How to address the accuracy of empirical magnetic field models?

Adv. Space Rex Vol. 28, No. 12, pp.1717~1726,200l 0 2001COSPAR. Published by Elsevier Science Ltd. All rights reserved

Pergamon

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0273-l 177/01$20.00+ 0.00

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PII: SO273-1177(01)00537-3

HOW TO ADDRESS THE ACCURACY OF EMPIRICAL MAGNETIC FIELD MODELS? T. I. Pulkkinenl 1Finnish

Meteorological

Institute,

Helsinki,

Finland

ABSTRACT Empirical magnetic field models are compared with high-altitude magnetic field measurements and results from an MHD simulation. Comparison of the T96 model and observations from GOES-8 and GOES-9 shows that if the observed solar wind and IMF parameters are used to compute the model field, the model field is more stretched than the observed field. On the other hand, if measurements made by one spacecraft are used to find the model parameters that give a best-fit field at that location, the RMS error can be reduced also at the other spacecraft four hours away in local time. Comparison of T96 model and MHD simulation results shows that the empirical models have a thinner current sheet than the MHD simulation, but that the lobe field values are quite similar to each other. Furthermore, a comparison of an event-oriented, modified T89 model and MHD simulation by Pulkkinen et al. [2000] reveals that if the empirical model is constructed by fitting to in-situ measurements, the resulting model is very similar to the MHD simulation magnetic field. These results indicate that an efficient method of utilizing the present-day empirical models is to select model parameters based on measurements from a few individual points. 8 2001 COSPAR. Published by Elsevier Science Ltd. All rights reserved.

1. INTRODUCTION Magnetic field models are widely used in magnetospheric research. Easy to use, analytically formulated magnetospheric models have been developed especially by N. A. Tsyganenko (Tsyganenko, 1989; Tsyganenko, 1990; Tsyganenko, 1995). These models have been used by the community to provide local magnetic field value and direction, needed for example in evaluation of plasma parameters such as particle Larmor radii (e.g., Pulkkinen et al., 1992). These models define the tail current sheet structure, which is an important parameter when, e.g., the tail stability is addressed (e.g., Biichner and Zelenyi, 1987). In the analysis of multi-spacecraft observations during the International Solar-Terrestrial Physics (ISTP) program, field models have been used to connect spatially (providing indirectly information on causality) observations from distant locations (e.g., Pulkkinen et al., 1998). Furthermore, field models have been utilized in comparisons of space-borne and ground-based measurements to provide magnetic connectivity with ionospheric auroral or convection signatures and dynamic processes in the tail (e.g., Lu et al., 1997). In the larger context, magnetic field models can be used to predict regions (solar wind, lobe, plasma sheet, etc.) where a particular spacecraft will reside during its orbit. Furthermore, large-scale models for the plasma and particle motions in the magnetosphere require background magnetic and electric field models, unless the computations are carried out self-consistently. The Tsyganenko models are based on a large set of magnetic field measurements from the magnetosphere, organized by activity indices such as Kp, Dst, interplanetary magnetic field (IMF) components (BY,IMF and BZJMF), and solar wind dynamic pressure (PDYN). The magnetospheric currents (cross-tail current, ring current, magnetopause currents, and other current systems) are represented by analytical functions containing several free parameters. The free parameters are then determined as functions of the organizing parameters by minimizing the error function created from the differences between the observed and model representation of the field, changing as the magnetic field values. This procedure leads to an analytical magnetospheric activity level or solar wind and IMF parameters change. The accuracy of magnetic field models can be addressed from at least three different viewpoints: The 1717

1718

T. 1.Pulkkinen

model accuracy can be evaluated by comparison with single-point measurements in space, by evaluating the correctness of magnetic field-aligned mapping between two points in space, or by evaluating the errors in the global field configuration in some average sense. This paper addresses the question of accuracy of the magnetic field models from three different perspectives: (1) local studies requiring accurate, instantaneous values for local field magnitude and direction; (2) multi-instrument studies requiring accurate connectivity often over some time period and limited region; and (3) large-scale studies requiring accurate, but often time-averaged, picture of the global field configuration. In section 2, the T96 (Tsyganenko, 1995) model, which is the most advanced and flexible of the presentday field models, is compared with observations from two geosynchronous spacecraft. After comparisons using the actual solar wind and interplanetary field parameters to drive the model, we search for those input parameters which produce the best fit with the local observations. In this comparison, each data point is evaluated independently, and hence successive configurations do not represent a dynamic evolution in the magnetosphere. Section 3 shows an event study (from Pulkkinen et al., 2000) during which the dynamic evolution is assumed to be known, and hence only one set of model parameters with a pre-determined time dependence is used to characterize the entire modeling period. In Section 4, the field models are compared with MHD models, which shows the differences and similarities given by these completely different approaches. The paper concludes with a short summary. 2. LOCAL MODEL EVALUATION For the local field evaluation, we use data from GOES-8 and GOES-9 spacecraft during January 2-8, 1997. GOES-8 and GOES-9 are geosynchronous satellites located at -75” and -135” geographic longitude and at 10.5” and 4.7” geomagnetic latitude, respectively. The chosen time period contained both quiet periods ss well as several large substorms, but no storm activity. Therefore, although the data set is too small for a statistical examination, the results should be representative of times without storm activity. Figure 1 shows the magnetic field intensity and polar angle for GOES-8 and GOES-S. The observations are shown in black. The grey curve shows the Tsyganenko 1996 model, computed using the observed Dst-index, solar wind dynamic pressure, and IMF By and Bz. The third and fourth panels show the errors computed as differences between the observed and model values. Note the daily variation in the errors, which indicates that in the dayside the observed field is larger than the model field, whereas in the nightside (shown with grey shading) the agreement is quite good. The nightside errors increase during substorm activity (see the panel showing the CL-index, which is a quasi-AL index created from measurements from the CANOPUS magnetometer chain). The same is true for both GOES-8 and GOES-g, even though they are at slightly different magnetic latitudes and local times. The polar angle (0 = tan-’ (Bz/sqrt(Bi + B$)) in SM coordinates where the 2 axis is along the dipole axis) also shows similar diurnal variation. Perhaps the dominant feature is the constant positive difference in the field angles, indicating that the T96 model predicts consistently smaller polar angles than were actually measured, Hence, the model field is more stretched than the observations would indicate. The model driver parameters, the observed Dst, PDYN, IMF By and BZ are shown in the bottom panels of both plots in Figure 1. For studies of magnetospheric dynamics and structure it is of importance to know how well a magnetic field value measured at one given point is representative of field values at other locations. The orbits of the two GOES spacecraft, separated by about four hours in local time, offer a good possibility to test that property: If the field value at one location is representative of the field values at neighboring locations, one would expect to find good correlation between the errors in the field values between GOES-8 and GOES-g. On the other hand, if the errors are scattered randomly, it would indicate that the field at any given time instant is spatially so variable that measurements at one location cannot be used to extrapolate information about other locations. Figure 2 shows the errors in field magnitude and polar angle, now plotted in the form of ~BCOES-8 vs ~BGOES_~ at every time instance. The nightside values (when both spacecraft were in the local time sector 21 < MLT < 03) are shown in grey. The field magnitudes show a correlation coefficient of 0.68, although there are several periods when the differences are quite large. The nightside values show slightly better correlation than the entire data set (R = 0.72), but the difference is not large. For the polar angles,

Accuracy of Field Models

1719

the differences are only weakly correlated, and the correlation coefficients for the entire data set and the nightside values even have opposite signs. The data set is thus dominated by the few instances where the fields were very different at the two locations.

T96 model and GOES observations

T96 model and GOES observations

2345678923456789 Days of 1997 Fig.

1.

(Left)

by the T96 model.

Comparison

of magnetic

Days of 1997 field magnitudes

Panels from top to bottom:

BGo,ysg - BTg6, CL-index,

Dst-index,

measured

by GOES-8 and GOES-9

and predicted

B at GOES-8, B at GOES-g, 6B = BGoESB - BTg6, 6B =

and solar wind dynamic

field polar angle measured by GOES-8 and GOES-9 and predicted

pressure Pdyn.

(Right)

by the T96 model.

Comparison

of magnetic

Panels from top to bottom:

8 at GOES-8, 8 at GOES-g, 619= 0 GOES~ - fh96, 68 = eGOES9 - &96, CL-index, IMF BY and IMF Bz. The T96 model values were computed using the measured activity parameters. All magnetic field quantities are given in units of nT, angles are given in degrees, and Pdyn is given in nPa. Periods when the spacecraft are in the nightside (X < 0) are shown with grey shading.

1720

T. I. Pulkkinen

20 5T 3 8

0 x iii

O-

0;

g

E

-20 -

-lo: R(nightside) -40 I. -40

Fig. 2. Correlation showing

( 0.719 .; 5; ..II..I **a III.lli -20 dE&ES8) 20

of errors between

the errors at GOES-8

both spacecraft

R = 0.352

=

40

measured

and model field magnitudes

vs the errors at GOES-9

were in the local time sector 21 <

and polar angles in a scatter

at the same time instances.

MLT < 03)are

shown

The

nightside

values

plot (when

in grey.

In the previous comparison, the magnetic field model was computed using the measured Dst, solar wind, and IMF values. However, it is also possible to search through the free parameter space (values of Dst, PDYN, and BZ,JMF)to look for a combination that gives the best possible fit to the measured magnetic field values. This was done in Figure 3, which shows the measured and model magnetic field magnitudes (in the same format as in Figure 1) when the model has been computed using values that give the best fit to the GOES-8 measurements. For shorter computing time (search through the parameter space for four free parameters requires of the order of lo4 field evaluations), the data have been averaged to 30-min averages. Furthermore, BY,IMFwas not varied, as it causes least changes in the field configuration. The ranges of variations were from -150 to 50 nT for Dst, from -20 to 20 nT for IMF Bz, and from 0.5 to 11.25 nPa for dynamic pressure and 20 steps were used for each parameter. Note that the fit at GOES-8 is nearly perfect, and that the fit for GOES-9 is also improved, even though the GOES-9 measurements were not used in the parameter search process. There is still a diurnal variation at GOES-g, which indicates that there are repeatable structural differences in the model and the actual magnetosphere. The same is true for the polar angle: the constant shift between the model and the observations has been removed, and the errors are small for GOES-8. For GOES-O, the errors are also much smaller, and now fluctuate around zero. The effects of averaging to 30-min values was examined using a subset of the entire interval, but the time resolution did not have an effect on the statistical results. Furthermore, the results were qualitatively similar when GOES-9 was used to determine the input parameters. Note how the overstretching is compensated by increase of Dst and increase of dynamic pressure when the spacecraft are on the dayside. When the spacecraft are on the nightside, for some reason the optimal configuration seems to be reached by strongly negative Bz and still higher than measured Dst. The dominance of PDYN in determination of the dayside model field, which is different from the nightside, means that if one of the spacecraft is on the dayside and the other on the nightside, parameter fitting does not provide as good results as if both spacecraft are either on the dayside or on the nightside. Figure 4 shows the correlation of the errors computed with the best-fit model for GOES-8 shown in Figures 3a and 3b. Correspondingly, the scatter in the horizontal axis is small, but it is reduced also in the vertical direction. Especially in the nightside (shown grey in the plot), the errors are actually quite small. This indicates that by choosing model values that give good representation in one local time sector in the nightside, the model also gives a realistic representation of the actual magnetic field at other nightside locations

a few hours away in local time.

1721

Accuracy of Field Models

T96 model and GOES observations: T96 model and GOES observations: Optimum input for GOES-8 Optimum input for GOES-8

CL (CANOPUS)

[nT]

lo0 -

-10 -

Dst (best-fit)[nTl

IMF By [nT]

147

2 Fig.

3.

3 (Left)

by the T96 model.

4

5 6 Days of 1997

Comparison

of magnetic

7

a

92

3

4

7

a

9

Dat.s of Ii97 field magnitudes

Panels from top to bottom:

measured

by GOES-8 and GOES-9 and predicted

23 at GOES-8, B at GOES-g,

6B = &oEss

- &g6,

6B =

CL-index, Dst-index, and solar wind dynamic pressure Pdyn. (Right) Comparison of magnetic &oEsa - &gs, field polar angle measured by GOES-8 and GOES-9 and predicted by the T96 model. Panels from top to bottom: f? at GOES-8, 8 at GOES-g, 68 = &OES~ T96 model values were computed time step independently.

- &96,

t%oEsg - oT9& CL-index,

in degrees, and Pdyn in nPa.

IMF

By, and

IMF

that gave a best fit to the GOES-8 observations

The IMF and solar wind parameters

that gave the best fit for the GOES-8 observations. shading.

68 =

using the parameters

(except for

All magnetic

Periods when the spacecraft

Bz. The at each

By) are not the measured, but those

field values are given in nT, angles are given

are on the nightside

(X

<

0) areshown

with

grey

1722

T. 1. Pulkkinen

Fig. 4. Correlation showing shown

3.

the errors

of errors between at GOES-8

measured and model field magnitudes

vs the errors at GOES-9

at the same time

and polar angles instances.

The

in a scatter

nightside

values

plot are

in grey.

MULTI-POINT

FIELD COMPARISONS

The high-altitude magnetic field measurements were used in the work of Pulkkinen et al. (1992) to further develop the Tsyganenko (1989) model in situations where the average field configuration did not give a good fit for in-situ measurements. To represent the substorm growth phase, they added a thin current sheet near the inner edge of the plasma sheet, and enhanced the existing current systems. These changes were selected so that they gave the best fit to in-situ measurements at several locations in the nightside tail. The temporal evolution of the tail was assumed linear such that with one set of parameters the entire growth phase evolution could be described. Kubyshkina et al. (1999) expanded these event-oriented models to include also the low-altitude isotropic boundaries as input data. Isotropic boundaries are the locations where a low-altitude, polar-orbiting spacecraft traveling from the equator toward the pole measures a change from a trapped, anisotropic distribution to isotropic distribution. This boundary can be used in determination of the magnetic field configuration at the equatorial plane, if it is assumed that the isotropy is reached by pitch-angle scattering (tail-like field) in the equatorial plane (see Sergeev et al., 1993). As the isotropic boundaries are only measured at one time instant, this model can only be constructed for one time step. Figure 5a (from Pulkkinen et al., 2000) shows results from such model fitting. The dashed lines represent the T89 model during a substorm that occurred on Dee 10, 1996. The thick solid lines show the eventoriented model by Pulkkinen et al., which included input data from GOES-8, GOES-g, GEOTAIL, and INTERBALL Tail Probe. The modeling was done for two periods separately, shown by the vertical dotted lines in the figure. The dark dots represent the Kubyshkina et al. results during two time instances when the isotropic boundary measurements were available. Both modified T89 models improve the fit during the growth phase quite substantially, as the strongly intensified currents are not present in the T89 model. Figure 5b shows the isotropic boundary locations as computed from the different models and compared with the observations. As can be seen, the Tsyganenko models cannot represent the very narrow range of latitudes where the transition from anisotropy to isotropy occurs for different-energy particles. T&mslated to the high-altitude field configuration, this means that the Tsyganenko models do not reproduce the very rapid decrease of the field as one moves away from the quasi-dipolar region. The Pulkkinen et al. method improves this quite substantially, but the best fit is naturally given by the model by Kubyshkina et al. (1999), which uses the isotropic boundaries as input data. From these results it can be concluded that the field models for a particular period can be made quite consistent with measurements. However, this requires understanding of the physical processes in the tail,

Accuracy of Field Models

1723

models are changed by additional currents. Detailed analysis of each event separately is required before such event-oriented models can be created. The accuracy of the models is of course best in the regions covered by observations, but 88 was shown in the previous section, the accuracy can be expected to be quite good within four hours of local time of the observations used to select model parameters. The Pulkkinen et al. method furthermore utilizes the time sequence of events in the modeling, because observations from the entire modeling period contribute to the parameter determination. Consequently, one set of parameters with the assumed time evolution can describe the evolution of the tail. In this case, the successive magnetic field configurations can be interpreted to represent the temporal evolution of the field configuration, which was not the case using the method described in the previous section where each observation point WSLSconsidered independently. as the

statistical

GOES 8 and GOES 9 : Dee lo,1996

40 20 0

Isotropic boundaries I

,““,“I

\69

Kp=2

-74

-20

: Dee 10, 1996 Q*‘*‘--

738 40 20

-62

.

i

-60 0630

0600

Fig.

5a.

field

modeling

Pulkkinen

Results from for

0830

and

(from

The dashed line shows the T89

The heavy solid line indicates

the modi-

et al. The solid dots

the model results by Kubyshkina

4. GLOBAL

magnetic

GOES-9

lo2

0900

The data are shown by

fied T89 model by Pulkkinen indicate

0800

event-oriented

GOES-8

et al., 2000).

thin solid lines. model.

0730 UT

0700

Kubyshkina et al. Pulkkinen et al.

fig.

5b. Isotropic

sured by NOAA various

I

, , , , , ,,

, , , , ,,

1

lo3 MV/Q

[nT km]

boundary

locations

spacecraft

lo5

IO4

both as mea-

and predicted

by the

models shown in Figure 5a (from

nen et al., 2000).

The isotropic

are shown as a function is directly

proportional

boundary

of particle

Pulkkilatitudes

rigidity,

which

to their energy.

et al.

CONFIGURATION The accuracy of the global magnetic field configuration is difficult to address, as there are no observations of the instantaneous field configuration over the magnetosphere. However, the magnetohydrodynamic (MHD) simulations have shown that they can in some cases reahstically represent dynamic magnetospheric events, so that comparisons with data during actual events show good results (e.g., Wiltberger et al., 2000). Here we compare each of the three models discussed in this paper (T89, T96, and the modified T89) with an MHD-simulation run for the event on Dee 10, 1996 (see Figure 5). The simulation was run with the Lyon-Fedder-Mobarry code (e.g., Fedder et al., 1995); the results are discussed in detail by Wiltberger et al. (2000). Figure 6a shows three frames in the noon-midnight meridian plane. The contours and grey shading show the magnetic field Bx component as given by the MHD simulation (top) and by the T96 model (middle). The bottom panel shows the differences between the model and the simulation. The Bx component was selected, as it is best for examination of the plasma/current sheet structurein the tail. It is

1724

T. I.Pulkkinen

evident from the contours in the top two panels that the field intensity decreases much more slowly in the MHD simulation as one crosses from the tail lobe to the center of the plasma sheet. This indicates that the MHD simulation predicts a much thicker current sheet than is present in the T96 model. Consequently, the differences between the simulation and model are small near the center of the current sheet, but become larger near the boundaries of the plasma sheet especially inside of about 30 RE. Figure 6b shows, in the same format, results for the T89 model. The results are basically similar, although the differences between the model and the simulation are slightly smaller for the T96 model than for the T89 model. This is mainly due to the larger lobe field in the T96 model, which arises from the better confined magnetopause. The current sheet structure in the T89 and T96 models is very similar.

15

Dee 10 1996 0644 UT MHD / T96

10 5 N

0

-5 -10 -15 15 10 5 N

0

-5 -10 -15 15 10 5 N

0

-5 -10 -15 0

Fig.

6a.

-10

-20

Comparison

-30 X

of MHD

-40

-50

model

0

-60

and T96

Fig.

-10

6b.

-20

Comparison

-30 X

of MHD

-40

-50

-60

model and T89

model field values at 0644 UT on Dee 10, 1996

model field values at 0644 UT on Dee 10, 1996

(during

(during

a quiet

period

before

substorm

growth

top to bottom: Bx from Bx from T96 model, difference BX,MHD - Bx,T~~. The contours are labeled in nT, and distances in X - 2 plane are in RE. phase).

Panels from

MHD simulation,

phase).

a quiet

period

Panels from

before

substorm

top to bottom:

growth

Bx from

Bx from T89 (Kp = 5) model, BX,MHD - Bx,T~~. The contours are labeled in nT, and distances in X- 2 plane in RE. MHD simulation,

difference

Figure 6c shows a comparison between the MHD simulation and the modified T89 model (see Figure 5) at 0730 UT, prior to the substorm onset. During this period there was an intense and thin current sheet in the inner magnetotail, and the tail was in a stretched configuration. The MHD model shows now a much thinner current/plasma sheet, shown in the figure as regions with < 5 nT field values. The modified T89 model specified a very thin current sheet, as is evident from the closely spaced contours at the current sheet center. The differences between the simulation and the model are quite small throughout the tail, except within the current sheet, where the differences arise from the fact that the MHD simulation does not produce as thin a current sheet as the modified T89 model. The reality is probably between the two: The MHD

1725

Accuracy of Field Models

simulation overestimates the current sheet thickness, as it cannot represent the non-MHD current carriers that can produce very thin current sheets. On the other hand, the empirical model allows the current to be concentrated in (maybe unrealistically) thin layers because it lacks representation of the plasma. 15

Dee lo,1996

0730 UT MHD / modif T89

15

10

10

5 N

5

0

N

0

-5

-5

-10

-10

-15 15

-15 0

-10

-20

-30 X

-40

-50

-60

10 5 N

0 -5 -10 -15 0

-10

-20

30

-40

-50

-60

Fig. 6c. Comparison of MHD model and modified T89 model field values at 0730 UT on Dee 10, 1996 (during a disturbed period before the substorm onset). Panels from top to bottom: modified T89 model, difference

The contours BX,MHD - Bx,,,,odif.

Bx

from MHD simulation, BX from

are labeled in nT, and in the X - 2 plane

distances are given in R,q.

SUMMARY In this paper we have compared empirical magnetic field models with high-altitude magnetic field measurements during an extended period (Jan 2-8, 1997) and results from an MHD simulation during one particular substorm event (Dee 10, 1996). The purpose of the study is to examine how well the empirical models represent the actual field configuration in the magnetosphere, and to determine the the most efficient method of using the models to provide best possible representation of the actual situation. We examined the T96 model using the actual measured solar wind, IMF, and activity parameters. Comparison with observations showed that there were systematic differences indicating that, especially in the dayside, the model field was more stretched than was observed. Thomsen et al. (1996) examined statistically the magnetic field direction determined from the LANL energetic particle measurements and compared those with the T89 model. They found that when the spacecraft were away from the equator, the models frequently produced overstretching, which is similar to that obtained in this study. Comparing other, more widely separated spacecraft in the magnetosphere leads to less favorable results: Magnetic field at a given point in the dayside depends mostly on solar wind pressure variations, which change the local magnetic field. On the other hand, for a given location in the magnetotail, the magnetic field is strongly dependent on the intensity of the cross-tail (and ring) currents, as well as on the location relative to the center of the current sheet. Therefore, in the tail much of the field variability comes from the tail flapping motion rather than from variability of the current intensities. Thus, the differences between model and observed values are only weakly correlated if the measuring spacecraft are in different regions of the magnetosphere, and especially if they are in the weak-field region of the magnetotail. We also tried to improve the fit by searching through the input parameter space to find the input parameters that gave the best fit for one spacecraft (GOES-8). This resulted in a nearly perfect fit for GOES-8, but improved substantially the fit also four hours away at GOES-g. Third, we discussed how event-oriented models can systematically be tuned to represent observations during some physical processes such as the 5.

1726

T.I. Pulkkinen

substorm growth phase (Pulkkinen et al., 2000). These results show that, for the most part, the magneto_ spheric configuration changes occur in a large scale such that if the field at one point is representative of the observations, it is likely that the same is true in a relatively large region of space. Time averaging used in either the input parameters or the actual data did not affect the results described above. In order to examine the global configuration at one time instant, we compared the empirical models with an MHD simulation. The results show that the empirical models have a thinner current sheet than the MHD simulation, but the lobe field values are quite similar to each other. Without the event-oriented modifications, the T96 model was more similar to the MHD simulation than the T89 model, although the differences were not large. The empirical models were constructed using observations, and the MHDsimulation in this case compared well with data. Thus, when the empirical models and MHD simulations produced relatively similar field configurations, there is good reason to believe that the empirical models represent the large-scale field configuration quite well, also in regions where they were not constrained by in-situ measurements. ACKNOWLEDGEMENTS The author would like to thank H. Singer for useful discussions on the GOES magnetic field measurements as well as for providing the data, M. Wiltberger for processing the MHD simulation results, R. Lepping and A. Lazarus for the WIND data, and T. Hughes and the Canadian Space Agency for the CANOPUS data. The NSSDC is thanked for maintaining and operating the CDAWeb data facility, which was utilized in collecting the various measurements used in this study. REFERENCES Biichner, J., and L. M. Zelenyi, Chaotization of the electron motion as the cause of an internal magnetotail instability and substorm onset, J. Geophys. Res., 92, 13456, 1987. Fedder, J. A., J. G. Lyon, S. P. Slinker, and C. M. Mobarry, Topological structure of the magnetotail as a function of interplanetary magnetic field direction, J. Geophys. Res., 100, 3613, 1995. Kubyshkina, M., V. A. Sergeev, and T. I. Pulkkinen, Hybrid input algorithm - Possibility to get a realistic event-oriented magnetospheric model, J. Geophys. Res., 104, 24977, 1999. Lu, G., G. Siscoe, A. Richmond, T. Pulkkinen, N. Tsyganenko, H. Singer, and B. Emery, Mapping of the ionospheric field-aligned currents to the equatorial magnetosphere, J. Geophys. Res., 102, 14467, 1997. Pulkkinen, T. I., D. N. Baker, R. J. Pellinen, J. Biichner, H. E. J. Koskinen, R. E. Lopez, R. L. Dyson, and L. A. Frank, Particle scattering and current sheet stability in the geomagnetic tail during the substorm growth phase, J. Geophys. Res., 97, 19,283, 1992. Pulkkinen, T. I., D. N. Baker, L. A. Frank, J. B. Sigwarth, H. J. Opgenoorth, R. Greenwald, E. FriisChristensen, T. Mukai, R. Nakamura, H. Singer, G. D. Reeves, M. Lester, Two substorm intensifications compared: onset, expansion, and global consequences, J. Geophys. Res., 103, 15, 1998. Pulkkinen, T. I., M. V. Kubyshkina, D. N. Baker, L. L. Cogger, S. Kokubun, T. Mukai, H. J. Singer, J. A. Slavin, and L. Zelenyi, Magnetotail currents during the growth phase and local aurora1 breakup, in Magnetospheric Current Systems, Geophysical Monograph 118, edited by Shin-i& Ohtani, Ryoichi Fujii, Michael Hesse, and Robert L. Lysak, p. 81, 2000. Sergeev, V. A., M. Malkov, and K. Mursula, Testing the isotropic boundary algorithm method to evaluate the magnetic field configuration in the tail, J. Geophys. Res., 98, 7609, 1993. Thomsen, M. F., D. J. McComas, G. D. Reeves, and L. A: Weiss, An observational test of the Tsyganenko (T89a) model of the magnetospheric field, J. Geophys. Res., 101, 24827, 1996. Tsyganenko, N. A., Magnetospheric magnetic field model with a warped tail current sheet, Planet. Space Sci., 37, 5, 1989. Tsyganenlq N. A., Quantitative models of the magnetospheric magnetic field: Methods and results, Space . Sci. Rev., 54, 75, 1990. Tsyganenko, N. A., Modeling the Earth’s magnetospheric magnetic field confined within a realistic magnetopause, J. Geophys. Res., 100, 5599, 1995. Wilt&ger, M., ‘I’. I. Pulkkinen, J. G. Lyon, and C. C. Goodrich, MHD simulation of the magnetotail during the December 10, 1996 substorm, J. Geophys. Res., fO5, 27649, 2000.