\documentstyle{spacekap}
\input{psfig.tex}
\begin{opening}
\title{MAGNETIC CONFIGURATION OF CORONAL\\ STREAMERS AND THREADS}
\author{S.Koutchmy}
\institute{Institut d'Astrophysique, CNRS, 98 Bis, Bd Arago, F-75014 Paris (France)}
\author{V.A.Koutvitsky}
\author{M.M.Molodensky}
\author{L.S.Solov'iev}
\institute{IZMIRAN, Moscow Region, Tro\"\i tsk, 142092 (Russia)}
\author{O.Koutchmy}
\institute{Universit\'e P. et M. Curie, Laboratoire d'Analyse Num\'erique,\\ Tour 55-65,
 4, Place Jussieu, F-75252 Paris (France)}                
\date{}
\end{opening}

\runningtitle{MAGNETIC CONFIGURATION OF CORONAL STREAMERS AND THREADS}
\runningauthor{S.~KOUTCHMY~ET~AL.}

\begin{document}

\begin{abstract}
We give a short account of the most prominent structures of the intermediate corona.
Then we propose an axially symmetrical model for coronal streamers, according to which 
charged particles move along magnetic surfaces whose sources are electrical currents
situated in the vicinity of the photosphere. The simplest current configuration is a
 pair of coaxial, coplanar, circular, and oppositely directed currents parallel to
 the photosphere. Magnetic surfaces for this current distribution exhibit a helmet-shaped
 separatrix and a saddle point. The temperature profile along the 
streamer can be predicted qualitatively if one takes into account the conservation of an 
adiabatic invariant in the drift theory of the charged particle motion.

\end{abstract}

\section{Introduction}
The structure of coronal streamers observed during a solar total eclipse (Koutchmy,
 1992) poses many theoretical problems that are  still unsolved :
\newline
\noindent i) the origin of the  typical helmet with a cusp or onion shape,\\
ii) the origin of the confinement of streams at distances from the surface\\  
exceeding the solar diameter and/or in the intermediate corona,\\
iii) the large excess density 
of particles on the visible surface of the streamer, etc.

The classical approach for solving these problems is to start with a simple magnetostatic configuration around 
the Sun, similar to a dipole field, and to try to solve the full MHD equations, assuming a strictly radial flow presumably 
due to the thermally driven ``wind''. Then the magnetic field lines are stretched out and the neutral sheet could be
 a streamer, although some additional forces are needed to maintain a significant density enhancement
in a neutral sheet. See Koutchmy and Livshits, (1992) for a recent review of ``classical'' models. Another empirical
method for computing the magnetic configuration of the corona in 3D is to introduce a so-called ``source surface'',
which considerably simplifies the computation, but leads to an artificial concept that does not have a physically
 plausible counterpart in the case of streamers.

Moreover, a modern approach for  analyzing the coronal structures convincingly shows that the intermediate corona
 is filled with  tiny and elongated structures like rays or threads as in Fig. 1. Some threads have obviously
a circular cross-section that they keep over many solar radii; they extend along a direction that is never exactly radial
: sometimes the deviation from the radial direction is as large as $60^\circ$\ (Koutchmy, 1969). Furthermore, these
structures were seen outside eclipses: YOHKOH images taken in SXR show that the intermediate corona, above the inner
 corona made of loops, is also filled with rays, especially in the vicinity of active regions. A beautifull example 
is given in Shibata et al.(1992).In SXR images, time sequences show that threads are jets, although 
the magnetic features channeling the jet may persist.
 The rays often have the shape of threads rising above a loop-like structure; sometimes a cusp is
 well identified. YOHKOH images show  more inner corona than eclipse images, but the fundamental loop-cusp-ray structure
 seems general. As to more quantitative aspects, little can be said from the observations of the 
temperature inside a range of 1 to 3 MK, but the density enhancements are at least one order of magnitude about the 
average background corona.

\begin{figure}[h]
\vspace{8 cm}
\caption{Processed WL coronal eclipse image of July 31, 1990 using the ``MadMax''
algorithm; note the long threads.}
\end{figure}


\section{Topological modelling and energy}
Some of the  above questions lead to the following model. We assume that 
charged particles from the Sun flow mainly along the separatrix magnetic surface, which has 
a singular X-point on the vertical axis. This magnetic surface with a sharp  top (cusp) is
interpreted as the visible surface of the coronal streamer.
The required magnetic configuration could be due to the superposition of a circular current-carrying
 prominence, or facula, surrounding the spot and having a field of the opposite
sign. The simplest current distribution is a pair
of circular, oppositely directed, coaxial, and coplanar currents.

Let $A_\rho ,\ A_\phi,\ A_{\rm z}$ be the components of the vector potential
in cylindrical coordinates $(\rho, \phi, z).$
The axial symmetry provides magnetic surfaces that are contour lines of 
$\Psi (\rho,z) = \rho\ A_{\phi}(\rho,z).$ Then the poloidal components of the magnetic field are
$B_{\rm z} = {1 \over \rho}\ {\partial \Psi \over \partial \rho},\ B_\rho = - {1 \over \rho}\ {\partial \Psi \over \partial z},\
B_\phi = 0.$

 Following Landau and Lifshits (1985), 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
 $$ \Psi (\rho,z) ={ {4\ A\ {\cal I}_{\rm A}}\over C } \left\lbrack G_{\rm A} (\rho,z) -{{a\ {\cal I}_{\rm a}} \over {A\ {\cal I}_{\rm A}}}
G_{\rm a} (\rho,z) \right\rbrack,\eqno(1) $$
where 
$$ G_{\rm x} (\rho,z) = {1 \over k_{\rm x}} {\left(\rho \over x\right)^{1/2} 
\left \lbrack \left( 1-{k^2_{\rm x} \over 2} \right)\ K(k_{\rm x})-E(k_{\rm x})
\right\rbrack},$$
with $ k^2_{\rm x} = 4\ x\ \rho\  {\left( (x+\rho)^2 +z^2\right)}^{-1},\ \ 
 K(k_{\rm x})$ and $E(k_{\rm x})$  full elliptic integrals of the $1$st and $2$nd kind respectively, and $C$
a constant.
\par
The magnetic field on the axis $z$ can be described using elementary functions:
$$ B_{\rm z} (0,z) = { {2\ \pi\ A^2\ {\cal I}_{\rm A} \over C} \left\lbrace {1 \over {(A^2 +z^2)^{3/2}}} -
{a^2 \ {\cal I}_{\rm a} \over A^2\ {\cal I}_{\rm A}}\ { 1 \over {(a^2 + z^2)^{3/2}}} \right\rbrace } .\eqno(2)$$

The following expansion in even powers of $\rho$ is valid for an arbitrary axially symmetrical 
configuration (Morozov and Solov'iev, 1963):
$$ \Psi (\rho,z) = {\rho ^2 \over 2} b(z) -{\rho ^4 \over 16} b^{\prime\prime} (z) + ... = \rho J_1 (\rho
{ \hat b}^\prime (z)), \eqno(3)$$
where $b(z) = {\hat b}^\prime (z) = B(0,z)$ is the field on the $z$-axis and $J_1(x)$ is 
Bessel function of the first kind.
\par
The cross-section of a magnetic surface for the configuration $(1)$ has a hyperbolic singular 
point $z_{\rm s}$ on the $z$-axis if the magnetic field $(2)$ is zero at $z_{\rm s}$, with
$$ {z^2_{\rm s} \over A^2} = {{(a/A)^{2 \lambda /3} -a^2 / A^2} \over {1 - (a/A)^{2 \lambda /3}}} =
{{(m_{\rm a}/m_{\rm A})^{2/3}-a^2/A^2} \over {1 - (m_{\rm a}/m_{\rm A})^{2/3}}}. \eqno(4)$$
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,
when $\lambda \rightarrow 3,\ z_{\rm s} \rightarrow 0.$ 
 When $\lambda \rightarrow 0,$ the magnetic moments and magnetic fields of both coils 
become equal, $m_{\rm A} \rightarrow ~-m_{\rm a},\ B_{\rm A} \rightarrow ~-B_{\rm a},$ and the ratio of the fields at the 
origin tends to the third power of $A/a$ :
$${ B_{\rm a} (0,0) \over B_{\rm A}(0,0)} = {{{a^2\ {\cal I}_{\rm a}} \over {A^2\ {\cal I}_{\rm A}}}\ \left({ A \over a} \right)^3 \
\rightarrow  \ \left({ A \over a} \right)^3} . \eqno(5)$$
\par
With $\lambda$ small enough, the top of a helmet corresponding to the point $\rho=0,\ z=z_{\rm s}$
occurs at an arbitrarily height  above the surface of the photosphere.
\par
The magnetic surfaces $\ \Psi (\rho,z) = const\ $ in the vicinity of the singular point
 $\rho=0,\ z=z_{\rm s}$ can be drawn using expansion $(3)$. Assuming that
$$ b(z) = {C\ B_{\rm z} (0,z) \over 2\ \pi\ A^2\ {\cal I}_{\rm A}} = {1 \over (A^2+z^2)^{3/2}}-{\mu \over (a^2+
z^2)^{3/2}},$$
so that $\mu = (a^2+z^2_{\rm s})^{-3/2}  (A^2+z^2_{\rm s})^{3/2},$ we obtain, to an accuracy of  $\rho ^4$,
$$\Psi={\rho ^2 \over 2} \left\lbrack {1 \over (A^2+z^2)^{3/2}}-{\mu \over (a^2+z^2)^{3/2}}
\right\rbrack
-{3 \rho ^4 \over 16} \left\lbrack {4 z^2-A^2 \over (A^2+z^2)^{7/2}}- {\mu(4 z^2-a^2 )
\over (a^2+z^2)^{7/2}}\right\rbrack .\eqno(6)$$
In the vicinity of $\rho=0$ and $z=z_{\rm s},\ \Psi = P\ \rho ^2\ \zeta ,$ with $\ \zeta =z-z_{\rm s}$ and $P$ a constant.
\par
If the Larmor radii of the particles are small compared to the caracteristic size of the
 configuration, then the drift approximation applies (Morozov and Solov'iev, 1963).
 If $B_\varphi = 0,$ then the drift
trajectories of particles lie on the magnetic surfaces $\ \Psi (\rho,z) = const.\ $ Particles drift in
the azimuthal direction, and some of them are reflected from regions of strong magnetic field, which act as
magnetic mirrors. Using the adiabatic invariant $I~=~\upsilon^2_{\bot}~/~B~=~const,$
the transverse and longitudinal energies of the particles are :
$$ \epsilon_\bot = \epsilon^0_\bot +{m I \over 2}\ ( B-B_0),\qquad  \epsilon_\| = \epsilon^0_\| -
{m I \over 2}\ (B-B_0). \eqno(7)$$
\par
When a particle moves into a strong field, its transverse energy, which determines the
 temperature, increases, but the longitudinal energy decreases, and vice versa. Now the
 dependence of the magnetic field on $z$ has a characteristic profile  and
Eq. (7) provides the radial temperature dependence in the corona. The deduced
behavior can be considered to be the result of the conservation of the transverse adiabatic invariant
in the drift motion of charged particles.
\par
To summarize, the coronal streamer in our model is the separatrix of an axially symmetric
 configuration, which arises as a result of the superposition of the magnetic fields of
 a facula surrounding the spot or of a filament surrounding  the facula. The tendency of charged particles to escape along the
 separatrix magnetic surface was established by Auerbach and Boozer (1980) and by Gribkov et al.(1984).
\par
In order to simulate an inclined streamer, while keeping the possibility of describing the
system with the function $\Psi$, we consider also a two-dimensional problem in cartesian
coordinates with $\partial \Psi / \partial y =0.$ We replace the system of circular currents 
 by a pair of linear conductors with oppositely directed currents. Then
$$\Psi =P_{\rm y} (x,z)={2 \over C}\sum_{\rm i} J_{\rm i} ln{ \left[ { {(x-x_{\rm i1})}^2-{(z-z_{\rm i1})}^2} \over
{ {(x-x_{\rm i2})}^2-{(z-z_{\rm i2})}^2} \right]}^{1/2},$$
where $(x_{\rm i1},z_{\rm i1})$ and $(x_{\rm i2},z_{\rm i2})$ are the coordinates of the conductors of the $i$-th pair.

\begin{figure}[htb]
\centerline{\hbox{ \hfill
\psfig{figure=carre.ps,height=6cm}
\hspace{-2.3cm}
\psfig{figure=carre.ps,height=6cm} \hfill} }
\vspace{.5cm}
\centerline{
\hbox{\small Fig.2.$B$--lines for $\lambda = 0.03,\ a /A = 0.1$}
\hspace{.1cm}
\hbox{\small Fig.3.$B$--lines in the plane configuration} }
\end{figure}


\section{Results and Discussion}
Fig. 2 shows the magnetic configurations calculated for selected parameters $a/A$
 and $\lambda\ $ from Eq. (4). The magnetic surface cross-sections shown in Fig. 2
have the helmet shape. With increasing height, magnetic surfaces in the vicinity of the
 separatrix become onion-shaped (cusp). The shape of the magnetic surfaces 
may vary in the vertical direction.
\par
A magnetic configuration with two saddle points in the plane $z=0$ is possible. This 
configuration can be considered for modelling the coronal polar rays or
 plumes and possibly also corresponds to the quiet Sun field in the corona.
 Fig. 3 shows an inclined ray in the plane approximation. Its behaviour at
 infinity is determined by superposing a uniform and a weak background magnetic field
$\ B_{\rm z}^0 = const.$ 
 The calculated and observed structures of Fig. 1 are similar. From the general appearance
 of helmets we can conclude that $0.001 <\lambda <1$ in these configurations, and the
best value for real structures is $\lambda \approx 0.03.$ This includes 
straight, very long, and slightly inclined ``rays''.

\section{Conclusion}
The ``separatrix model'' of coronal streamers rests on the assumption of negligible influence of the proper currents on 
the motion of charged particles along magnetic surfaces. Magnetic surfaces are assumed to be created by external
sources placed on the solar surface. The smallness of the magnetic fields does not prevent, in principle, the
 existence of a singular $X$-point on the axis where the magnetic field tends to zero. The 
boundaries of the helmets and cusps are mathematically isolated separatrix surfaces. The problem of the filling of 
the boundary with charged particles or plasma requires further work. The geometry of the magnetic field lines of 
the helmet configuration is analogous to that of streamlines around a sphere in a stream of fluid. In our case the
non-existing lower hemisphere can be considered as a reflection of the MHD-configuration in equilibrium in the
perfectly conducting plane of the photosphere. The nested meridional cross-sections of the toroidal magnetic
 surfaces observed inside the helmets are typical of the equilibrium MHD-configurations in TOKAMAKS inside the
separatrix surface.

%\clearpage

\begin{thebibliography}{}
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\reftitle{Gribkov V.M., Morozov A.I., and Pogytche O.P.}{1984}{Plasma flow in toroidal systems
with separatrix}{Fizika plazmy}{23}{82}
\article{Koutchmy,S.}{1969}{Astrophys. Letters}{3}{215}
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\article{Loucif M.L. and Koutchmy S.}{1989}{A.A. Suppl. Series}{77}{45}
\reftitle{Morozov A.I., and Solov'iev L.S.}{1963}{Geometry of the magnetic field}
{Plasma Theory}{2}{}
\reftitle{Morozov A.I., and Solov'iev L.S.}{1963}{Motion of charged particles in electromagnetic
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\article{Shibata, K. and the YOHKOH team}{1992}{PASJ}{44}{L173}
\end{thebibliography}{}


\end{document}