current continuity without FACs at the HD.

When comparing our results to the sketch of thelarge-scale electrodynamics of the HD presented byKoskinen and Pulkkinen (1995; their Fig. 7), a generalagreement with our results can be stated, except for twopoints: (1) The FACs connected with the Pedersencurrents are shown in their sketch as being restricted tothe immediate environment of the HD and spatiallyseparated from the FACs connected with the Hallcurrents. Our results indicate that due to the conduc-tance gradient perpendicular to the HD, the FACsconnected with the Pedersen currents are fairly evenlydistributed over a broad area south of the electric HD,overlapping with the location of the maximum Hallcurrent FACs. The latter are located south of the electricHD (regarding the Hall currents of the eastwardelectrojet), in agreement with the picture of Koskinenand Pulkkinen (1995). (2) Since the westward electrojetis very weak in the situation we studied, its contributionto FACs is negligible. This is in contrast to thesymmetric picture of Koskinen and Pulkkinen (1995)regarding the position and strength of the jets and itsFACs to both sides of the HD. However, we admit thatthese results may be a feature of the special situationunder study, and it is not clear whether they can begeneralized. It may well be that a stronger westwardelectrojet exists north of our analysis region, inaccordance with the results of Lampen (1985).

Another question raised by Koskinen and Pulkkinen(1995) is why the break-up arcs are often closer to themagnetically de®ned HD than to the electrically de®nedone. Although a complete answer to this question is notpossible from ionospheric results only, it is interesting tonote that the FACs connected to the Hall currents(Fig. 6f) peak exactly in the region between theelectrically and the magnetically de®ned HD, closer tothe latter. Since the substorm current wedge is believed tobe mainly constituted by Hall currents (e.g., Kamide andBaumjohann, 1993), it seems reasonable that the break-up starts where they are fed from the magnetosphere.

Finally, another question is why the magnetic HD (asde®ned by ground magnetometers) is typically located


1±2� south of the electric one. By recalculating theground magnetic ®eld from our results by Biot-Savartintegration, we found that if the e�ect of the horizontalionospheric currents only is included, the magnetic HDwould shift even north of the electric one, due to themuch larger strength of the eastward electrojet com-pared to the westward one in our event. The picturechanges when we include the magnetic e�ect of theFACs: although the calculation of ~BG from the currentsinside our analysis region only cannot completelyregenerate the true ~BG, since the e�ect of the currentsoutside that region is missing, it became clear that themain e�ect in the southward shift of the magnetic HDcomes from the east-west-aligned sheet of FACsconnected to the Hall currents. This e�ect is mostintense if there is a positive west-east gradient in absolutevalue of the magnitude of these FACs, as seen in Fig. 6f.

We would like to stress that we do not claim ourresults obtained from a single snapshot analysis to benecessarily general for the whole HD event on 27October 1977, or for all HD events. However, concern-ing the orientation of the HD, all available snapshots ofthe electric ®eld for our event show the same goodalignment with geomagnetic latitude as described in thepreceding.

From the data available for this event, it is di�cult tobuild up a time-series of similar analyses as presented inSect. 3, as this would require the study of the dynamicevolution of the HD. For later times than studied here,the HD soon moves out of the STARE ®eld of view tothe south, and for earlier times it is soon located northof the SMA network. Moreover, the amount of back-scatter received by STARE varies. Such a time-seriesanalysis could thus only be performed if electric ®elddata would be available on approximately the same areaas the magnetometer data. At the moment, e�orts aremade to use SuperDARN data in such a way, andresults will be published in the near future. In that kindof analysis, the advantage of the use of sphericalgeometry will become more obvious, too. The eventstudied in this paper could still be handled in planargeometry, but it was chosen in order to compare ourmethod with the earlier ``trial and error'' method, andbecause of the interesting physical situation. Moreover,with our study we have shown that the method ofcharacteristics in spherical geometry can be regarded asa superset of the planar version in the sense that it canalso be applied to smaller areas as earlier typicallyanalyzed in planar geometry.

Future work will include the application of ourmethod to di�erent types of ionospheric electrodynamicsituations and to larger regions of study.

5 Summary

Regarding the methodical aspects of this paper, we cansummarize our results as follows:

1. The method of characteristics is available in sphericalgeometry for use on large analysis areas in the auroral

zone, including tools for magnetic ®eld separation,upward continuation, and calculation of ionosphericequivalent currents. The spherical version of themethod can be used for smaller regions as previouslystudied by planar geometry methods as well.

2. The amount of assumptions and modeled input isclearly reduced as compared to the earlier ``trial anderror'' method, and the output distributions revealmore details, thus allowing a more detailed physicalinterpretation. Since the method of characteristics is aforward method, it is also much faster than any ®ttingmethod. However, since divergence and curl of inputquantities (or quantities derived from them) have tobe estimated, good two-dimensional input datacoverage of ~E and ~BG is required for instantaneoustime-step analyses.

The main results of the application of our method toa Harang discontinuity situation on 27 October 1977,17:39, UT are:

1. In general, our study supports the results of earlierones in that the conductances in the HD region andnorth of it are clearly reduced compared to theeastward electrojet region south of it, and that mostof the current is diverted into FACs at the HD,although plasma ¯ow is observed through it.

2. The FACs connected with the Pedersen currents showa nearly uniform distribution south of the electric HDwith magnitudes around 1A=km2. In contrast, theFACs connected with the Hall currents peak in theregion between the electric and magnetic HD, closerto the magnetic one, with magnitudes up to1:5A=km2, in the same area where initial break-upsare often observed.

3. The FACs connected with the Hall currents form alatitudinally aligned sheet and are mainly responsiblefor the southward shift of the magnetically de®nedHD with respect to the electrically de®ned one by1±2� of latitude. That shift is enhanced if a west-eastgradient in absolute value of the magnitude of thoseFACs is present.

4. The HD shows no noticeable tilt against geomagneticlatitude.

Acknowledgements. The author likes to thank K.H. Glaûmeier(Braunschweig) for his steady encouragement and help during thiswork. He is also grateful to B. Inhester (Katlenburg-Lindau), H.Koskinen and A. Viljanen (both Helsinki), and J. Untiedt(MuÈ nster) for valuable discussions or comments. The STAREdata used in this paper have kindly be provided by E. Nielsen(Katlenburg-Lindau) whose support is gratefully acknowledged.This work was supported by a DAAD fellowship HSP II ®nancedby the German Federal Ministry for Research and Technology anda grant from the German Science Foundation.

Topical Editor D. AlcaydeÂ thanks V.O. Papitashvili and M.Lester for their help in evaluating this paper.

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method of characteristics in spherical geometry applied to a harang

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