Solar Wind Streamlines in the Magnetospheric Noon-Midnight Plane

The magnetosphere formed in the interaction between the magnetized solar wind and the earth's magnetic field has a magnetopause that sepa­ rates interplanetary and terrestrial plasmas. The self-closed magnetopause is covered by the solar wind streamlines that are closest to the earth. Solar wind streamlines are delineated by intersection curves of electric and mag­ netic equipotential surfaces. The front part of the magnetopause is an elec­ tric equipotential surface that interfaces between oppositely directed inter­ planetary and closed field lines. The rear part is a magnetic equipotential surface that has open field lines passing through. Thus the magnetopause current is intense on the former closedness part of the magnetopause, which is a tangential discontinuity, and insignificant on the latter openness part, which is a rotational discontinuity. The part of closedness magnetopause has a pair of T-type magnetic neutral points on the exterior/interior face of the current sheet. The part of openness magnetopause has an X-type mag­ netic neutral line, in the studied case of southward interplanetary field. The neutral line is manifested by its cross point in the noon-midnight plane in our two-dimensional elucidation. In the noon-midnight plane, which is an electric equipotential surface, solar wind streamlines are delineated by traces of magnetic equipotential surfaces. (


A B STRACT
The magnetosphere formed in the interaction between the magnetized solar wind and the earth's magnetic field has a magnetopause that sepa rates interplanetary and terrestrial plasmas.The self-closed magnetopause is covered by the solar wind streamlines that are closest to the earth.Solar

INTRODUCTION
Formation as well as sustainment of the earth's magnetosphere in the continual interac tion between the magnetized solar wind and the earth's magnetic field is a complicated mag netohydrodynamic process.It involves not only the diversion of the solar wind flow by the earth's magnetic field but also the reconnection of the interplanetary field lines with the terres-trial field lines in a quasi-steady manner.The result is a finite-volume cavity that excludes the solar wind from getting closer to the magnetized earth.This magnetospheric cavity is filled with terrestrial plasma.It constitutes the magnetosphere proper, the boundary surface of which is the magnetopause in entirety.Thus the magnetopause, interfacing between the interplan etary and terrestrial plasmas, is covered by the solar wind streamlines that come in contact with the terrestrial plasma.
Following Chapman and Ferraro's pioneering work, the early theories of the magneto sphere were built on variant models of a completely closed magnetosphere (see Chapman (a) (b) x .- The magnetopause of a partially open magnetosphere has a closedness part in the front and an openness part in the rear.The closedness magnetopause is an electric equipotential surface and the openness magnetopause is a magnetic equipotential surface.In this paper we shall educidate the above-mentioned magnetopause structure by a two-dimensional treatment.
The obtained results reveal the noon-midnight meridional features of a corresponding three dimensional magnetosphere.It should be noted that the diverted flow of the solar wind is inherently three-dimensional (see Fig. 1).Global continuity may require some of the segmen tal planar streamlines in the noon-midnight plane to be connected by off-plane streamlines.
The necessitated off-plane diversion is accounted for by sink and source of mass and flux flows in the two-dimensional context.

MAGNETOSPHERIC MAGNETIC FIELD
Consider a solar wind plasma that moves toward the earth with a velocity UJ =-lxU1 (1) in the sun-to-earth direction and carries an interplanetary magnetic field (2) in the southward direction.In the remote region of undisturbed solar wind, the interplanetary electric field ( 3 ) given by -u1XB1 is in the dawn-to-dusk direction.We use rectangular coordinates (x, z) for the noon-midnight plane, with the origin at the earth's center, the x-axis pointing sunward and the z-axis pointing northward.We shall also use polar coordinates (r, 9) with x=r sine, z=r cose, so the polar axis is along the z-axis, the noon plane has positive e and the midnight plane has negative e.The solar wind will interact with the earth's dipole field.In the noon-midnight plane the earth's magnetic field will be approximated by which is the magnetic field of a two-dimensional dipole with a southward moment of strength M o.
The current induced in the interaction produces a perturbation magnetic field, to be denoted B P , viz., B �x t) in the exterior region of interplanetary plasma and B �n t) in the interior region of terrestrial plasma.With this large perturbation to the quasi-uniform interplanetary field and dipolar terrestrial field the resultant magnetic field is the magnetospheric magnetic field.More precisely, the exterior field B( ext)=B1+B �e x t) +B0 is the magnetic field in the magnetosheath and the interior field B(in t )=B0+B gn t) +B1 is the magnetic field in the magnetosphere proper.The discontinuity of Br across the interface be tween the two regions accounts for the magnetopause current ) Here ln denotes the unit vector outwardly normal to the magnetopause and µ0 is the magnetic permeability.
In terms of currents, the magnetospheric magnetic field is attributable to three currents.
The earth's core current, which produces the dipole field B0, has a dipole moment at the origin with other negative-order moments (such as quadrupole and octupole moments) being insignificant.The distant heliospherical current, which sustains the quasi-uniform field B1, has a linear moment with other positive-order moments (such as quadratic and cubic moments) being insignificant.The induced magnetopause current, which produces the perturbation field B P , has magnetic moments of various orders, whatever spatial distribution it may have.Al though any current associated with field line reconnection is ignored, the occurrence of reconnection does affect B P .
In fact, open field lines that interconnect the earth and interplan etary space will show up in the superposition of the three constituent fields if the additive sum B1+B0is not cancelled out by a part of B P .
In the idealization that the induced magnetopause current is concentrated as a current sheet, the magnetospheric magnetic field is piecewise current-free.In the current-free (viz., VxB=O by Ampere's law in differential form) regions the magnetospheric magnetic field has a scalar magnetic potential .Q(x), which is related to the magnetic field by B=-V Q.The mag netic potential is discontinuous at the points where the current is not zero.This is readily seen because the line integral of V Q will not be zero if the contour of integration encloses a current (by Ampere's law in integral form).The dipole field B0 has the magnetic potential As to the induced field B P , its magnetic potential Q P is a piecewise harmonic function that vanishes at infinity and remains finite, say zero, at the origin.In the exterior region, Q P has the multipole expansion (9) with certain coefficients c_2, c _3, • • •, in terms of the two-dimensional harmonic functions of negative-order moment: On the other hand, in the interior region Q P has the Taylor expansion (10) with matching coefficients c1, c2, ••• , in terms of the two-dimensional harmonic functions of positive-order moment: roO) = -z =r cos 0, m(2) = -xz = -r2 sine cose,' .. .Note that ro(-2) represents a positive (viz., northward) dipole, ro(-3 ) a positive quardrupole, co< 1) a positive (viz., northward) constant field, and m(2) a field with positive constant gradient.
The crux of the problem is the induced magnetic field.Neither the geometric position of the magnetopause nor the spatial distribution of the magnetopause current is amenable to simple algorithms of calculation that involve considerations of force balance and flux continuity.
Nevertheless, physically meaningful representations for B P can be d D vised by suitable pre scriptions based on consideration of flux continuity alone.With Q� x t prescribed, by a suit able choice of the harmonic coefficients c_2, c_3, • • •, the topological structure ofB(e xt ) is readily ascertained by the neutral points and neutral line it contains.The magnetopause is then to be identified with a self-closed surface, in the external magnetic field B (e x t ) , that encloses the origin.The matching coefficients c1, c2, ••.,for the internal magnetic field B (i nt ) is to be determined by the continuity of magnetic flux through the magnetopause.Magnetic field lines are to be calculated by integration of the differential equation dx/d't=B(x), with 't as a variable that specifies constituent points of a field line.Alternatively, we may prescribe the geometry of the magnetopause, both its closedness and openness parts.Then the exterior magnetic po- ) at the origin as far as the ex terior magnetic field is concerned and is equivalent to a current at infinity that sustains a northward constant field of strength B 1 +MJR 2 as far as the interior magnetic field is concerned.As the radius R varies from 0 to oo, the dipole moment M 0 +B 1R2 increases from M0 to+= and the linear moment B1+M0/R2decreases from +oo to Br The choice of R=(M 0 /B 1) 1 12will yield the prototype closed magnetosphere given by B ( e x t ) =B1-B0 and B (i n t) =B0-Br The magnetopause current is 1,xlx4µ 0 -1 B1 sin0.
On the other hand, if a circle of radius R centered at the origin is prescribed as the magne topause in full openness, the solution of the Dirichlet-Neumann problem is which has a value equal to zero on the whole circle r=R.Thus Q�x t) =(M0-B1R2)r l cos0.The magnetic field ( 18) vanishes at r=R, 0=±tn.Since an,( e xt);ar has the value of 2B 1 cos0 on the circular magneto pause in full openness, the solution for the matching Neumann problem is Thus Q� nt) =(B1-MJR2)rcos 0. The corresponding magnetic field ( 1 9) has a poloidal component in addition to the required radial component on the circle r=R.The discontinuity in the poloidal magnetic field incurs a magnetopause current, which can be rather small.It follows from B �ex t) =(-l+B1R2/M0)B0 and B � n t ) =(l-M0/B1R2)B1 that the magnetopause current iP=l,xl,µ 0 -1 (2MJR2-2B1)sin8 has an internal image, for the exterior magnetic field, that consists of a dipole of moment 1,(M0-B1R2) at the origin and an external image, for the interior magnetic field, at infinity that sustains a constant field 1,(MJR2-B1).As the radius R varies from 0 to oo, the dipole moment M0-B1R2 decreases from M0 to -oo and the linear moment MJR2-B1decreases from+= to-Br The choice of R = (MJB1) 1 12 will yield the proto type open magnetosphere given by B=B1+B0 everywhere.The magnetopause current happens to be zero.

MAGNETOSPHERIC ELECTRIC FIELD
The steady-state electric field E( x) in the magnetosphere is curl-free (viz., VxE=O by Faraday's law).Thus, it has an electric potential <P( x), which is related to the electric field by E=-V <P.In the remote region of the xz-plane Eis in the direction perpendicular to the plane.
There <P has the same asymptotic value, say zero.
With a perfect electrical conductivity for space plasma the sum E+uxB, which signifies the electric field in a local coordinate frame that moves with the plasma's flow velocity u, is zero.Namely, E =-uxB. ( By virtue of B• E=O, the electric field has no component aligned with the magnetic field.It follows from (B• V)<P=O that the electric potential is invariant on individual magnetic field lines.Hence the electric potential remains to have the value of zero on all magnetic field lines, in the noon-midnight plane, that map from infinity.So the part of the xz-plane in the exterior region is an electric equipotential surface.There the electric field is perpendicular to the xz plane unless B becomes zero to nullify the direction of E.

DRIFT MOTION OF SOLAR WIND PLASMA
The bulk motion of a plasma in a magnetic field can be described by a flow velocity

MAGNETOPAUSE
The earth's magnetic field deflects the oncoming solar wind.So the interplanetary plasma, carrying the frozen-in interplanetary magnetic field lines, flows around the earth at some distances.The various points of closest approach to the earth constitute a self-closed surface that encloses all terrestrial plasma.This surface, separating interplanetary and terrestrial plasmas, is the magnetopause in entirety.In other words, the magnetopause comprises points where the interplanetary plasma makes contact with the terrestrial plasma.These contact points of inter planetary plasma are grouped into segments of solar wind streamlines.Some of the streamline segments are on an electric equipotential surface.Others are on a magnetic equipotential surface.
The former solar wind streamlines cover the exterior face of the current sheet which is sand wiched by interplanetary and terrestrial field lines.The latter solar wind streamlines run from a site of field line reconnection to another site.Open field lines are formed by reconnection between interplanetary and closed field lines approaching to each other.At subsequent reconnection between opposite segments of open field lines interplanetary and closed field lines are reverted back.
The part of magnetopause with an invariant electric potential, passed through by electric field lines, is also covered by magnetic field lines in perpendicularity to the streamlines.On the closedness magnetopause the solar wind streamlines carry interplanetary field lines that slip on the exterior face of the magnetopause current sheet.On the other hand, the part of magnetopause with an invariant magnetic potential, passed through by magnetic field lines, is also covered by electric field lines in perpendicularity to the streamlines.On the openness magnetopause the solar wind streamlines carry open field lines that traverse the magnetopause.
When the openness part of the magnetopause is non-existent (as if shrinking to the antisolar point) so the whole magnetopause is in closedness, the magnetosphere is completely closed.In the other limit, when the closedness part of the magnetopause is non-existent (as if shrinking to the subsolar point) so the whole magnetopause is in openness, the magnetosphere is fully open.

PROTOTYPE CLOSED MAGNETOSPHERE
Before we deal with partially open magnetospheres, it is instructive to examine two pro totype magnetospheres.One of them is completely closed, with .Q P( e xt J =-200 and n Jm t ) = -2.Qr The other is fully open, with .Q P =O.Both of them have a circular magnetopause, as the xz-trace of a corresponding spherical magnetopause in three-dimensional setting.
First, we consider the prototype closed magnetosphere.It has the magnetic potential Q ( e x t) = (B1r+ 1:°) cose,

PROTOTYPE OPEN MAGNETOSPHERE
Next, we consider the prototype open magnetosphere.It has the magnetic potential in both the exterior and interior regions.This magnetic potential has a value of zero on the circle r=(M0/B/12 as well as on the equatorial line z=O.The circle, with magnetic flux passing through, will be shown the magnetopause.It is a magnetic equipotential curve that encloses the earth at the origin.The magnetic field is symmetric about the z-axis.It vanishes on the intersection of the circular and straight lines of zero magnetic potential.The two intersection points r=(M0/B1)112, 0=±tn are the xz-trace of a magnetic neutral line in the three-dimensional setting.The flux function is The whole plane is partitioned by a two-branch separatrix (Bl+M0/r)sin0=2B/12M0112 into three magnetic cells, which are filled with closed, open, or detached interplanetary field lines, respectively.The three cells make contact at the cross points of magnetic neutral line, which are X-type in local field line topology.The magnetic field is radial, equal to -lr2B1cos0, at the magnetopause, through which open field lines pass.The magnetopause current happens to be zero because BP is zero.There is no electric equipotential curve enclosing the origin.In these configurations the circular magnetopause has a geometry of high symmetry.So the exterior and interior harmonic functions due to the magnetopause current are simple terms with a magnetic moment of the lowest order.The coefficient c _2, equal to M 0 +B�2 in the case of closed magnetosphere or equal to M0-B1R2 in the case of open magnetosphere, for the dipole moment ranges between -oo and +oo when the free parameter R for the radius of circle varies from 0 to oo.And the coefficient c1, equal to B1+M0/R2 in the case of closed magneto sphere or equal to--B1+M0/R2 in the case of open magnetosphere, for the linear moment ranges between -B 1 and +oo when the free parameter R varies from 0 to oo.Accordingly, if c 1 is less than --B I ' the magnetopause can not be circular in either complete closedness or full openness.
In that case, the harmonic function due to the magnetopause current must have additional terms of higher order and the magnetopause is non-circular.
To examine closed or open magnetospheres with a non-circular magnetopause, we con sider upon the use of 1, •B ( ext) being zero.The curve of zero electric potential, formed by the mag netic field lines that emanate from the north neutral point and terminate at the south neutral point, will be self-closed to enclose the origin if c_2 is greater than M0 and c_ 3 has a magnitude not too large.This self-closed curve of zero electric potential is a non-circular magnetopause of a completely closed magnetosphere.
Otherwise, if c_ 3 has a sufficiently large magnitude, neither the curve of zero magnetic potential nor the curve of zero electric potential will be self-closed to enclose the origin.So the magnetopause is neither in full openness nor in complete closedness.In other words, the magnetosphere has to be partially open.

PARTIALLY OPEN MAGNETOSPHERES
The magnetopause of a partially open magnetosphere comprises a part with an invariant electric equipotential in the front and a part with an invariant magnetic equipotential in the rear.The front part is formed by the field lines that emanate from the north magnetic neutral point and/or terminate at the south magnetic neutral point.The rear part contains points of magnetic neutral line.The front and rear parts seamed together enclose the origin.An example of partially open magnetosphere is one having a circular magnetopause with its front half in closedness and its rear half in openness.
To see the features of a partially open magnetosphere with a non-circular magnetopause we shall examine the harmonic expansion (34) and its magnetic field ----:.: :_ ::j
3. -1 -1.5 L----'------'------' '-----'------'------'------' -1.5 -1 -.5  induced in the interaction.So are the size and shape of the magnetopause, depending on the induced current.For a magnetosphere to be completely closed, the magnetopause current must have a positive (viz., northward) dipole moment large enough to overpower the earth's nega tive (viz., southward) dipole moment and also have a positive linear moment large enough to overpower the negative linear moment of the interplanetary magnetic field.The larger the magnetopause current, the smaller the size of the closedness magnetopause will be.On the other hand, for a magnetosphere to be fully open, the magnetopause current can not have a positive dipole moment too large as to overpower the earth's negative dipole moment.An eastward magnetopause current will make the size of the openness magnetopause large whereas a westward magnetopause current will make it small.
wind streamlines are delineated by intersection curves of electric and mag netic equipotential surfaces.The front part of the magnetopause is an elec tric equipotential surface that interfaces between oppositely directed inter planetary and closed field lines.The rear part is a magnetic equipotential surface that has open field lines passing through.Thus the magnetopause current is intense on the former closedness part of the magnetopause, which is a tangential discontinuity, and insignificant on the latter openness part, which i s a rotational discontinuity.The part of closedness magnetopause has a pair of T-type magnetic neutral points on the exterior/interior face of the current sheet.The part of openness magnetopause has an X-type mag netic neutral line, in the studied case of southward interplanetary field.The neutral line is manifested by its cross point in the noon-midnight plane in our two-dimensional elucidation.In the noon-midnight plane, which is an electric equipotential surface, solar wind streamlines are delineated by traces of magnetic equipotential surfaces.(Key words: Solar wind streamlines, Magnetopause current, Closedness magnetopause, Openness magnetopause, Partially open magnetosphere)

Fig. 1 .
Fig. 1.Solar wind streamlines around a spherical magnetopause (a) in a closed magnetosphere and (b) in an open magnetosphere.Solid lines are mag netic field lines and dotted lines are electric field lines.Dashed lines are streamlines.

._
µ �lJ X (B (e xt) B(int) can be calculated by �olv�ng a �aplace e3uation sub1ect to the boundary condition that g (ext ) has• a zero normal denvative (viz., an< e xt /an=O) on the prescribed closedness magnetopause and an invariant value (say, ,n<ext)=O) on the pre scribed openness magnetopause.Correspondingly, the interior magnetic potential .Q (int>, which becomes n0 near the origi � , can be calculated by solving a Laplace equation subject to the boundary condition that ,n<m t) has a normal derivative equal to the normal derivative of g (ext) (viz., a,n ( int)/an=an<e x t ) /an) on the whole magnetopause.In other words, the magnetic poten tial due to the ) magnetopause current can be determined by solving a Dirichlet-Neumann proband then solvrng a Neumann problem for .Qp :For instance, the prescription of c_2=M0+B1R2, c_3=0, C_/=0, ... for .Q� xt ) =(M0+B1R2)r 1 case renders ,n <ext) = B1(r + � 2 ) cos8,(13)which has a radial derivative equal to zero on the whole circle r== R. The corresponding mag netic field (14) has a pair of neutral points at r=R, 0=(!±t)rc.The boundary condition on that circle as a mag netopause in complete closedness is a.Q� n t) /ar=-(B1+Ma1R2)cos0.The solution for the match ing Neumann problem is .Q�n t) =-(B1+Ma1R2)rcos 8. Namely, c1=B1+Ma1R2, c 2 =0, C3=0, u=u ..L +u1 1 that comprises a cross-field drift velocity and a field-aligned slide velocity .It is the drift motion that carries frozen-in magnetic field lines to cross magnetic separatrices in enter ing or exiting magnetic cells of different field line connectivity.Thus, we may neglect the slide motion, which passively accommodates the continuity of mass flow.The drift velocityExBu ..L = B•B (22)is perpendicular to both the electric and the magnetic fields.The perpendicularity means that the drift velocity vector lies on an electric equipotential surface and also on a magnetic equipo tential surface.Hence, various intersection curves of electric and magnetic equipotential sur faces are streamlines of drift motion.Moreover, each electric equipotential surface is covered by various streamlines, so is each magnetic equipotential surface.Accordingly, the xz-traces of various magnetic equipotential surfaces are streamlines in the noon-midnight plane of zero electric potential.More important are the solar wind streamlines on the magnetopause.The streamlines with zero magnetic potential constitute the openness magnetopause.As to the streamlines on the closedness magnetopause, they are non-planar curves of zero electric potential.These streamlines with zero electric potential have xz-trace points.The trace points appear as sink or source of mass and flux flows in our two-dimensional treatment.
Both exterior and interior magnetic potentials have a normal derivative of zero on the circle r= (M0 / B1)112• The circle, without magnetic flux passing through, will be shown the magnetopause.It is an electric equipotential curve that encloses the earth at the origin.The magnetic field (25) B(int) = tr(-Mo + B1) cose + le(-Mo -B1) sine r2 r2 (26) is symmetric about the z-axis.The exterior magnetic field has a pair of neutral points at z = ±(M0 / B1)112 of the z-axis on the exterior face of the current sheet, which is covered by semi circular magnetic field lines that emanate from the north neutral point and terminate at the south neutral point.The interior magnetic field has a pair of neutral points at the pole points z=±(MiB/12 of the z-axis on the interior face of the current sheet, which is covered by semi circular magnetic field lines that emanate from the south neutral point and terminate at the north neutral point .These magnetic field lines have the same electric potential as the neutral points.The magnetic neutral points on the exterior and interior faces of current sheet are T type in local field-line topology.The flux function \f'(ext) = ( -B1r+ 1:°) sine, '¥(int) = (-Mo + B1 r) integral Jo B/r, e)r de for the magnetic flux is invariant on individual magnetic field lines.The two semi-circular field lines, emanating from the north neutral point and termi nate at the south neutral point, have an electric potential of zero.They constitute a self-closed curve that encloses the origin.The whole plane is partitioned by the magnetopause into two magnetic cells of interplanetary or closed field lines.There is no cell of open field lines.The two cells share the magnetopause as a common wall, which is a current sheet.The magneto pause current, being sandwiched by _!} (ex t ) =102B1sin0 on the exterior face and B ( i n t) =-l02B1sin0 on the interior face, is ip"=l,xlx4µ0 B1sin0.There is no magnetic equipotential curve enclos ing the origin.

Figure 2
Figure 2 shows the flow of solar wind that passes around the magnetopause.The solar wind streamlines, delineated by magnetic equipotential lines [1 +M0/B/x2+z2)]z=z=, are orthogonal to the magnetic field lines [1-MiB/x2+z2)]x=X00• The solar wind does not flow across the circle of zero electric potential at all.This electric equipotential curve interfaces between interplanetary and terrestrial plasmas.It constitutes the magnetopause of the completely closed magnetosphere.Upon incidence on the magnetopause in the noon plane the solar wind stream- Fig. 2. Proto � pe closed magneto � phere with circular magnetopause, represented by .Q e xt ) =Q1-Q0 and ,Q(m t )=Q0-Qr Solid lines are magnetic field lines.Dashed lines are solar wind streamlines.

Figure 3
Figure 3 shows the flow of solar wind that passes around the magnetopause.The solar wind streamlines, delineated by magnetic equipotential lines [l-M0/B 1 (x2+z2)]z=z"", are orthogonal to the magnetic field lines [1 +M0/B/x2+z2)]x=X"".The solar wind does not flow across the circle of zero magneic potential at all.This magnetic equipotential curve interfaces between interplanetary and terrestrial plasmas.It constitutes the magnetopause of the fully open magnetosphere.Upon incidence at the subsolar point x=(M0/B1)112 on the positive x-axis the solar wind streamlines split to leave the equatorial line.The diverted streamlines, staying on the magnetopause in the noon-midnight plane, become confluent to leave the magneto-

IA
southward uniform field immersing a positive quadrupole, represented by Q ( ex t) =B1z+c_ 3 xz/(x2+z2)2.Thick solid line is curve of zero electric potential and thick dashed line is curve of z ero magnetic potential.Solid lines are magnetic field lines.Dashed lines are solar wind streamlines.

Fig. 7 .
Fig. 7.A southward dipole immersed in a non-uniform magnetic field, repre sented by g(int)::: -M0z/(x 2 +z 2 )-c2xz.Thick solid line is curve of zero electric equipotential and thick dashed line is curve of zero magnetic potential.Solid lines are magnetic field lines.Dashed lines are solar wind streamlines.