\documentstyle [12pt] {article}

\begin{document}
\title{
 Axisymmetric Magnetic Fields : Geometry and Magnitude } 
\author{ Olga Koutchmy \\
        Universit\'e P. et M. Curie \\
        Laboratoire d'Analyse Num\'erique, CNRS \\
        Tour 55-65, 5e etage \\
        4, Place Jussieu, 75252 Paris Cedex 05 (France) \\
        fax: 33 1 44 27 72 00 \\
        e-mail: koutchmy@ann.jussieu.fr \\
   \and
         Serge Koutchmy \\
        Institut d'Astrophysique, CNRS \\
        98 Bis, Bd Arago, 75014 Paris (France) \\
        fax: 33 1 44 32 80 00                 \\
        e/mail: skoutchmy@solar.stanford.edu }
\maketitle

\begin{abstract}
 Convenient formulae describing  the vector magnetic field in the simple case  
of axisymmetric configurations ($2.5$ D problem) are given. We first consider the 
classical problem of the coil to test the precision of our calculations.
 Then we compute both the direction 
( magnetic field lines ) and especially the strength  of the magnetic field for the most
representative cases. 

To allow a global evaluation of the results,
 maps with both field lines and lines of equal strength are proposed.  
 Ring currents are considered 
using their moments to reduce the number of free parameters. Graphes are 
given to allow a quick evaluation of the behavior of the magnetic field in
cases which are beleived  to be of astrophysical interest.  
\end{abstract}

\section{Introduction}
The topology and the strength of the magnetic field is of prime importance to guide in the
understanding of the equilibrium state as well as the dynamics of astrophysical 
plasmas. In a laboratory MHD set-up the same is true but usually the effects  
produced by the gravity are negligeable.There electric currents are used to  
control the magnetic field and, eventually, the resulting magnetic (Amp\`ere) force. 

 In solar physics magnetic phenomena are the main actors of the stellar atmosphere
 activity;
unfortunately, no direct measurements of electric currents are yet possible, 
although the field is well measured at the surface. Therefore extrapolations have been 
largely used to guess its topology above the surface, where currents are 
significantly smaller due to the exponential decrease of the plasma 
density. Currents under or at the surface are obviously needed to produce and 
 maintain the field in a quasi- stationnary state, 
following the Maxwell's equations. They cannot be neglected above the surface in case 
when the interactions are important, like the case of resistive dissipations or 
when a twist of the field is involved. At the surface, they are
mainly produced by the differential motions of charge particles of the "neutral"
plasma, due to flows. Alfven, for example, \cite{kn:alf1,kn:alf2} stressed at many occasions that in
the field of MHD, it is much preferable to think in terms of currents, rather then in terms
of magnetic field lines.

 A safe way to consider the extrapolations of the field is to start from a current, 
compute the field at several levels, including the level where observations 
provide measurements, and then compare and discuss the implications on the surrounding plasma by 
considering the pressure balance (the so called magneto-static case). Due to 
the gravity stratification, currents are beleived to be mainly horizontal; then the 
field at the surface is mainly vertical in case when the  currents are not too far from 
the surface. This is well observed in the umbra of sunspot, which is the structure of 
the solar surface showing the largest concentration of magnetic flux. 

 To roughly consider the magneto-static equilibrium problem of solar features
  called 
flux tubes, large assumptions have been introduced starting with the so called 
'thin-flux-tube' approximation. Transverse components of the field are then neglected, 
as well as the variations of the strength of the field over the diameter of the 
tube and finally the field is zero outside. The Maxwell's equations cannot be 
satisfied and obviously there is no need to talk about currents. This approximation 
is very usefull to look at the thermo-dynamical and radiative transfert problems 
but the following step, which takes into account {\em variation} of the field, is not 
less interesting. This would allow to evaluate effects due to the Amp\`ere force
$\vec J \times \vec B$ 
which is often considered equivalent to the effect of the gradient of an 
isotropic magnetic pressure  minus the effect of a tension force 
 $(\vec B \vec \nabla) \vec B$, see 
a good discussion of this decomposition in the classical textbooks
\cite{kn:lan,kn:pri,kn:lor}. 
Accordingly, knowing the topology of the field and/or its curvature and, 
simultaneously, the magnitude or strength  of the field, we should be capable at a first glance 
to discuss the equilibrium of the configuration and compare it with observed 
phenomena. This is the aim of this elementary computer simulation for  the  case of ring 
currents. We also beleive it includes the case of the most simple magnetic plasmoid.   
We then assume an axisymmetric configuration which allows the closure of 
electric currents and can still be treated based on existing analytic solutions
\cite{kn:lan}.  

 We notice that the topology of the magnetic field  was already
investigated  in solar case \cite{kn:sak} using a "composite" solenoid model; however the magnetostatic
equilibrium has been mainly considered only in the most simple case of a "thin" flux
tube \cite{kn:ste}. Removing this condition has been also attempted using an empirical description
of the topology of the magnetic field \cite{kn:piz}.


\section {The Ring Current Case} 

\subsection {Basic formulae}

Following \cite{kn:lan} we use the practical Gauss system of units to write the basic formulae and the solutions
used to describe the magnetic field. Formulae written in the officially recommended system MKS can be
found in the textbooks \cite{kn:pri,kn:lor}. Then we write the Maxwell's equation in the form:
\begin{eqnarray}
    rot \vec B & = { 4 \pi \over c }  \vec j
                div \vec B & = 0 \\
\end{eqnarray}
with $\vec B = ( B_x, B_y, B_z ) $ the intensity of the magnetic field, 
 $\vec j$ the density of the electric current and $c $ a constant (velocity of light).

When
$| \vec B |$ is expressed in Gauss, it is the field induced by a current of $5$ amp\`eres
intensity seen at a distance of $1$ cm.

To find $\vec B$ the usual method consists of introducing a vector potential $\vec A$ such that:
\begin{equation}
                \vec B = rot\ \vec A
\end{equation}
which is always possible provided the condition $(2)$ is fulfilled. To have the uniqueness of 
$\vec A$ an additional condition is introduced:
\begin{equation}
               div\ \vec A = 0
\end{equation}
Then the vector potential $\vec A$ can everywhere be deduced from the Poisson's equation:
\begin{equation}
                \Delta \vec A = - { 4 \pi \over c }\ \vec j 
\end{equation}
obtained from from $( 1 )$ by using $(3)$ and
because of \[ rot\ rot\ \vec A = \nabla\ div\ \vec A - \Delta \vec A = - \Delta \vec A \]
The solution of $(5)$ which vanishes at infinity is known:
\begin{equation}
                \vec A = {1 \over c} \int {\vec j\ dv \over R },\quad
 dv = dx dy dz
\end{equation}
where $R$ is the distance from a point where $\vec A$ is computed to un elementary volume $dv$.
Instead of considering a distribution of currents $\vec j$ over a volume, it is more convenient
to use a substitute which is a "linear" current $J$ such that :
 \[ \vec j \ d v \rightarrow  J \ \vec {d l} \]
where $J$ is the total current flowing inside the conductor, then :
\begin{equation}
                \vec A = { J \over c} \int {\vec {d l} \over R }
\end{equation}
We consider first the ring currents of radius $a$. The cylindrical system of coordinates 
 $( \rho, \phi, z )$ is naturally used to write $\vec B$ in this axisymmetric problem. The vector
potential  has then only one component $A_\phi = A( \rho,z)$ and following $(7)$
\begin{equation}
                A_\phi = { J \over c } \oint { {cos \phi \ dl} \over R } = 
{ 2 J \over c } \int_0^\pi { {a\ cos \phi\ d\phi} \over { (a^2 +\rho^2 +z^2 -2 a\ \rho\ cos \phi
)^{1/2} } }
\end{equation}
Analytical solutions to compute $A_\phi$ are given in \cite{kn:lan}:
\begin{equation}
                A_\phi = { 4 J \over c } {1 \over k}  {\left( a \over \rho \right) }^{1/2} \left\lbrack
 \left( 1 - {k^2 \over
2} \right) K(k) - E(k)\right \rbrack 
\end{equation}
with  $k^2 = { 4 a \ \rho \ {( ( a + \rho)^2 + z^2 )}^{-1}},\  K \hbox{\ and } E$ the
  complete elliptic integrals of the first and  the second kind respectively.

Usually the topology of the magnetic field is described using magnetic field lines.
The magnetic field vector is everywhere tangent to field lines, so that for a point
 $M(\rho,\phi,z)$ on a field line   we have:
\begin{equation}
                {d \rho \over B_\rho} =  {\rho\ d \phi \over B_\phi} = {d z \over B_z} 
\end{equation}
with:
\begin{eqnarray}
                B_\rho &= { {1 \over \rho } {\partial A_z \over \partial
\phi } - {\partial A_\phi \over \partial z}} \nonumber \\
                B_\phi &= { {\partial A_\rho \over \partial z}
 - {\partial A_z \over \partial \rho}} \\
                B_z &= {{1 \over \rho}   {\partial \over \partial \rho }{( \rho\ A_\phi )}
   - {1 \over \rho} {\partial A_\rho \over \partial \phi}} \nonumber \\
\end{eqnarray}
Taking into account the axial symmetry we obtain:
\begin{equation}
                \rho\ A_\phi ( \rho, z ) = const
\end{equation}
which represents indeed a magnetic surface with the same symmetry as $\vec B$.

From $(9)$ we deduce:
\begin{equation}
                \rho \ A_\phi (\rho, z) =
{ 4 J \over c } {1 \over k}  {( a \ \rho ) }^{1/2} \left\lbrack \left( 1 - {k^2 \over
2} \right) K(k) - E(k)\right \rbrack = const
\end{equation}

A development in serie proposed in \cite{kn:rad}  for computing $K$ and $E$
 was first tested.

Close to the  conductor, $k \simeq 1$. Keeping the first $2$ terms of the development when $1/2 < k < 1$,
the magnetic surfaces can be described  with:
\begin{equation}
                \rho \ A_\phi ( \rho, z ) \simeq {2 J \ a \over c}\ \left( \ln ({ 8 a \over r}) -
2\right) = const, \quad r^2 = ( a - \rho  )^2 + z^2
\end{equation}

Far enough from the current, the well known dipolar approximation can easily be recovered.
Keeping  the first $2$ terms of development when $0 < k < 1/2$,
the magnetic surfaces are described there with:
\begin{equation} 
                \rho \ A_\phi (\rho, z) = { \pi J \over c}   { a^2 \rho^2 \over {( a^2 + \rho^2 )^{3/2}}} =
const
\end{equation}

The last convenient formulae gives the potential close to the symmetry axis, for $\rho \ll a$
\begin{equation}
  A_\phi ( \rho,z ) = {\pi \ J \over c} {a^2 \ \rho \over \lbrack (a + \rho)^2 + z^2 \rbrack 
^{3/2} } 
\end{equation}

 We found indeed more practical to use
the approximate polynomial formulae given in \cite{kn:abr}; in the appendix we give useful numerical
expressions which were finally introduced in our calculations.

It is important to realize that good approximations should be used in order to avoid problems
appearing when several currents are used as well as when computing the field close to the currents, see
further in $2.2.$

The components of $\vec B$ are given in \cite{kn:lan} in a general form:
\begin{eqnarray} 
B_\phi &= 0 \nonumber \\
B_\rho &=  {2 J \over c}\ { z \over {\rho
 \ \sqrt {(a+\rho)^2 + z^2} }}
\left [ -K( k ) +{{a^2 + \rho^2 + z^2} \over {(a-\rho)^2 +z^2} }\ E( k )\right ] \\
B_z &=  {2 J \over c}\
 {1  \over { \sqrt {(a+\rho)^2 + z^2} } }
\left [ K( k ) +{{a^2 - \rho^2 - z^2} \over {(a-\rho)^2 +z^2} }\ E( k )\right ] \nonumber 
\end{eqnarray}

$K$ and $E$ are computed using the formulae given in the appendix.

On the symmetry axis  $z$ where $\rho = 0$ we found:
\begin{eqnarray}
B_\phi &= 0 \nonumber \\
B_\rho &= 0 \\
B_z    &= {2 \pi \ J \ a^2 \over {c\ (a^2 + z^2)^{3/2} }} \ \sim z^{-3} \ \ \hbox{ for } z \gg a 
\end{eqnarray}

\subsection{ Test of Numerical Results}
Before considering in more details the solutions, it is important to test the precision of our program
using the classical case of a coil of ring currents. This problem has been considered in the past
with insufficient numerical precision \cite{kn:alt}, which of course leads to questionable results.
Although correct analytic formulae were used in this paper, the numerical procedure to compute
the complete elliptic integrals and avoid the singularity close to the ring current was too approximate,
so artifacts resulted. We recomputed the configuration proposed in  \cite{kn:alt}, with our formulae, see
Fig.$1$,  using exactly the same geometric parameters.

This configuration results from the superposition of $11$ equally spaced ring currents
supposed to match the system producing a solar sunspot \cite{kn:sak} .
 Field lines  are shown in dotted lines; lines of
equal strength $| \vec B | = const$ are shown in solid lines. Because of the symmetry about the 
vertical axis, we show only half of the set of curves in a cut through the symmetry axis. 
As far as the distribution of field lines  is concerned, an excellent agreement is seen with the
classical pattern given in \cite{kn:lor} . It is difficult to find in a textbook the distribution of the
strength of $B$. To evaluate more carefully our solution, we show on Fig.$2$ the distribution of 
 $| \vec B | $ close to the ring currents. Comparing with the solution shown on Fig.$1$ of
\cite{kn:alt}, we immediately see the difference resulting from the use of a better resolution: no
anomaly in the form of magnetically isolated tores appear in our solution. We beleive they were
artifacts of calculations in \cite{kn:alt}. In our case artifacts seems to appear at distances $d_a$
 from the
neutral lines situated between individual currents of order of $1/3$ of that distance which is a half
value of the distance $d_e$ of separation between currents. As $d_c = r_c /10$ with $r_c$ beeing the
radius of individual currents, the critical distance is then $d_a \simeq r_c /60$, which corresponds to
the resolution of our calculation. Further we will avoid discussions of phenomena at such small distance
from currents.

A last point should be mentionned. Although we first did our computations of magnetic fields using the
system "Mathematica", we found more suitable and especially more easy to do it in the frame of the "IDL" (
Interactive Data Language ), without loosing any precision. All graphs presented here were obtained with the "IDL".

\subsection{Magnetic Structure of a ring current}

It is of interest to discuss the most simple case of a ring current using our solution which seems
precise enough and does not produce artifacts. On Fig.$3a$ and $3b$ we show these solutions at $2$
different scales. Again only a quadrant is shown with both field lines  and lines of 
 $| \vec B | = const$ . It is certainly rather naive to think that such model is close to real
situation met in cosmic plasma; however we found this "zero order" approximation rather usefull to
understand the most simple features of an isolated magnetic system, starting from a sunspot: the main
directions of magnetic lines, the weak values of reversed field strength outside the spot, the large
values of the magnetic pressure inside and above the ring current beleived to flow under the photosphere
at a depth of order of its radius. A more detailed comparison with observed values is outside the scope
of this paper and will be presented in a forthcoming paper. Noticeable on Fig.$3b$ is the behavior of
the field at large distances, similar to a dipolar field, with an increasing with distance  "scale
height" of variation of   $| \vec B |$.

\section{The Double Ring Current Case} 
We now consider the more general case of a configuration occuring with $2$ coaxial currents flow in opposite
directions. Singular or null points are then appearing, so the use of analytical formulae 
to perform the  numerical calculations
should be carefully done.

Following \cite{kn:lan}, the function $\Psi$ for a pair of  coils with radii $a$ and $A$,
 and currents $-{\cal I}_{\rm a}$ and ${\cal I}_{\rm A}$, can be written as
\begin{equation}
               \Psi (\rho,z) = \Psi_a + \Psi_A ={ {4\ A\ {\cal I}_{\rm A}}\over C } \left\lbrack G_{\rm A} (\rho,z)
 -{{a\ {\cal I}_{\rm a}} \over {A\ {\cal I}_{\rm A}}}
G_{\rm a} (\rho,z) \right\rbrack
\end{equation}
where 
\begin{equation}
 G_{\rm x} (\rho,z) = {1 \over k_{\rm x}} {\left(\rho \over x\right)^{1/2} 
\left \lbrack \left( 1-{k^2_{\rm x} \over 2} \right)\ K(k_{\rm x})-E(k_{\rm x})
\right\rbrack},\quad x = a, A
\end{equation}
with $ k^2_{\rm x} = 4\ x\ \rho\  {\left( (x+\rho)^2 +z^2\right)}^{-1},\ \ 
 K(k_{\rm x})$ and $E(k_{\rm x})$ the complete elliptic integrals of the $1$st and  $2$nd kind respectively, and $C$
a constant. 
 
The magnetic field on the axis $z$ can be described using elementary functions:
\begin{equation}
 B_{\rm z} (0,z) = { {2\ \pi\ A^2\ {\cal I}_{\rm A} \over C} \left\lbrace {1 \over {(A^2 +z^2)^{3/2}}} -
{a^2 \ {\cal I}_{\rm a} \over A^2\ {\cal I}_{\rm A}}\ { 1 \over {(a^2 + z^2)^{3/2}}} \right\rbrace } 
\end{equation}
This function is easy to plot and  usefull for choosing the parameters when a numerical simulation is planned.

The cross-section of a magnetic surface for the configuration $(21)$ has a hyperbolic singular 
point $z_{\rm s}$ on the $z$-axis if the magnetic field $(23)$ is zero at $z_{\rm s}$, with
\begin{equation}
 {z^2_{\rm s} \over A^2} = {{(a/A)^{2 \lambda /3} -a^2 / A^2} \over {1 - (a/A)^{2 \lambda /3}}} = 
{{(m_{\rm a}/m_{\rm A})^{2/3}-a^2/A^2} \over {1 - (m_{\rm a}/m_{\rm A})^{2/3}}}
\end{equation}
 On Fig.$4a$ and $4b$ we give $2$ examples of solutions computed with different values of $\lambda$ ;
 symbols are the same as in the case of a single ring current. The field  lines  going to the singular point are called
 the separatrix; in $2.5 D$ they corresponds to  a magnetic surface.
Here $m_{\rm a}/m_{\rm A} \equiv (a/A)^\lambda$ is the ratio of the magnetic moments of the rings and $m_{\rm a}/
m_{\rm A} = a^2\ {\cal I}_{\rm a} / ( A^2\ {\cal I}_{\rm A}) .$ The singular point can exist if $\ 0 < \lambda < 3 .$
\par 
When $\lambda \rightarrow 0,\ z_{\rm s} \rightarrow \infty $ and, alternatively,
when $\lambda \rightarrow 3,\ z_{\rm s} \rightarrow 0.$ 

The singular point, which is a null point for the strength of the magnetic field plays an important role in MHD.
It can be regarded as a region of both trapping and storage of plasma particles; it is also a privileged region where
waves can be dissipated, where the turbulence is enhanced and  where magnetic annihilation mechanisms could
eventually initiated.
Finally, on Fig.$5$ we show the result of calculations performed over a large field of view, for the case of $\lambda =
1$ already shown on Fig.$4a$. Fig.$5$ illustrates how the magnetic field of the external ring dominates indeed at large
distances, although the current of the inner ring is stronger than in the external ring in the same proportion as
 the radii.  


\section{Conclusions}  
The computation of both the topology of the magnetic field and its strength 
 has been done with good precision for axisymmetric currents. The case of one
single current ring can be easily modified by adding an external field of constant value
and of arbitrary direction to simulate some large scale effect. In addition, a  more
general case of $2$ superposed axisymmetric fields due to ring currents was treated.
Applications of these results, as far as the topology alone of the field is concerned,
were already discussed in \cite{kn:kou}. Here we show how the strength of the field
is distributed; lines of $ | \vec B | = const$ are also the lines of equal magnetic
pressure. The variation of the magnetic pressure along the separatrix can be followed.
The trapping region around the singular point is apparent. An obvious extension of
our calculations is to consider a time-dependant problem putting particles in the
system. The topology of the  magnetic fields can also be easily visualized making a movie, by
changing smoothly the initial conditions, e.g. parameters  describing the ring
currents, and  repeating the calculations. The program written for the IDL is
 available on request from the first author. We think it could be  usefull for $2.5 D$
 MHD simulations.

\section{Acknowledgments}  
We are indebted to M. M. Molodensky, L. Solov'ev and V. Koutvitsky for providing usefull
references and for extended discussions.
Inspiring discussions with P. Lorrain as well as his continuous interest to this work are highly appreciated.
 Part of this work were made during a stay at
NSO - Sacramento Peak Observatory (USA) and at N.A.O.J. Mitaka, Tokyo (Japan); we thank these
Observatories for their hospitality and for providing ressources.

\section{Appendix}
\begin{eqnarray}
K( m ) = (a_0 +a_1 m_1 +a_2 m_1^2) +(b_0 +b_1 m_1 +b_2 m_1^2)\ \ln ( 1 / m_1 ) + \epsilon ( m ),  \nonumber \\
| \epsilon ( m ) | \leq 3\ 10^{-5} \nonumber 
\end{eqnarray}
\begin{eqnarray}
 a_0&=&1.3862944  \quad b_0=0.5 \nonumber \\
 a_1&=&0.1119723  \quad b_1=0.1213478   \nonumber \\ 
 a_2&=&0.0725296  \quad b_2=0.0288729   \nonumber 
\end{eqnarray}
\begin{eqnarray}
K( m ) = (a_0 +a_1 m_1 +a_2 m_1^2 +a_3 m_1^3 +a_4 m_1^4) 
  + (b_0 +b_1 m_1 +b_2 m_1^2 \nonumber \\
 +b_3 m_1^3 +b_4 m_1^4)\ \ln ( 1 / m_1 ) + \epsilon ( m ), \quad
 | \epsilon ( m ) | \leq 2\ 10^{-8} \nonumber 
\end{eqnarray}
\begin{eqnarray}
a_0&=&1.38629436112  \quad b_0=0.5   \nonumber \\
a_1&=&0.09666344259  \quad b_1=0.12498593597  \nonumber \\
a_2&=&0.03590092383  \quad b_2=0.06880248576  \nonumber \\
a_3&=&0.03742563713  \quad b_3=0.03328355346   \nonumber \\
a_4&=&0.01451196212  \quad b_4=0.00441787012  \nonumber 
\end{eqnarray}
\begin{eqnarray}
E( m ) = (1 +a_1 m_1 +a_2 m_1^2) +(b_1 m_1 +b_2 m_1^2)\ ln ( 1 / m_1 ) + \epsilon ( m ),  \nonumber \\
| \epsilon ( m ) | \leq 4\ 10^{-5} \nonumber
\end{eqnarray}
\begin{eqnarray}
a_1&=&0.4630151  \quad b_1=0.2452727   \nonumber \\
a_2&=&0.1077812  \quad b_2=0.0412496   \nonumber 
\end{eqnarray}
\begin{eqnarray}
E( m ) = (1 +a_1 m_1 +a_2 m_1^2 +a_3 m_1^3 +a_4 m_1^4)
 + (b_1 m_1 +b_2 m_1^2 \nonumber \\
 +b_3 m_1^3 +b_4 m_1^4) \ \ln ( 1 / m_1 ) + \epsilon ( m ), \quad
| \epsilon ( m ) | \leq 2\ 10^{-8} \nonumber 
\end{eqnarray}
\begin{eqnarray}
a_1&=&0.44325141463   \quad b_1=0.24998368310   \nonumber \\
a_2&=&0.06260601220   \quad b_2=0.09200180037   \nonumber \\
a_3&=&0.04757383546   \quad b_3=0.04069697526 \nonumber \\
a_4&=&0.01736506451   \quad b_4=0.00526449639   \nonumber 
\end{eqnarray}
with $m = k^2, m_1 = 1 - m.$

\newpage
\begin{thebibliography}{kndpabc}

\bibitem{kn:alf1} H. Alfv\`en, 
in {\em Physics of the Hot Plasma in the Magnetosphere,}
 edited by
 B. Hultqvist {\em et al.} (Plenum Press, New York, 1975).

\bibitem{kn:alf2} H. Alfv\`en, 
 Cosmic Plasma, 
{\em  Astrophys. and Sp. Sc. Lib.}
 {\bf 82} (1981). 

\bibitem{kn:lan} L. D. Landau and E. M. Lifshits,
{\em Electrodynamics of Continuous Media} 
(Nauka, Moscow, 1985)
 
\bibitem{kn:pri} E. R. Priest,
{\em Solar Magnetohydrodynamics}
(Reidel, Dordrecht, 1982)

\bibitem{kn:lor} P. Lorrain, D. P. Corson and F. Lorrain,
{\em Electromagnetic Fields and Waves}
(Freeman, New York, 1988)

\bibitem{kn:sak} T. Sakurai and Y. Uchida,  
{\em Solar Physics} {\bf 52}, $397-416$ (1977)

\bibitem{kn:ste} J. Stenflo, {\em AA Rev.}, {\bf 1}, 3--48 (1989)

\bibitem{kn:piz} V. J. Pizzo, {\em Ap. J.} {\bf 302}, 785-808 (1986)

\bibitem{kn:mor} A. I. Morozov and L. S. Solov'ev,
Geometry of The Magnetic Field, in
{\em Plasma Theory}, {\bf 2}, 32, (1963)

\bibitem{kn:rad} Brigitte Radon,
 Sviluppi in serie degli integrali ellittici, in
{\em Atti Accad. Naz. Lincei,
 Mem. Scienze Fisiche} {\bf S. VIII, vol.II}, 69--109
(1950)

\bibitem{kn:abr} M. Abramowitz and I. Stegun,
{\em Handbook of Mathematical Functions},
(Dover Publications, 1972)

\bibitem{kn:alt} M. D. Altschuler, {\em Solar Physics} {\bf 1}, 377--388 (1967)

\bibitem{kn:kou}  S. Koutchmy, V. Koutvitsky, M. Molodensky, L. Solov'iev, and O. Koutchmy,
{\em Space Science Reviews} {\bf 70}, 283--288 (1994)

\end{thebibliography}
\newpage
\section{Captions For Figures} 
\par
Fig.$1.$ Distribution of field lines (dotted lines) and of lines of equal strength of $B$ (solid lines) for the
case of a coil made of $11$ ring currents superposed on the same axis. Units of $B$ are relative units in a logarithmic
scale. The map corresponds to a cut throught the symmetry axis.

Fig.$2.$ Detailed distribution of lines of $B$ close to the currents to show the absence of anomalies in shape of closed
magnetic tores at the scale of the coil.

Fig.$3a.$ Distribution of field lines (dotted lines) and of lines $| \vec B | = const$ ( solid lines) for a single
ring current. Note the dip of  $| \vec B | $ inside the ring of radius $0.2$.

Fig.$3b.$ As before over a larger field of view to show the assymptotic behavior.

Fig.$4a.$ As on Fig.$1$ but for the case of $2$ ring currents of radii $0.1$ and $1.$ with $\lambda = 1.$ The dotted
field lines corresponding to the separatrix is enhanced.

Fig.$4b.$ As Fig.$4a$ for $\lambda = 3/2$ and the same scales ( spatial scale and scale of strength )

Fig.$5.$ As on Fig.$4a$ but over a field of view $5$ times larger. Note the dominance of the magnetic field of the external
ring at large distances.

\end{document}