- Email: [email protected]
Modelling of magnetic field near the magnetopause
ARTICLE IN PRESS
Planetary and Space Science 53 (2005) 127–131 www.elsevier.com/locate/pss
Modelling of magnetic field near the magnetopause E. Romashetsa,, M. Vandasb, T. Nagatsumac a
Institute of Terrestrial Magnetism, Ionosphere, and Radio Wave Propagation, Russian Academy of Sciences, Troitsk, Moscow Region 142190, Russia b Astronomical Institute, Academy of Sciences, Bocˇnı´ II 1401, 14131 Praha 4, Czech Republic c National Institute of Information and Communications Technology, 4-2-1 Nukui-Kita, Koganei, Tokyo 184-8795, Japan Accepted 12 September 2004
Abstract There are many reliable phenomenological and theoretical methods to model conditions on the outer and inner sides of magnetopause during quiet and disturbed periods. We present here, in this paper, an approximate analytical description of magnetic field distribution between the bow shock and the magnetopause, and on the inner side of the magnetopause. The approach is the following: (i) the field near the Earth’s surface is equal to one of an inclined dipole located at the center; (ii) on the magnetopause the normal component of the magnetic field is zero; (iii) on the bow shock the normal component is continuous. The solution, obtained in coordinates of a paraboloid of rotation, is a function of dipole orientation and distance to the bow shock, which is determined by the solar wind parameters. Magnetic fields induced by the ring current are also considered. The intensity of the ring current is determined by a magnetic field jump at the magnetopause. r 2004 Elsevier Ltd. All rights reserved. Keywords: Magnetosphere; Solar wind; Magnetopause; Geomagnetic storms
1. Introduction Spherical harmonic analysis was applied by Schulz and McNab (1996) to calculate the magnetospheric field. The approach treated field distribution as current-free and given by a number of potential functions. Corresponding coefficients can be found from agreement with observations. Another approach is to study the evolution of current systems inside the magnetosphere and to calculate their contributions to the ground magnetic field variations (Alexeev and Feldstein, 2001). Many other models of the magnetosphere have been developed recently. Some are based on data analysis (Tsyganenko, 1995), whereas others take into account various effects, e.g., those resulting from reconnection at the magnetopause (Kalegaev, 2000); earlier methods and models
Corresponding author.
E-mail address: [email protected] (E. Romashets). 0032-0633/$ - see front matter r 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.pss.2004.09.036
were described in the review by Tsyganenko (1990). We propose using an analytical approach to calculate the modification of inner and outer fields by assuming a scalar potential outside the magnetopause and a vector potential for a modified dipole filled inside; expressions for a field induced by the ring current are also introduced.
2. Modification of the dipole field by the magnetopause A problem of note is how to model a dipole field that is bounded by a cavity shaped by a paraboloid of rotation (along the x axis) and filled with low-density plasma. The paraboloid is substituted for the shape of the magnetopause. The resulting magnetic field must have no normal component on the inner side of the magnetopause. Let us assume that the dipole makes an angle of g0 ffi 23 with the z axis and its projection makes the angle b0 with the xy plane ð0 pb0 o360 Þ: Then with these assumptions the scalar potential of
ARTICLE IN PRESS E. Romashets et al. / Planetary and Space Science 53 (2005) 127–131
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the dipole is x y C0 ¼ D0 3 sin g0 cos b0 þ D0 3 sin g0 sin b0 r r z þ D0 3 cos g0 ; ð1Þ r where D0 scales the magnetic field, B ¼ grad C0 (it is D0 8:3 1015 T m3 for the Earth), and r is the radius vector. Three dimensional parabolic coordinates, s; t; and j; are defined as x ¼ 12 ðs2 t2 Þ;
(3)
z ¼ st sin j;
(4)
the potential C0 of dipole (1) can be expressed as st C0 ¼ 8D0 ðcos g0 sin j 2 ðs þ t2 Þ3 þ sin g0 sin b0 cos jÞ s2 t2 sin g0 cos b0 : ðs2 þ t2 Þ3
ð5Þ
tð5s2 t2 Þ
ðs2 þ t2 Þ þ sin g0 sin b0 cos jÞ 16D0
Bt ¼ 8D0
Bs js¼s0 ¼ 8D0
sðs 2t Þ ðs2 þ t2 Þ4þ1=2
sin g0 cos b0 ;
16D0
ðs2 þ t2 Þ4þ1=2
ðcos g0 sin j
s0 ðs20 2t2 Þ ðs20 þ t2 Þ4þ1=2
sin g0 cos b0 :
ð13Þ
So, if we add to the dipole field an additional field equal to (13) with the opposite sign at the magnetopause, the total field will have no normal component to the surface. To construct the additional field we use a vector potential A:
At ¼ 8D0
(14) st2 ð5s2 t2 Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðcos g0 cos j ðs20 þ t2 Þ4 s2 þ t2
sin g0 sin b0 sin jÞ; ðs20
B~ s ¼ 8D0
st sin g0 cos b0 : þ t2 Þ 3
ð15Þ (16)
tð5s2 t2 Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðcos g0 sin j ðs20 þ t2 Þ4 s2 þ t2
þ sin g0 sin b0 cos jÞ þ 16D0
2
tð2s t Þ
ðs20 þ t2 Þ4þ1=2
þ sin g0 sin b0 cos jÞ
ð6Þ
ðcos g0 sin j ðs2 þ t2 Þ4þ1=2 þ sin g0 sin b0 cos jÞ 2
tð5s20 t2 Þ
e ¼ rot A; In parabolic coordinates the additional field, B has components
2
sðs2 5t2 Þ
16D0
with the nose at x0 ¼ The magnetic field component s on the magnetopause surface is
Aj ¼ 8D0
ðcos g0 sin j 4þ1=2
2
(12)
s20 =2:
As ¼ 0;
The magnetic field s; t; and j components given by potential (5) are Bs ¼ 8D0
2s20 x þ y2 þ z2 s40 ¼ 0
(2)
y ¼ st cos j;
þ 4D0
The magnetopause surface is defined by s ¼ s0 ¼ const:; i.e., by the equation
sin g0 cos b0 ;
1 ðcos g0 cos j ðs2 þ t2 Þ3 sin g0 sin b0 sin jÞ:
B~ t ¼ 16D0
ð8Þ
The relationship between Cartesian and parabolic components is s t Bx ¼ Bs pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Bt pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; (9) 2 2 2 s þt s þ t2
t s Bz ¼ Bs pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sin j þ Bt pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sin j 2 2 2 s þt s þ t2 þ Bj cos j:
sðs20 2t2 Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sin g0 cos b0 ; þ t2 Þ4 s2 þ t2
ð7Þ
Bj ¼ 8D0
t s By ¼ Bs pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi cos j þ Bt pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi cos j 2 2 2 s þt s þ t2 Bj sin j;
ðs20
B~ j ¼ 8D0
ðs20
þ
t pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sin g0 cos b0 ; s2 þ t2
t2 Þ 3
ð17Þ
(18)
t2 ð15s2 t2 Þ ðcos g0 cos j ðs20 þ t2 Þ4 ðs2 þ t2 Þ
sin g0 sin b0 sin jÞ:
ð19Þ
Superposition of the fields given by potential (5) and formulas (17)–(19) satisfies the boundary conditions: (i) near the Earth magnetic field modification is small; (ii) the normal component to the magnetopause is zero.
ð10Þ 3. Magnetic field outside the magnetopause ð11Þ
As with the magnetopause, the bow shock shape is approximated by a paraboloid. Eq. (12) with the index 0
ARTICLE IN PRESS E. Romashets et al. / Planetary and Space Science 53 (2005) 127–131
changed to 1, holds, i.e., the nose of the bow shock is at x1 ¼ s21 =2: To calculate the field between the magnetopause and the bow shock, methods of potentials (Romashets and Nagatsuma, 2002; Romashets et al., 2003) can be used. The same method was also used to calculate the magnetic field near a coronal mass ejection (CME) in interplanetary space (Romashets and Vandas, 2002). The boundary conditions assumed here are (i) continuity of the normal component to the bow shock surface and (ii) absence of this component at the magnetopause. We shall assume that, aside from the Earth’s bow shock (i.e., upstream), the interplanetary magnetic field (IMF) is uniform and has (constant) components: B0x ; B0y ; and B0z : The IMF near 1 AU and during quiet conditions has a direction of around 45 with respect to a Sun–Earth line and has a small component Bz ; that is, B0x B0y and B0z 0: B0z may be comparable to the other components or may be larger when magnetic clouds or other disturbances approach the Earth’s orbit. The upstream potential function is C0 ¼ B0x x þ B0y y þ B0z z ¼ B0x 12ðs2 t2 Þ þ ðB0y cos j þ B0z sin jÞst:
ð20Þ
Next, we find the potential of the field lying between the bow shock and the magnetopause (i.e., inside the magnetosheath), which must satisfy the above conditions (i)–(ii), in the form
1 2 s 2 C1 ¼ K B0x ðs t Þ þ ex0 ln 2 s0 ox0 t þ ðB0y cos j þ B0z sin jÞ st þ : s
129
Fig. 1. Calculated field lines in the solar wind, in the magnetosheath, and inside the magnetopause for a winter season. The interplanetary magnetic field (IMF) far from the Earth has negative X and Z components; the Y component is zero. The figure is a cross section of the three dimensional magnetosphere, cut by the plane Y ¼ 0 (XYZ are the geocentric solar ecliptic coordinates). One can see that the Earth’s dipole field is modified slightly near the Earth’s surface and more strongly near magnetopause. The magnetopause is not penetrated by field lines. On the outer side the Bz component may be strong enough to trigger reconnection process, even if the component is relatively weak in the incoming solar wind flow.
The components of magnetic field given by C1 are ð21Þ
Bs ¼
x1 2x0 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 2 s ðx1 x0 Þ s2 þ t2 ½B0x s þ ðB0y cos j þ B0z sin jÞt ;
ð25Þ
The constants e and o are found from condition (ii), @C1 ¼ 0; @s s¼s0 and the constant K from condition (i), @C1 @C0 ¼ : @s s¼s1 @s s¼s1
(22)
(23) Bj ¼
Eqs. (22)–(23) yield e ¼ 2; o ¼ 2; and K ¼ x1 =ðx1 x0 Þ: Thus (21) simplifies to C1 ¼
x1 1 s B0x ðs2 t2 Þ 2x0 ln 2 s0 x1 x0
2x0 þ ðB0y cos j þ B0z sin jÞst 1 þ 2 : s
Bt ¼
ð24Þ
x1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi B0x t þ ðB0y cos j ðx1 x0 Þ s2 þ t2
2x0 þ B0z sin jÞs 1 þ 2 ; s
ð26Þ
x1 2x0 ð B0y sin j þ B0z cos jÞ 1 þ 2 : ðx1 x0 Þ s ð27Þ
The components Bx ; By ; and Bz are calculated using (9)–(11). Figs. 1 and 2 show the resulting fields inside and outside the magnetopause. The field lines for the inner magnetosphere are given by Eqs. (6)–(8) and (17)–(19)
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130
by^ ¼ m0 j B 4p Z 2p 0
bz^ ¼ m0 j B 4p Z
2p 0
^ z^rc sin j ^ dj; ^ þ ðy^ rc sin jÞ ^ 2 þ z^2 3=2 ½ðx^ rc cos jÞ ð32Þ 2
^ c cos j^ þ yr ^ c sin j ^ r2c xr ^ dj: 2 ^ þ ðy^ rc sin jÞ ^ 2 þ z^2 3=2 ½ðx^ rc cos jÞ
ð33Þ
Fig. 2. The same as in Fig. 1, but for a summer season.
and for the magnetosheath by (25)–(27). The strong summer/winter asymmetry is evident. The integrated jump of the tangential component on the magnetopause during the periods of negative Bz of IMF is stronger in summer seasons (Fig. 2). 4. The ring current field and its modification by the magnetopause This section derives an approximate field contribution caused by the ring current. The ring current is modelled as the current of intensity j along a circle of radius rc in the x^ y^ plane, with its center at the point ð0; 0; 0Þ in ^ y; ^ and z^; where the z^ axis is Cartesian coordinates x; along the dipole axis. That is, this system is related to the geocentric solar ecliptic (GSE) system x, y, and z by
Integrals cannot be expressed in analytical functions, but they can be quite easily evaluated numerically. Here ^ z system. The all vector components are written in the x^ y^ bx ; B by ; and B bz (in the GSE magnetic field components, B system) as functions of x, y, and z, are determined by transformations based on (28)–(29). bx ; B by ; and B bz from the condition of Next we modify B having a zero normal component at the magnetopause (s ¼ s0 ), and introduce a vector potential of an additional field, A ¼ ð0; 0; Aj Þ in the stj system. Aj is determined by the expression ðrot AÞs js¼s0 1 @ bs js : ðhj Aj Þjs¼s0 ¼ B ¼ 0 ht hj @t
ð34Þ
Thus Aj is the following: Z 1 bs js Þ dt: ht hj ð B Aj ¼ (35) 0 hj bs js ; we apply transformations (9)–(11) and To find B 0 bx ; B by ; and B bz : Components of the additional (2)–(4) to B field are bs js ; B¯ s ¼ ðrot AÞs ¼ B 0 B¯ t ¼ ðrot AÞt ¼
1 @ hs hj @s
(36) Z
bs js Þ dt; ht hj ð B 0 ð37Þ
x ¼ x^ cos g0 cos b0 þ y^ sin b0 þ z^ sin g0 cos b0 ; ð28Þ
B¯ j ¼ ðrot AÞj ¼ 0:
(38)
y ¼ x^ cos g0 sin b0 y^ cos b0 þ z^ sin g0 sin b0 ; ð29Þ z ¼ x^ sin g0 þ z^ cos g0 :
(30)
The exact formulas for the magnetic field induced by the ring current are given by the Biot–Savart law bx^ ¼ m0 j B 4p Z 2p 0
z^rc cos j^ ^ dj; ^ þ ðy^ rc sin jÞ ^ 2 þ z^2 3=2 ½ðx^ rc cos jÞ 2
ð31Þ
5. Conclusions This paper uses methods of scalar and vector potentials to develop an analytical description of the magnetospheric fields. The corresponding potential functions were found from boundary conditions for the magnetopause and the Earth’s bow shock. The model field for internal and external distributions is determined mainly from two parameters: the distance from the Earth to the magnetopause, x0 ; and to the bow shock, x1 : If conditions in incoming solar wind are
ARTICLE IN PRESS E. Romashets et al. / Planetary and Space Science 53 (2005) 127–131
known, these two parameters can be calculated using methods developed in a number of studies, including, e.g., Farris and Russell (1994), Cairns and Grabbe (1994), Neˇmecˇek and Sˇafra´nkova´ (1991). Calculation results show a summer/winter asymmetry in geoeffectiveness of disturbances. For southward IMF, geoeffectiveness of geomagnetic storms are expected to be stronger in summer seasons.
Acknowledgements This work was supported by EU-INTAS-ESA grant 99-00727, RFBR grant 03-02-16340 and CRDF grant TGP-944, and by project S1003006 from the Academy of Sciences of the Czech Republic and by grant 205/03/ 0953 from the Grant Agency of the Czech Republic. References Alexeev, I.I., Feldstein, Y.I., 2001. Modeling of geomagnetic field during magnetic storms and comparison with observations. J. Atm. Sol.-Terr. Phys. 63, 431–440.
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Cairns, I.H., Grabbe, C.H., 1994. Towards an MHD theory for the standoff distance of Earth’s bow shock. Geophys. Res. Lett. 21, 2781–2784. Farris, M.N., Russell, C.T., 1994. Determining the standoff distance of the bow shock: Mach number dependence and the models. J. Geophys. Res. 99, 17180–17681. Kalegaev, V.V., 2000. Magnetosheath conditions and magnetopause structure for high magnetic shear. Phys. Chem. Earth 25, 173–176. Neˇmecˇek, Z., Sˇafra´nkova´, J., 1991. The Earth’s bow shock and magnetopause position as a result of the solar wind–magnetosphere interaction. J. Atm. Terr. Phys. 53, 1049–1054. Romashets, E., Nagatsuma, T., 2002. Modification of interplanetary magnetic field by the Earth’s magnetosphere. ESA SP-477, 475–477. Romashets, E., Vandas, M., 2002. Modelling of IMF draping around supersonic magnetic cloud. ESA SP-477, 297–300. Romashets, E., Vandas, M., Nagatsuma, T., 2003. Evolution of geoeffective disturbances in interplanetary space. In: Proceedings of Space Weather Workshop Looking Towards a European Space Weather Space Programme. ESTEC, The Netherlands, pp. 59–62. Schulz, M., McNab, M., 1996. Source–surface modelling of planetary magnetospheres. J. Geophys. Res. 101, 5095–5118. Tsyganenko, N., 1990. Quantitative models of the magnetospheric magnetic field: methods and results. Space Sci. Rev. 54, 75–186. Tsyganenko, N.A., 1995. Modelling of the Earth’s magnetospheric magnetic field confined within realistic magnetopause. J. Geophys. Res. 100, 5599–5612.
Planetary and Space Science 53 (2005) 127–131 www.elsevier.com/locate/pss
Modelling of magnetic field near the magnetopause E. Romashetsa,, M. Vandasb, T. Nagatsumac a
Institute of Terrestrial Magnetism, Ionosphere, and Radio Wave Propagation, Russian Academy of Sciences, Troitsk, Moscow Region 142190, Russia b Astronomical Institute, Academy of Sciences, Bocˇnı´ II 1401, 14131 Praha 4, Czech Republic c National Institute of Information and Communications Technology, 4-2-1 Nukui-Kita, Koganei, Tokyo 184-8795, Japan Accepted 12 September 2004
Abstract There are many reliable phenomenological and theoretical methods to model conditions on the outer and inner sides of magnetopause during quiet and disturbed periods. We present here, in this paper, an approximate analytical description of magnetic field distribution between the bow shock and the magnetopause, and on the inner side of the magnetopause. The approach is the following: (i) the field near the Earth’s surface is equal to one of an inclined dipole located at the center; (ii) on the magnetopause the normal component of the magnetic field is zero; (iii) on the bow shock the normal component is continuous. The solution, obtained in coordinates of a paraboloid of rotation, is a function of dipole orientation and distance to the bow shock, which is determined by the solar wind parameters. Magnetic fields induced by the ring current are also considered. The intensity of the ring current is determined by a magnetic field jump at the magnetopause. r 2004 Elsevier Ltd. All rights reserved. Keywords: Magnetosphere; Solar wind; Magnetopause; Geomagnetic storms
1. Introduction Spherical harmonic analysis was applied by Schulz and McNab (1996) to calculate the magnetospheric field. The approach treated field distribution as current-free and given by a number of potential functions. Corresponding coefficients can be found from agreement with observations. Another approach is to study the evolution of current systems inside the magnetosphere and to calculate their contributions to the ground magnetic field variations (Alexeev and Feldstein, 2001). Many other models of the magnetosphere have been developed recently. Some are based on data analysis (Tsyganenko, 1995), whereas others take into account various effects, e.g., those resulting from reconnection at the magnetopause (Kalegaev, 2000); earlier methods and models
Corresponding author.
E-mail address: [email protected] (E. Romashets). 0032-0633/$ - see front matter r 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.pss.2004.09.036
were described in the review by Tsyganenko (1990). We propose using an analytical approach to calculate the modification of inner and outer fields by assuming a scalar potential outside the magnetopause and a vector potential for a modified dipole filled inside; expressions for a field induced by the ring current are also introduced.
2. Modification of the dipole field by the magnetopause A problem of note is how to model a dipole field that is bounded by a cavity shaped by a paraboloid of rotation (along the x axis) and filled with low-density plasma. The paraboloid is substituted for the shape of the magnetopause. The resulting magnetic field must have no normal component on the inner side of the magnetopause. Let us assume that the dipole makes an angle of g0 ffi 23 with the z axis and its projection makes the angle b0 with the xy plane ð0 pb0 o360 Þ: Then with these assumptions the scalar potential of
ARTICLE IN PRESS E. Romashets et al. / Planetary and Space Science 53 (2005) 127–131
128
the dipole is x y C0 ¼ D0 3 sin g0 cos b0 þ D0 3 sin g0 sin b0 r r z þ D0 3 cos g0 ; ð1Þ r where D0 scales the magnetic field, B ¼ grad C0 (it is D0 8:3 1015 T m3 for the Earth), and r is the radius vector. Three dimensional parabolic coordinates, s; t; and j; are defined as x ¼ 12 ðs2 t2 Þ;
(3)
z ¼ st sin j;
(4)
the potential C0 of dipole (1) can be expressed as st C0 ¼ 8D0 ðcos g0 sin j 2 ðs þ t2 Þ3 þ sin g0 sin b0 cos jÞ s2 t2 sin g0 cos b0 : ðs2 þ t2 Þ3
ð5Þ
tð5s2 t2 Þ
ðs2 þ t2 Þ þ sin g0 sin b0 cos jÞ 16D0
Bt ¼ 8D0
Bs js¼s0 ¼ 8D0
sðs 2t Þ ðs2 þ t2 Þ4þ1=2
sin g0 cos b0 ;
16D0
ðs2 þ t2 Þ4þ1=2
ðcos g0 sin j
s0 ðs20 2t2 Þ ðs20 þ t2 Þ4þ1=2
sin g0 cos b0 :
ð13Þ
So, if we add to the dipole field an additional field equal to (13) with the opposite sign at the magnetopause, the total field will have no normal component to the surface. To construct the additional field we use a vector potential A:
At ¼ 8D0
(14) st2 ð5s2 t2 Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðcos g0 cos j ðs20 þ t2 Þ4 s2 þ t2
sin g0 sin b0 sin jÞ; ðs20
B~ s ¼ 8D0
st sin g0 cos b0 : þ t2 Þ 3
ð15Þ (16)
tð5s2 t2 Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðcos g0 sin j ðs20 þ t2 Þ4 s2 þ t2
þ sin g0 sin b0 cos jÞ þ 16D0
2
tð2s t Þ
ðs20 þ t2 Þ4þ1=2
þ sin g0 sin b0 cos jÞ
ð6Þ
ðcos g0 sin j ðs2 þ t2 Þ4þ1=2 þ sin g0 sin b0 cos jÞ 2
tð5s20 t2 Þ
e ¼ rot A; In parabolic coordinates the additional field, B has components
2
sðs2 5t2 Þ
16D0
with the nose at x0 ¼ The magnetic field component s on the magnetopause surface is
Aj ¼ 8D0
ðcos g0 sin j 4þ1=2
2
(12)
s20 =2:
As ¼ 0;
The magnetic field s; t; and j components given by potential (5) are Bs ¼ 8D0
2s20 x þ y2 þ z2 s40 ¼ 0
(2)
y ¼ st cos j;
þ 4D0
The magnetopause surface is defined by s ¼ s0 ¼ const:; i.e., by the equation
sin g0 cos b0 ;
1 ðcos g0 cos j ðs2 þ t2 Þ3 sin g0 sin b0 sin jÞ:
B~ t ¼ 16D0
ð8Þ
The relationship between Cartesian and parabolic components is s t Bx ¼ Bs pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Bt pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; (9) 2 2 2 s þt s þ t2
t s Bz ¼ Bs pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sin j þ Bt pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sin j 2 2 2 s þt s þ t2 þ Bj cos j:
sðs20 2t2 Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sin g0 cos b0 ; þ t2 Þ4 s2 þ t2
ð7Þ
Bj ¼ 8D0
t s By ¼ Bs pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi cos j þ Bt pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi cos j 2 2 2 s þt s þ t2 Bj sin j;
ðs20
B~ j ¼ 8D0
ðs20
þ
t pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sin g0 cos b0 ; s2 þ t2
t2 Þ 3
ð17Þ
(18)
t2 ð15s2 t2 Þ ðcos g0 cos j ðs20 þ t2 Þ4 ðs2 þ t2 Þ
sin g0 sin b0 sin jÞ:
ð19Þ
Superposition of the fields given by potential (5) and formulas (17)–(19) satisfies the boundary conditions: (i) near the Earth magnetic field modification is small; (ii) the normal component to the magnetopause is zero.
ð10Þ 3. Magnetic field outside the magnetopause ð11Þ
As with the magnetopause, the bow shock shape is approximated by a paraboloid. Eq. (12) with the index 0
ARTICLE IN PRESS E. Romashets et al. / Planetary and Space Science 53 (2005) 127–131
changed to 1, holds, i.e., the nose of the bow shock is at x1 ¼ s21 =2: To calculate the field between the magnetopause and the bow shock, methods of potentials (Romashets and Nagatsuma, 2002; Romashets et al., 2003) can be used. The same method was also used to calculate the magnetic field near a coronal mass ejection (CME) in interplanetary space (Romashets and Vandas, 2002). The boundary conditions assumed here are (i) continuity of the normal component to the bow shock surface and (ii) absence of this component at the magnetopause. We shall assume that, aside from the Earth’s bow shock (i.e., upstream), the interplanetary magnetic field (IMF) is uniform and has (constant) components: B0x ; B0y ; and B0z : The IMF near 1 AU and during quiet conditions has a direction of around 45 with respect to a Sun–Earth line and has a small component Bz ; that is, B0x B0y and B0z 0: B0z may be comparable to the other components or may be larger when magnetic clouds or other disturbances approach the Earth’s orbit. The upstream potential function is C0 ¼ B0x x þ B0y y þ B0z z ¼ B0x 12ðs2 t2 Þ þ ðB0y cos j þ B0z sin jÞst:
ð20Þ
Next, we find the potential of the field lying between the bow shock and the magnetopause (i.e., inside the magnetosheath), which must satisfy the above conditions (i)–(ii), in the form
1 2 s 2 C1 ¼ K B0x ðs t Þ þ ex0 ln 2 s0 ox0 t þ ðB0y cos j þ B0z sin jÞ st þ : s
129
Fig. 1. Calculated field lines in the solar wind, in the magnetosheath, and inside the magnetopause for a winter season. The interplanetary magnetic field (IMF) far from the Earth has negative X and Z components; the Y component is zero. The figure is a cross section of the three dimensional magnetosphere, cut by the plane Y ¼ 0 (XYZ are the geocentric solar ecliptic coordinates). One can see that the Earth’s dipole field is modified slightly near the Earth’s surface and more strongly near magnetopause. The magnetopause is not penetrated by field lines. On the outer side the Bz component may be strong enough to trigger reconnection process, even if the component is relatively weak in the incoming solar wind flow.
The components of magnetic field given by C1 are ð21Þ
Bs ¼
x1 2x0 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 2 s ðx1 x0 Þ s2 þ t2 ½B0x s þ ðB0y cos j þ B0z sin jÞt ;
ð25Þ
The constants e and o are found from condition (ii), @C1 ¼ 0; @s s¼s0 and the constant K from condition (i), @C1 @C0 ¼ : @s s¼s1 @s s¼s1
(22)
(23) Bj ¼
Eqs. (22)–(23) yield e ¼ 2; o ¼ 2; and K ¼ x1 =ðx1 x0 Þ: Thus (21) simplifies to C1 ¼
x1 1 s B0x ðs2 t2 Þ 2x0 ln 2 s0 x1 x0
2x0 þ ðB0y cos j þ B0z sin jÞst 1 þ 2 : s
Bt ¼
ð24Þ
x1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi B0x t þ ðB0y cos j ðx1 x0 Þ s2 þ t2
2x0 þ B0z sin jÞs 1 þ 2 ; s
ð26Þ
x1 2x0 ð B0y sin j þ B0z cos jÞ 1 þ 2 : ðx1 x0 Þ s ð27Þ
The components Bx ; By ; and Bz are calculated using (9)–(11). Figs. 1 and 2 show the resulting fields inside and outside the magnetopause. The field lines for the inner magnetosphere are given by Eqs. (6)–(8) and (17)–(19)
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by^ ¼ m0 j B 4p Z 2p 0
bz^ ¼ m0 j B 4p Z
2p 0
^ z^rc sin j ^ dj; ^ þ ðy^ rc sin jÞ ^ 2 þ z^2 3=2 ½ðx^ rc cos jÞ ð32Þ 2
^ c cos j^ þ yr ^ c sin j ^ r2c xr ^ dj: 2 ^ þ ðy^ rc sin jÞ ^ 2 þ z^2 3=2 ½ðx^ rc cos jÞ
ð33Þ
Fig. 2. The same as in Fig. 1, but for a summer season.
and for the magnetosheath by (25)–(27). The strong summer/winter asymmetry is evident. The integrated jump of the tangential component on the magnetopause during the periods of negative Bz of IMF is stronger in summer seasons (Fig. 2). 4. The ring current field and its modification by the magnetopause This section derives an approximate field contribution caused by the ring current. The ring current is modelled as the current of intensity j along a circle of radius rc in the x^ y^ plane, with its center at the point ð0; 0; 0Þ in ^ y; ^ and z^; where the z^ axis is Cartesian coordinates x; along the dipole axis. That is, this system is related to the geocentric solar ecliptic (GSE) system x, y, and z by
Integrals cannot be expressed in analytical functions, but they can be quite easily evaluated numerically. Here ^ z system. The all vector components are written in the x^ y^ bx ; B by ; and B bz (in the GSE magnetic field components, B system) as functions of x, y, and z, are determined by transformations based on (28)–(29). bx ; B by ; and B bz from the condition of Next we modify B having a zero normal component at the magnetopause (s ¼ s0 ), and introduce a vector potential of an additional field, A ¼ ð0; 0; Aj Þ in the stj system. Aj is determined by the expression ðrot AÞs js¼s0 1 @ bs js : ðhj Aj Þjs¼s0 ¼ B ¼ 0 ht hj @t
ð34Þ
Thus Aj is the following: Z 1 bs js Þ dt: ht hj ð B Aj ¼ (35) 0 hj bs js ; we apply transformations (9)–(11) and To find B 0 bx ; B by ; and B bz : Components of the additional (2)–(4) to B field are bs js ; B¯ s ¼ ðrot AÞs ¼ B 0 B¯ t ¼ ðrot AÞt ¼
1 @ hs hj @s
(36) Z
bs js Þ dt; ht hj ð B 0 ð37Þ
x ¼ x^ cos g0 cos b0 þ y^ sin b0 þ z^ sin g0 cos b0 ; ð28Þ
B¯ j ¼ ðrot AÞj ¼ 0:
(38)
y ¼ x^ cos g0 sin b0 y^ cos b0 þ z^ sin g0 sin b0 ; ð29Þ z ¼ x^ sin g0 þ z^ cos g0 :
(30)
The exact formulas for the magnetic field induced by the ring current are given by the Biot–Savart law bx^ ¼ m0 j B 4p Z 2p 0
z^rc cos j^ ^ dj; ^ þ ðy^ rc sin jÞ ^ 2 þ z^2 3=2 ½ðx^ rc cos jÞ 2
ð31Þ
5. Conclusions This paper uses methods of scalar and vector potentials to develop an analytical description of the magnetospheric fields. The corresponding potential functions were found from boundary conditions for the magnetopause and the Earth’s bow shock. The model field for internal and external distributions is determined mainly from two parameters: the distance from the Earth to the magnetopause, x0 ; and to the bow shock, x1 : If conditions in incoming solar wind are
ARTICLE IN PRESS E. Romashets et al. / Planetary and Space Science 53 (2005) 127–131
known, these two parameters can be calculated using methods developed in a number of studies, including, e.g., Farris and Russell (1994), Cairns and Grabbe (1994), Neˇmecˇek and Sˇafra´nkova´ (1991). Calculation results show a summer/winter asymmetry in geoeffectiveness of disturbances. For southward IMF, geoeffectiveness of geomagnetic storms are expected to be stronger in summer seasons.
Acknowledgements This work was supported by EU-INTAS-ESA grant 99-00727, RFBR grant 03-02-16340 and CRDF grant TGP-944, and by project S1003006 from the Academy of Sciences of the Czech Republic and by grant 205/03/ 0953 from the Grant Agency of the Czech Republic. References Alexeev, I.I., Feldstein, Y.I., 2001. Modeling of geomagnetic field during magnetic storms and comparison with observations. J. Atm. Sol.-Terr. Phys. 63, 431–440.
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