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Journal of Atmospheric and Solar-Terrestrial Physics 63 (2001) 431­440

www.elsevier.nl/locate/jastp

Modeling of geomagnetic Úeld during magnetic storms and comparison with observations
I.I. Alexeeva ; , Ya.I. Feldsteinb
b a Institute of Nuclear Physics, Moscow State University, Moscow, Russia Institute of Terrestrial Magnetism, Ionosphere and Radio Wave Propagation of the Russian Academy of Science, Troitsk, Moscow Region, Russia

Received 15 November 1999; accepted 21 February 2000

Abstract This paper discusses: (a) development of the dynamic paraboloid magnetospheric Úeld model, (b) application of this model for the evaluation of a variety of magnetospheric current systems and their contribution to the ground magnetic Úeld variations during magnetic storms, (c) investigation of auroral electrojet dynamics and behavior of plasma precipitation boundaries, and (d) usage of the paraboloid magnetospheric Úeld model for revealing relationships between geomagnetic phenomena at low altitudes and the large-scale magnetospheric plasma domains. The model's input parameters are determined by the solar wind plasma velocity and density, the IMF strength and direction, the tail lobe magnetic ux F , and the total energy of ring current particles. The auroral particle precipitation boundaries are determined from, the DMSP particle observations; these boundaries are used to calculate the value of F . The in uence of the Úeld-aligned tail, and ring currents on the magnetospheric Úeld structure is studied. It is found that the polar cap area is strongly controlled by the tail current. The paraboloid magnetospheric Úeld model is utilized for the mapping of the auroral electrojet centerlines and boundaries into the magnetosphere. Analysis of the magnetic Úeld variations during magnetic storms shows that the contributions of the ring current, tail current, and the magnetopause currents to the Dst variation are approximately equal. c 2001 Elsevier Science Ltd. All rights reserved. Keywords: Magnetospheric magnetic Úeld model; Magnetosphere

1. Introduction The Earth's magnetosphere is strongly disturbed during magnetic storms. Storm intervals are interesting for study and are important for the safety of geosynchronous satellites and many other practical needs. However, the average magnetospheric Úeld model usually used for the modeling e orts is too crude for the realistic description of the disturbed magnetosphere. The magnetospheric magnetic Úeld model developed by Tsyganenko, (1995) is widely used in many studies (see also Tsyganenko and Stern, 1996). The latest version of that

Corresponding author. Tel.: 007-095-939-1036; fax: 007-095939-3553. E-mail address: alexeev@dec1.npi.msu.su (I.I. Alexeev).

model T96 uses the observed Dst to parameterize the level of activity in the magnetosphere. This input has replaced the Kp -index used in the previous version T89 of the model. The major limitation of the validity of T96 (as it was noted by the author himself in the program reference manual) is the restriction 20 nT ¿Dst ¿ - 100 nT. Thus, this model cannot be utilized for strong magnetic storms because it is based on spacecraft data collected over many years but does not include intervals of strong storms. In this database, highly disturbed time intervals occupy a small fraction of the entire time interval covered by the data. In contrast to the empirical models, the paraboloid model of Alexeev et al. (1996) is a dynamic model of the magnetosphere which is able to reproduce well shorter time scales (1 h) in Dst . This model has no limitations due to the strength of the storm and, therefore, can be used even for super storms.

1364-6826/01/$ - see front matter c 2001 Elsevier Science Ltd. All rights reserved. PII: S1364-6826(00)00170-X


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The paraboloid model of Alexeev et al. (1996) is based on the solution of the Laplace equation inside the magnetopause which is a paraboloid of revolution (see also Alexeev, 1978). As it was pointed out by Alexeev and Shabansky (1972) in the case of the earth's dipole source, the magnetospheric magnetic Úeld can be evaluated that has a vanishing normal component at the magnetopause. Similar problem was solved by Alexeev et al. (1975) for the magnetotail plasma sheet source. A version of the magnetospheric model in which magnetopause was a paraboloid of revolution was constructed by Greene and Miller (1990). The paraboloid modeling technique was described by Stern (1985). We adopt a version of the paraboloid model of Alexeev et al. (1996) which determines the magnetospheric magnetic Úeld by solar wind plasma as well as magnetospheric plasma dynamics. The magnetospheric dynamics is approximated by a sequence of static models of magnetospheric current systems ("wire" approach) which are driven by input parameters that are varied with time. Kamide et al. (1998) have presented a common view of many investigators on the problem of magnetic storms. Here we will consider a very important issue that was not discussed by Kamide et al. (1998); this issue concerns the contribution of the tail current sheet magnetic Úeld to the Dst variations in the course of a magnetic storm. After Burton et al. (1975), the following formula is used to account for the magnetopause current contribution in the observed Dst : Dst = DR + DCF - Dst (0): (1) Here DR is the pressure corrected Dst which describes the symmetric part of the disturbed ring current, DCF is the Úeld of the magnetopause Chapman­Ferraro currents, and Dst (0) = +22 nT is a constant deÚned by the DR and DCF Úelds during magnetically quiet intervals. In the framework of the paraboloid magnetospheric magnetic Úeld model, Alexeev et al. (1992, 1996) have shown that during the main phase of a magnetic storm the magnetotail current Úeld DT (as observed at the Earth's surface) is of the same order as DR. A similar result was obtained by Maltsev et al. (1996), and indeed, the conclusion that "the large magnetic disturbance shows characteristics more of a magnetospheric tail sheet current than of a ring current" was made earlier by Campbell (1973). (Note that the triangulated hypothetical current method used by Campbell (1973) is too rough for numerical estimations.) Kamide et al. (1998) discussed magnetospheric tail dynamics only in terms of unsteady convection and the associated plasma heating. It is known that the energy transfer and= or energy storage in the magnetotail control the energy budget associated with a geomagnetic storm. For this reason, evaluation of the direct contribution of the tail current magnetic Úeld to the Dst variation is very important. If the DT Úeld has roughly the same strength as the DR Úeld during the main phase of a magnetic storm, it is necessary to revise a number of fundamental parameters of the ring cur-

rent obtained on the basis of ground geomagnetic Úeld variation data. This will require modiÚcation of the model of DR dynamics in the course of a magnetic storm. If the DT Úeld is essential in the equation of energy balance, we must recalculate the injection function and the ring current decay parameter. Obviously, the relationships between the di erent contributors to the total magnetospheric energy should also be revised. Below, based on the study of two magnetic storms (6 ­11 February and 23­27 November 1986), we will demonstrate the relationships between the contributions from di erent sources to the Dst variation, using the paraboloid model of the magnetospheric Úeld by Alexeev (1978) and Alexeev et al. (1996). The paraboloid model input is determined by independent data obtained from magnetic observatories and AMPTE= CCE and DMSP F6-F7 satellites (Dremukhina et al., 1999). The model calculations show that the Dst variation is not only related to the ring current; in considering contributions to Dst , other magnetospheric current systems (in particular, the magnetotail currents and the magnetopause currents) should not be overlooked. 2. Model of the magnetospheric magnetic field during magnetic storms 2.1. Accuracy of the paraboloid model approach to magnetopause conÚguration In our study, the magnetopause is represented by a paraboloid of revolution. First, we discuss the accuracy of the paraboloid approach. Fig. 1 (adapted from Kalegaev and Lyutov (2000)) shows a meridional cross section of the paraboloid of revolution and points were a number of spacecraft orbits crossed the magnetopause. These magnetopause crossing points was calculated by Kalegaev and Lyutov (2000) from data collected by Sibeck et al. (1991). Solar magnetospheric coordinates of the crossing points were multiplied by the factor (p0 =psw )1=6 . It describes a magnetosphere scaling by solar wind dynamic pressure. Here p0 is the average value calculated for all data set, and psw is a current dynamic pressure at magnetopause crossing time. All points are placed on the GSM x­ plot where = y2 + z 2 . One can see that for x¿ - 30 RE distances between the crossings and the paraboloid are typically less than 3 RE . As seen, a least-squares Út to all empirical points (a thin line) is very close to the meridional cross section of the paraboloid of revolution (a heavy line) (Kalegaev and Lyutov, 2000). 2.2. Magnetospheric magnetic Úeld sources To better explain our approach, we provide below a short description of the paraboloid magnetospheric Úeld model. Utilizing the paraboloid approach, we can construct a time-dependent model of all known magnetospheric


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Fig. 1. The two-dimensional cross-section of the magnetopause approximated by the paraboloid of revolution and the points where the satellite orbits crossed the magnetopause determined from measured particles ux and magnetic Úeld vectors. The coordinates of the data points by Sibeck et al. (1991) was corrected by Kalegaev and Lyutov (2000) taken into account solar wind plasma dynamic pressure. Shown are the points in X ­ plane of GSM coordinates ( = y2 + z 2 ).

currents in the ionosphere, in the magnetosphere, and on the magnetopause. 5. IMF penetration into the magnetosphere. The continuity equations for the magnetic Úeld and the electric current density div B = 0 and div j =0; are true for all the model calculations. The magnetic Úeld vector Bm can be calculated by summing the Úelds of magnetospheric origin
Fig. 2. Current systems used for the calculation of the magnetic Úeld in the magnetosphere. All current systems are closed and one can see the closed currents' loops. If we go from equator (from Earth) to the pole (to the distant tail) we meet: (1) Ring current; (2) Tail current and closure magnetopause current; (3) Partial ring current and Region 2 Úeld-aligned current; and (4) on the day side one can see Region 1 Úeld-aligned currents closed by the magnetopause current.

Bm (t )= Bd ( )+ Bcf ( ; R1 )+ Bt ( ; R1 ;R2 ;F ) +Br ( ; Br )+ Bsr ( ; R1 ;Br ) +Bfac ( ; R1 ;F ;I0 )+ b(Rm ; BIMF ):

(2)

current systems; however, we note that every current system has its own time scale. The main contributors to the magnetospheric magnetic Úeld are the following (see Fig. 2): 1. The intrinsic geomagnetic (dipole) Úeld, as well as the shielding magnetopause currents, which conÚne the dipole Úeld inside the magnetosphere (Chapman­Ferraro currents). 2. The tail currents and their closure currents on the magnetopause. 3. The symmetric ring current and the corresponding shielding magnetopause current, whose contributions are mostly important during the magnetic storms. 4. The three-dimensional current systems representing the Regions 1 and 2 Úeld-aligned currents and their closure

Here Bd ( ) is the dipole magnetic Úeld; Bcf ( ; R1 ) is the Úeld of currents on the magnetopause shielding the dipole Úeld; Bt ( ; R1 ;R2 ;F ) is the Úeld of the magnetospheric tail current system (cross-tail currents and closure magnetopause currents); Br ( ; Br ) is the Úeld of the ring current; Bsr ( ; R1 ;Br ) is the Úeld of currents on the magnetopause shielding the ring current Úeld; Bfac ( ; R1 ;F ;I0 ) is the Úeld-aligned currents; b(Rm ; BIMF ) is the part of the interplanetary magnetic Úeld penetrating into the magnetosphere. To make the magnetospheric magnetic Úeld (calculated from Eq. (2)) time-dependent, we have to deÚne the time-dependent input parameters: the geomagnetic dipole tilt angle , the geocentric distance to the subsolar point R1 , the geocentric distance to the earthward edge of the magnetospheric tail current sheet R2 , the tail lobe magnetic ux F , the ring current magnetic Úeld strength at the Earth's center Br , the total strength of the Region 1 Úeld-aligned current I0 , the interplanetary magnetic Úeld vector BIMF , and the magnetic Reynolds number of the solar wind ow Rm which determines the part of the IMF penetrating into the magnetosphere. The ratio of the IMF penetrating into the


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magnetosphere to the solar wind IMF was found by Alexeev (1984, 1986) from an analytic solution of the problem of the conducting uid owing past paraboloid of revolution. Modeling of the Dst variation was limited to the Úrst Úve terms on the right-hand side of Eq. (2), because Bfac does not contribute much to the symmetric ground disturbances, and b is only about 0.1 times the IMF. Therefore, only the limited number of parameters required to specify the above-mentioned current systems has been used. However, it may be possible in principle to determine all contributions by measuring and inputting all the solar wind and magnetospheric parameters. To calculate the magnetic Úeld disturbances using the paraboloid model, it is necessary to deÚne only Úve time-dependent input parameters: the geomagnetic dipole tilt angle , the geocentric distance to the subsolar point R1 , the geocentric distance to the earthward edge of the magnetospheric tail current sheet R2 , the tail lobe magnetic ux F , and the ring current magnetic Úeld strength at the Earth's center Br . The projection of the geomagnetic dipole tilt angle on the XZ plane of the solar-magnetospheric coordinate system is a known function of UT (e.g. Alexeev et al., 1996). The value R1 was determined by the dynamic pressure of the solar wind Psw (nPa) and the IMF Bz component (nT) as given by Shue et al. (1997): R1 =(Psw )
- 1 6:6

magnetic Úeld pressure at the subsolar magnetopause point. These R values have been used as input parameters for the 1 subsequent calculations of the model's magnetic Úeld variations at the Earth's surface. The di erence between R1 and R values is small (tenths of RE ). During the periods with 1 southward IMF R ¡R1 , and R ¿R1 during the periods 1 1 with Bz ¿ 0. 2.2.1. The dipole Úeld and the Úeld of the magnetopause shielding currents The dipole Úeld Bd = -Ud , where Ud = RE R
3

B0 (z cos

+ x sin ):

The magnetic Úeld of the magnetopause shielding currents, Bcf , has been calculated by Alexeev and Shabansky (1972). The normal to the magnetopause component Bn of the total Úeld B = Bd + Bcf equals zero. The potential Ucf (Bcf = -Ucf ) of the magnetopause shielding currents has been calculated as a solution of the Laplace equation with the boundary condition B · n =0 or Bcf · n = -Bd · n:

R

E

11:4+0:013B 11:4+0:140B

z z

for Bz ¿ 0; for Bz ¡ 0:

(3)

The value Br was determined by Dessler­Parker­Sckopke relation (Dessler and Parker, 1959; Sckopke, 1966) 2K Br = B 0 : (4) 3d Here B0 is the dipole Úeld at the Earth's equator (0:3 Gauss=3 â 104 nT), K is the total kinetic energy of the ring current particles, d is the geodipole magnetic Úeld energy above the Earth's surface (Carovillano and Siscoe, 1973). Other input model parameters are determined by the following relationships: RE R2 = ; F =2B0 R2 sin2 ám : (5) E sin2 án Here án is the midnight colatitude of the equatorward boundary of the auroral oval, ám is the angular polar cap radius, and RE is the Earth's radius. In order to obtain the values án and ám , we used the DMSP satellite data on the locations of di erent auroral precipitating particle patterns in the high-latitude region. The K values are calculated from the total ion energy obtained by the AMPTE= CCE satellite inside the radiation belt. The procedure of these model parameters calculation based on the satellite data was described by Dremukhina et al. (1999). Values of the parameter R1 where obtained from Shue et al. (1997), but these values were then recalculated by the iteration method using the balance condition between the solar wind dynamic pressure and the paraboloid model

Here n is the normal to the magnetopause. As a consequence of the paraboloid axial symmetry, the potential Ucf has a simple representation in spherical coordinates R; #; . The polar axis of this coordinate system is the Earth­Sun line, # being the polar angle (cos # = x=R), and the azimuthal angle is equal to zero in the X ­Z plane of the solar-magnetospheric coordinates. In these coordinates, the scalar potential Ucf is written as Ucf = - B0 R R2 1
3 E n=1

R R1

n

[dn sin

Pn (cos #) (6) d Pn : dx

1 +d cos cos Pn (cos #)]; n

where Pn (x)= 1 d n (x2 - 1) 2 n! d xn
n n

and

1 Pn (x)=

1-x

2

The Úrst six dimensionless coe cients dn and d are listed n in the second and third columns of Table 1; these coe cients describe the magnetic Úeld of the currents induced on the
Table 1 The coe cients of expansion of the potential Ucf in spherical harmonics (d ;dn ) and in the Bessel functions (Dn ;Gn ) n n 1 2 3 4 5 6 d
n

d

n

D

n

G

n

0.6497 0.2165 0.0434 -0:0008 -0:0049 -0:0022

0.9403 -0:4650 0.1293 -0:0148 -0:0160 -0:0225

6.573368 31.07137 79.88151 158.0693 269.9342 --

0.670460 2.947181 6.039411 9.771301 14.04944 --


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435

the current sheet (Alexeev et al., 1975; Alexeev and Bobrovnikov, 1997). The current sheet is placed at ¿ 0 and 0 ¡Ú ¡Úc ( ), where the function Úc ( ) is d d ; for |cos | ¿ 0 R1 |cos | 0 R1 Úc ( )= (10) d 1 ¿ |cos |: for 0 R1 Here d is the half thickness of the current sheet. Inside of the current sheet, the magnetic Úeld of the tail current system is a sum of two terms Bt = B2 - bt R1 Ut1 :
Fig. 3. Chapman­Ferraro magnetopause current which shields the dipole Úeld. The normal (to the magnetopause) component of the ~ total Úeld Bn = 0. Outside of the magnetopause, the magnetic Úeld equals zero.

(11)

Here bt is the magnetic Úeld strength in the tail lobe at the inner edge of the tail current sheet. This value is deÚned by the model parameters R1 ; R2 ; and F : bt = 2F : R2 0 1 (12)

magnetopause by a dipole perpendicular and parallel to the solar wind ow, respectively. The expansion parameter of (6) is R=R1 , therefore it can be used only up to R 6 R1 . Over the right-hand side, it is more convenient to represent the sum of potentials Ud + Ucf in parabolic coordinates as an expansion in the Bessel functions Ud + Ucf =
B0 R3 E R2 n=1 1

B2 is found here as a partial solution of the vector potential problem: B2 = b
t 0

Ú Úc (0)

cos ; 2 + Ú2

B2Ú =0;

B

2

=0:

(13)

[sin

D n J0 (
1n

0n

Ú)e
1n

0n

K0 (
1n

0n

) (7)

+ cos cos Gn J1 (

Ú)e

K1 (

)]:

Here the parabolic coordinates ; Ú; are deÚned through the (x; y; z ) solar-magnetospheric Cartesian coordinates Ú2 - Ú sin Ú cos
2

+1=2x=R1 ; = y=R1 ; = z=R1 : (8) of the equations will use (6) for the 0 . The value of 0 inner edge of the (9)

The current density vector is proportional to â B2 . It is tangential to the paraboloid = const and parallel to the equatorial plane. The scalar potential Ut1 (see Eq. (11)) deÚnes a component of the tail magnetic Úeld which is perpendicular to the equatorial plane. The potential Ut1 can be written as


Ut1 ( ; Ú; )=
k; n=1

cnk cos n Jn (

nk

Ú)Kn (

nk

):

(14)

In Eq. (7), 0n ; 1n are the solutions J0 (x) = 0 and J1 (x) = 0, respectively. We case ¡ 0 , and (7) for the case ¿ is determined by the distance to the geomagnetic tail current sheet, R2 :
0

Here nk is the k th solution of Jn (x) = 0. Outside of the current sheet, the scalar potential Ut of the magnetic Úeld of the tail current system is
1

bnk cos n Jn (
k; n=1

nk

Ú)In (

nk

) (15)

=

2R2 1+ : R1

Ut = bt R

In numerical calculations, we used the Úrst six coe cients Dn and Gn , presented in the fourth and Úfth columns of Table 1. Fig. 3 shows the magnetopause currents (heavy lines) which shield the dipole Úeld, and the magnetic Úeld lines (thin lines) of the magnetospheric Úeld which go to the magnetopause. 2.2.2. Magnetic Úelds of the tail current system We used a model of the tail current system magnetic Úeld which takes into account the Únite thickness of



for
0

¡

0

; 1 ¿ Ú¿ 0;

ln sign( 2 -| |)+ Ut1 ( ; Ú; ) for ¿
0

; 1 ¿ Ú ¿ Úc ( ):

In Eqs. (14) and (15), the coe cients bnk and cnk are deÚned by fnk as bnk =2
nk

fnk [1 + (

2 nk In

(

nk 0

)Kn ( )

nk 0

)];

cnk =2f

3 nk nk In

nk 0

)In (

nk 0


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Table 2 Numerical values of the coe cients f the tail current system k=n 1 2 3 4 5 1 2.0635 0.108665 0.029803 0.012946 0.006536 3 -0:4437 -0:053383 -0:017021 -0:008451 -0:004620

nk

of scalar potential Ut of 5 0.2949 0.041799 0.012939 0.006415 0.003708 7 -0:280 -0:04171 -0:01203 -0:00537 -0:00309

Fig. 5. Field lines of the tail currents system in the noon­midnight meridian plane.

Fig. 4. Tail currents system: shown are the current lines (heavy curves) and the magnetic Úeld lines (thin ones). The normal (to ~ the magnetopause) component of the total Úeld Bn = 0. Outside of the magnetopause, the magnetic Úeld equals zero (Alexeev and Bobrovnikov, 1997).

and fnk =
-

to the magnetopause currents that shield the dipole Úeld. Therefore, the subsolar magnetopause distance decreases during the disturbed time for two main reasons: (1) an increase in the solar wind pressure, and (2) a decrease in the magnetospheric Úeld at the subsolar point as a consequence of the increase of the tail current system. An increase in the density or velocity of the solar wind entails a decrease in R1 . During the strong geomagnetic disturbances, the earthward edge of the tail current sheet moves closer to the Earth (i.e., R2 decreases), and the size of the polar cap or the value of the tail lobe magnetic ux increases (i.e., F grows).
1 Úc (')

cos n {cos =Úc (0)

Úc ( ) 0

Jn ( (
2 nk

nk

Ú)Ú d Ú + sign( = 2 -| |)
2 2 n nk

Jn (

nk

Ú)Ú d Ú} d '

- n )J (

)In (

nk 0

)

:

Numerical values of fnk are presented in Table 2 for 0 = 2:4 (R2 =0:7R1 ) and n=2m+1. For n =2m the coe cients fnk are zero. Fig. 4 shows current (heavy lines) of the tail current system and the magnetic Úeld lines (thin lines) of this current at the magnetopause for quiet conditions. The summary Úeld in the magnetosphere B = -(Ud + Ucf ) + Bt will be determined by the four parameters: ; R1 ;R2 ;F . The last three parameters change with the level of geomagnetic activity, the Bz component of the IMF, and the solar wind plasma dynamic pressure. The mean values of these parameters used in our numerical calculations are R1 = 10 RE ; R2 = 7 RE and F = 380 MWb. These values correspond to bt = 40 nT. Fig. 5 shows the magnetopause and the magnetic Úeld lines of the tail current system in the noon­midnight meridian plane. It is a very important characteristic of the tail current system that the tail currents from the inner part of the tail current sheet are closed at the subsolar magnetopause near noon. The direction of this current at the subsolar magnetopause is opposite

2.2.3. Magnetic Úeld of the ring current The ring current is created by trapped energetic particles. Their drift in the geomagnetic Úeld produces a westward azimuthal current. During magnetic storms energetic particles are intensively convected and injected into the inner magnetosphere. The total energy of trapped particles increases and the ring current contribution to the magnetic Úeld at the Earth's surface can reach 1­2% of the dipole Úeld. In the region of the ring current maximum (at R 3 RE ), its contribution becomes essential. During quiet conditions, the ring current intensity decreases by a factor of 10 or more. Measurements of the magnetic Úeld during the magnetic storms show strong evidence of signiÚcant asymmetry in the ring current, which is especially large during the storm main phase. However, during most of the storm, especially during the recovery phase, the ring current can be treated as axially symmetric. It is convenient to introduce the magnetic Úeld vector potential A (B = curl A). The external boundary, where the current becomes zero, coincides with the


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437

distance to the inner edge of the geomagnetic tail current sheet, R2 (R2 7 RE ). The distance to the current maximum is equal to 0:5 R2 . In spherical coordinates (where the polar axis is anti-parallel to the dipole moment vector and the azimuthal angle is zero at magnetic local noon), the ring current density vector has only one non-zero component -j' (R; á; '). The radial and latitudinal components of the current density are zero. The dependence of the azimuthal current density on the geocentric distance R and on the distance above the equatorial plane z = R cos á is given by the formula j' = 15gR 2 R2 R sin á 20 R 0
-7 rc

for 0 6 R 6 R2 ; for R¿R2 ;

(16)

where Rrc (R)= R2 + R2 = 2, and gR is the magnetic mo2 ment of the ring current. The azimuthal component of the vector potential A' is written as 2 - 1 for 0 6 R 6 R ; 2 3 Rrc R3 2 A' = gR R sin á (17) 1 for R¿R2 : R3 The ring current magnetic Úeld R5 Rrc Bd +2B0 gR for 0 6 R 6 R2 Br = ME B d for R¿R2 : vector Br is written as RE R2 ;
3

R2 Rrc

5

Fig. 6. Ring current system: shown are the magnetic Úeld lines of the ring current and the shielding current. The normal (to the ~ magnetopause) component of the total Úeld Bn = 0. Outside of the magnetopause, the magnetic Úeld equals zero.

-1 e

z

(18)

3. Magnetic storms of 23 --27 November 1986 The paraboloid magnetospheric Úeld model allows us to calculate contributions of a number of magnetospheric current systems to the Dst variation during magnetic storms. Fig. 7a shows contributions to the ground geomagnetic variations from the geotail current system, Bt , the Chapman­ Ferraro currents, Bcf , and the ring current Úeld, Br during the magnetic storm of 23­27 November, 1986 (Dremukhina et al., 1999). Here the magnetic Úeld of the induced currents inside the Earth has also been taken into account by multiplying factor 1.5 of the model horizontal perturbation Úeld. The estimation of this factor by Langel and Estes (1985) gave 1.3 because the Earth is not perfectly diamagnetic. But we use 1.5 for excluding of undeÚnity connected with Earth's conductivity. Values of Br have been calculated based on the values of the total ion energy measured by the AMPTE= CCE satellite (Dremukhina et al., 1999). As one can see, values of Bcf , Br and Bt have comparable and large values during the main phase of the storm. The time behaviors of Bcf and Bt are similar, but the directions of these vector are opposite to each other. During the recovery phase Bcf and Bt decreased faster than Br . Fig. 7b shows comparisons between the model's magnetospheric magnetic Úeld Bm - Bd and the Dst index. As calculated for the entire storm interval, the correlation coe cient between Dst and Bm - Bd is equal to r = 0:82, and the standard deviation is =16:1 nT.

In (18), Bd is the Earth's dipole Úeld, ME = B0 R3 is the E Earth's dipole moment, and ez is a unit vector along the Z -axis. The strength of the ring current magnetic Úeld at the Earth's center (approximately equal to the surface Úeld perturbation) can be calculated as Br = |Br (0)| = 2g R (1 - 4 2): R3 2 (19)

This value Br deÚnes the total energy of ring current particles (see Eq. (4)). The ring current magnetic moment gR and the total ring current value I as functions of Br and R2 are gR =0:1M and I =5:34 MA R2 Br : 100 nT 7 RE (20)
E

Br 100 nT

R2 7R

3 E

Fig. 6 shows the magnetic Úeld lines in the noon­midnight meridian plane. These Úeld lines show the magnetic Úeld of the ring currents and of the magnetopause currents that shield the ring current Úeld.


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Fig. 7. (a) During the magnetic storms of 23­27 November 1986 the contributions of magnetospheric current systems to the ground Dst variations as calculated by using the paraboloid model of the magnetosphere. Triangles -- Chapman­Ferraro currents Bcf , asterisks -- the geotail current system Bt , and solid curve -- the ring current Úeld Br . (b) Comparison of the model Úeld Bm - Bd = Bcf + Bt + Br with the observed Dst index.

4. Magnetic storm of 6 --11 February, 1986 Kozyra et al. (1998) have pointed out that there is a rel atively large discrepancy between the Dst (where Dst is the ring current contribution to the disturbance magnetic Úeld Dst ) inferred from satellite measurements and that estimated from the Dst index for storm 6 ­11 February 1986; neglecting the magnetotail current system can cause such a discrepancy. Many investigators have studied this storm. Hamilton et al. (1988) estimated the ring current ion energy in the course of this strong magnetic storm with a minimum hourly averaged value of Dst = -300 nT. Unfortunately, solar wind data are unavailable during the most intense part of the storm main phase, and deÚnitive model calculations of the contributions from the di erent magnetospheric current systems to Dst are impossible. However, we can compare the observed Dst with the total kinetic energy of the ring current ions Up , obtained from the AMPTE= CCE CHEM observations and presented by Hamilton et al., 1988. In Table 3 (see Hamilton et al. (1988)), Up is the total kinetic energy of the ring current ions. In the third column, the predicted B (nT)

was calculated by using the Dessler­Parker­Sckopke relationship (Dessler and Parker, 1959; Sckopke, 1966); in the fourth column, the observed Dst values are listed. In the next two columns of Table 3, the ratios of the predicted Dst to the observed Dst as calculated by Hamilton et al. (1988) and by Feldstein (1992) are presented. The last column of Table 3 shows the results of our calculations, which include crude estimates of the tail current contribution. The ratio of the predicted B to the observed Dst , given in the 5th column of Table 3 according to Hamilton et al. (1988), varies from 0.24 to 0.84; in all cases, the ratio is less than unity. During the recovery phase (when the ring current might be expected to become more symmetrical), the ratio is close to 0.5. Feldstein (1992) has discussed the reasons for a large discrepancy between the predicted and observed magnetic Úeld values. Feldstein (1992) has corrected the observed values by taking into account the magnetopause current contribution and the e ects of the induced currents in the solid Earth. As one can see from the ratios listed in the 6th column, they increase up to about 0.8 during the recovery phase. As we showed in a case study (Alexeev et al., 1996; Dremukhina et al., 1999), the magnetopause current contribution DCF and the tail current contribution DT are approximately equal in magnitude but they have opposite signs. For that approach, the 7th column lists the ratios of predicted B to the observed Dst . As one can see, these ratios are close to unity during the recovery phase. Therefore, the relatively large discrepancy between DR inferred from satellite measurements and that estimated from the Dst index (observed by Kozyra et al., 1998) disappeared after we took into account the tail current contributions. It is one of the results of our model study that main contributors to Dst DR, DT , and -DCF are all of the same order of magnitude. The observed Dst variation (storm 6 ­11 February 1986) corresponds with good accuracy to the symmetric ring current contribution which is determined by the total trapped ion kinetic energy. 5. Tail lobe magnetic field Let us compare the paraboloid model calculation of the magnetic Úeld with observations in the distant tail. Feldstein et al. (1999) mapped the center, equatorward and poleward boundaries of the eastward and westward electrojets from ionospheric altitudes to the distant magnetotail. The eastward electrojet was mapped to the inner magnetosphere in the ring current region, but the westward electrojet was mapped to the entire plasma sheet, from its inner to its outer boundary. The magnetic Úeld strength B was also determined from the model Úeld. During the magnetic storms of 10 May 1992 (Dst = -200 nT) and 6 February 1994 (Dst = -120 nT), mapping of the polar boundary of the westward electrojet to distances near X = -50 RE produces a calculated magnetic Úeld strength of 34 nT at 18:36


I.I. Alexeev, Ya.I. Feldstein / Journal of Atmospheric and Solar-Terrestrial Physics 63 (2001) 431­440 Table 3 Comparison of ring current ion energy content with observed and predicted onground Dst variation Pass AMPTE= CCE 1 1 2 2 3 3 4 4 5 5 6 6 7 7 8 8 in out in out in out in out in out in out in out in out Total Up 1030 keV for L =2­7 3.67 3.33 3.98 3.59 8.68 22.1 12.4 12.4 (47.6) 41.5 25.2 21.8 17.2 16.9 14.8 12.6 Predicted B; nT -14 -13 -15 -14 -34 -88 -50 -50 -189 -165 -100 -87 -68 -67 -59 -50 :6 :2 :9 :3 Observed Dst ;nT +0:7 +2:7 -7:2 +1:2 -83 -88 -115 -131 -257 -244 -131 -133 -112 -106 -92 -73 Predicted= observed Hamilton et al. (1988) -- -- -- -- 0.24 0.84 0.31 0.27 0.68 0.62 0.66 0.55 0.49 0.50 0.49 0.49 Predicted= observed Feldstein (1992) -- -- -- -- 0.33 0.8 0.4 0.3 0.86 -- 0.88 0.85 -- -- 0.8 0.72

439

Predicted= observed -- -- -- -- 0.62 1.5 0.65 0.57 1.11 1.02 1.15 0.98 0.91 0.94 0.97 1.04

UT of 10 May for the Úrst storm and 27 nT at 21:24 UT of 6 February for the second storm. The usual strength of the tail Úeld at these geocentric distances is about 10 nT. The model predicts enhancements of the tail lobe magnetic Úeld several times during the magnetic storm. These results are supported by Geotail data (Kokubun et al., 1996); the data show that the geotail lobe magnetic Úeld strength at geocentric distances 100 RE is several tens of nanoteslas during magnetic storms. The tail lobe Úeld increases when the Dst variation increases. However, at these distances, the tail lobe Úeld strength varies (with distance) very slowly. For this reason, the agreement between the model calculations and the data is good enough.

ues served for the determination of the tail lobe magnetic ux and the distance to the inner edge of the tail currents. The tail current dynamics during the studied magnetic storms is strongly supported by geotail observations in the distant tail obtained by Kokubun et al. (1996). Our general conclusion indicates that the depression of the magnetic Úeld strength at the Earth's equator due to the development of the ring current reaches the same value as the tail current magnetic Úeld. Acknowledgements The authors acknowledge the members of the NPI MSU and IZMIRAN teams -- Elena Belenkaya, Vladimir Kalegaev, Sergey Bobrovnikov, Anatoly Levitin, Ludmila Gromova, and Lidia Dremukhina -- who took part in the paraboloid model calculations and the data analysis. The authors also thank Vladimir Papitashvili of the University of Michigan for a careful and helpful review of the manuscript. They thank the referee for useful comments and corrections of the manuscript. The work was supported by the INTAS-RFBR Grant 95-0932. I.I.A. acknowledges partial support from RFBR Grant 98-05-64784 and from the international supplement to NSF Grant ATM-9801941. Y.I.F. acknowledges partial support from the RFBR Grant 99-05-65611 and from the ISSI. References
Alexeev, I.I., 1978. Regular magnetic Úeld in the Earth's magnetosphere. Geomagnetism and Aeronomy (Engl. Transl.) 18, 447.

6. Conclusion Studying magnetospheric dynamics in the course of a magnetic storm, we have shown that the tail current contribution to Dst is essential. Utilizing the paraboloid model of the magnetosphere, we were able to reproduce short time scale (1 h) variations in Dst . The comparison of the model results with measured Dst for the 23­27 November 1986 and 6 ­11 February 1986 magnetic storms indicates that the tail currents may a ect the storm-time Dst as much as the ring current. This result is supported by the independent proxy data for the total ring current ion energy estimated from AMPTE= CCE measurements. We use the solar wind data (Bz IMF and dynamic pressure of the plasma ow) as model input. The tail current dynamics has been determined from DMSP particle data. The electron precipitation boundaries were used for the calculation of the polar cap magnetic ux and the midnight equatorial polar oval boundary. These val-


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I.I. Alexeev, Ya.I. Feldstein / Journal of Atmospheric and Solar-Terrestrial Physics 63 (2001) 431­440 Hamilton, D.C., Gloekez, G., Ipavich, F.M., Studeman, W., Wilken, B., Kzemsez, G., 1988. Ring current development during the great magnetic storm of February 1986. Journal of Geophysical Research 93, 14,343. Kalegaev, V.V., Lyutov, Yu.G., 2000. The solar wind control of the magnetopause. Advances in Space Research 23 (7-8), 1489. Kamide, Y., et al., 1998. Current understanding of magnetic storms: storm ­substorm relationships. Journal of Geophysical Research 103, 17,705. Kokubun, S., Fzank, L.A., Hayashi, K., Kamida, Y., Lepping, R.P., Mukai, T., Nakamura, R., Paterson, W.R., Yamamoto, T., Yamoto, K., 1996. Large Úeld events in the distant magnetotail during magnetic storms. Journal of Geomagnetism and Geoelectricity 48, 561. Kozyra, J.U., Foz, M.C., Sanches, E.R., Evans, D.S., Hamilton, D.C., Nagg, A.F., 1998. The role of precipitation losses in producting the rapid early recovery phase of the Great Magnetic Storm of February 1986. Journal of Geophysical Research 103, 6801. Langel, R.A., Estes, R.H., 1985. Large-scale, near-Úeld magnetic Úelds from external sources and corresponding induced internal Úeld. Journal of Geophysical Research 90, 2487. Maltsev, Y.P., Azykov, A.A., Belova, E.G., Gvozdevsky, B.B., Sadargaleev, V.V., 1996. Magnetic ux redistribution in the storm time magnetosphere. Journal of Geophysical Research 101, 7697. Sckopke, N., 1966. A general relation between the energy of trapped particles and the disturbance Úeld over the Earth. Journal of Geophysical Research 71, 3125. Shue, J.-H., Chao, J.K., Fu, H.C., Russel, C.T., Song, P., Khuzana, K.K., Singer, M.J., 1997. A new functional form to study the solar wind control of the magnetopause size and shape. Journal of Geophysical Research 102, 9497. Sibeck, D.G., Lopez, R.E., Roelof, E.C., 1991. Solar wind control of the magnetopause shape, location, and motion. Journal of Geophysical Research 96, 5489. Stern, D.P., 1985. Parabolic harmonics in magnetospheric modeling: the main dipole and ring current. Journal of Geophysical Research 90, 10,851. Tsyganenko, N.A., 1995. Modeling the Earth's magnetospheric magnetic Úeld conÚned within a realistic magnetopause. Journal of Geophysical Research 100, 5599. Tsyganenko, N.A., Stern, D.P., 1996. Modeling the global magnetic Úeld the large-scale Birkeland current systems. Journal of Geophysical Research 101, 27,187.

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