A mathematical magnetospheric field model with independent physical parameters

Planet Slmee Sei., Vol. 29, 1~. 1-20 ~Petgamon Press Ltd., 1981. Printed in Northern Ireland

0032--0633/81/0101-0001502.00/0

A MATHEMATICAL MAGNETOSPHERIC FIELD MODEL W1TH I N D E P E N D E N T P H Y S I C A L P A R A M E T E R S GERD-HANNES VOIGT

Department of Space Physics and Astronomy, Rice University, Houston, TX 77001, U.S.A.

(Received in final form 16 July 1980) Abstmet--A quantitative magnetospheric magnetic field model has been calculated in three dimensions. The model is based on an analytical solution of the Chapman-Ferraro problem. For this solution, the magnetopanse was assumed to be an infinitesimally thin discontinuity with given geometry. The shape of the dayside magnetopanse is in agreement with measurements derived from spacecraft boundary crossings. The magnetic field of the magnetopause currents can be derived from scalar potentials. The scalar potentials result from solutions of Laplace's equation with Neumann's boundary conditions. The boundary values and the magnetic flux through the magnetopause are determined by all magnetic sources which are located inside and outside the magnetospheric cavity. They include the Earth's dipole field, the fields of the equatorial ring current and tail current systems, and the homogeneous interplanetary magnetic field. In addition, the flux through the magnetopause depends on two constants of interconnection which provide the possibility of calculating static interconnection between magnetospheric and interplanetary field lines. Realistic numerical values for both constants have been derived empirically from observed displacements of the polar cusps which are due to changes in the orientation of the interplanetary field. The transition from a closed to an open magnetosphere and vice versa can be computed in terms of a change of the magnetic boundary conditions on the magnetopause. The magnetic field configuration of the closed magnetosphere is independent of the amount and orientation of the interplanetary field. In contrast, the configuration of the open magnetosphere confirms the observational finding that field line interconnection occurs primarily in the polar cusp and high latitude tail regions. The tail current system reflects explicitly the effect of dayside magnetospheric compression which is caused by the solar wind. In addition, the position of the plasma sheet relative to the ecliptic plane depends explicitly on the tilt angle of the Earth's dipole. Near the tail axis, the tail field is approximately in a self-consistent equilibrium with the tail currents and the isotropic thermal plasma. The models for the equatorial ring current depend on the Dst-parameter. They are self-consistent with respect to measured energy distributions of ring current protons and the axially symmetric part of the magnetospheric field. 1. INTRODUCTION

A quantitative magnetospheric B-field model is a mathematical formalism which reproduces the observed magnetic field vector in dependence of physical parameters at any point within the magnetospheric cavity. A model of this sort must include at least the field of the Earth's dipole and the fields of three different magnetospheric current systems which contribute essentially to the large-scale configuration of the magnetospheric B-field. The first current system contains the ChapmanFerraro currents. They flow entirely on the magnetopause and shield the magnetospheric cavity against the magnetosheath and the interplanetary space. These boundary currents form two extended vortexes which circle around the northern and southern cusp regions. The currents of the second system flow in the magnetospheric tail region from dawn to dusk.

These currents are concentrated in the plasma sheet and are closed over the northern and southern tail magnetopause. Their specific configuration depends on the plasma population in the tail (Bird and Beard, 1972; Birn et al., 1977; Fuchs and Voigt, 1979). On the earthward edge, the tail currents merge continuously into the equatorial ring current (Sugiura, 1972; Sugnira and Poros, 1973). The third current system is the equatorial ring current itself. This current consists mainly of protons in the energy range between 10 and 50 keY (Frank, 1967). Those protons are trapped in the region of closed field lines and drift around the Earth into westward direction. In an ideal model, these three magnetospheric current systems and the corresponding B-field configuration would be a self-consistent response to the solar wind-magnetosphere interaction. In other words, .the currents which we need in order to compute the magnetic field would be an answer

2

GERD-HAmcES VOIGT

gained from the model and not an input into the model. Unfortunately, we are far away from a self-consistent model of this ideal type. It is not yet possible to calculate simultaneously the currents, together with the plasma and the magnetic field throughout the whole magnetosphere. The only simplified treatment in this sense is restricted to two dimensions and to an isotropic plasma distributed throughout the magnetosphere (Fuchs and Voigt, 1979). Therefore, most of the existing magnetospheric models (including the model outlined in this paper) contain the three current systems mentioned above as more or less empirical inputs. The strength and location of each current system can be varied separately in those models in order to be able to reproduce the real magnetospheric field configuration derived from spacecraft measurements. Details of individual magnetospheric models with variable current systems are explained in Walker's (1976, 1979) review articles. In general, a realistic magnetospheric model must be three dimensional in order to describe local-time dependent phenomena. It must also be valid for all dipole tilt angles in order to reproduce daily and seasonal variations of the magnetic field. Finally, the model must include physical input parameters which can be related to measurable quantities like the solar wind pressure, the polarity of the interplanetary magnetic field, and magnetic disturbance indices. Variations of those input parameters should reproduce quasi-static changes of the magnetospheric B-field during quiet and moderately disturbed periods. It is the purpose of this paper to outline the mathematical formalism for a magnetospheric Bfield model which meets these general criteria. The mathematical method employed is similar to that used in an early version of the model (Voigt, 1972). In this calculation, the magnetopanse was assumed as an idealized tangential discontinuity. Therefore, the model was 'dosed' since no magnetic field could penetrate the magnetopause. In contrast, the formalism of the present model is improved by the versatility of the magnetic boundary conditions: the normal component of the total B-field is not required to be zero at the magnetopause. Thus, the model is capable of reproducing the macroscopic influence of the interplanetary magnetic field on the magnetospheric B-field configuration. In this respect, the present model is distinguished from the other magnetospheric models reviewed by Walker (1976, 1979). The mathematical method for the solution of the Chapman-Ferraro problem requires some physical

a pr/or/ assumptions which will be discussed in Section 2. Following these assumptions, the general formalism for the magnetic fields inside and outside the magnetosphere will be outlined in Section 3. There, we shall see that the 'dosed' magnetosphere is a very singular case in a generalized class of 'open' models. Section 4 contains the description of the magnetopause geometry, the list of the model parameters, and the definitions of the coordinate systems used. This information is needed for the explicit formulation of the magnetic fields inside and outside the magnetosphere in Section 5. The formalism of the tail current system in Section 6 and the description of the equatorial ring current model in Section 7 complete the mathematical formulation of the magnetospheric model. Some aspects of the 'open' magnetosphere as far as they result from static field line interconnection at the magnetopause will be outlined in Section 8. The advantages and the limitations of the validity of the model will be summarized in the concluding Section 9. 2. PHYSICAL A P P R O X I M A T I O N S

In order to accomplish the mathematical representation of the B-field throughout the whole magnetosphere, we must first discuss some physical approximations. (a) We know that reconnection of the magnetosheath and magnetospheric fields is irregular and not in a steady state (Haerendel ei al., 1978). At present, however, it is impossible to compute the time-dependent process of field line merging in the context of a global magnetospheric model. Let us therefore assume that local and time averages of statistically distributed reconnection events at the magnetopause can be described in terms of a static configuration of interconnected field lines. This approximation allows us to calculate the actual distribution of the normal component B~ of the total field on the magnetopause (Roederer, 1977). (b) Although the understanding of field line interconnection requires the knowledge of the microscopic properties of the magnetopause (Willis, 1978), let us assume that the model magnetopause is an infinitesimally thin discontinuity in the 'open' as well as in the 'closed' model. This approximation allows us to describe the transition from a 'closed' to an 'open' magnetosphere in terms of a change in the magnetic boundary conditions at the magnetopause. (c) Let us replace the self-consistent Mead-Beard type of the magnetopause (Mead and Beard, 1964) by a boundary with a predetermined geometry.

A mathematical magnetospheric field model This approximation enables us to solve the Chapman-Ferraro problem not only for the Earth's dipole (Mead, 1964; Choe and Beard, 1974; Olson and Pfitzer, 1974) but also, with high numerical accuracy, for any additional magnetospheric current system. W e need the high accuracy of the magnetic field components since the magnetopause of the 'open' magnetosphere deviates only minimally from a pure tangential discontinuity. (d) Let us ignore a specific distribution of the thermal plasma throughout the magnetosphere. The problem of computing the magnetic field in three dimensions is then reduced to the conceptually simple solution of Laplace's equation with Neumann's boundary values specified on the predetermined magnetopause. (e) The last approximation implies that the magnetic field of the Chapman-Ferraro currents can be derived from scalar potentials. However, the potential theory is a static theory. Therefore, the model does not involve any time dependency. Thus, we must restrict ourselves to the description of quasi-static changes in the B-field configuration which can be simulated by a series of static model states. It should be mentioned that this argument also applies to the other published magnetospheric models reviewed by Walker (1976, 1979).

the constant of interconnection in (3.2) is the same for all internal fields specified in (3.1). A corresponding argumentation holds in the magnetosheath where the interplanetary field B~,~f is shielded against the magnetospheric cavity by the Chapman-Ferraro field Be*. Since we anticipate different plasma processes on both sides of the magnetopause, we call Cimf another 'constant of interconnection', so that the proportion Cimf of the normal component ft- B~,~f penetrates the magnetopause. The corresponding boundary condition is therefore given by ft" (B,mf + B~*~)= Cimfft " Bi,,f.

Since the model magnetopause is assumed as an infinitesimally thin discontinuity, both unknown Chapman-Ferraro fields, B~, and B*o, can be derived from scalar potentials, B~, = - V U~f,

(3.4)

B~ - -V U~.

(3.5)

Introducing (3.4) into (3.2), and (3.5) into (3.3), we find Neumann's boundary conditions for Uc~ and Uc~ at the magnetopause as follows: O

O~ U,f. = +(1 -

3. GENERAL FORMALISM

The physical approximations outlined in the last section enable us to obtain the magnetic field of the Chapman-Ferraro currents from the solution of Neumann's boundary value problem of the potential theory. Let us assume that B~ represents the field of the Earth's dipole Ba plus the fields of the tail currents and the equatorial ring current, B i and B=, respectively. The field of the sources located within the magnetospheric cavity is then given by B~ = Ba +Bj +B=.

(3.1)

In order to shield the internal field B~ against the interplanetary space, we need an additional field B*, which is due to the Chapman-Ferraro currents. Let Ca be a 'constant of interconnection', so that only a certain proportion Ca of the normal component ft. B, can penetrate the magnetopause. The boundary condition at the magnetopause is then given by fi- (B~ + B~*¢~)= Carl- B~.

(3.2)

Here, fi denotes the unit vector perpendicular to the magnetopause. W e have assumed implicitly that

(3.3)

Cd)ft • B.

0 O--nUco = +(1 -Cim~)fi • B,,,f.

(3.6)

(3.7)

Moreover, both potentials must meet Laplace's equations

ucf, = 0

(3.8)

A U,o = 0

(3.9)

A

everywhere in the interplanetary space and in the magnetosphere. Let us choose the nomenclature Uc~, --- U ~ and Uc~ ~ U ~ -~ when we consider Ucf, and U~ in the region outside the magnetosphere. In the same way, we call Ucf, =- Ui~ and Uc~ ~- Uie~t when we consider both potentials inside the magnetosphere. Note that both the internal (int) and external (ext) solutions of (3.8) and (3.9) must meet the boundary conditions (3.6) and (3.7), respectively. The boundary value problem is entirely defined by the equations (3.6) to (3.9). It is evident that the configurations of the shielding fields (3.4) and (3.5) are determined by the configurations of the initially given fields Bs and Bimt via the boundary conditions (3.6) and (3.7).

4

GERD-HAUNES VOIGT

The numerical values of both constants of interconnection lie independently in the ranges of 0-
(3.10)

0 -< Cimf-----1.

(3.11)

The closed magnetosphere without any field line intereonnection at the magnetopause is simply determined by C d = C ~ = 0 . We can derive the Chapman-Ferraro potential for the inner space U ~ from the solution uof, of the closed model (Voigt, 1972) by the substitution U ~ = (1 - Cd) u~f,.

(3.12)

The same argument is valid in the outer space where U ~ ' can be substituted by Uce~ t =

(1 - Ci~) u~.

(3.13)

The Chapman-Ferraro fields of the closed model are then given by B~, = -Vu~f~

(3.14)

B~ = -Vuc~.

(3.15)

The solutions of (3.8) and (3.9) are unique with respect to the boundary values (3.6) and (3.7). Therefore, we find immediately V U ~ t = + ( 1 - Cd)B~

(3.16)

V U ~ ~= +(1 - C~)Btmf.

(3.17)

The total fields in the magnetosphere, B~,,, and in the magnetosheath, B,~,, are formally given by S,~, = B , - V ui¢~ + B,.,f - V U ~ ~

(3.18)

B,~, = B, - VU~;t + B , , , f - V U ~ ~.

(3.19)

By introducing (3.6) and (3.7) into (3.18) and (3.19), we can prove that both the internal and external fields, B~, and B,~,, meet the same boundary condition B , (Xmp)= fi" [CdB, + C~B~,,f].

(3.20)

B~(x~p)=fi "B~., =ft.B.~, is the total flux density at a specific location x.tp at the maguetopause. In other words, equation (3.20) is capable of describing the observed or calculated distribution of the normal component of the total field B.(x.~p) on the magnetopause. The difference in the tangential components on both sides of the magnetopause determines the configuration of the whole Chapman-Ferraro current system which is given by 1

Lt(x,,~) = +~--~fi • [B,~, - n , , , ] .

(3.21)

The magnetopause unit vector h points out of the magnetosphere. Introducing (3.12), (3.14), and (3.17) into (3.18), and (3.13), (3.15), and (3.16) into (3.19), we obtain the final field configuration inside and outside the magnetospheric cavity: Bi~, = B~ + (1 - Cd)Bcfs + CI,~B~,,~

(3.22)

B.x, = B~,.f + (1 - Ci~)Bc~ + C d B s.

(3.23)

The magnetospheric model is then determined by equations (3.1) and (3.20) to (3.23). It is presumed that the fields Bs and Bi,,f are given a pr/or/. The fields of the boundary currents, Bcf~ and B~, must be calculated via the boundary conditions (3.6) and (3.7). This procedure will be carried out explicitly in Section 5. Let us note down some properties of the model outlined in this section: It is evident from (3.22) and (3.23) that any changes in the orientation of the interplanetary field Bi,,f, or changes in the dipole tilt angle or in the ring current and tail field configurations in (3.1) affect the topology of the Chapman-Ferraro current system (3.21). The amount of the current density Ija(x,~p)[ depends on the numerical values of both constants of interconnection, Cd and C~=f. The boundary condition (3.20) shows again that the 'closed' model is specified by Cd = C ~ = 0. The normal component B,(x,,p) of the total field vanishes at the boundary in this case, and the magnetopause is a pure tangential discontinuity. W e can therefore conclude that the 'closed' magnetosphere is a very special case in a general class of 'open' models. The problem of finding realistic numerical values for C~ and Ci=f in the case of an 'open' model will be discussed in Section 8. 4. MAGN]ffFOPAUSE GF_.OM~t-K¥ AND

MODI~ PARAMEf~tcS We cannot specify explicitly the boundary values (3.6) and (3.7) for the Chapman-Ferraro potentials Ucf, and Uc~ unless we have defined the magnetopause geometry and the coordinate system. The dayside part of the magnetopause is approximated by a half-sphere which is connected continuously to the tail magnetopause. The tail magnetopause is represented by a semi-infinite cylinder with constant radius. This radius R is identical for both the half-sphere and the cylinder. Figure l(a) shows the coordinate system: the center of the half-sphere (S) is displaced by b Earth's radii from the Earth's center (D) into the antisolar direction. The stand-off distance ro to the

A mathematical magnetospheric field model (GSE) coordinates is given by

Zose~

z

XosE = - ( z + b)1 YGSE= +y

~

(4.2)

ZGSE ~ -['X.

(a) d~= 0 ° ~I. dawn X~E~dus k

X=0° x~;O0~

midnight

(c)YGsE

The axis of the Earth's dipole coincides with the zo-axis of the (xD, YD, zo) coordinate system in Fig. l(b). The orientation of this system relative to the (GSE) system is specified by the dipole tilt angle. Figures l(b) and 1(c) show that an arbitrary tilt angle is defined by the polar angle 0 measured from the zosE-axis, and by the angle X measured from the X~sE-axis in the ecliptic plane. The transformation between the (GSE) and (XD, YD, Zt,) coordinates is given by YD = A(0, X) "|YGsE ] Z

(b}

PIG.

(4.3)

~ZGsE/

and [cos0cosx cos0sinx A(~, X) = ~ -sin X cos X \ s i n ~ cos X sin 0 sin X

1. COORDINATE SYSTEMS:

(a) the parameter b indicates the displacement of the Earth's dipole (D) relative to the center (S) of the halfsphere. The untilted dipole (D) is situated in the origin of the geocentric-solar-ecliptic (GSE) system. (b) and (c) the angles 0 and X define an arbitrary tilt angle of the earth's dipole relative to the (GSE) system. subsolar point is then given by

ro=R-b.

(4.1)

The eccentric position of the dipole (D) relative , to the center of the sphere (S) enables us to fit the curvature of the dayside magnetopause to spacecraft boundary crossings as they have been reported by Fairfield (1971). Numerical values for magnetic quiet conditions are R = 16 Re and b = 5 Re. The stand-off distance is then given by ro = 11 Re, according to (4.1). Walker (1976) compared the model boundary based on these parameters with the magnetopause obtained from Fairfield's (1971) measurements and found a good agreement on the dayside. The mathematical formulas in the following Sections 5 and 6 will be expressed with respect to the (x, y, z) coordinate system depicted in Fig. l(a). The center of the half-sphere is located in the origin of this system. Its z-axis coincides with the axis of the tail cylinder. The transformation into geocentric-solar-ecliptic

-so~

cos d//" (4.4) Note that the (XD, YD, ZD) system coincides with the solar magnetic (SM) coordinates in the case of X = 0 °. The (GSE) and (SM) coordinates are defined according to Russell (1971). The list of model parameters is ordered with respect to their influence on the magnetospheric field topology: (1) to: stand-off distance to the subsolar point, measured in earth radii. (2) R: radius of the magnetospheric tail at xGs~ = - 1 0 Re, measured in earth radii. (3) ~, X: two angles defining an arbitrary tilt angle of the Earth's dipole relative to the (GSE) coordinate system. (4) D,,: index denoting the perturbation of the magnetospheric field at the Earth's surface caused by the axi-symmetric part of the equatorial ring current, measured in nT (see Section 7 for explanation). (5) Ca,-C~: two constants of interconnection which describe the amount of the magnetic flux at the magnetopause (see Section 3 for explanation). (6) X: factor which describes the decreases of the tail current density and the magnetic field down the tail (see Section 6 for explanation). In addition to these parameters, we must introduce two further parameters in order to specify the interplanetary magnetic field B~f:

6

GERD-HANNES VOIGT

(7) Ho: strength of the homogeneous interplanetary magnetic field Bi,,f, measured in nT. (8) ~: orientation of the interplanetary field (IMF), measured from the ZOSE-axis into the antisunward direction. Some special cases for information: = 0:

I M F northward,

= ~r/2:

I M F antisunward,

= ~r:

I M F southward, 5.1. Solution in the dayside magnetosphere

= - i r / 2 : I M F sunward. The model parameters can be used to adjust the model to variable magnetospheric situations. The stand-off distance ro and the tail radius R are particularly convenient parameters for simulating changes of the overall field configuration which are due to magnetospheric compression by the solar wind (Voigt, 1976). Daily and seasonal variations of the magnetic field can be reproduced by corresponding variations of both angles ~b and X (Voigt, 1974). If one is interested in the fine structure of the B-field in the inner magnetosphere, the D~,-index can be related to the corresponding ring current field. Details of the ring current model will be explained in Section 7. We can study the influence of the interplanetary magnetic field on the field configuration in high latitude polar cusp or tail regions by fitting both constants of interconnection, Co and C ~ , to real magnetospheric situations derived from observations (Voigt and Fuchs, 1979). The factor ), is introduced into the model in order to match the tail field configuration to magnetic measurements as they have been reported by Behannon (1968) and Mihalov and Sonett (1968). Details will be explained in Section 6. 5. EXPLICIT FORMALISM

The only unknown quantities in the final field configuration are the Chapman-Ferraro fields Bcfs in (3.22) and B¢~ in (3.23). Their corresponding scalar potentials in (3.14) and (3.15) are determined by the boundary values (3.6) and (3.7) by means of equations (3.12) and (3.13), respectively. In other words, we can restrict ourselves in the following analysis to the solutions u~f~ and u~g for the 'closed' model with the boundary conditions 0 On u ~ = +fi , B,

m

The boundary values (5.1) and (5.2) are distributed on the magnetopause described in Section 4. The Chapman-Ferraro fields (3.14) and (3.15) can therefore be expanded into spherical harmonics in the spherical part of the magnetosphere (dayside) and into Fourier-Bessel series in the cylindrical part (tail region). These expansions will be outlined in the following Subsections 5.1 and 5.2.

(5.1)

The position of the dayside magnetopause is given by r = R in a spherical coordinate system (r, @, q~) which corresponds to the (x, y, z) system shown in Fig. l(a). Let us assume that f(O, ~0) is a function of the boundary values specified in (5.1) or (5.2). This function can be expanded into spherical harmonics (e.g. Wyld, 1976):

comYo (o, ).

(5.3)

n=O m=O

The Chapman-Ferraro potential u~¢s for the inner space (r-< R) is then given by n

u~r,(r,O,q~)= ~ n=l

r n

~ 1C~"Y."(O,q~).R._ ~. m=O

(5.4)

~

In the space outside the magnetosphere (r--> R), the potential ucfi is given by u ~ ( r , O, ~ ) = -

=

"

1

C myd~(O ,~).

r "÷~ .

(5.5) Note that Neumann's boundary value problem requires that the summations in (5.4) and (5.5) start with n = 1. Both expansions (5.4) and (5.5) satisfy the boundary values (5.3) at the magnetopause

r=R. The earth's dipole field Ba is the dominant term in (3.1) which contributes to the boundary values (5.1). With respect to the dipole-fixed (Xo, YD, ZD) system in Fig. l(b) and (c), the dipole potential ud is given by ZD 1Aa

-M[xr,2 + yo 2 + zo2]3/2

(5.6)

Here, M = 8 x 1021 (Tcm 3) denotes the magnetic moment of the Earth's dipole. We find the corresponding expression in the spherical (r, @, q~) coordinates by means of the transformations (4.2) to (4.4):

ua(r, O, ~o) = - M " ~ A,[_fl(~o, X, ~b) . P~l(cos @) n=l

0

- - u~f, = + f i - B,m~. 0N

(5.2)

- f 2 ( x , ~,)" n .

1 V.(cos O)] .+, r

(5.7)

A mathematical magnetospheric field model P, are Legendre's polynomials, and P~ are the associated Legendre polynomials. The coefficients in (5.7) are given by

.4. = (-1)~+'b"-L

(5.8)

The quantity b is explained in Section 4. The functions :x and [2 contain the coordinate angle , , and both angles X and ¢ which define the dipole tilt angle, fx(~0, X, q') = cos ~0 cos ~ + s i n ~o sin ~k sin X'[

f2(x, ¢) = sin ¢ cos X.

(5.9)

J

The derivative of (5.7) with respect to r, specified at the magnetopause r = R, results in the boundary values ~ . Ba. Thus, we obtain

Bi,,f,, = +Ho(cos/j sin 0 cos q~+sin ~ cos 0 ) ) /

Bimf.o = +Ho(cos ~ cos O c o s , - s i n / j sin O)~ (5.13) /

Bi,,f.~ = - H o cos ~ sin re.

j

The expansion (5.3) is determined by the B~,,,f.rcomponent. Thus, we obtain ft. B,.~ = f(O, ~o) = Ho[cos ~ cos q~PxX(cosO) + sin ~,P~(cos 0)]. (5.14) In consideration of (5.5), the Chapman-Ferraro potential u~ is given by

~" Ba = f(O, ¢) = - M " ~ (n + 1)A,[fl • p 1 (cos O) n~l

1 - f 2 " n" P, (cos O)] R,+=.

source which contributes to the boundary values (5.2). The components of Bi,~f in the spherical (r, 19, q~) coordinate system can be obtained following the definitions and transformations (4.2) to (4.4) in Section 4.

(5.10)

u~(r, O, ~o) = -½Ho[cos ~ cos ~Pll(COs O ) + sin ~Px(cos ag)] ~-~.

(5.15) This is the explicit form of the expansion (5.3) which we need for calculating (5.4). Therefore, we obtain the Chapman-Ferraro potential u~f,,

u,f,(r, ag, ¢) = - ~. B , . [f~(~o, X, ~k) " P,* (cos O) n=l

- f 2 ( x , ¢ ) " n . / ' . ( c o s O)]. r".

(5.11)

This solution is valid in the region 0-----r-
(5.12)

The Chapman-Ferraro field B,f, which we need in (3.22) can be obtained from (5.11) and (3.14). In Sections 6 and 7, we shall discuss the fact that the additional consideration of the tail field B i and the ring current field B,~ in (5.1) will result in a numerical modification of the coefficients (5.12). This modification cannot be obtained analytically but must be calculated numerically by means of the relations of orthogonality for the spherical harmonics (e.g. Wyld, 1976). If one is interested in an approximation within a range of accuracy of about 10-20%, it is sufficient to use the coefficients (5.12) for calculating the field of the Chapman-Ferraro currents in the dayside magnetosphere. The homogeneous interplanetary field B~,,f in the space outside the magnetospheric cavity is the only

This solution is valid in the region r-> R. The Chapman-Ferraro field II,~ which we need in (3.23) can be obtained from (5.15) and (3.15). Note that this field leads to a compression and deformation of the initially assumed homogeneous interplanetary field ll,~f in the region of the magnetosheath near the magnetopause. 5.2. Solution in the tail region The position of the tailmagnetopause is given by p = R in a cylindrical coordinate system (p, q~,z) which corresponds to the (x, y, z) system shown in Fig. l(a). Let us first extend the potential ua, in (5.11) into the cylindrical tail region. For this purpose, it is convenient to replace the boundary condition (5.1) by the equivalent form 0-'n ul = 0--~(ua + ua,) = 0

(5.16)

at the magnetopause p = R. Keep in mind that we have excluded in this first step the fields B i and B,~ in (5.11) and (5.16). The substitution (5.16) enables us to expand the potential ul = ua + u~, into Fourier-Bessel series at the plane z = 0. Let us therefore specify the dipole potential (5.6) and the Chapman-Ferraro potential (5.11) at the plane z = 0. Hence, we obtain the expressions M

ua(P, rC, z=O)=[p2+bZ]3/2

(b'[2-p'fx)

(5.17)

8

GERD-HANNESVOIGT

and

tained from (3.14) by recalculating ua, from (5.26),

ucf,(p, q~, z = 0 ) = ~ B,[n. ~2" P , ( 0 ) - f l " P,~(0)]p". n~l

(5.18) Both functions/1 a n d / 2 are noted down in (5.9). Collecting terms with respect to [~ and [2, we obtain two radial functions gl and go:

pM

ucr,(p, ~, z) = ul(p, ~, z ) - ud(p, ~, z). (5.27) Here, ua(p, to, z) is the dipole potential (5.6) transformed into cylindrical coordinates. The transformation of (5.13) into cylindrical (p, ¢, z) coordinates, Bimf.p = +Ho cos ~ cos q~)

t- ~.. B,P,:(O)p"

Bi~f. z = +H0 sin ~

g~(P) = "~[pZ + b213/2 ,=1

bM

I- ~. nB,Pn(O)p n. (5.20)

These radial functions can be expanded into Fourier-Bessel series,

shows that the p-component of Bi,~f is independent of z. Therefore, we choose the z-independent solution of Laplace's equation for calculating u~ in the space outside the magnetospheric cavity: u~(p, ~0) = -

gl(P) = ~.. adl(xlp/R)

(5.21)

i=l

go(p) = ~ bdo(y,p/R).

(5.22)

According to the boundary conditions (5.16), the quantities x~ and y~ must be the roots of the derivations of the Bessel-Functions,

~ Jl(x~p/R)

=0 }

(5.23)

at the magnetopause p = R, arranged by ascending values. The coefficients in (5.21) and (5.22) are calculated by means of the relations of orthogonality: 2

1

pg~(p)J~(x~p]R)dp (5.24)

2 1 fR b, =/~2 [jo~y,)]2 Jo pgo(p)Jo(y~p/R) dp.

(5.25)

The potential u~ in the tail region is then given by

u~(p, ~, z) =

~_

R "÷1 1 (Dmcosm~o+E,.sinm~o)----~ m=lm P (5.29)

With respect to the Bimf.: c o m p o n e n t in (5.28), the boundary condition (5.2) requires m = 1 with E1 = 0 and D1 = +/40 cos ~. The final solution is therefore given by R2

uc~(O, tO)= -14o cos 6 cos q~" - - . P

o Jo(y,p/R) = 0, Op

a~ = R2 (x2 - 1)~Jo(X~)]2

(5.28)

B,~f.~ = - H o cos ~ sin

(5.19) go(P) = -I [p2+ b213n .=1

l

-[1(~, X, ~) X adl(x~p/R)e-X:/~ +/2(X, ~k) ~. bdo(Y,p/R)e -':/R.

(5.26)

This solution is valid in the region p<--R. The Chapman-Ferraro field B,f, in (3.22) can be ob-

(5.30)

This solution is valid in the region p ~ R . The Chapman-Ferraro field B,~ in (3.23) can be obtained from (5.30) and (3.15). It must be emphasized that (5.30) represents the solution of Laplace's equation for the infinitely extended cylinder. Therefore, the potentials (5.15) and (5.30) differ at the plane z = 0. The field Bc~ derived from (5.15) is consequently valid on the dayside of the magnetosphere, whereas the field derived from (5.30) is valid in the tail region beyond XGSE= --10 Re. In the region near the z = 0 plane, both solutions must be regarded as approximations. Physical reasons outlined in Section 8 suggest, however, that the error in the total field outside the magnetospheric cavity is small, since the constant C~,~ in (3.23) is approximately equal to unity. A similar argument is valid inside the magnetospheric cavity if we consider the field Bc~ derived from (5.11) or (5.27). Due to the combination of two different coordinate systems, Laplace's equation is not separable in the whole magnetosphere. The separation is only possible for either the sphere or the cylinder. Consequently, a discontinuity of the z-component of the field Bc~, occurs at the plane z = 0. Since the sphere and the cylinder have

A mathematical magnetospheric field model a common region of intersection, a rapidly converging procedure could be carried out in order to improve the coefficients (5.12), (5.24), and (5.25). This procedure is described in detail in Appendix 1 of the early paper (Voigt, 1972), referred to in the Introduction. In the following sections, the magnetospheric model will be completed by the field B~ of the tail current system, and the field B,~ of the equatorial ring current. Both fields modify the coefficients (5.12) of the potential (5.11) in the dayside part of the magnetosphere, and the coefficients (5.24) and (5.25) of the potential (5.26) in the tail region.

field Bf at the earthward edge of the tail currents. Let us call ul* the potential (5.26) modified by the potential of this additional vacuum field. Then, the tail field, excluding the field of the equatorial ring current, can be formally written as B~=l = - V U l * + B I.

(6.2)

Following the explanation outlined above, the explicit expressions for the three components of (6.2) are given as follows: B,.,.o

x,,)x

=

i~1 x

[Jo(r~plR)-

J~(~plR+f2(x, )])

V,)

6. T H E T A I L C U R R E N T SYSTEM

The vacuum tail field B~ = - V u 1 calculated from (5.26) decreases too rapidly down the tail compared to tail field observations reported by Behannon (1968) and Mihalov and Sonett (1968). W e must therefore introduce an additional field Bj which is due to distributed tail currents. Instead of creating a more or less sophisticated tail current sheet, let us simply stretch out the field lines of the vacuum field B1 into a tail-like configuration. W e shall see below that this procedure will result in a flexible tail current sheet which reflects all structural information of the vacuum field, especially that of the dipole tilt angle and that of magnetospheric compression. A quantitative way to stretch out the tail field lines is to multiply both the 0- and 4~-components of B1 by a factor k < 1. The resulting new components can be regarded as the O- and ~-components of Bj. Thus, the components Bj.o and B j . , are in the order of k, whereas the component Bj. ~ and the three components B~ are in the order of unity. Moreover, the three components of B~ must decrease clown the tall according to exp (-Xx~ • and exp (-~,y~ • in order to meet the condition V. Bj = 0. The numerical value of k lies in the range

z/R)

z/R)

0-
(6.1)

and can be fitted to observations (see Fig. 2). In the context of Figs. 3 and 4, we shall justify that the scaling described by equation (6.1) results in a reasonable tall field model and plasma sheet configuration. The magnetic field in the tall region consists of the field B1 = - V u l derived from (5.26), plus the field Bj of the tail currents, plus an additional vacuum field which provides the continuity of the

x~

([)tVJz)+½8,(z)]xJl(y,p/R))

i=1

Bt,a~., = - ( s i n ~0 cos 0 - c o s
x~

([Xc~(z)+½/3,(z)]

(6.3)

12(~plR)]) ~ ([2o~(z)+/3Jz)]

x [Jo(xdplR) +

Btail.z =--fl((~O,~, I / / ) X

x Jl(xtp/R)) + f2(X, 0)

The functions [1 and h are noted down in (5.9), and the z-dependent functions are given by ahx~

~i(z) =~-~ exp (-Xx, zlR) 13,(z) = ~

(1 - )t) exp

y~(z) = ~

exp

(-x~zlR) (6.4)

(-)ty~z/R)

@(z) = _~2 (1 - A) exp

(-y,z/R)

The coefficients a~ and bi are determined by the relations (5.24) and (5.25), respectively. The quantities x~ and Yi are defined by (5.23). In (6.3), the functions /31(z) and 8i(z) are due to the vacuum field, - V u l * , and the functions t~(z) and T~(z) are due to the tail current field B i. The reader may prove that the tail field (6.3) meets the requirement V . B,,a, = 0. We can calculate the components of the tail current density

10

GERD-HANNES VOIGT

ZOSE

X=l.0

.20

.....(~

XosE ,,20

/:,0

)

-,0 /

j/"/

/

.201

Z GSE

~.: 0.15 .2O

*20

-20'

•=0IzosE

~

XOSE

°20

~

X: 0.0

2...... -30

-&O

![ ;50

.20¸ FIO. 2. TAIL FIELD CONFIOURATION OF THE CLOSED MODEL. T h e tail field d e c r e a s e s d o w n the tail in d e p e n d e n c e of the f a c t o r A. X = 1.0: v a c u u m field w i t h o u t tail currents. k = 0.15: r e a l i s t i c tail field configuration. = 0.0: tail c u r r e n t s are i n d e p e n d e n t of Z or X o s ~ , r e s p e c t i v e l y .

A mathematical magnetospheric field model

Note that the terms /3~(z) and 8~(z) vanish in the components of the tail currents (6.5) since these terms are due to the vacuum field derived from

Zose

.20 XGSE

-20

.20IZGSE

FIG. 3. FIELD LINE CONFIGURATION OF THE CLOSED MODEL. Field lines are plotted in the noon-midnight meridian plane for the untilted dipole (top) and the dipole tilted in the noon-midnight meridian, t~ = - 3 5 ° and X = 0°, (bottom). The dipole tilt results in a displacement of the neutral sheet out of the ecliptic plane. J(0, cb, z) from (6.3) by means of Maxwell's equation V× B~al= 41rj:

io(p, ,c, z)

1

= 4-"~(sin ~0cos tk - c o s ~0 sin ~ sin X)(1 - X2)

x i=l~ (~(z) R [Jo(xip]R)+J2(xlp[R)]) iv(p, ~, z) =~

fl(1-A2) ~

(o~(Z)R [.lo(x~o/R)-J2(x,olR)])

1 2 ® +4"-'~ f 2 ( 1 - ~ )i~l

/,(p, , , z)=O.

11

('i(Z)RJI(yip]R)) (6.5)

UI*. For reasons of continuity, an additional vacuum field must appear on the dayside of the magnetosphere in the case of X < 1. This field is in analogy to that which changes ul into ul*; it modifies the coefficients (5.12) and is responsible for any changes in the dayside field configuration whenever the tail currents are changed by the factor A. The factor A indicates the decreases of the magnetic field (6.3) and the tall currents (6.5) down the tail. ~, = 1 results in the limiting situation of the vacuum field without any contribution of the tail currents. The tail currents (6.5) vanish in this case, and the potential u~* changes into u~ noted down in (5.26). Consequently, the field (6.3) changes into B~ = -Vu~ in the case of X = 1. The other limiting situation, X = 0, results in a configuration similar to the Harris sheet. The tail currents (6.5) are independent of z in this case, and no field line can penetrate the plasma sheet in the tall region (see the bottom picture of Fig. 2). In order to fit the model tail to the measured tall field decrease reported by Mihalov and Sonett (1968), the value of X =0.15 is required for magnetically quiet conditions. It should be noted that this value has been derived empirically. The three situations discussed above are shown in Fig. 2: the vacuum configuration (A = 1) is compared with the realistic tail field decrease (A = 0.15) and with the configuration of the Harris sheet (A =0). In general, the numerical value of A is responsible for the magnetic flux which penetrates the plasma sheet. As mentioned above, the tail currents (6.5) reflect the influence of the dipole tilt angle by means of both functions fi(4~, X, $) and f2(X, $)Figure 3 shows the field line configuration of the 'closed' magnetosphere in the noon-midnight meridian plane for the untilted dipole (X = $ = 0°), and the dipole tilted in the noon-midnight meridian (X = 0°; ~b= - 3 5 ° ) . We can see qualitatively from this figure that the dipole tilt results in a remarkable displacement of the neutral sheet out of the ecliptic plane. In the near earth tail region, the dipole field dominated part of the magnetosphere determines the tail field structure. The plasma, and consequently the tall currents, are adjusted to this field structure. Therefore, the plasma sheet is shifted below the solar ecliptic plane during the northern

12

GERo-HAnNES VOIGT Z0$E

YGSE

XOSE

FIG. 4. PosrnoN OF THE PLASMASHEET IN THE YGSE-Z G S E PLANE.

The circular tail cross section has a radius of R = 22 RE at XGSE=--35RE. The current streamlines forming the plasma sheet have been calculated from equations (6.5). In the upper picture, the geometry of the plasma sheet is due to the untilted Earth's dipole, in the lower picture due to the dipole tilted by tO=+35 °, and x =0° (tilt in the noon-midnightmeridian plane). Note that the plasma sheet crosses the ecliptic plane when the dipole is tilted. hemisphere winter, and is raised above this plane during the northern hemisphere summer. The detailed structure of the plasma sheet is plotted in Fig. 4. However, a tail radius of R = 22 Earth's radii was assumed at the geocentric distance of XosE = - 3 5 Earth's radii. Plotted are tall current stream lines calculated from equation (6.5). The central stream lines in both pictures indicate the stream lines of the maximum current density.

The 'thickness' of the plasma sheet is indicated by the upper and lower stream lines. These stream lines indicate the half of the maximum current density. The upper picture in Fig. 4 shows the plasma sheet structure for the untilted Earth's dipole. Note that the plasma sheet becomes broader near the flank's of the tail. The lower picture in Fig. 4 shows the plasma sheet structure for the dipole tilted by ~ = +35 ° and X = 0° in the noon-midnight meridian. It is interesting to note that the plasma sheet crosses the ecliptic plane whenever the dipole is tilted in the n o o n midnight meridian. This result confirms a recent finding of Fairfield (1980) who has derived the position of the neutral sheet by measurements of the tail magnetic field polarity. Keep in mind that the radius of the tail is constant in the present magnetospheric model. Therefore, the plasma sheet must cross the ecliptic plane in order to meet the requirement of magnetic flux equality in both tall lobes. It should be noted additionally that the tail current density decreases down the tall according to the A-dependent exponential terms in (6.5). But the 'thickness' of the plasma sheet increases down the tall due to the same exponential factors. In Fig. 5, the position of the neutral sheet is plotted in the midnight meridian up to the distance of 60 Earth's radii down the tail. The tilt angle, = +35 °, is the same as in the lower picture of Fig. 4. For this calculation, the position of the neutral sheet was defined by the change of the tail magnetic field polarity. The lower part of Fig. 5 depicts the dependence of the neutral sheet's position on the tail current decrease indicated by A: the more the tail current density decreases down the tail, the more the neutral sheet tends to return to the solar ecliptic plane. The tail currents (6.5) reflect also variations of the dayside field configuration which are due to magnetospheric compression. Magnetospheric compression is described in the model by both parameters ro (stand-off distance) and R (tall radius). These two quantities affect the coefficients a~ and b~ in the z-dependent functions ai(z) and ~/~(z) noted down in (6.4). The upper part of Fig. 5 shows the effect of magnetospheric compression by means of the variation of the stand-off distance: the displacement of the neutral sheet related to the quiet magnetosphere (to = 12 Re) exceeds that related to the compressed state (to=8 Re) in the region between

A mathematical magnetospheric field model

13

ZGSE ro=12 RE

3 2

ro= 8R E

1

-10 '

-Lo

2'o

-6'o

ZGSE

X=0.2

i

-t0 FIG.

5. POSITION

I

i

-20

-30

OF THE NEUTRAL

~

-~

*

-50

i

X

=0.&

x

o.e _

-60

SHEET IN THE NOON-MIDNIGHT

XGS E MERIDIAN.

The displacement of the neutral sheet out of the ecliptic plane is plotted in dependence of the stand-off distance r o (top) and the taft field decrease indicated by the factor k (bottom). The Earth's dipole tilt angle is given by ¢ = + 3 5 ° and X = 0 ° in both pictures.

XGSE = --20 and - 6 0 Earth's radii. One can conclude that the neutral sheet tends to be nearer the ecliptic plane when the magnetosphere is compressed or the averaged magnetic activity is increased. The results of Figs. 3, 4 and 5 can be summarized as follows: the dipole tilt causes the displacement of the neutral sheet out of the ecliptic plane. During the northern summer, the neutral sheet is shifted above the ecliptic plane in the noon-midnight meridian, but is depressed below this plane near the flanks of the tail. It should be emphasized again that this displacement is not introduced artificially into the model. Rather, it is a consequence of the explicit dependence of the tail currents on the dipole tilt angle. In this respect, the present model is distinguished from other tilt dependent magnetospberic models (Alexejev and Shabansky, 1972; Tsyganenko, 1976) which contain the neutral sheet displacement by means of empirical formulas. The formalism outlined in this section provides a causal relationship between the configuration of the dayside magnetosphere and the density and configuration of the tail currents. The tail field itself can be fitted by the factor A either to the observed tail field or to the measured magnetic flux which penetrates the plasma sheet. Fairfield's (1980) statement that the shape and position of the neutral sheet do not vary significantly with distance down the tall is consistent with this magnetospheric model if one assumes the factor A to be in the range of 0 ~
7. THE EQUATORIAL RING CURRENT

The ring current field B~ introduced into the magnetospheric model has been derived from stationary, axially symmetric ring current models calculated by Sckopke (1972). The ring currents have been computed for the vacuum magnetic field of the Earth's dipole Bd shielded against the interplanetary space by the corresponding ChapmanFerraro field B~,. The ring currents are selfconsistent with respect to the axis-symmetric parts of both fields. The calculation of the steady-state equilibrium between the ring current plasma and the magnetic field is based on the assumption that convection velocities are small in the ring current region. In addition, E-fields which are due to charge separation have been neglected. Under these assumptions, the situation is described by the static M H D force equation j,~ × (Ba +Bct~ + B,~) = V • P.

(7.1)

The ring current density j,~ is related to the ring current field by Maxwell's equation V X (Bd +Bcfs +Brc) -- 4wj,~.

(7.2)

Inserting (7.2) into (7.1), we obtain a non-linear equation for B,~, (V × B,~) x (Bd + B~f, + B,~) = 4wV • P,

(7.3)

which can be solved iteratively (Sckopke, 1972), assuming that field aligned currents do not exist. Of

14

GERD-HANNES VOIGT

- 20+....___...,-"'~. _ . ~ ~

r/R E

,,,/ 80Tf I00"~-/

'2°1 1/.0-t

\ /

/V/

2 Dst=- 33'y 3 O=t=- 50y /* Dst =-100~'

ring current field

FIG. 6. RaNGCURRENTFIELDB,~ IN THE EOUATORIALPLANE. The ring current field is plotted versus the radial distance from the Earth. It is generated by Sckopke's (1972) axially symmetric self-consistent ring current models.

course, the solution of (7.3) must meet the condition V . B,~ = 0.

(7.4)

The vacuum fields B~ and Bcf, in (7.3) can be derived from the corresponding scalar potentials (5.7) and (5.11), respectively. However, the selfconsistent solution of (7.3) and (7.4) is restricted to the first order terms (n = 1) in the series expansions of ud and u,f, since these terms describe the axially symmetric parts of the dipole field Bd and the Chapman-Ferraro field Bcfs. The elements of the pressure tensor P in (7.3) are related to the kinetic energy densities of the ring current particles parallel and perpendicular to the magnetic field. The energy density distribution must be given a pr/oti in the equatorial plane (i.e. a plane which is intersected by all field lines). The shape of the distribution function can be chosen arbitrarily as long as the stability of the particle distribution is not being considered. However, the energy distributions of the particles in the model belts were approximated to energy distributions of ring current protons measured by Frank (1967). Therefore, the ring current models reflect the main characteristics of the plasma distribution in the ring current region during quiet and moderately disturbed periods. The ring current field B,~ in the equatorial plane produced by those model belts is plotted in Fig. 6 for different magnetic disturbances indicated by the D,,-parameter. The gradients of IB,~[ between 1 and 3 Earth radii depend on the shape of the low-

energy end of the model energy density profiles and are physically insignificant. We obtain from Fig. 6 that the maximum of the ring current density (i.e. the 'position' of the ring current) is located between 6 and 7 Earth radii in the equatorial plane. However, the B,, profiles show that the maximum of the ring current density increases and moves earthward for increasing negative D,,-values. Moreover, we can see that the ring current field decreases like a dipolar field beyond 7 Earth radii. In other words, for calculating the coefficients of the spherical harmonics expansion (5.12) and the Fourier-Bessel expansions (5.24) and (5.25), the ring current could be involved by increasing the numerical value of the Earth's dipole moment M. For example, the D= = - 3 2 n T and D= = - 1 0 0 nT ring currents increase the dipole moment by 10.6% and 19.0%, respectively. Sugiura (1972), and Sugiura and Poros (1973) pointed out that the equatorial ring current contributes essentially to the magnetic field in equatorial regions between 4 and 8 Earth radii. Figure 7 shows equatorial AB-profiles (AB = total magnetospheric field minus reference dipole field) in the noonmidnight meridian plane. Dotted lines represent observed AB-profiles for two magnetic activity groups, K p = 0 - 1 and K p = 2 - 3 , taken from Sugiura and Poros (1973). Solid lines represent AB-profiles calculated from the present magnetospheric field model. Here, AB is the total field of the 'closed' model, B~,,, noted in (3.22) minus the geomagnetic dipole field, Ba, derived from the scalar potential (5.7). Two different ring currents,

A mathematical magnetospheric field model

~

15

70 F 60~

model calculation (~ Ost=-32V

so ~ 1.0~::=~

@ o=t = - soy (~) without ring current

20

. . . . . . .-

1

O-10

' L'~.C¢"'10

'."

8

ml-20 <3-30 -40 -50 -60 -70

16

®

\

x--A"1 ~

FiG. 7. AB VERSUSX o s E

:

o,,,,,a,on, (~ (~)

kp-O-i kp=2-3

AT TIlE EQUATOR IN THE NOON-MIDNIGHT MERIDIAN.

&B indicates the total magnetospheric field minus the geomagnetic dipole field. Solid lines show model calculations for the untilted Earth's dipole and the dosed magnetosphere. Dotted lines represent observations taken from Sugiura and Poros (1973).

D,, = - 3 2 nT and D= = - 5 0 nT, have been chosen in order to match the model to the observed ABprofiles as closely as possible. Equatorial ABprofiles obtained from the Olson-Pfitzer model (Olson and Pfitzer, 1974) and other magnetospheric models can be found in Fig. 26 of Walker's (1976) review. The model magnetospheric AB-field without any ring current is also shown in Fig. 7. This curve clarifies again that the quiet-time ring current must be inserted into a quantitative magnetospheric field model in order to reproduce the real magnetic field structure during magnetically quiet conditions (Sugiura and Poros, 1973; Olson and Pfitzer, 1974). 8. ASPECTS OF THE OPEN M A G N E T O S P I ~ J t E

It is necessary to verify that the magnetospheric model outlined in the previous sections can reproduce the macroscopic influence of the interplanetary magnetic field (IMF) on the magnetospheric field in high latitude polar cusp and tail regions. Since the model is not time-dependent, any test must be restricted to those phenomena which can be described in terms of static field line interconnection at the magnetopause. There is some indirect evidence for interconnection between magnetospheric and interplanetary field lines: Aubry et al. (1970) were able to demonstrate that a reversal of the vertical component of the I M F from northward to southward results in an earthward motion of the dayside magnetopause

during periods of constant pressure of the solar wind. In this case, the decrease of the stand-off distance is associated with an equatorward shift of the dayside polar cusps (Burch, 1973; Kamide et al., 1976). These effects have been derived on the basis of statistical averages of spacecraft data; they reflect therefore the macroscopic influence of the I M F on the m a g n e t o s p h e r i c B-field mentioned above. Considering those macroscopic phenomena, the term 'open magnetosphere' will be used in the context of our model in order to describe time averaged (static) situations of interconnected field lines. In the case of an open magnetosphere, both 'constants of interconnection', Cd and C ~ , are greater than zero. In order to demonstrate the ) effect of various boundary conditions on the magnetospheric field topology, let us simplify the situation in the first step saying that both constants are identical, CO = C ~ = Co. Figures 8 and 9 show magnetospheric field lines in the noon-midnight meridian plane in dependence of three different values of Co and I M F orientations, respectively. Field lines are spaced every 2 ° in latitude, starting at 66 ° . Only field lines with one or two ends on the Earth have been plotted; pure interplanetary field lines without connection to the dipole have been omitted. The model parameters in both figures refer to a quiet magnetosphere; they are given by ro = 11 Re, R = 1 6 R e , ¢ , = X = 0 °, D , , = 0 n T , and X=0.15. The amount of the IMF is H a = - 5 nT. In Fig. 8, the 'closed' model (Co = 0) is compared

16

GERD-HANNES VOIGT | ZGSE

dosed

magnetosphere

Co • ao

• 2o;

XGSE 9s- _+~_ _ _ -20

-ZO,

~Zos~

,2o',I.

~

IMF IOUth

/

co, o.l

',,%

FIG. 8. CLOSED AND OPEN MAGNETOSPHERICFIELD LINE TOPOLOGIES. Field lines are plotted in the noon-midnight meridian plane. The upper picture is identical with the upper picture in Fig. 3. top: Co = Cd = C~a = 0.0 middle: C O=Cd=Ci~=O. 1 bottom: C O= Cd = Cimf = 0.2. The closed model without any field line interconnection at the magnetopause is independent of the IMF strength and orientation. The open model allows high latitude polar cusp and tail field lines to interconnect with the IMF. The I M F points southward in this figure. Its amount is H o = 5 nT.

A mathematical magnetospheric field model

17

Z GSE

i XGSE *20

~ZGsE

~

/

,.20

ZGS E IMF,~ ,20

• 2o

I\~,\~.

~----~-~__--~J~-3o

~

.,oi

-s-o'

-20

FIG. 9. INFLUF_~C]5OF THE ~ DIRF_,C'I~ONON THE POLAR CUSPTOPOLOGY. T h e I M F direction is indicated by the arrows. Its a m o u n t is H o = 5 nT. T h e constants of interconnection are given by C o = C d = C i m ~ = 0 . 1 . North--south asymmetries of the polar cusp topologies arise whenever the I M F is n o t strongly southward directed.

18

GERD-HANNES VOIGT

with two different 'open' configurations (Co = 0.1 and Co = 0.2) in the special case of a southward directed IMF. The 'closed' model is independent of the IMF strength and direction since the magnetopause is a pure tangential discontinuity in this case (see equation (3.20)). The more the flux through the magnetopause increases by means of an increase of Co, the more high latitude polar cusp and tail field lines are interconnected with the IMF. This result confirms the findings of Haerendel et al. (1978) that field line reconnection occurs mainly in the cusp region rather than at the subsolar magnetopause. Moreover, the field line topologies of the 'open' model show that IMF interconnected field lines located near the equatorward edges of the polar cusps have been eroded from the dayside. In Fig. 9, the influence of the southward directed IMF is compared with the influence of other IMF orientations marked by the arrows. Asymmetries between the cusp topologies in the northern and southern hemisphere arise whenever the IMF deviates from the northward or southward direction. Keep in mind that in reality north-south asymmetries of phenomena observed in the polar ovals or the cusp regions are due as well to seasonal variations as to changes of the IMF orientation. It is impossible at present to calculate Cd and C~,a on the basis of any reconnection theory. Nevertheless, it is possible to derive realistic values for both constants by fitting empirically the magnetospheric model to measured displacements of the polar cusps which are due to changes of the IMF orientation. A close agreement with IMF related polar cusp displacements reported by Burch (1973) could be attained with CO=0.12 and Cimf=0.93 (Voigt and Fuchs, 1979). In other words, 12% of the magnetospheric magnetic flux and 93% of the IMF penetrate the magnetopause during average magnetospheric and interplanetary conditions. Similar values for Ca and -C~ have been successfully used to calculate the effect of field line erosion on the dayside magnetopause when the vertical component of the IMF turns from northward to southward: during quiet solar wind conditions, the model magnetosphere is confined to subsolar distances of ro = 10.4 Re and ro = 11.2 Re in response to a southward and northward directed IMF, respectively (Voigt, 1979). These values for the stand-off distance correspond reasonably well to the average values of r0 = 10.5 Re and ro = 11.6 Re which were found by Fairfield (1971) with respect to both IMF orientations. For the present, the model calculations cited above suggest that in reality both 'constants of

interconnection' hold the relations (Cd)2<< 1 (1 - C ~ ) 2 << 1.

(8.1) (8.2)

It is reasonable to assume that these relations are valid for periods of quiet magnetospheric and interplanetary conditions. 9. S U M M A R Y A N D DISCUSSION

The physical characteristics of the magnetospheric model described in this paper will be summarized in this section in order to discuss the model's range of application. The model is based on a three-dimensional solution of the Chapman-Ferraro problem. For this solution, the magnetopause is assumed to be an infinitesimally thin discontinuity with a fixed geometry. On the dayside, the model geometry is in agreement with measurements derived from spacecraft boundary crossings, as outlined in Section 4. Because of the thin discontinuity approximation, we cannot describe microscopic phenomena within the magnetopause which are relevant for the understanding of field line reconnection and plasma entry processes. On the other hand, this macroscopic model enables us to calculate the steady-state magnetic flux through the magnetopause. It is discussed in Section 3 that the flux through the boundary is determined on the one hand by all magnetic sources which are located inside and outside the magnetospheric cavity, and on the other hand by two 'constants of interconnection' which provide the possibility of calculating an 'open' or a 'closed' magnetosphere. The 'closed' magnetosphere without any connection between magnetospheric and interplanetary field lines results when the normal component of the total B-field is equal to zero at the magnetopause. This situation is a very special case in a generalized class of 'open' models. It is not yet possible to calculate the 'open' magnetosphere or both 'constants of interconnection' in terms of any reconnection theory. Therefore, the model does not reflect explicitly the plasma processes which may result in a configuration of interconnected field lines. However, one can derive realistic numerical values for both constants by fitting empirically the model either to displacements of the polar cusps or to displacements of the stand-off distance which independently are observed with respect to various IMF orientations. This is outlined in Section 8. The success of this

A mathematical magnetospheric field model empirical fit suggests that the model is capable of reproducing the macroscopic influence of the I M F on the magnetospheric field in the polar cusp and high latitude tail regions. It must be pointed out that the model does not include electric fields. Therefore, we cannot describe phenomena which are necessary for the understanding of magnetospheric-ionospheric coupling effects. Despite the physical approximations discussed in Section 2, the model confirms the following observations: (a) The Chapman-Ferraro currents flow entirely on the magnetopause and form two extended current loops which circle around the northern and southern cusp regions. The cusps themselves are funnel-shaped. (b) Field line interconnection occurs primarily in the cusp regions. The field line topology of the open model shows that I M F interconnected field lines located near the equatorward edges of the polar cusps are eroded from the dayside. Consequently, the topologies of the polar cusps and the position of the subsolar stagnation point change in response to various I M F orientations. Both phenomena have been observed (see references in Section 8) and are reproduced reasonably well by this model. The present model and most of the other published models referred to in Walker's (1976, 1979) reviews rely essentially on magnetic field observations. However, it must be emphasized that a complete understanding of the physics of the magnetosphere requires the development of models which are self-consistent at least with respect to the plasma and the magnetic field. A first step to meet this requirement was done by including Sckopke's (1972) self-consistent equatorial ring currents into the mathematical formalism of this model. It is outlined in Section 7 that these ring currents are based on observed energy distribution functions of ring current protons. However, the self-consistency of the ring currents is restricted to that part of the magnetospheric field which is axially symmetric relative to the Earth's dipole axis. Therefore, the model should be used only for modelling the magnetosphere during quiet and moderately disturbed periods when the ring current in nature is approximately symmetric. Despite the fact that the model ring currents are not self-consistent with respect to the total nonsymmetric magnetospheric field, they reflect the characteristics of the real plasma distribution in the ring current region. In this respect, the ring current models can be regarded as a first approximation for

19

calculating the self-consistent equilibrium between the anisotropic ring current plasma and the real magnetospheric B-field. The list of model parameters in Section 4 contains the D,,-index which is a measure for that part of the magnetospheric field which is generated by the equatorial ring current. The self-consistent equilibrium problem is also relevant in the tail region. Birn et al. (1977) have calculated that the solution of both static equations [V × Btail] X Btail - 4arVP

V" B~il = 0

(9.1) (9.2)

yields realistic magnetotail configurations under quiet conditions. The thermal pressure in (9.1) was assumed to be isotropic in this case. Unfortunately, the theory of Birn et al. (1977) is restricted to models of the distant tail, where the vacuum magnetic field, i.e. the influence of the geomagnetic main field and the field of the dayside C h a p m a n Ferraro currents, can be neglected. In contrast to the tail models of Birn et al. (1977), the tail field of the present magnetospheric model includes this vacuum field. It is outlined in Section 6 that the tail current system noted down in (6.5) reflects explicitly the effect of dayside magnetospheric compression which is due to changes in the solar wind conditions. In addition, the position of the plasma sheet relative to the ecliptic plane is explicitly dependent on the tilt angle of the Earth's dipole axis. The view of the tail cross section in Fig. 4 shows that the plasma sheet is curved and crosses the ecliptic plane, whenever the Earth's dipole is tilted in the noon-midnight meridian plane. This theoretical result confirms Fairfield's (1980) argument that the neutral sheet crosses the solar-magnetospheric equatorial plane rather than simply approaching it on the flanks of the tail. However, the tail field noted down in (6.3) meets the condition (9.2), but is self-consistent only in the region near the tail axis. This implies that the thermal pressure P in (9.1) is not exactly constant along the magnetospheric model field lines. Nevertheless, the formalism outlined in Section 6 provides a causal relationship between the configuration of the dayside magnetosphere and the density and configuration of the tail currents. The list of model parameters in Section 4 contains the factor A by means of which the model tail field can be fitted either to the observed tail field or to the measured magnetic flux which penetrates the plasma sheet. Another important aspect of the geomagnetic tail

20

GERD-HANNES VOIGT

is its stability (Schindler, 1979). T h e plasma and magnetic field configurations d e p e n d essentially on the pressure function P which is assumed a pr/or/ on the tail axis. Birn (1979) was able to find a pressure function which gives rise to an unstable dynamic d e v e l o p m e n t of the tail field configuration. In contrast, the tail field configuration of the present m o d e l remains stable if one chooses an isotropic pressure function which meets the condition (9.1) as closely as possible. Therefore, we cannot describe substorm events and related phenomena. REI~llENCY_~

Alexejev, I. I. and Shabanskij, V. P. (1972). A model of a magnetic field in the geomagnetosphere. Planet. Space Sci. 20, 117-133. Aubry, M. P., Russell, C. T. and Kiveison, M. G. (1970). Inward motion of the magnetopause before a substorm. J. geophys. Res. 75, 7018-7031. Behannon, K. W. (1968). Mapping of the earth's bow shock and magnetic tail by Explorer 33. J. geophys. Res. 73, 907-930. Bird, M. K. and Beard, D. B. (1972). The self-consistent geomagnetic tail under static conditions. Planet. Space Sci. 20, 2057-2072. Birn, J. (1979). Gleichgewichtsstruktur des Magnetosph~renschweifs und dynamische Struktur~inderungen. Kleinheubacher. Bet. 22, 203-212. Birn, J., Sommer, R. and Schindler, K. (1977). Selfconsistent theory of the quiet magnetotail in three dimensions. J. geophys. Res. 82, 147-154. Burch, J. L. (1973). Rate of erosion of dayside magnetic flux based on a quantitative study of the dependence of polar cusp latitude on the interplanetary magnetic field. Radio Sci. 8, 955-961. Choe, J. Y. and Beard, D. B. (1974). The compressed geomagnetic field as a function of dipole tilt. Planet. Space Sci. 22, 595-608. Fairfield, D. H. (1971). Average and unusual locations of the earth's magnetopause and bow shock. J. geophys. Res. 76, 6700-6716. Fairfield, D. H. (1980). A statistical determination of the shape and position of the geomagnetic neutral sheet. J. geophys. Res. 85, 775-780. Frank, L. A. (1967). On the extraterrestrial ring current during geomagnetic storms. J. geophys. Res. "/2, 37533767. Fuchs, K. and Voigt, G. H. (1979). Self-consistent theory of a magnetospheric B-field model, in Quantitative Modeling of Magnetospheric Processes (Edited by W. P. Olson), pp. 86-95, Geophys. Monogr. Ser., vol. 21, AGU, Washington, D.C. Haerendel, G., Paschmann, G., Rosenbauer, H. and Hedgecock, P. C. (1978). The frontside boundary layer of the magnetosphere and the problem of reconnection. J. geophys. Res. 83, 3195-3216. Kamide, Y., Burch, J. L., Winningham, J. D. and Akasofu, S. I. (1976), Dependence of the latitude of the cleft on the interplanetary magnetic field and substorm activity. J. geophys. Res. 81, 698-704. Mead, G. D. (1964). Deformation of the geomagnetic

field by the solar wind. J. geophys. Res. 69, 1181-1195. Mead, G. D. and Beard, D. B. (1964). Shape of the geomagnetic field solar wind boundary. J. geophys. Res. 69, 1169-1179. Mihalov, J. D. and Sonett, C. P. (1968). The cislunar geomagnetic tail gradient in 1967. J. geophys. Res. 73, 6837-6841. Olson, W. P. and Pfitzer, K. A. (1974). A quantitative model of the magnetospheric magnetic field. J. geophys. Res. 79, 3739-3748. Roederer, J. G. (1977). Global problems in magnetospheric plasma physics and prospects for their solution. Space Sci. Rev. 21, 23-70. Russell, C. T. (1971). Geophysical coordinate transformations. Cosmic Electrodyn. 2, 184-196. Schindler, K. (1979). Theories of tail structures. Space Sci. Rev. 23, 365-374. Sckopke, N. (1972). A study of self-consistent ring current models. Cosmic Electrodyn. 3, 330-348. Sugiura, M. (1972). Equatorial current sheet in the magnetosphere. J. geophys. Res. 77, 6093-6103. Sugiura, M. S. and Poros, D. J. (1973). A magnetospheric field model incorporating the O G O 3 and OGO 5 magnetic field observations. Planet. Space Sci. 21, 1763-1773. Tsyganenko, N. A. (1976). A model of the cislunar magnetospheric field. Ann. Geophys. 32, 1-12. Voigt, G. H. (1972). A three dimensional, analytical magnetospheric model with defined magnetopause. Z. Geophys. 38, 319-346. Voigt, G. H. (1974). Calculation of the shape and position of the last closed field line boundary and the coordinates of the magnetopause neutral points in a theoretical magnetospheric field model. J. geophys. Res. 40, 213-228. Voigt, G. H. (1976). Influence of magnetospheric parameters on geosynchronous field characteristics, last closed field lines and dayside neutral points. In The Scienti~c Satellite Programme During the International Magnetospheric Study. (Edited by Knott and Battrick), pp. 381-396, Reidel, Dordrecht, Holland. Voigt, G. H. (1979). Influence of the interplanetary magnetic field on the position of the dayside magnetopause. In Proceedings of Magnetospheric Boundary Layers Con[erence, (Edited by Battrick and Mort,) pp. 315-321, European Space Agency, ESA SP-148. Voigt, G. H. and Fuchs, K. (1979). A macroscopic model for field line interconnection between the magnetosphere and the interplanetary space. In Quantitative Modeling of Magnetospheric Processes, (Edited by W. P. Olson) pp. 448-459, Geophys. Monogr. Ser., vol. 21, AGU, Washington, D.C. Walker, R. J. (1976). An evaluation of recent quantitative magnetospheric magnetic field models. Rev. Geophys. Space Phys. 14, 411-427. Walker, R. J. (1979). Quantitative modeling of planetary magnetospheric magnetic fields. In Quantitative Modeling of Magnetospheric Processes. (Edited by W. P. Olson) pp. 9-34. Geophys. Monogr. Ser., vol. 21, AGU, Washington, D.C. Willis, D. M. (1978). The magnetopause, microstructure and interaction with magnetospheric plasma. J. Atmos. Ten'. Phys. 40, 301-322. Wyld, H. W. (1976). Mathematical Methods for Physics. pp. 96-99, W. A. Benjamin, Inc. Reading, Mass., USA.