- Email: [email protected]
Quantitative magnetospheric magnetic field modelling with toroidal and poloidal vector fields
Planer. Space Sci., Vol. 32, No. 8. pp. 965-974, Printed in Great Britain.
0032~633/84$3.00+0.00 Pergamon Press Ltd.
1984
QUANTITATIVE MAGNETOSPHERIC MAGNETIC FIELD MODELLING WITH TOROIDAL AND POLOIDAL VECTOR FIELDS
Division
Mathematiques,
J. C. KOSIK C.N.E.S., Av. E. Belin, 31400 Toulouse,
France
(Received in_finalform 6 January 1984) Abstract-Toroidal and poloidal vector fields allow divergence free magnetic field representations in regions where currents flow. We derive genera1 magnetospheric magnetic fields using combinations of spherical harmonic expansions of the toroidal and poloidal fields. Adding restrictive conditions like the field line topology symmetry or the magnetic field measurements, more specific magnetospheric magnetic field models can be derived. Two examples of this technique are given : an axisymmetric mode1 with a ring current in the equatorial region and a time-dependent mode1 of the Earth’s magnetosphere. Our results are compared with the Olson-Pfitzer model.
VECTOR
INTRODUCTION
EXPANSIONS
A general vector field B can be constructed as a sum of toroidal and poloidal components (Stern, 1975):
Since 1960 several magnetospheric field models have been constructed for the magnetospheres of the Earth and the planets. The simplest models like the model of Mead (1964) did not take into account the internal current distributions like the ring-current. More recent models like those of Olson and Pfitzer (1974, 1979), Mead and Fairfield (1975), Hedgecock and Thomas (1975), Tsyganenko and Usmanov (1982), take into account the tail and ring currents. Such current distributions lead to the following conditions : V/\B=pJ,
FIELD
B = V A $,r+V
A V A $*r
(1)
where the first term on the right hand side is the toroidal component and the other term the poloidal component. tjl and $Z are the generating functions and r a unit vector. To derive the three components of B we use the following spherical harmonic expansions for $I and $2:
V-B=O.
$1 = CC IT;(r)PYYQ(a,_ cos m9 + b,,,, sin +>
This last condition is usually satisfied by adding some constraints on the coefficients and their analytical expansions. However, due to the least squares method, this condition is not always satisfied as was shown recently by Walker and Southwood (1982). To avoid these difficulties Stern (1975) and Granzow (1983), have shown that it is simpler to use general vector fields having toroidal and poloidal components. In the first part of this paper we use this technique associated with spherical harmonics expansions to define a class of quantitative magnetospheric magnetic field models. Such a class can be split into different sub-classes depending on the symmetry conditions for the field line topology. Then each sub-class can lead to different models depending on the quantitative requirements. Two applications of this method are given in the second part of the paper. The first one is quite simple and is an example of an axisymmetric topology. The second application is the Earth’s magnetosphere with a tilt-dependent model obtained using the AB contours from Sugiura and Poros (1973).
$2 = CC S&KY@{ c,,,, cos rn4 + d,_ sin m4).
(2)
In these expansions T(r) and S(r) are general scalar functions of r and Py are the well-known Legendre polynomials. Replacing (2) in (1) we get for the three components of B the following expansions : B, = cc
x
- T
[cl, cos rncj + d,_ sin mQl]
tan-‘(Q)% +!?!&mip;
sin2 0
as,ap;n 1
S,(r) ap;l
43 = CCcosm4 Cl,rae+Br%T [
i
+mb,~~(r~~]+sinrn~[d~~{~~
-ma,,,,7;(r)g > 965
I
(3)
J. C.
966
KOSIK
-0.61a,,T,(4-15~0~~0)
-a,_7;(r),,
1
ap;l
-c2,
X
1.
-0.61b,,T,(4-
These formulae can be developed further using the definitions ofthe Legendrepolynomials. For the sake of simplicity, each component is split into symmetric and antisymmetric parts (subscripts s and as respectively) and given only for the lowest order terms. We thus obtain : B,, = 2;c,,
B,,, =
2:c,,
sin B+ ~r:,,,(6cos2
sin 0+6&?c,,
B,= = -c,,sin8(Sr!
O-2)
sin 0 cos ~9 cos 4
+ $)-c,,sin28e
+ z)
BOnS=cos+IIcosO(++~)+b,,TI
+c,,~cos20 sin 26 sin Tz+0.6lb,,T,
J;J +2b,,
x (5 cos2 O-l)
I
+sin 4[d,,
+d,,$cos20
X
-a,,T,-a2,~cos0
cos B
(
:+$
T,
coso s, 1
-0.61a,,T,(5cosZO-1)
B, = d, 7 - + aloTt sin 0 smO r +d2,(3
co52 0 - 1) S, 3sine ; + a,,T,sin20
B~~~=cosi[d,,~+~)-a,,T;cosD +d,,&osH
+sin$
x [--c,,($++)-b,,T,cosH
+sinm$
-b,,,,7;(r)z
1
5 + % -a2,T,,,/%os28 ( r )
)
15~0~~0)
1
A CLASS OF QUANTITATIVE MAGNETOSPHERES
Formulae (3) satisfy the conditions V A B = PJ and V *B = 0 and thus can represent any type of magnetic field, with or without current distributions. No further conditions are applied to the expansions : S(r) and T(r) are still unspecified and the different coefficients a,,,,, b,_, cl,, d,_ can have any numerical values. As they stand, formulae (3) represent a large class of magnetospheric models. This class can be split into different sub-classes by the introduction of qualitative and quantitative conditions : symmetry or asymmetry in the field line topology and distribution of the field intensity. Axisymmetric magnetospheres will require the following coefficients cr r, b, 1, czl, b,,, . . to vanish, i.e. all the coefficients with subscript m # 0 will vanish. Non-axisymmetric magnetospheres will contain a priori all the coefficients. Each of these two classes can lead to different models depending on some peculiar features: ring current, corotation, plasma sheet, etc. Complex models can be represented by a family of expansions (2) as recommended by Stern (1975). In the last step we have to take into account the experimental data measurements which are more or less abundant depending on the magnetosphere. In the following part of the paper we would like to present two possible applications of the technique. The first application will be devoted to a general axisymmetric corotating magnetosphere and the other one will lead to a first approximation to the Earths magnetosphere.
CASE OF AN AXISYMMETRIC MAGNETOSPHERE
Let us build an axisymmetric magnetosphere having its field lines inflated by a ring current located in the equatorial region. We assume also that this model is symmetric with respect to the equatorial plane. Symmetry with respect to the Equator implies that when the colatitude f3 changes n-0, we have the following changes : B, + -B,, B, + B,, B, + -B,.
Quantitative magnetospheric magnetic field modelling with toroidal and poloidal vector fields Axisymmetry implies that when longitude 4 changes to 4 +A, we have the following changes : B, + B,, B, -+ BB and B, + B,. If we apply these conditions to expansions (3) we are left with a series of nonvanishing coefficients: cl,,, d,,,, u2,,, Cam, d3,, and the three components of the field are (up to the second order) : B,, = 2~cl,cos6 r
961
These three components are still quite general as five coefficients remain to be chosen and the functions Sand T to be defined. These components should be added to the internal field components. If one assumes that the internal field is that of a dipole and that our magnetosphere possesses an equatorial ring current, a possible topology is shown in Fig. 1 obtained with the following conditions :
-S”cjO i-
S, = A,r’ exp (- kr’)
x cos 8(33 -45 cos* 8)
cl0 = 0.02,
A, = - 1.5,
k = 0.03
(6)
c30 = dlo = a,, = d,, = 0. CASE OF THE EARTH’S MAGNETOSPHERE
case s, B, = d,,? - + aZoT2sin26 sin0 r
(5)
We will first develop a zero-tilt magnetosphere assuming symmetry with respect to the Equator and the noon-midnight meridian. These symmetries imply the following conditions: when @changes to n - 0, we have B,-* -B,, Bg-+ -B, and B,-+ -B,; when I#I changes to -C/J, we have B, + B,, B, + B, and B, +
ZSM
1’5.
\
\
\
‘\
\
I
.
\\ ‘.
\
/I ’
.’
i
:
/ /’ / /I /
-4’ ;;
---4
\-
a-15. FIG. ~.FIELD LINEPATTERNOFANAXISYMMETRIC
MAGNETOSPHERE
WITHAN
Dotted lines represent dipole field lines.
EQUATORIALRING
CURRENT.
968
J. C.
-B,. Applying these conditions to equations (4), the non-vanishing coefficients are: ctO, cal, b,,, b,,. Thus the non-vanishing components of the external field are : BIS = ~c2~,,S,--c,,S,(l8-25cos’8)] Bos= -sinli[e,,(++$)
KOSIK
analytical expansions, the Si and rj can have quite different analytical forms, i.e. S, and S3 are different. It is thus possible to make an adequate choice ofthe couples [S(r), P,(@)]and [I’$+),P,(0)] in order to achieve a “good modelling”. Here the analytical expressions of the Si and q were chosen in order to obtain a good fit of the AB contours derived by Sugiura and Poros (1973). The analytic expressions of the Si and Tj are : S, = A,r3exp(-,&2),
8,
S3 = rz exp (- kr’)
= 0
Tl = rexp(-kg’),
B,,, = 10.4c,, 2 cos (bsin 0 cos @
r
B,_ = 1.732~~~cos28cos~
+[b,,T,
S, = r3exp(-k,r2)
3 + $ i r
c >
+0.61b,,(S co? 0--l)T,]cos
4
(7)
B+,,. = - 1.732~~ cos Osin 4
-~os~[~~~~~+0.61b~~(4-15sinzU)~~]sin#. In these formulae the Si and 7jarefunctions ofthe radial distance r. As we construct the model as a sum of
(8) T, = rexp(-kg*).
The numerical coefficients which do not vanish are : et0 = 1, c2, = 0.11, b,, = 1, A, = -1.5, k = 0.04, k, = k, = 0.003. The AS contours obtained with these formulae are shown in Fig. 2 and can be compared with the experimental results of Sugiura and Poros (Fig. 3) and the model of Olson and Pfitzer (1974) in Fig. 4. The field line pattern obtained with our model (Fig. 5) can be compared with the result of Olson and Pfitzer (Fig. 6). To improve our model we have added the tilt effects. The appropriate formulae were obtained using the following conditions : when 0 changes to rc- 0 and the
MIDNIGHT
UNITS
: GAMMA
/
b--
12
b.2.3.ABCONTOUROBTAINEDBY
SUGIURAA~POROS~OM~XP~RI~NTDATA.
969
+si\rl 15. /
FIG. 5.FIELD LINE PATTERN~NTHE
./
/
NOON-MIDNICNT
“----
MERlDlAN CDRRFSPONUINC~TQ OUR ZERO-TILT MODBL.
Quantitative
magnetospheric
magnetic
field modelling
tilt angle I,+changes to -I,+, we have B, + - B,, B, + B,, B,-+ -B,. The non-vanishing components of the “tilt contribution” are :
with toroidal
and poloidal
971
vector fields
angle $ expressed in degrees. Here again we have selected a few coefficients and the expressions for the SF and Tj* are: S: = r3exp(-k*r’),
Sz = r2exp(-k*r2) (10)
Tfj = r exp ( - Qr2).
B,, = c2,~(6cos20-2)
The non-vanishing numerical values :
cz. = 0.06,
k* =
coefficients 0.08,
b,,
have =
0.01,
the following kf =
0.003.
To obtain the magnetic field of our tilted magnetosphere we add the tilt components (9) multiplied by the tilt angle $ expressed in degrees to the zero-tilt components (7). Figure 7 shows the AB contours obtained with this model which can be compared with the result obtained by Olson and Pfitzer (Fig. 8). The field line patterns can also be compared (Figs. 9 and 10).
DISCUSSION
+ b,,fiTz
1
cos 26 sin 4.
The asterisk * indicates that the S: and Tj* are a new set of functions which should be multiplied by the tilt
FIG.~.ABCONTOUR
These two examples illustrate the possibilities of the formulation in terms of toroidal and poloidal fields. As mentioned earlier, one of the advantages is that this representation is automatically divergence-free and this property enables the calculations to be checked
OBTAINEDWITHOURTILTMODELFOR
A TILTANGLEOF 35 DEGREES.
912
J. C. KOSIK
), 10
+40
4
0
FIG.& AB CONTOUROBTAINEDWITHTHEMODELOF~LSON-PFITZER
FIG.~.FIELD LINEPATTERNOBTAINEDWITHOURMODELFORA
FOR A 35" TILT ANGLE.
35" TILT.
Quantitative
magnetospheric
magnetic
field modelling
with toroidal
and poloidal
FIG. ~~.FIELDLINEPATTERNOBTAINED WITHTHEOLSON-PFITZERMODELFORA
easily. Another advantage is the large choice in the radial functions. However it can also be a disadvantage when the radial or angular behaviour is not properly chosen. The difficulty increases also when the number of coefficients increases. In particular, it is easier to model the near-Earth field (up to 7-8 RJ than the distant field, like the tail region. In our model of the Earth’s magnetosphere there is in fact no ad hoc or physically calculated magnetopause : the AB regions plotted in Figs. 2 and 7 extend to the entire plane and are bounded only if the external field contributions vanish when r increases. Our single constraint was to obtain a reasonably good fit to the AB contours of Sugiura and Poros in a limited region of space, i.e. inside the magnetopause, neglecting the numerical behaviour of AB outside this region. In fact al1 the radial functions we use are of the exponential type and vanish more or less rapidly. It was thus possible to represent a zero-tilt magnetosphere with seven coefficients only and a tilted model was obtained adding four other coefficients. The model magnetosphere we get is not perfect, especially in the tail region, but the results are quite similar to those obtained by Olson and Pfitzer with 300 coefficients. Our next efforts will be to improve this model and also
vector fields
973
35” TILT.
apply this technique to other magnetospheres and to more fundamental problems like the momentum balance topologies in the tail region.
Acknowledgements-We gratefully acknowledge Roger Gendrin for his support and encouragement of our efforts in our part-time activity on modelling. We would like to thank David Stern for interesting comments on his paper.
REFERENCES Granzow, K. D. (1983) Spherical harmonic representation of the magnetic field in the presence of a current density. Geophys. .I. R. astr. Sot. 14,489. Hedgecock, P. C. and Thomas, B. T. (1975) Heos observations of the configuration of the magnetosphere. Geophys. J. R. astr. Sot. 41, 391 Mead, G. D. (1964) Deformation of the geomagnetic field by the solar wind. J. geophys. Res. 69, 1169. Mead, G. D. and Fairfield, D. H. (1975) A quantitative magnetospheric model derived from spacecraft magnetometer data. J. geophys. Res. 80,523. Olson, W. P. and Pfitzer, K. A. (1974) A quantitative model of themagnetosphericmagneticfield. J. geophys. Res. 79,3739. Olson, W. P. and Pfitzer, K. A. (1979) In Quantitative Modelling ofMagnetospheric Processes (Edited by Olson, W. P.). A.G.U. monograph 1979.
914
J. C. KOSIK
Stern, D. P. (1975) Representations of magnetic fields in space. Rev. Geophys. Space. Phys. 14, 199. Sugiura, M. and Poros, D. J. (1973) A magnetospheric field model incorporating the OGO 3 and 5 magnetic field observations. Planet. Space. Sci. 21, 1163. Tsyganenko, N. A. and Usmanov, A. V. (1982) Determination of the magnetospheric current system parameters and
development of experimental geomagnetic field models based on data from IMP and HEOS satellites. Planet. Space. Sci. 30, 985. Walker, R. J. and Southwood, D. J. (1982) Momentum balance and flux conservation in model magnetospheric magnetic fields. J. geophys. Res. 87, 7460.
0032~633/84$3.00+0.00 Pergamon Press Ltd.
1984
QUANTITATIVE MAGNETOSPHERIC MAGNETIC FIELD MODELLING WITH TOROIDAL AND POLOIDAL VECTOR FIELDS
Division
Mathematiques,
J. C. KOSIK C.N.E.S., Av. E. Belin, 31400 Toulouse,
France
(Received in_finalform 6 January 1984) Abstract-Toroidal and poloidal vector fields allow divergence free magnetic field representations in regions where currents flow. We derive genera1 magnetospheric magnetic fields using combinations of spherical harmonic expansions of the toroidal and poloidal fields. Adding restrictive conditions like the field line topology symmetry or the magnetic field measurements, more specific magnetospheric magnetic field models can be derived. Two examples of this technique are given : an axisymmetric mode1 with a ring current in the equatorial region and a time-dependent mode1 of the Earth’s magnetosphere. Our results are compared with the Olson-Pfitzer model.
VECTOR
INTRODUCTION
EXPANSIONS
A general vector field B can be constructed as a sum of toroidal and poloidal components (Stern, 1975):
Since 1960 several magnetospheric field models have been constructed for the magnetospheres of the Earth and the planets. The simplest models like the model of Mead (1964) did not take into account the internal current distributions like the ring-current. More recent models like those of Olson and Pfitzer (1974, 1979), Mead and Fairfield (1975), Hedgecock and Thomas (1975), Tsyganenko and Usmanov (1982), take into account the tail and ring currents. Such current distributions lead to the following conditions : V/\B=pJ,
FIELD
B = V A $,r+V
A V A $*r
(1)
where the first term on the right hand side is the toroidal component and the other term the poloidal component. tjl and $Z are the generating functions and r a unit vector. To derive the three components of B we use the following spherical harmonic expansions for $I and $2:
V-B=O.
$1 = CC IT;(r)PYYQ(a,_ cos m9 + b,,,, sin +>
This last condition is usually satisfied by adding some constraints on the coefficients and their analytical expansions. However, due to the least squares method, this condition is not always satisfied as was shown recently by Walker and Southwood (1982). To avoid these difficulties Stern (1975) and Granzow (1983), have shown that it is simpler to use general vector fields having toroidal and poloidal components. In the first part of this paper we use this technique associated with spherical harmonics expansions to define a class of quantitative magnetospheric magnetic field models. Such a class can be split into different sub-classes depending on the symmetry conditions for the field line topology. Then each sub-class can lead to different models depending on the quantitative requirements. Two applications of this method are given in the second part of the paper. The first one is quite simple and is an example of an axisymmetric topology. The second application is the Earth’s magnetosphere with a tilt-dependent model obtained using the AB contours from Sugiura and Poros (1973).
$2 = CC S&KY@{ c,,,, cos rn4 + d,_ sin m4).
(2)
In these expansions T(r) and S(r) are general scalar functions of r and Py are the well-known Legendre polynomials. Replacing (2) in (1) we get for the three components of B the following expansions : B, = cc
x
- T
[cl, cos rncj + d,_ sin mQl]
tan-‘(Q)% +!?!&mip;
sin2 0
as,ap;n 1
S,(r) ap;l
43 = CCcosm4 Cl,rae+Br%T [
i
+mb,~~(r~~]+sinrn~[d~~{~~
-ma,,,,7;(r)g > 965
I
(3)
J. C.
966
KOSIK
-0.61a,,T,(4-15~0~~0)
-a,_7;(r),,
1
ap;l
-c2,
X
1.
-0.61b,,T,(4-
These formulae can be developed further using the definitions ofthe Legendrepolynomials. For the sake of simplicity, each component is split into symmetric and antisymmetric parts (subscripts s and as respectively) and given only for the lowest order terms. We thus obtain : B,, = 2;c,,
B,,, =
2:c,,
sin B+ ~r:,,,(6cos2
sin 0+6&?c,,
B,= = -c,,sin8(Sr!
O-2)
sin 0 cos ~9 cos 4
+ $)-c,,sin28e
+ z)
BOnS=cos+IIcosO(++~)+b,,TI
+c,,~cos20 sin 26 sin Tz+0.6lb,,T,
J;J +2b,,
x (5 cos2 O-l)
I
+sin 4[d,,
+d,,$cos20
X
-a,,T,-a2,~cos0
cos B
(
:+$
T,
coso s, 1
-0.61a,,T,(5cosZO-1)
B, = d, 7 - + aloTt sin 0 smO r +d2,(3
co52 0 - 1) S, 3sine ; + a,,T,sin20
B~~~=cosi[d,,~+~)-a,,T;cosD +d,,&osH
+sin$
x [--c,,($++)-b,,T,cosH
+sinm$
-b,,,,7;(r)z
1
5 + % -a2,T,,,/%os28 ( r )
)
15~0~~0)
1
A CLASS OF QUANTITATIVE MAGNETOSPHERES
Formulae (3) satisfy the conditions V A B = PJ and V *B = 0 and thus can represent any type of magnetic field, with or without current distributions. No further conditions are applied to the expansions : S(r) and T(r) are still unspecified and the different coefficients a,,,,, b,_, cl,, d,_ can have any numerical values. As they stand, formulae (3) represent a large class of magnetospheric models. This class can be split into different sub-classes by the introduction of qualitative and quantitative conditions : symmetry or asymmetry in the field line topology and distribution of the field intensity. Axisymmetric magnetospheres will require the following coefficients cr r, b, 1, czl, b,,, . . to vanish, i.e. all the coefficients with subscript m # 0 will vanish. Non-axisymmetric magnetospheres will contain a priori all the coefficients. Each of these two classes can lead to different models depending on some peculiar features: ring current, corotation, plasma sheet, etc. Complex models can be represented by a family of expansions (2) as recommended by Stern (1975). In the last step we have to take into account the experimental data measurements which are more or less abundant depending on the magnetosphere. In the following part of the paper we would like to present two possible applications of the technique. The first application will be devoted to a general axisymmetric corotating magnetosphere and the other one will lead to a first approximation to the Earths magnetosphere.
CASE OF AN AXISYMMETRIC MAGNETOSPHERE
Let us build an axisymmetric magnetosphere having its field lines inflated by a ring current located in the equatorial region. We assume also that this model is symmetric with respect to the equatorial plane. Symmetry with respect to the Equator implies that when the colatitude f3 changes n-0, we have the following changes : B, + -B,, B, + B,, B, + -B,.
Quantitative magnetospheric magnetic field modelling with toroidal and poloidal vector fields Axisymmetry implies that when longitude 4 changes to 4 +A, we have the following changes : B, + B,, B, -+ BB and B, + B,. If we apply these conditions to expansions (3) we are left with a series of nonvanishing coefficients: cl,,, d,,,, u2,,, Cam, d3,, and the three components of the field are (up to the second order) : B,, = 2~cl,cos6 r
961
These three components are still quite general as five coefficients remain to be chosen and the functions Sand T to be defined. These components should be added to the internal field components. If one assumes that the internal field is that of a dipole and that our magnetosphere possesses an equatorial ring current, a possible topology is shown in Fig. 1 obtained with the following conditions :
-S”cjO i-
S, = A,r’ exp (- kr’)
x cos 8(33 -45 cos* 8)
cl0 = 0.02,
A, = - 1.5,
k = 0.03
(6)
c30 = dlo = a,, = d,, = 0. CASE OF THE EARTH’S MAGNETOSPHERE
case s, B, = d,,? - + aZoT2sin26 sin0 r
(5)
We will first develop a zero-tilt magnetosphere assuming symmetry with respect to the Equator and the noon-midnight meridian. These symmetries imply the following conditions: when @changes to n - 0, we have B,-* -B,, Bg-+ -B, and B,-+ -B,; when I#I changes to -C/J, we have B, + B,, B, + B, and B, +
ZSM
1’5.
\
\
\
‘\
\
I
.
\\ ‘.
\
/I ’
.’
i
:
/ /’ / /I /
-4’ ;;
---4
\-
a-15. FIG. ~.FIELD LINEPATTERNOFANAXISYMMETRIC
MAGNETOSPHERE
WITHAN
Dotted lines represent dipole field lines.
EQUATORIALRING
CURRENT.
968
J. C.
-B,. Applying these conditions to equations (4), the non-vanishing coefficients are: ctO, cal, b,,, b,,. Thus the non-vanishing components of the external field are : BIS = ~c2~,,S,--c,,S,(l8-25cos’8)] Bos= -sinli[e,,(++$)
KOSIK
analytical expansions, the Si and rj can have quite different analytical forms, i.e. S, and S3 are different. It is thus possible to make an adequate choice ofthe couples [S(r), P,(@)]and [I’$+),P,(0)] in order to achieve a “good modelling”. Here the analytical expressions of the Si and q were chosen in order to obtain a good fit of the AB contours derived by Sugiura and Poros (1973). The analytic expressions of the Si and Tj are : S, = A,r3exp(-,&2),
8,
S3 = rz exp (- kr’)
= 0
Tl = rexp(-kg’),
B,,, = 10.4c,, 2 cos (bsin 0 cos @
r
B,_ = 1.732~~~cos28cos~
+[b,,T,
S, = r3exp(-k,r2)
3 + $ i r
c >
+0.61b,,(S co? 0--l)T,]cos
4
(7)
B+,,. = - 1.732~~ cos Osin 4
-~os~[~~~~~+0.61b~~(4-15sinzU)~~]sin#. In these formulae the Si and 7jarefunctions ofthe radial distance r. As we construct the model as a sum of
(8) T, = rexp(-kg*).
The numerical coefficients which do not vanish are : et0 = 1, c2, = 0.11, b,, = 1, A, = -1.5, k = 0.04, k, = k, = 0.003. The AS contours obtained with these formulae are shown in Fig. 2 and can be compared with the experimental results of Sugiura and Poros (Fig. 3) and the model of Olson and Pfitzer (1974) in Fig. 4. The field line pattern obtained with our model (Fig. 5) can be compared with the result of Olson and Pfitzer (Fig. 6). To improve our model we have added the tilt effects. The appropriate formulae were obtained using the following conditions : when 0 changes to rc- 0 and the
MIDNIGHT
UNITS
: GAMMA
/
b--
12
b.2.3.ABCONTOUROBTAINEDBY
SUGIURAA~POROS~OM~XP~RI~NTDATA.
969
+si\rl 15. /
FIG. 5.FIELD LINE PATTERN~NTHE
./
/
NOON-MIDNICNT
“----
MERlDlAN CDRRFSPONUINC~TQ OUR ZERO-TILT MODBL.
Quantitative
magnetospheric
magnetic
field modelling
tilt angle I,+changes to -I,+, we have B, + - B,, B, + B,, B,-+ -B,. The non-vanishing components of the “tilt contribution” are :
with toroidal
and poloidal
971
vector fields
angle $ expressed in degrees. Here again we have selected a few coefficients and the expressions for the SF and Tj* are: S: = r3exp(-k*r’),
Sz = r2exp(-k*r2) (10)
Tfj = r exp ( - Qr2).
B,, = c2,~(6cos20-2)
The non-vanishing numerical values :
cz. = 0.06,
k* =
coefficients 0.08,
b,,
have =
0.01,
the following kf =
0.003.
To obtain the magnetic field of our tilted magnetosphere we add the tilt components (9) multiplied by the tilt angle $ expressed in degrees to the zero-tilt components (7). Figure 7 shows the AB contours obtained with this model which can be compared with the result obtained by Olson and Pfitzer (Fig. 8). The field line patterns can also be compared (Figs. 9 and 10).
DISCUSSION
+ b,,fiTz
1
cos 26 sin 4.
The asterisk * indicates that the S: and Tj* are a new set of functions which should be multiplied by the tilt
FIG.~.ABCONTOUR
These two examples illustrate the possibilities of the formulation in terms of toroidal and poloidal fields. As mentioned earlier, one of the advantages is that this representation is automatically divergence-free and this property enables the calculations to be checked
OBTAINEDWITHOURTILTMODELFOR
A TILTANGLEOF 35 DEGREES.
912
J. C. KOSIK
), 10
+40
4
0
FIG.& AB CONTOUROBTAINEDWITHTHEMODELOF~LSON-PFITZER
FIG.~.FIELD LINEPATTERNOBTAINEDWITHOURMODELFORA
FOR A 35" TILT ANGLE.
35" TILT.
Quantitative
magnetospheric
magnetic
field modelling
with toroidal
and poloidal
FIG. ~~.FIELDLINEPATTERNOBTAINED WITHTHEOLSON-PFITZERMODELFORA
easily. Another advantage is the large choice in the radial functions. However it can also be a disadvantage when the radial or angular behaviour is not properly chosen. The difficulty increases also when the number of coefficients increases. In particular, it is easier to model the near-Earth field (up to 7-8 RJ than the distant field, like the tail region. In our model of the Earth’s magnetosphere there is in fact no ad hoc or physically calculated magnetopause : the AB regions plotted in Figs. 2 and 7 extend to the entire plane and are bounded only if the external field contributions vanish when r increases. Our single constraint was to obtain a reasonably good fit to the AB contours of Sugiura and Poros in a limited region of space, i.e. inside the magnetopause, neglecting the numerical behaviour of AB outside this region. In fact al1 the radial functions we use are of the exponential type and vanish more or less rapidly. It was thus possible to represent a zero-tilt magnetosphere with seven coefficients only and a tilted model was obtained adding four other coefficients. The model magnetosphere we get is not perfect, especially in the tail region, but the results are quite similar to those obtained by Olson and Pfitzer with 300 coefficients. Our next efforts will be to improve this model and also
vector fields
973
35” TILT.
apply this technique to other magnetospheres and to more fundamental problems like the momentum balance topologies in the tail region.
Acknowledgements-We gratefully acknowledge Roger Gendrin for his support and encouragement of our efforts in our part-time activity on modelling. We would like to thank David Stern for interesting comments on his paper.
REFERENCES Granzow, K. D. (1983) Spherical harmonic representation of the magnetic field in the presence of a current density. Geophys. .I. R. astr. Sot. 14,489. Hedgecock, P. C. and Thomas, B. T. (1975) Heos observations of the configuration of the magnetosphere. Geophys. J. R. astr. Sot. 41, 391 Mead, G. D. (1964) Deformation of the geomagnetic field by the solar wind. J. geophys. Res. 69, 1169. Mead, G. D. and Fairfield, D. H. (1975) A quantitative magnetospheric model derived from spacecraft magnetometer data. J. geophys. Res. 80,523. Olson, W. P. and Pfitzer, K. A. (1974) A quantitative model of themagnetosphericmagneticfield. J. geophys. Res. 79,3739. Olson, W. P. and Pfitzer, K. A. (1979) In Quantitative Modelling ofMagnetospheric Processes (Edited by Olson, W. P.). A.G.U. monograph 1979.
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Stern, D. P. (1975) Representations of magnetic fields in space. Rev. Geophys. Space. Phys. 14, 199. Sugiura, M. and Poros, D. J. (1973) A magnetospheric field model incorporating the OGO 3 and 5 magnetic field observations. Planet. Space. Sci. 21, 1163. Tsyganenko, N. A. and Usmanov, A. V. (1982) Determination of the magnetospheric current system parameters and
development of experimental geomagnetic field models based on data from IMP and HEOS satellites. Planet. Space. Sci. 30, 985. Walker, R. J. and Southwood, D. J. (1982) Momentum balance and flux conservation in model magnetospheric magnetic fields. J. geophys. Res. 87, 7460.