R. Lukianova and O. Troshichev
Arctic and Antarctic Research Institute, St. Petersburg, Russia
Yu. Galperin and N. Jorjio
Institute of Space Research, Moscow, Russia
Received July 15, 2000
The basic scheme of FAC distribution in the dayside polar region has been yielded by Iijima and Potemra [1976]. This scheme shows availability of three current sheets: Region 2 FAC at low-latitude edge of the auroral oval, Region 1 FAC at high-latitude edge of the oval, and the cusp FAC region located poleward of Region 1. The original scheme of Iijima and Potemra [1976] was not specified the influence of the IMF By component and Region 1 FAC and currents poleward of Region 1 were regarded as two independent systems. McDiarmid et al. [1979] put forward a specific model in which the poleward sheet of field-aligned currents was considered as an extension of Region 1 across the noon under influence of IMF azimuthal component. The difference between these two basic patterns is of fundamental importance: in the first case ( Iijima and Potemra [1976]) the cusp FACs and Region 1 FAC have different sources, while in the second case ( McDiarmid et al. [1979]) their source is the same. All subsequent FAC patterns in the dayside cusp region succeed these two patterns [D'Angelo, 1980; de la Beaujardiere et al., 1993; Doyle et al., 1981; Friis-Christensen et al., 1985; Levitin et al., 1982; Ohtani et al., 1995a, 1995b; Saunders, 1992; Taguchi et al., 1993; Troshichev et al., 1982, 1996; Watanabe et al., 1996; Yamauchi et al., 1993]. In spite of escalating number of spacecraft measurements of the field-aligned current effects in the cusp region the determination of the FAC structure continues to be ambiguous up to now because of two main reasons: 1) FAC patterns are reconstructed on the basis of isolated spacecraft traverses through cusp/cleft region, and 2) only zonal (east-west) component of the magnetic field perturbation is generally taken into account on the implicit assumption that a satellite orbit is perpendicular to extended sheets of currents.
Meanwhile the results of the simulation analysis of Lukianova [1997] have shown that different patterns of the field-aligned currents in the cusp/cleft region, proposed by Saunders [1992], Yamauchi et al. [1993], Taguchi et al. [1993], and Troshichev et al. [1996], produce a very similar distribution of the meridional (north-south) component of electric field Eq, which is responsible for the zonal part of convection flows. The zonal (east-west) component of the electric field Ej responsible for meridional part of the convection flows turned out to be more sensitive to choice of FAC pattern. It means that the meridional component of the transverse magnetic perturbations should be taken into account in analyses of FAC pattern in addition to zonal component. In this article we present the results of simulation analysis (profiles of zonal BE and meridional BN components of the transverse magnetic perturbations), obtained for specific structures of sources, the effects of near-edge traverse of current sheets being taken into account.
General regularities of BE and BN profiles, typical for different current structures, give us a background for reconstruction of FAC, observed in passes of AUREOL spacecraft in the day-time cusp region. The reconstruction is realized by method of model calculation and successive approximation of calculated magnetic disturbances to observed ones. The multi-sheet assumption with consideration for edge effects makes the best guess of FAC pattern from only one path of a satellite.
To derive a set of typical BE and BN signatures we examine the magnetic field perturbations produced by model FAC structures.
Magnetic field can be expressed in terms of vector-potential as B= curl A with an additional condition div A=0. If current is assumed to flow normally to thin spherical shell of radius r (geocentric altitude of satellite in our case) the vector-potential has only component Ar in spherical coordinates q (latitude), j (longitude) and r. From the expression B= curl A (q,j) er, where er is the unit vector outward-directed from center of the Earth, the zonal and meridional components of magnetic field are defined as
| (1) |
Using a network of points with Dj in longitude and Dq in latitude for the unit contour round the every point of grid (i,j) the components Bj and Bq are calculated respectively
 | (2) |
If current flows through the contour, it will be related to magnetic field as
 | (3) |
where j is current density, l is perimeter and s is area of the unit contour. By combining (2) and (3), the equation for every grid point is derived. Equations are solved numerically by a finite difference scheme over a network of points spaced Dq=0.5o in latitude and Dj =1o in longitude. The main expression for iteration technique is
 | (4) |
where f is left-hand side of (3), w is over-relaxation parameter, and n is number of iterations. Once the distribution of vector potential is obtained, we can derive the magnetic field from (1).
Effects of finite extension of one, two, three, and four
current sheets are examined for cases when spacecraft
crosses these sheets at right angle. Multisheet patterns
are schematically represented in Figures 1, 2, 3, and
4. The following
simplifying assumptions are
taken in the calculation.
All FACs sheets are parallel
to each other. Currents in
the sheets are of uniform density.
The edges of current sheets are of
rectangular form. Figures 1, 2, 3, and
4
show behavior of the meridional north-south
BN (solid lines) and zonal
east-west
BE (dotted lines)
components of magnetic perturbation
along the ascending spacecraft orbit
in case of one (Figure 1), two (Figure 2),
three (Figure 3),
and four (Figure 4) current
sheets. The computation was carried out
in the spherical coordinates,
but current sheets are
presented for simplicity as
rectangular in Figures 1-4. It is common knowledge that just east-west
magnetic component determines latitudinal position of sheets, their
polarity and intensity. If spacecraft moves toward the pole the
positive (negative) trend of
BE indicates downward (upward)
currents, latitudinal width of the trend being indicated a
width of current sheet. Variations of east-west magnetic component are
irrespective of whether satellite crosses the current sheets
far from edges (track 0 in all Figures) or close to them (for example,
tracks 1 and 2).
In contrast the north-south component BN is strongly dependent on the edge positions. BN is close to zero far from the sheet edges (track 0), and this peculiarity is solid evidence of the azimutally extended current sheets. BN shows the distinctive features of opposite sign at the morning and evening edges of the current sheets (tracks 1 and 2). One peaked wave is typical of one current sheet: it can be negative or positive depending on sort of edge and direction of currents. The sine wave is typical of two current sheets; the double-humped positive or negative wave is for three sheets; and two sine waves occurring against the background of increasing or decreasing magnetic field are typical of four current sheets. Thus the meridional magnetic component BN evidences on the sheet edges in addition to information about number and polarities of current sheets.
Examples of two and three consecutive and parallel sheets with displaced edges are given in Figure 2, tracks 3-8. As before, BE component (solid line) shows polarity of current sheets: the negative peaked wave is observed for upward/downward current sheets, and positive peaked wave is for downward/upward current sheets. The BN component (dotted line) takes form of a smoothed wave in this case. It testifies the location of the current sheet edges (dawn or dusk), the maximum in BN component being displaced relative to maximum in BE component. The opposite regularity is true for inverse location of the sheet edges. Sign of the magnetic BN component is determined by mutual disposition of the upward and downward field-aligned currents. Figure 3, tracks 3-8, shows behavior of zonal BE and meridional BN components of magnetic perturbations in case of three current sheets with displaced edges.
The specific case of orthogonal track without crossing of current sheets can be observed when spacecraft moves along the meridian just in gap between oppositely directed current sheets. One can see that in this case only BN component shows a distinguishing characteristic whereas component BE is held close to zero (Figure 1, track 10). If spacecraft crosses the current sheets far from edges but at obtuse or acute angle, the BE component shows generally the same regularity as for orthogonal crossing, although the differentials are flatten out. But BN component in this case increases or decreases along the track, depending on angle of inclination between the spacecraft orbit and current sheets.
Therefore the east-west component provides us with reliable information about number and polarity of the current sheets irrespective of spacecraft orbit angle and the proximity of the sheet edges. The edge effects (or effects of increasing current density) can be identified solely from the north-south component BN if spacecraft crosses the current sheets not far from edges. Thus examination of distinctive changes in both BE and BN magnetic components, observed along the spacecraft track, pro vides us with reliable signatures for identification of the current sheet structures: number and polarity of the current sheets, mutual disposition of the current edges, and spacecraft orbit inclination to current sheets. In doing so it should be borne in mind that disturbances, observed by satellites, are produced mainly by currents flowing in the nearest vicinity of the satellite orbit.
General regularities in variations of BE and BN components of magnetic field, typical of the different structures of current sheets, give us a background for recognition of FAC patterns observed in course of spacecraft passes through the day-side cusp region. The recognition is realized by method of model calculation and successive approximation of calculated patterns of magnetic disturbances to the observed patterns. The problem of the best fitting of the curves calculated for model patterns to the actual observed profiles for meridional (or zonal) component of the magnetic field perturbation is solved by finding the smallest value for function:
 | (5) |
where
N is a number of one-dimension grid
points along the spacecraft trajectory
projection, and
{x} is a set of unknown
parameters of
1, 
, n current sheets.
Bi( exp) is a magnitude of actual observed
magnetic field disturbances ( BE or
BN components) at
a point
(i) and
Bi(x) is a computing one. The function
Bi(x) can be written as Taylor series
 | (6) |
where x(0) is a zero-order approximation of parameters. The computing and actual profiles of magnetic disturbance are fitted while minimizing the function
 | (7) |
Minimization of function (7) is reduced to solution of the system of n linear algebraic equations for local minimums when xi=ji, Dxi=ji-ji(0) and ji is current density in the i sheet. As a result a set of optimal current parameters providing the best fitting of the experimental and calculated profiles is determined. A practical inversion algorithm for determination of current structures is based on numerical solution of two-dimensional equation (3) and consists of three consecutive steps (or approximations), as follows: 1) The distribution of only zonal component of magnetic disturbance BE is analyzed. As a result of computer simulation the current structure is adopted that provides the best fitting of observed and calculated trends of BE along the satellite path. Parameters of hypothetical current sheets such as polarity and latitudinal width are stated on the assumption that these sheets are extended from dusk to dawn. Relative intensity of currents in each sheet is obtained from expression (7). 2) The only meridional component BN is analyzed. The number of current sheets is specified from results of the previous step, the each current sheet being cut in the vicinity of the spacecraft trajectory. The position of current sheet edges ensuring the best fitting of experimental and calculated profiles of BN is chosen as actual. 3) The additional current sheets uncrossed by the spacecraft trajectory are included in the analysis to improve the fitting of BN profiles. Parameters of additional sheets are selected to ensure the minimum discrepancy between experimental and calculated BN profiles.
Four consecutive AUREOL 3 tracks on October 14, 1981
when spacecraft crossed the northern day-time cusp region are
examined. Figure 5
shows the IMF parameters in period of these
crossings. The
By IMF was negative in course
of two first crossings
and close to zero in course of two last crossings, whereas vertical
component
Bz was close to zero for the first track, southward for
second track, and positive for two last tracks. Spacecraft moved
under a very little angle to the noon-midnight meridian.
Figures 6,
7,
8,
and 9
(left part) illustrate the
satellite-measured (solid lines) and calculated
(dotted lines) profiles of zonal
BE and meridional
BN components of magnetic perturbations for tracks
307N, 308N, 309N, and 310N, correspondingly. Indications
of universal time (UT), height ( h ), magnetic local time
(MLT), and invariant latitude (ILAT) are given for each
track as well. The right part of the Figures 6-9 shows
the corresponding FAC patterns ensuring the best fitting
of experimental and calculated profiles. Options a), b), c)
are for the first, second and third approximations in
framework of the described algorithm. Only large-scale
structures of the field-aligned currents are examined
in our analysis, although there is no question that
spacecraft data are evidence for smaller-scale current
layers as well (see, for example, orbits 307N and 308N).
Some parameters of calculated large-scale current patterns
(number of current sheets and relative intensity of currents)
are given in Table 1. To derive density the 0.2 (mkA m
-2 )
coefficient might be applied.
Profile of magnetic perturbation in case of orbit 307N consists of one negative peaked wave in BE and one negative peaked wave in BN . These waves are rather flat and change almost synchronously. These peculiarities of the profile are described by three current sheets with displaced edges, the flowing up currents being located in the equatorward sheet in the afternoon sector, and the flowing down currents being located in the intermediate sheet in the pre-noon sector and in the poleward sheet in the afternoon sector. As Figure 6 shows the best fitting of the observed and calculated profiles of magnetic perturbations is attained if one more the wide sheet with the flowing up weak currents is suggested to be located outside of the spacecraft trajectory in the dawn sector. Magnetic signatures for orbit 308N (Figure 7) are typical of two up/down current sheets crossed near their westward edge: one negative peaked wave in BE and the sine wave in BN. In general the FAC structures derived for tracks 307N and 308N are rather similar, their main feature in the afternoon sector is the flowing up currents located at 75-80( ILAT and the flowing down currents located at higher latitudes. The FAC structure of this kind is typical of patterns of the field-aligned currents in the northern cusp region under conditions of the southward and large negative IMF components [Iijima and Potemra, 1976; McDiarmid et al., 1979; Troshichev et al., 1982].
The FAC structure observed for tracks 309N and 310N
is of larger interest, since in this case we deal
with conditions of the northward IMF. Unfortunately,
data from track 309N are not provided for latitudes
higher
82o ILAT, whereas data from track 310N are
not available at latitudes lower
72o ILAT. However,
we can regard data from these tracks as complement
each other, bearing in mind that they were obtained
under the similar IMF conditions. Track 309N (Figure 8)
shows signature of two main sheets in the
afternoon sector, with the flowing up currents at
equatorward side ( <79 ILAT) and the flowing down
currents at poleward side ( >79 ILAT). Besides of
it two additional sheets in the prenoon sector with upflowing
current at 80 ILAT and downflowing one at
higher latitudes ensure the best agreement
between the calculated and experimental
profiles of
BN, profile
BE being practically
invariable. Track 310N indicates the flowing
down currents at latitudes less 80 ILAT and
flowing up currents at latitudes 80-84
ILAT, the current sheets being crossed
near their westward edge. The best fitting
of the observed and calculated profiles of
magnetic perturbations for track 310N is
attained if we assume availability of two additional
sheets with oppositely directed currents in the pre-noon
sector (see Figure 9). Therefore we conclude that
spacecraft AUREOL 3, crossing the northern day-time
cusp region under conditions
Bz 
0,
By 
0 approximately
along the noon meridian, met three current sheets.
The most equatorward sheet,
with the flowing up currents
in the afternoon sector observed
at orbit 309N, evidently falls into
the category of Region 1 FAC. The
further intermediate sheet, located
at latitudes about
80o (orbit 309N) or 76-79
o (Orbit 310N)
with currents opposite
in direction to those
in Region 1, can be
regarded as the proper
cusp FAC system. And
the most poleward sheet
of currents observed at
orbit 310N, flowing up
in the afternoon sector
and the flowing down in
the prenoon and afternoon sectors, is evidently the
mantle FAC system. The pairs of oppositely directed
currents in the last two sheets are separated by gap
in the field-aligned currents located at the noon meridian.
Patterns of the field-aligned currents, generated in the northern day-time cusp region under influence of azimuthal By or northward Bz IMF components, were presented in recent years by Saunders [1992], and Yamauchi et al. [1993], Taguchi et al. [1993], Ohtani et al. [1995a, 1995b], Troshichev et al. [1996]. The FAC pattern given by Erlandson et al. [1988] is evolution of concept presented in studies of Iijima and Potemra [1976], D'Angelo [1980] and Troshichev et al. [1982]. The pattern consists of the traditional Region 1 and Region 2 field-aligned currents and cusp currents located poleward of Region 1 and named "mantle" current. Mantle current flows into the ionosphere in the afternoon sector and away from the ionosphere in the pre-noon sector irrespective of the sign of By. The influence of the IMF azimuthal component shows itself mainly in the partial extent of the region 1 and mantle currents, shifted in pairs across local noon in association with By. The similar current pattern is presented in study of [Ohtani et al., 1995a], where term "Region 0" is used to refer to any FAC system poleward of Region 1 currents. The pattern of Taguchi et al. [1993] includes, along with regions 1 and 2, the additional two layers of Birkeland currents in the noon sector with flow directions specified by By. This double current system is always located poleward of Region 1. As a result, the spacecraft intersecting the polar region in the pre-noon or afternoon hours should meet four current sheets. Indeed, the four-current sheet structure was observed in the pre-noon sector, when IMF By component was negative [Ohtani et al., 1995b]. The model of Troshichev et al. [1996], obtained on the basis of ground magnetic data, can be regarded as an intermediate between patterns proposed by Erlanson et al. [1988] and Taguchi et al. [1993]. This model assumes that low-latitude cusp current sheet in the additional pair of Birkeland currents [see Taguchi et al., 1993] is adjacent to the Region 1 on the morning or on the evening side depending on the sign of By.
The patterns of Saunders [1992], Yamauchi et al. [1993], and de la Beaujardiere et al. [1993] develop the concept proposed at first by McDiarmid et al. [1979]. These patterns consider the field-aligned currents observed in the cusp region as an extension of the Region 1 currents to the noon sector under influence of IMF By.
Let us compare the patterns mentioned above with the FAC structures derived from observations of magnetic perturbations on board AUREOL 3 spacecraft under conditions of the northward IMF and By close to zero. In the first place it is necessary to note that FAC systems, presented in Figure 8 and Figure 9, suggest the FAC pattern with quasi-symmetrical distribution of pairs of oppositely directed currents relative to the noon meridian in each sheet. It implies that currents in all sheets in the cusp region change their direction when crossing the noon meridian, so that the gap in the field-aligned currents takes place at the noon for By=0. It would appear reasonable to infer that this gap displaces under influence of the nonzero By towards dawn or dusk. This feature is of crucial importance for testing of the FAC models in the daytime cusp region. It means that concept of the cusp-current sheet as the westward or eastward extension of the Region 1 currents [McDiarmid et al., 1979] is inapplicable.
Secondly, the FAC systems, presented in Figure 8 and Figure 9,
suggest the FAC pattern with two sheets containing the
pairs of oppositely directed currents poleward from
traditional Region 1 FAC. It means that a two-layer
current structure is available instead of a single zone of
"mantle" currents
[Erlandson et al., 1988]
or "Region 0"
[Ohtani et al., 1995a].
Taking into account Region 2 FAC observed at
latitudes lower
75o in the dawn and dusk sectors but invisible at noon
meridian we can speak about four-layer current structure. It would be
noted that a four-layer structure with the oppositely directed currents
(Region 2, Region 1 and two additional layers poleward of Region 1)
has been presented in patterns of
Taguchi et al. [1993]
and
Ohtani et al. [1995b].
However, this structure
was regarded as typical
of only one, dawn or dusk,
MLT sector, depending on
sign of
By. Our results,
obtained for conditions
of
By=0, suggest that a
two-layer current structure poleward of Region 1 can be
observed in the prenoon
and afternoon MLT sectors
of the polar region simultaneously.
The FAC pattern in the day-time cusp
region summarizing our results for
conditions
Bz>0,
By 0 is presented
in Figure 10.
The current patterns presented in this study were constructed on the basis of isolated spacecraft traverses through the cusp/cleft region like to overwhelming majority of patterns in other studies mentioned here. Adoption of statistically significant set of data is required to approve the obtained results. However, there is no question that use of the meridional component of magnetic perturbations in analyses of the FAC current structure provides us with a new and important information about the edge effects and gaps in the currents sheets. This information is especially important for determination of FAC pattern in the day-time cusp region.
The algorithm for interpretation of spacecraft observations of the large-scale field-aligned currents is proposed, the zonal and meridional components of the magnetic field perturbations being used in the analysis. While a zonal component specifies the number, polarity, and latitudinal width of FAC sheets, a meridional component is an evidence of edges and gaps in the current sheets. The last circumstance is especially important for the day-time cusp region where Region 1 FAC are terminated in vicinity of the noon meridian whereas current sheets located poleward of Region 1 (i.e., cusp currents) are evidently confined to limited MLT sector. The algorithm ensures automated derivation of "two-dimensional" FAC structure in the region of spacecraft trajectory in assuming that the current sheets are located along the latitude. Four consecutive AUREOL 3 spacecraft intersections of the northern day-time cusp region have been used as experimental basis for the analysis. Conclusion is made that the specific FAC system located poleward of Region 1 is typical of conditions Bz>0 and By=0. The system consists of two current sheets with pairs of oppositely directed currents separated by gaps at the noon meridian, and, therefore can not be regarded as an extension of the Region 1 currents from other local times.
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