Magnetospheric Configuration with Magnetotail Current

Magnetospheric configuration with magneto tail current is studied on a two-dimensional formulation. The planar magnetic field is represented by an analytic function of a complex variable that represents points of the. Noon-midnight plane. In our treatment, which accounts for both-the Chapman-Ferraro magnetopause current and the magnetotail current, the function that describes the ffl3gnetospheric magnetic field is a rational function. Thus, the magnetic neutral points, which characterize the magnetic topology of the field lines, can be simply ascertained by finding the roots of a polynomial function. The function for the magnetic field is the derivative off a complex potential, the real part of which is a flux function and the imaginary part a magnetic potential of the magnetic field. The inverse function of the complex potential describes the position variable in terms of the complex potential. Field lines are described by a parametric equation obtainable from the afore-mentioned inverse function. Thus, no integration off the differential equation for field lines is needed.


INTRODUCTION
The earth's magnetosphere is sustained by continual interaction between the earth's mag netic field and the solar wind.The interaction involves induction of magnetopause current and reconnection of geomagnetic 'and interplanetary field lines.These two features are accounted for in a recent modeling by Yeh (199 7) .The illustrated field-line configuration is three-dimen sional.It accounts for the Chapman-Ferraro magnetopause current by an image dipole placed beyond the dayside magnetopause (cf., Chapman, 1963).And it accounts for the field-line reconnection by a simple superposition of the two magnetic fields (Dungey, 1963).
""'The Chapman-Ferraro magnetopause current is caused by the solar wind's frontal im pingement (Chapman and Ferraro, 1931).In addition to the Chapman-Ferraro current the in-1 Institute of Space Science, National Central University, Chung-Li, Taiwan, ROC teraction between the solar wind and the earth also induces a magnetotail current, which is caused by lateral pinching of the bypassing flow of the solar wind.If is desirable to include the magnetotail current in the modeling of the magnetosphere.Inclusion of the magnetotail cur rent in a three-dimensional configuration is rather formidable.Instead, we may consider a two-dimensional configuration as a preliminary.The planar magnetic field in the two-dimen sional configuration amounts to the noon-midnight profile of the three-dimensional magneto sphere.
The mathematics for planar magnetic fields in current-free regions is facilitated by the usage of complex variables.The two-component magnetic field is to be represented by the derivative of a complex potential which is a function of a complex variable that represents points of the plane.The complex potential is made up of a flux function as its real part and a magnetic potential as its imaginary part.Such usages of analytic functions of a complex vari able were fruitfully explored by Dungey (1961) and Hurley (1961) in obtaining the magneto pause profile of a closed magnetosphere.However, their adoption of Ferraro (1960)' s ap proximation of specular reflection for the incident corpuscular particles as the pressure bal ance at the magnetopause led to the spurious elongation of the closed magnetosphere to infin ity (see Yeh, 1999).Such a spurious geometric openness also appear in Unti and Atkinson (1968)'s inclusion of a neutral sheet current in a two-dimensional closed magnetosphere.
In this paper we study the field-line configuration on the noon-midnight plane in a par tially open magnetosphere that includes the magnetotail current.This magnetospheric mag netic field is assumed to be due to four current sources.The part of magnetic field that is due to distant heliospheric currents is accounted for by a uniform magnetic field in southward direction.The part due to the earth's core current is accounted for by a magnetic dipole of southward moment.The part due to the magnetopause current is accounted for by an image dipole of southward moment, which is greater than the earth's dipole moment.The part due to the magnetotail current is accounted for by a line current in duskward direction for its cross tail current and an image line current in opposite direction for its return current.Each of these four partial magnetic fields has a known complex potential.The resultant complex potential as an analytic function of the complex variable for the position points has a multi-branch inverse function.Field lines are described by a parametric equation obtainable from the afore-men tioned inverse function.Thus, no integration of the diff erential equation for field lines is needed.The crux of the problem is to ascertain the magnetic neutral points thaL�haracterize the mag netic topology of the field lines.In our consideration the function that represents the magneto spheric magnetic field is arational function.Thus, the magnetic neutral points are simply given by the roots of a polynomial function .

PLANAR MAGNETIC FIELD
Generally speaking, a planar magnetic field lxBx+ lYBY in a plane (x, y) can be described by a flux function 'P(x, y).The relationship (1) assures the satisfaction of the magnetic solenoidality, viz., aB/ax+aB / ay=O.On the other hand, in a current-free region the magnetic field can also be described by a magnetic potential .Q(x, y).The relationship an an Bx=--, By=-ax ay (2) assures the satisfaction of the current-free condition, viz., oB /ox--aB /dy=O.It follows from equations (1) and (2) that 'P and .Q as functions of x and y satisfy Cauchy-Riemann equations o'P/ox=oQ/oy, o'P/oy=-dQ/ox.Hence, 'P and Q satisfy Laplace equation (a2tox2+02/oy2)'P=O, (o2!Ch2+o2/oy2).Q=0.Accordingly, the planar magnetic field can be represented by a complex function B(z)=B (x, y)+iB (x, y) of the complex variable z=x+iy.Indeed, a current-free planar y x magnetic field can be written B=-d<f>/dz in terms of a complex potential cJ.>(z)='P(x, y)+i.Q(x, y).Furthermore, cJ.>(z) is an analytic function of z.So is its negative-signed derivative B(z). ) / (z-z0)2 with a complex potential given by M 0 exp i(K+8 0 ) /(z-z0).Here µ 0 stands for the 2 2 magnetic permeability.

MAGNETOSPHERIC MAGNETIC FIELD IN NOON�MIDNIGHT PLANE
To study the magnetospheric magnetic field in the noon-midnight meridional plane, we choose the origin of the rectangular coordinates (x, y) to be at the earth's center, the x-axis sunward and the y-axis northward.A partially open magnetosphere in two-dimensional plane is represented by the magnetic field (3) The first term M0/z2 for a magnetic field due to a southward magnetic moment -1 Y M0 at the origin z=O accounts for the earth's magnetic field.The second term -B 1 for a southward mag The null points of this magnetospheric magnetic field are located at the positions that satisfy the sixth-degree algebraic equation (4) Two of the six roots have positive real parts.Their imaginary parts will be non-zero when neither B1 nor Ly is too large.The remaining four roots are two negative and two positive.The pair of complex-conjugate roots represent north/south neutral points on the dayside magneto pause.The two negative roots represent x-axis crossing points of two equatorial neutral lines in the _ tail region.The two positive roots are physically meaningless because they are null points beyond the subsolar point.In fact, the method of images restricts the validity of equa tion (3) to a domain not beyond the magnetopause.These features follow from the reasoning explained below.
Equation (4) has real-valued coefficients.Its complex-valued roots must be in conjugate pair.From the relationships between coefficients and roots the six roots of equation (4) have a positive sum of 2xc+XT+XT' and a positive product of M0xe(-XT)XT' /Br The positive sum means that at least one of the six roots has a posiive real part, and accordingly the positive product means that at least two of the six roots have positive real parts.The two roots that have positive real parts do have non-zero imaginary parts when both B1 and I T are zero.More details are revealed in the following two limiting cases.In the limit case of B1 ---7 0, two of the six roots become ±00 and hence the remaining four roots become xc(M0±iVMoMc)/(M0+Mc), x T , XT' when I T is zero.In the limit case of Mc ---7 M0 and xT +x T , -7x c , the sixth-degree equation becomes a cubic equation 4 2 B, is not too large, the sum is greater than xc? and the product is negative.Hence one of the three roots for (z_l_xc)2 is negative and two are positive.The negative root for (z_l_xc)2, vary-2 2 ing from ixe 2 to 0 when B1and1r increase from 0 to larger values that make B1+µ0IT/ n (1-xc-4 2 x T ) equal to 8M0/ xe 2 , yields a pair of complex-conjugate roots for z.One of the two positive roots for (z_l_xc)2, varying from +oo to lxe 2 when B1 and IT increase from 0 to +oo, yields a 2 4 negative and a positive roots for z.So does the other positive root for (z_l_xc)2, which varies 2 .
from (1-xc-xT)2to lxe 2 .The pair of complex-conjugate roots for z represent the north/south 2 4 neutral points.They move from 1-xc±ilxc to coalesce at lx c +iO.The former negative root for  (10) The two-branch function   n 0 to -oo and the south cusp-to-earth field line has n varying from positive -n 0 to +oo.Like wise, the parametric function z l<I>='I'o.+mdescribes the four field lines that emanate from or terminate at the crossing point.Two of them have negative n varying from 0 to -oo, the other two have positive n varying from 0 to +oo. Figure 2 shows the field lines for the obtained patially open magnetosphere with M/M0=10 and B1xe!M0=20.

A CLOSED MAGNETOSPHERE WITH MAGNETOTAIL CURRENT
Now we include the magnetotail current.In the absence of the interplanetary magnetic field equation (3) reduces to It describes a closed magnetosphere with magnetotail current.The null points of this magnetic field for a closed magnetosphere can be found from the quartic equation Figure 3 shows the field lines for the obtained closed magnetosphere with M J M0=10 and (µoIT/27t)x jM 0 =5, x/xc=-0.33,�.=+00.There is a magnetic island of field lines that encircle the magnetotail current.These isolated field lines do not link through the earth.

A PARTIALLY OPEN MAGNETOSPHERE WITH MAGNETOTAIL CURRENT
The most general case of our ultimate interest is a partially open magnetosphere with magnetotail current in the presence of the magnetotail current as well as the interplanetary magnetic field.
Without loss of generality we shall ignor the image current (viz., xr--7+00).The ignoring makes one of the physically meaningless roots of equation (4) recede to +00.The simplified magnetic field    In our study of a two-dimensional magnetosphere, the cross-tail current is represented by a line current.It is encircled by field lines, that do not link through the earth.These isolated field lines constitute a magnetic island.The existence of the magnetic island incurs a near earth neutral line in the magnetotail.The near-earth neutral line delineates the edge of the plasma sheet in the tail region.The neutral sheet of the magnetotail is nothing but a magnetic island in the form of a thin sheet.When the magnetic island is deformed into a neutral sheet, the cross-tail current must spread with suitable spatial distribution so that the isolated field lines are confined to a thin sheet.In the limit of zero width for the magnetic island, the isolate field lines appear as if they were non-existent.
Finally we remark that in our study we represent the Chapman-Ferraro magnetopause current by a magnetic dipole as an image current.The image dipole has a negative-order moment.So it is located outside the magnetopause on which the actual current resides.There fore, the obtained magnetic field is valid only in the interior region inside the closed surface formed by the entire magnetopause.The magnetopause in a closed magnetosphere is delin eated by the field lines that emanate from the south neutral point and terminated at the north neutral point (see Fig. 1 and Fig. 3).As for a partially open magnetosphere, the front part of the magnetopause is covered by field lines emanated from or terminated at neutral points.The rear part of the magnetopause is an equipotential surface of zero magnetic potential.
Since the inverse function of an analytic function of a complex variable is analytic too, the function z(cJ.>)inverse to <t>(z ) is an analytic function of <I>.The inverse function z(cJ.>) can be multi-valued.Discretion must be exercised to choose the proper branch of the multi-branched function z(cJ.>) to obtain the field lines or equipotential lines.The branch points occur at where the derivative dcJ.>/dz becomes zero.Thus, the values of cJ.> at magnetic null points are necessar ily branch points of the function z(cJ.>).Suppose f(z, <I>) is a known function that serves to define z in terms of cJ.> implicitly by the equation f(z, cJ.>)=0.Then, in addition to satisfying the equation f(z, <1>)=0 the branch-point values of <I> must also satisfy the equation of!dz=O in view that both of/oz+(of!a<l>)(d<l>/dz) and d<I>/dz vanish there.When the function f(z, <I>) is a polynomial in the first argument z, the branch-point values of cJ.> as determined by the simulta neous equations f(z, <l>)=O and of/oz=O will render the discriminator of the polynomial vanish ing.In this formulation on analytic functions of complex variable, a uniform magnetic field by a constant function B 0 exp i(� 8 0 ) with a complex potential given by a linear function -[B0 exp i(� 8 0 )] z.A magnetic field due to a line current lxXli0 located at the position z 0 is represented by a interplanetary magnetic field.The third term M/(z-zc)2 for a magnetic field due to an image dipole of southward magnetic moment -lyMc, with Mc>M0, placed at z=xc>O on the sunward side of the earth accounts for the Chapman-Ferraro magneto pause current.The fourth term for a magnetic field due to a duskward current -1,xl} T at z=x T <O on the tail ward side of the earth and the image of its return current lxxlyI T placed at z=xr>O on the sun ward side accounts for the magnetotail current.The choice of xr -Hoo will amount to ignoring the return part of the magnetotail current.

z
Fig.I.A closed magnetosphere without magnetotail current.MJM0 = 10.Sub solar and antisolar traces of magnetopause as well as cusp-to-dipole field lines are indicated by thick lines.
They are the north/south neutral points on the dayside magnetopause of the closed points x0±iy 0. The inverse function z(<I>) can be obtained explicitly from the quadratic equation <l>z2-(Mo+Mc + x c<l>)z + Moxc = 0.
at <l>='P0±iQ0• There the quadratic discriminator, which is a quadratic in <I>, vanishes.The two-branch parametric function zl<l>= 'l' o+i n with Q varying from positive -Q.0 to negative Q.0 describes the separatrix field lines, which constitute the noon-midnight meridi onal trace of the entire magnetopause of the closed magnetosphere.The subsolar separatrix field line is given by one branch of the parametric function and the antisolar separatrix field line is given by another branch, both from the south neutral point to the north neutral point.The subsolar/antisolar points xc/(1±,JMcfMo)+iO are given by the two values of zl<I> = 'l' o+i O• with Q equal to 0. The two cusp-to-earth field lines are given by one of the two branches of the parametric function z l <P = 'l'o +i n .The north cusp-to-earth field line has negative Q varying from .Q0 to -oo and the south cusp-to-earth field line has positive Q varying from -Q.0 to +oo (seeYeh, 1999).
23)has five null points.They are located at the positions that satisfy the fifth-degree equationThe five roots have a positive product M0xJ(-XT)/Br Hence one of the roots is positive, being greater than x c .Once this physically meaningless positive root (or alternatively one of the negative roots for the crossing points of neutral lines) is numerically found, equation (24) can be factored to yield a quartic equation.The resulting quartic equation, which can be solved by Ferrari's quartic formula, has two complex-conjugate roots and two negative roots if neither B1 nor 1r is too large.The complex-conjugate roots x0±iy0 represent the north/south neutral points on the sunward magnetopause of the partially open magnetosphere.The two negative roots represent two crossing points of neutral lines in the tail region.The farther crossing point x0.+i0 is on the tail ward side of the line current whereas the near-earth crossing point x0 .. +iO is between the line current and the earth's dipole.The simplified complex potential <l?( z ) = Mo + B1z + Mc + µ 2 °1T log(z -XT).

Figure 4
at the north/south neutral points and the two crossing points of neutral lines.On the x-axis the flux function varies as MJx+B1x+Mc!(x-xc)+_l_µ o lT logQx-x1Mx-xT' I ).It 2 it attains maximal values at the two crossing points and becomes -oo at the cross-tail current line.The two crossing points will be connected by two field lines, that delineate a magnetic island, when the flux function has the same value at them.This requires a certain compatibility constraint shows the field lines for the obtained partially open magnetosphere with Mc/ M 0 =10, B1x(IM0=20 and ( µo ly/27t)xc/M0=5, x/xc=-0.33There is a magnetic island of iso lated field lines that encircle the magnetotail current.

Fig. 4 .
Fig.4.A partially open magnetosphere with magnetotail current.McfM0=10, BiXtfM0=20 and (µoh/2n)xcfM0=5, x/xc=-0.33.Traces of separatrix surfaces are indicated by thick lines.Dashed lines indicate where B vanx ishes and dotted lines where B vanishes.North/south neutral points and y crossing points of two equatorial neutral lines are at intersections of dashed and dotted lines.