<figure><img src="image_69.png" /></figure>

**Figures 5** : Schématisation de la formation des protubérances dans une configuration Inverse.

La formation d'une protubérance dans une nappe de courant est représentée schématiquement sur la figure 5a. Le "tearing mode" forme des îlots de champ magnétique, un seul étant représenté sur la figure. Ces îlots coalessent par attraction magnétique pour former une ligne de courant: figure 5b. Elle est maintenue contre la gravité par l'ancrage des lignes de champ dans la photosphère. Kuperus et Raadu (1974).

<figure><img src="image_70.png" /></figure>

**Figures 6** : Configurations Inverses obtenues avec des fonctions holomorphes.

La configuration 6a est soutenue par un champ magnétique ouvert, cette configuration est cependant peu réaliste car elle doit être soutenue par le vent solaire. La configuration 6b présente un îlot, avec une singularité de courant au sommet en $z = q$; seule la partie inférieure de la protubérance est soutenue. La dernière figure, 6c, est la plus réaliste: la protubérance est formée dans une région quadrupolaire. Malherbe et Priest (1983a).

une forte compression du plasma par le champ magnétique, perpendiculairement au plan de la protubérance, est possible; de plus, le plasma est bien isolé de la couronne chaude par les îlots magnétiques formés dans la nappe de courant, et il est donc plus apte à se condenser. Cependant, ce modèle pose aussi de nombreux problèmes que l'on discutera à la fin de ce chapitre: paragraphe 2.4. Kuperus et Raadu supposent l'existence initiale d'une nappe de courant, avec un champ vertical. Le plasma froid formé ne peut alors pas être supporté par le champ magnétique vertical (figure 5a) et il s'effondre à basse altitude tout en diminuant son extension verticale (à cause de l'attraction des courants de la protubérance de même sens): figure 5b. Il est alors stabilisé par l'ancrage des lignes de champ dans la photosphère dense. Cela se modélise, en champ potentiel, par une image de la protubérance par rapport au plan de la photosphère. Ce courant image est de signe opposé et il donne lieu à une force de répulsion: $F = \mu_0.I^2/4\pi h$, qui équilibre la force de gravité, et l'action du champ magnétique (donné par la distribution photosphérique du flux) sur la protubérance. Ce modèle a été étendu par Van Tend et Kuperus (1978) en incluant un champ coronal plus réaliste.

Malherbe et Priest (1983a) ont appliqué la méthode complexe décrite pour les configurations Normales (figures 2), aux configurations Inverses. La figure 6a montre une configuration magnétique soutenue à grande hauteur: cela pourrait être réalisé si le vent solaire était suffisamment important. Mais, les protubérances solaires ayant un champ axial important, le vent solaire doit alors avoir une importante composante horizontale de la vitesse, de direction opposée de part et d'autre de la protubérance. Cela semble en contradiction avec les connaissances actuelles: Anzer (1984, 1988). La figure 6b donne une configuration fermée, qui nécessite une singularité de courant en $z = q$. De plus, le champ magnétique ne supporte pas la partie supérieure de la protubérance! En revanche, la figure 6c représente une configuration plausible, mais elle nécessite une région quadrupolaire. Il est à noter que, dans cette configuration, le support est réalisé de la même façon que dans les configurations Normales. On peut considérer cette configuration comme Normale avec une polarité "parasite" à la base.

Anzer (1984) et Anzer et Priest (1985) ont étudié analytiquement la formation d'une protubérance dans une configuration similaire à la figure 6a: figure 7a. La protubérance se contracte alors sous l'effet de l'attraction des courants, et donne une ligne de courant, figure 7b. L'équilibre de cet état final n'est cependant pas trouvé et ils concluent que des configurations plus
<figure><img src="image_71.png" /><figcaption>**Figures 7 :** Formation d'une protubérance dans une configuration inverse. Une nappe de courant est supposée présente au début: figure 7a. Le système ne peut pas être en équilibre, les forces d'attraction entre les courants de la protubérance contractent alors cette dernière en une ligne de courant: figure 7b. En supposant la conservation du courant total de la protubérance, un équilibre final ne peut être atteint. La formation ne peut s'effectuer dans une telle configuration. Anzer et Priest (1985).</figcaption></figure>

<figure><img src="image_72.png" /><figcaption>**Figures 8 :** Champ potentiel à trois dimensions. Une configuration magnétique tridimensionnelle supportant une protubérance est obtenue en superposant à un champ potentiel symétrique tridimensionnel une nappe de courant bidimensionnelle dans le plan $x = 0$. La figure 8a représente la densité surfacique de la protubérance: des régions plus denses sont présentes, on peut les associer aux pieds des protubérances. La figure 8b montre une représentation tridimensionnelle des lignes de champ magnétique. Le plan grisé dans cette dernière figure schématise la protubérance et les points noirs montrent l'intersection des lignes de champ avec le plan $z = 0$. Wu et Low (1987).</figcaption></figure>
<figure><img src="image_73.png" /></figure>

<figure><img src="image_74.png" /><figcaption>**Figures 9** : Modélisation de la structure magnétique par un champ sans force linéaire bidimensionnel.</figcaption></figure>

La figure 9a définit les conditions aux limites sur le potentiel magnétique $A(x, z)$. La protubérance est schématisée par la ligne de courant d'intensité I: les figures 9b et 9c donnent deux exemples de configurations, respectivement Normale et Inverse. La composante du champ normale à la figure n'est pas représentée: les isocontours de cette composante sont identiques aux lignes de champ tracées. Amari et Aly (1988).

complexes sont nécessaires pour expliquer les observations de configurations Inverses.

L'étude des configurations magnétiques que l'on peut déduire des observations a été réalisée par Anzer (1972), avec l'hypothèse d'une symétrie de translation, pour une configuration Normale (paragraphe précédent). Cette étude a été généralisée aux configurations Inverses dans les articles A5 et A12. L'ensemble des solutions possibles a été envisagée dans le cadre d'un champ potentiel (article A5) puis ces résultats ont été généralisés à un champ sans force linéaire (article A12).

### 1.1.3 Modèles récents

#### a. Modèle potentiel à 3 dimensions

Les protubérances solaires rejoignent périodiquement la chromosphère par des "pieds". Un modèle réaliste de protubérance nécessite donc une troisième dimension. Cependant, même dans le cas potentiel, la construction de tels modèles est très difficile. Wu et Low (1987) ont utilisé un champ à 3 dimensions qui n'a pas de composante verticale dans le plan Oyz. Su-perposé à une nappe de courant à deux dimensions dans le plan x = 0, cela modélise une protubérance avec une densité surfacique dépendant des deux coordonnées y et z. L'exemple qu'ils ont développé, donne des concentra-tions de matière périodiques à basse altitude, qui peuvent représenter les pieds des protubérances: figure 8a. Les lignes de champ correspondantes sont représentées à la figure 8b. Cependant, comme aucune contrainte ne peut être mise sur les conditions aux limites connues du champ magnétique dans la protubérance, il est difficile de comparer, quantitativement, ce modèle aux observations. De plus, le champ étant potentiel, l'influence de la com-posante du champ magnétique le long de l'axe de la protubérance ne peut être étudiée.

Un modèle analytique de champ magnétique sans force et linéaire à 3 dimensions a été développé dans l'article A9, Démoulin et al (1989c). Les conditions aux limites sont imposées au niveau photosphérique, où le champ est périodiquement concentré par les cellules de convection du réseau de super-granulation. L'équilibre des lignes de courant, modélisant une pro-tubérance, est alors étudié dans un tel champ et la relation entre les pieds des protubérances et le réseau de supergranulation est discutée.
<figure><img src="image_75.png" /><figcaption>**Figures 10** : Evolution schématique d'un tube de flux torsadé par des mouvements photosphériques.</figcaption></figure>

La figure 10a représente le tube non torsadé et 10b montre le tube pour la torsion critique $\Phi_{crit}$ au-delà de laquelle la courbure est concave vers le haut au sommet du tube. De la matière froide peut alors se déposer pour former une protubérance: figure 10c. Enfin, lorsque la torsion est trop grande, la protubérance donne lieu à une éruption: figure 10d. Priest et al (1989).

### b. Champ sans force ("force free field")

Amary et Aly (1988) ont résolu analytiquement le problème du champ sans force dans une configuration à deux dimensions de la figure 9a. Le champ magnétique est décrit par son potentiel $A(x,z)$ et se met sous la forme:

$$B = (\partial A/\partial z, \alpha A, -\partial A/\partial x)$$

Les conditions aux limites sont traduites sur le potentiel $A$, et la protubérance est modélisée par une ligne de courant massive. Un résumé détaillé de ces travaux est présenté dans le paragraphe 2 de l'article A4, Démoulin et Priest (1988a). L'intérêt de ces travaux réside dans la description d'une nouvelle méthode permettant la prise en compte des trois composantes du champ magnétique. De nombreuses extensions sont possibles et seront décrites au chapitre D dans l'article A4. Des configurations Normale et Inverse sont présentées par les figures 9b et 9c à titre d'exemples.

### c. Tube de champ torsadé

Priest et al (1988) ont proposé récemment un nouveau modèle pour la structure magnétique des protubérances solaires. Une représentation générale de l'évolution est donnée par les figures 10: un tube de flux est torsadé à la base par des mouvements photosphériques de cisaillement et de torsion. La rotation différentielle, mais surtout la force de Coriolis au niveau photosphérique sont de bons "candidats" pour torsader les tubes de flux de la couronne. Lorsque la torsion $\Phi$ augmente, les lignes de champ forment un creux au sommet du tube de champ, pour une valeur de $\Phi$ supérieure à une valeur critique $\Phi_{crit}$: figure 10b. Ce creux permet de retenir toute matière froide formée ou transportée au sommet du tube de champ: figure 10c. Le modèle conduit donc naturellement à la formation d'une protubérance, si de la matière froide est présente au sommet des lignes de champ. Cette évolution de la structure magnétique nécessite des mouvements tridimensionnels mais seule une étude analytique simplifiée a été effectuée: aucun cisaillement n'a été introduit et seul l'effet de la torsion des lignes de champ a été étudié. Le modèle devra être complété par l'étude du cisaillement, car les observations montrent que les protubérances ne peuvent se former qu'en présence d'un fort cisaillement! Priest et al ont aussi étudié le support de la matière dense dans une telle configuration: lorsque l'on ajoute de la masse dans le creux des lignes de champ, ces dernières se déforment et créent un courant qui permettra de supporter la protubérance par la force de Laplace. Priest et al ont étudié la formation: la perturbation introduite par la matière froide est faible. De plus,
<figure><img src="image_76.png" /><figcaption>Figure 11 : Profil de température moyen déduit des observations pour la région de transition protubérance-couronne pour deux protubérances. Les raies utilisées sont représentées par des points. Schmahl et Orall (1980)</figcaption></figure>

<figure><img src="image_77.png" /><figcaption>Figures 12 a et b</figcaption></figure>

seule la réponse du champ au voisinage de la protubérance a été modélisée. Les premiers calculs exposés dans cet article montrent un modèle très riche en possibilités, incluant la description de l'évolution des protubérances de la formation à l'éruption. Cependant, la structure du champ magnétique étant à trois dimensions, l'analyse complète du modèle reste à effectuer. Il est en particulier important d'étudier l'influence du cisaillement sur la torsion $\Phi_{crit}$. Les propriétés de ce modèle seront discutées plus amplement au chapitre C, dans la partie résumant les connaissances sur la formation des protubérances.

## 1.2 Structure magnétique et thermique interne

### 1.2.1 Observations

Observées avec une résolution supérieure à 1", les protubérances sont de fins feuillets (~ 5000 km) de plasma froid plongés dans la "fournaise" coronale (T~ $10^6$ K). Comment de telles structures peuvent-elles résister au chauffage produit par les régions coronales voisines? Le champ magnétique fournit une partie importante de la réponse car il canalise le flux de chaleur le long des lignes de champ, évitant ainsi le contact direct de la protubérance avec les régions avoisinantes. Comme pour le passage de la chromosphère à la couronne, une zone de transition de faible taille existe entre la protubérance et la couronne. La mesure du gradient de température peut être déduite de l'intensité relative des raies, par la mesure du coefficient d'émission différentielle. La figure 11 donne deux exemples de profils moyens obtenus par Schmahl et Orall (1980). La zone de transition est semblable à celle de la transition chromosphère-couronne, avec cependant un gradient de température moins fort: Engvold et al (1987).

Les observations de haute résolution (~ 0·5") de Dunn (1960) et Engvold (1976), au limbe solaire, montrent que les protubérances quiescentes sont formées d'une structure fine composée de fins filets froids: ceux-là ont une épaisseur de l'ordre de quelques 0·5". Ils sont en moyenne orientés verticalement et évoluent en un temps court (8 minutes). Ces structures sont également visibles en projection sur le disque: article A1, Simon et al (1986). Le champ magnétique n'a, en revanche, pas des variations spatiales et temporelles aussi rapides: Leroy (1985). Bien que diverses interprétations aient été proposées, ces structures fines gardent en partie leurs mystères: aussi, je
<figure><img src="image_78.png" /><figcaption>Figure 12 : Modèle d'équilibre interne d'une protubérance.</figcaption></figure>

La température est supposée constante et les paramètres ne sont fonction que de x. La figure (a) représente le champ vertical Bz, (b) la pression gazeuse. L'abscisse est normalisée à l'échelle gravitationnelle du plasma: $H_g = k_B T_0 / m \cdot g$. La figure (c) représente les lignes de champ. Kippenhahn et Schlüter (1957).

<figure><img src="image_79.png" /><figcaption>Figures 13 : Exemples de configurations magnétiques obtenues par un modèle à deux dimensions.</figcaption></figure>

Les lignes de champ sont en pointillés, les limites de la protubérance sont indiquées par les traits continus. Ce modèle généralise, en deux dimensions, celui des figures 12. Ballester et Priest (1987).

présenterai d'abord des modèles considérant le filament comme homogène, puis je discuterai les tentatives de modélisation de la structure fine des protubérances.

### 1.2.2 Modèle de la structure interne des protubérances

La modélisation de la structure interne des protubérances, même dans l'hypothèse de l'absence de structure fine, comporte de nombreuses difficultés. Aucun modèle incorporant l'ensemble des aspects n'a encore été construit. J'exposerai ainsi successivement l'interaction du plasma avec le champ magnétique, puis l'aspect thermique lié à l'équation de l'énergie, pour terminer sur les problèmes de transfert de rayonnement.

Kippenhahn et Schlüter (1957) ont modélisé le support interne de la protubérance par le champ magnétique, en considérant que le plasma est à une température uniforme et en supposant une dépendance spatiale des paramètres uniquement orthogonalement au plan de la protubérance (axe Ox). L'équilibre de la protubérance est donné par l'équilibre des forces:

$$-\nabla p - \rho g e_z - \nabla (B^2/2\mu_0) + (\mathbf{B} \cdot \nabla) \mathbf{B} / \mu_0 = 0$$

Les conditions aux limites sont: $p \to 0$ et $Bz \to Bz_\infty$ lorsque $x \to \infty$, et par symétrie $Bz = 0$ en $x = 0$. La solution de ce problème est:

$$ 
\begin{align*} 
Bx &= \text{constante} \\ 
Bz &= Bz_\infty \tanh \left( \frac{Bz_\infty \cdot x}{Bx \cdot H_g} \right) \\ 
P &= \frac{Bz_\infty^2}{2\mu_0} \sech^2 \left( \frac{Bz_\infty \cdot x}{Bx \cdot H_g} \right) 
\end{align*} 
$$

où $H_g = k_B T / mg$ est la hauteur gravitationnelle du plasma. Ces grandeurs sont représentées sur les figures 12a et 12b, et la figure 12c schématise la configuration magnétique obtenue au voisinage de la protubérance. L'épaisseur de la protubérance est approximativement:

$$d \approx \frac{4 \cdot H_g}{\operatorname{tg} \delta}$$

où $\delta$ est l'angle entre les lignes de champ et l'horizontale, loin de la protubérance. En utilisant l'angle $\delta = 30^\circ$ déduit des observations par Bommier
<figure><img src="image_80.png" /><figcaption>Figure 14 : Profils de température modélisant la structure thermique d'une protubérance.</figcaption></figure>

Ils sont obtenus en résolvant l'équation correspondant à l'équilibre des forces et celle de l'énergie sans transfert de rayonnement. Les tirets indiquent un modèle avec une fonction de rayonnement atténuée à faible température, les tirets pointés donnent un modèle sans chauffage dans la protubérance. Milne et al (1979).

et al (1986) et l'échelle de hauteur gravitationnelle pour un plasma à 7000 K (soit $H_g \sim 200$ km), on obtient: $d \sim 1400$km. Cela est en remarquable accord avec la largeur observée pour les structures fines. Poland et Anzer (1971) ont généralisé ce modèle pour une distribution de température donnée en fonction de x. Ballester et Priest (1987) ont étendu ces calculs à deux dimensions, en permettant aux paramètres de varier lentement selon la hauteur z. La largeur de la protubérance est trouvée décroître avec la hauteur, lorsque l'intensité du champ croît légèrement, en accord avec les observations. Les figures 13 donnent deux exemples de configurations magnétiques. La principale hypothèse de ces modèles est d'imposer un profil de température. Pour lever cette hypothèse, la résolution de l'équation de l'énergie est nécessaire.

Milne et al (1979) ont couplé la magnétostatique à une équation de l'énergie, où le transfert de rayonnement est négligé. Le plasma est ainsi supposé optiquement mince, et décrit localement par une fonction de rayonnement, $L_r$, fonction seulement des propriétés locales du plasma. Le chauffage H de la matière est une inconnue majeure: il est choisi, par simplicité, uniforme par unité de masse: $H = H_0 \cdot \rho$. La conduction thermique s'effectue seulement selon les lignes de champ car la conduction orthogonale aux lignes de champ nécessiterait une échelle de longueur très faible ($\sim 1$ km) pour être de l'ordre de grandeur de la conduction parallèlement au champ. Milne et al ont résolu les équations de l'équilibre des forces, conjointement à l'équation de l'énergie:

$$ \frac{d}{ds} \left( \kappa_0 \cdot T^{5/2} \frac{dT}{ds} \cdot \frac{Bx^2}{B^2} \right) = L_r - H \quad (1) $$

Ils supposent que le système est symétrique par rapport au plan de la protubérance ($x=0$) et que les conditions aux limites sont imposées, en $x = \pm Hg_c$: $P = Pc$, $T = Tc$ et $L_r = H$ ($Pc$ et $Tc$ sont des grandeurs caractéristiques de la couronne, et $Hg_c = k_B \cdot Tc/m \cdot g$ est l'échelle gravitationnelle de la couronne). L'équilibre dépend principalement de deux paramètres: du $\beta$ du plasma et de l'angle de cisaillement $\phi$ ($\beta = \frac{2\mu_0Pc}{Bx^2}$ et $\phi = \text{tg}^{-1}(\frac{By}{Bx})$). Le champ magnétique ne peut soutenir la protubérance que si $\beta$ est inférieure ou de l'ordre de l'unité. Une condensation froide est obtenue lorsque l'angle de cisaillement $\phi$ appartient à un intervalle de quelques degrés autour de la valeur de $82^\circ$. Cela doit provenir des conditions aux limites ($T = Tc$ et $L_r = H$) qui sont arbitrairement placées en $x = \pm Hg_c$. Une cune modification de cette condition aux limites n'a été envisagée: elle me
<figure><img src="image_81.png" /><figcaption>Figure 15 : Profils de température modélisant la structure thermique d'une protubérance.</figcaption></figure>

Ils sont obtenus en résolvant le calcul du transfert de rayonnement dans une structure fibreuse. L'abscisse est la colonne densité; $m \approx 10^{-4}$ représente une épaisseur de l'ordre de 300 km. Heasley et Mihalas (1976).

semble cependant très importante. Il serait particulièrement intéressant de fixer cette condition au niveau de la chromosphère, et d'étudier l'influence de la géométrie du champ magnétique qui intervient fortement en canalisant le flux de chaleur le long des lignes de champ. De plus, le profil de température obtenu, figure 14, ne correspond pas aux observations: la température et la taille de la région froide sont très faibles: elles sont respectivement de seulement 3000 K et 25 km! Différentes modifications ont été apportées au modèle initial. L'absence de chauffage dans la protubérance élargit un peu la taille, mais la température centrale est encore plus froide ($T = 400$ K!). L'effet d'une épaisseur optique importante est de réduire les pertes radiatives. Cela a été simulé en diminuant fortement la fonction de refroidissement $L_r$ à basse température. Une taille de l'ordre de 700 km peut alors être obtenue, mais la température centrale reste cependant trop froide: $T \approx 2000$ K. Ces modifications montrent bien l'influence des différents paramètres, mais un profil réaliste de température ne peut être obtenu.

Une approche radicalement différente a été utilisée par Poland et Anzer (1971) qui ont développé un modèle de transfert utilisant un atome d'hydrogène avec deux niveaux et le continuum. La protubérance est irradiée par les rayonnements photosphérique, chromosphérique et coronal. Le transfert de rayonnement a été effectué en modélisant la protubérance par une nappe de plasma de température uniforme ($T = 7000$ K). La zone de transition protubérance-couronne a été modélisée en supposant simplement que le flux thermique est constant dans cette région, ce qui est en accord avec les observations. Le modèle est globalement en équilibre énergétique, mais il ne résout pas le problème du transport vers et du rayonnement dans la protubérance, de l'énergie apportée par le flux conductif de la zone de transition.

Le transfert de rayonnement a été étudié plus en détail par Heasley et Mihalas (1976), mais la conduction thermique est négligée: les calculs ne s'appliquent donc qu'au coeur des protubérances. Cependant, le calcul du rapport d'intensité des raies est en désaccord avec l'observation des spectres des protubérances. Le résultat a été beaucoup amélioré en utilisant une structure hautement filamenteuse (taux de remplissage de 0·1), ce qui permet au rayonnement photosphérique de pénétrer plus profondément dans le plasma froid. La température centrale obtenue est faible: $T \approx 5000$ K, figure 15 (voir le chapitre précédent pour plus de détail). Ces modèles fibreux permettent de mieux interpréter les observations; une contradiction interne devra cependant être résolue: les calculs de transfert de rayonnement sont effectués
dans l'hypothèse d'une géométrie plan parallèle, alors que manifestement une
structure très inhomogène est présente.

En conclusion, les divers modèles actuels de structure interne des pro-
tubérances ont développé un ou plusieurs aspects du problème, mais sans
prendre en compte simultanément les différents phénomènes. Pour simplifier
ce difficile problème, il me semble justifié d'étudier des plasmas de faible β.
La structure magnétique pourra alors être imposée presque indépendamment
de la répartition spatiale du plasma. Le problème, encore complexe, sera alors
de résoudre le transfert de rayonnement pour les régions internes (T ≈ 7000
K), et de coupler cette description à la zone de transition et à la couronne
qui sont décrites par l'équation de l'énergie (1). Le transfert de rayonnement
dans le milieu optiquement épais de la protubérance devrait permettre de
thermostat er la région interne de la protubérance vers T ≈ 7000 K, quel que
soit l'apport de chaleur conductif de la région de transition protubérance-
couronne. Ce problème est analogue à celui posé par la transition brutale de
la chromosphère à la couronne: le gradient de température doit brusquement
s'annuler au bas de la zone de transition, pour donner une région d'environ
2000 km où la température est presque uniforme. Un modèle de chromo-
sphère, autre que semi-empirique, n'existe cependant pas encore, ce qui est
dû à la difficulté du problème: le transfert de rayonnement est couplé à la
répartition générale de la matière car le plasma n'est ni optiquement mince,
ni épais et un grand nombre de raies interviennent.

1.2.3 Modèles de structures fines

De nombreux modèles ont été proposés pour interpréter les observations. Les
structures observées ressemblent parfois à des arches, lorsqu'on les observe
sur le disque, mais elles apparaissent en majorité verticales au limbe. Poland
et Tandberg-Hanssen (1983) ont proposé leur modélisation par des filets de
matière froide isotherme; Kundu (1986) et Gary (1986) modélisent ces struc-
tures par des filets froids entourés d'une gaine de plasma chaud. Cependant,
le problème de l'équilibre de ces filets le long des lignes de champ est diffi-
cile à résoudre car l'échelle de hauteur gravitationnelle n'est que de 200 km,
et donc beaucoup plus faible que la hauteur des structures (> 5000 km).
Cette difficulté peut être résolue si les filets ne dessinent plus les lignes de
champ magnétique: le plasma est alors piégé dans un creux des lignes de
champ. Cette supposition est en accord avec les observations de Leroy et
al (1983-1984) qui donnent un champ horizontal dans les protubérances. La

<figure><img src="image_82.png" /><figcaption>Figure 16 : Déformation des lignes de champ par la force de gravité.
Cela donne une interprétation pour la direction, à dominante verticale, de la
structure fine. Poland et al (1987).</figcaption></figure>
direction, en moyenne verticale, des structures fines peut alors être créée
par la déformation gravitationnelle de la structure magnétique: figure 16,
*Poland et al* (1987). La condensation étant initialisée dans un creux des
lignes de champ au centre de la structure (par exemple), les lignes de champ
inférieures (respectivement supérieures) seront déformées par la pression (re-
spectivement dépression) magnétique, d'où la création d'une structure al-
longée dans une direction presque verticale. Cela suppose alors une variation
de la température, le long des lignes de champ, identique à celle décrite
au paragraphe précédent pour les modèles de la structure globale. *Rabin*
(1986) a cependant montré qu'un modèle dominé par la conduction thermique
n'était pas suffisant à basse température, car cela conduit à une région froide
de faible taille et donc à un rayonnement de trop faible intensité pour ces
températures. Il propose alors un modèle comprenant 1000 filets froids dans
l'épaisseur d'une protubérance, soient des filets de quelques kilomètres de
diamètre! Une autre possibilité est de créer une région froide de plus grande
taille où la conduction thermique est négligeable: on revient au problème
discuté à la fin du paragraphe précédent.

Les modèles expliquant la formation de la structure fine sont rares. *Kuperus et Tandberg-Hanssen* (1967) ont proposé un modèle de formation dans une nappe de courant qui est illustré par les figures 5 de ce chapitre et les figures 19 du chapitre C. L'interaction de l'instabilité thermique et de l'intabilité résistive dans une nappe de courant (ou "tearing mode") provoque alors des filets de plasmas froids. Cette étude a été précisée par *Van Hoven et Mok* (1984) puis *Tachi et al* (1985). Les calculs de *Malherbe et al* (1987), *Forbes et al* (1988), montrent clairement la formation d'îlots magnétiques entourant des condensations, bien que la résolution du code numérique ne permette pas de décrire des tailles aussi faibles que 1000 km: figures 22 du chapitre C. Cependant, la formation des filets dans ces calculs bidimensionnels sans gravité conduit à des structures horizontales. Les structures fines pourraient aussi résulter d'une instabilité MHD ("ballooning mode"): Hood, communication privée; mais le champ mesuré par *Leroy* (1988) est relativement uniforme!

Les modèles d'évolution des structures fines n'existent pas. Les obser-
vations montrent que leur apparence est modifiée sur des temps de l'ordre
de la dizaine de minutes. Est-ce dû à un déplacement de la matière le long
des lignes de champ ou à la destruction de la condensation suivie de sa re-
formation? Une légère modification de la torsion dans le modèle de Priest

et al (1988) pourrait expliquer cette évolution par le déplacement du creux,
donc du plasma froid, dans la structure magnétique. Les structures fines
restent actuellement mystérieuses. Nous avons abordé le problème de leur
stabilité thermique dans l'article A2, *Démoulin et al* (1987a), en incluant les
conductions thermiques parallèle et orthogonale au champ magnétique.

## 2 Résumé des travaux de thèse

Quatre articles peuvent être regroupés dans ce chapitre. Dans les articles A5 et A12, nous avons calculé le champ magnétique des protubérances solaires en utilisant des conditions aux limites données par les observations dans la photosphère et dans les protubérances. Dans l'article A9, nous proposons un modèle tridimensionnel donnant l'ensemble de la structure magnétique des protubérances en relation avec les concentrations du champ magnétique chromosphérique. Cela permet de localiser l'emplacement des pieds des protubérances dans le réseau de super granulation. Enfin, dans le dernier article (A2), nous étudions la structure et la stabilité des structures fines des protubérances quiescentes. Un cinquième article, en préparation, concerne l'équilibre des configurations Inverses modélisées par une nappe de courant de taille finie. Les problèmes de non équilibre de la partie supérieure de la nappe sont exposés brièvement et des solutions sont discutées.

### 2.1 **Champ magnétique des protubérances solaires quiescentes calculé à partir des conditions aux limites observationnelles.**

Deux méthodes ont été utilisées. La première, partiellement analytique, suppose un champ coronal sans courant (ou potentiel). La deuxième est purement numérique, mais permet d'avoir un champ coronal traversé par des courants; ainsi une importante composante du champ magnétique selon l'axe de la protubérance peut être introduite.
### 2.1.1 Champ magnétique coronal potentiel: article A5

Le but de cet article est d'utiliser les données d'observations du champ magnétique, dans la photosphère et la protubérance, pour en déduire le champ magnétique régnant dans le voisinage d'une protubérance. Cela conduit à résoudre un problème aux conditions aux limites mixtes car la composante verticale du champ est mesurée dans la photosphère, alors que la composante horizontale est connue dans la protubérance. Ce travail a été commencé par Anzer (1972) dans un cas particulier: figures 3; ici nous généralisons cette étude. En premier lieu, la protubérance est supposée être à une hauteur finie, P, au-dessus de la photosphère (figure 1a de l'article A5): cela conduit à reformuler le problème en fonction du champ magnétique (et non de son potentiel comme dans le travail d'Anzer). En second lieu, les calculs sont étendus à la configuration Inverse ($\beta < 0$). Enfin, l'ensemble des solutions possibles est décrit. Le principal résultat concerne l'indétermination de la solution. Les hypothèses faites sont pourtant importantes: la protubérance est une nappe de courant verticale, plongée dans un champ magnétique potentiel symétrique à deux dimensions; mais les seules données observationnelles conduisent à une infinité de solutions. Une solution unique de ce problème nécessite en fait des conditions aux limites qui resteront inobservables même avec des magnétomètres vectoriels tels que Thémis. Elle nécessiterait de mesurer le champ magnétique au-dessus et au-dessous de la protubérance, dans une région coronale. Cette indétermination est, en partie, levée en limitant les configurations possibles par des arguments physiques: les configurations sont astreintes à ne pas avoir de singularité de champ magnétique. Des exemples de configurations Normale et Inverse sont alors calculés, pour des conditions aux limites analogues à celles d'Anzer (figures 3a et 3b). Le champ normal à la protubérance est cependant pris plus faible car on observe que le champ magnétique est très incliné sur le plan de la protubérance. Le paramètre $\beta$ d'Anzer (qu'il a fixé au voisinage de 0.7) doit donc être réduit à des valeurs de l'ordre de 0.2. Des configurations entièrement supportées par le champ magnétique ont été trouvées pour les deux classes de configuration magnétique (Normale et Inverse). Il est cependant à noter que seules des configurations Inverses de topologie identique à celle de la figure 6c peuvent être soutenues entièrement par le champ magnétique. Je détaillerai ce point au paragraphe 2.4.

### 2.1.2 Champ magnétique coronal sans force, linéaire: article A12

L'utilisation des fonctions holomorphes, décrites dans l'article précédent, ne permet pas d'incorporer la composante du champ magnétique parallèle à la protubérance: l'introduction de cette composante fera l'objet de cet article. Le champ magnétique est supposé être sans force et linéaire: $rotB = \alpha B$ ($\alpha$ = constante). Les conditions aux limites sont similaires à celles de l'article A5, si ce n'est que le champ vertical photosphérique est décomposé en série de Fourier: figure 1 de l'article. Le problème est résolu numériquement en discrétisant les équations décrivant le potentiel magnétique d'équilibre $A(x, z)$ sur une grille à deux dimensions. On suppose que le champ décroît vers zéro à grande distance: aussi les conditions aux limites au sommet de la grille sont-elles placées à grande distance, grâce à un maillage non uniforme, pour ne pas influer sur les résultats. Le cisaillement est décrit par le paramètre non dimensionné $\alpha$ normalisé à la taille de la région: $\alpha'$. Ce dernier est inférieur à $2\pi$ pour ne pas créer d'îlots de champ magnétique non physiques.

De même que dans l'article A5, le champ magnétique n'est pas totalement déterminé par les données observationnelles. Ici, l'indétermination se limite à deux paramètres: $A'(0, P)$ et $\alpha'$ (car on ne considère que des champs tendant vers zéro à l'infini). Le paramètre $A'(0, P)$ est le potentiel magnétique à la base de la protubérance, normalisé au potentiel maximum de la configuration; il représente aussi le flux du champ magnétique entre la protubérance et la photosphère (plan $z = 0$) normalisé au flux total traversant la photosphère pour $x > 0$. L'étude de diverses configurations permet de tirer des conclusions générales pour les configurations Normale et Inverse. L'influence des paramètres sur la configuration magnétique, et donc sur la masse supportée par la protubérance, est, dans l'ordre décroissant des effets: $A'(0, P)$, le rapport de l'intensité du champ de la protubérance à celui du champ photosphérique moyen, l'amplitude des harmoniques du champ de base, et enfin le paramètre de cisaillement $\alpha'$. Cette étude justifie donc les hypothèses faites dans le cas d'un champ potentiel: article A5. Bien que les protubérances aient un champ presque colinéaire à leurs axes, seule, la composante normale du champ horizontal joue un rôle important pour le problème aux conditions aux limites définies par la figure 1 de l'article. Le paramètre principal étant $A'(0, P)$, qui est très mal connu, la déduction, à partir des observations, de la configuration magnétique est donc limitée. Une ligne de champ est décrite par $A'(x, z)$ = constante, $A'(0, P)$ permet de localiser le point d'ancrage photo-
<figure><img src="image_83.png" /><figcaption>**Figure 17** : Calcul avec une densité surfacique de masse imposée dans la protubérance, pour une configuration Normale.</figcaption></figure>

La courbe 1 donne la densité obtenue en imposant $Bx = 2$ G dans la protubérance. Cette densité est prise comme état initial du calcul qui consiste à imposer la densité de masse protubérantielle. Les courbes 2 à 5 donnent la densité de masse, imposée après diverses itérations. La topologie de la configuration magnétique passe alors de celle de 2a à celle de 2b avec formation d'un îlot.

sphérique de la ligne de champ qui part du bas de la protubérance ($z = P$). Cela peut fournir une méthode d'estimation de cet important paramètre: lors de l'activation d'une protubérance, de la matière froide peut s'écouler le long des lignes de champ et, ainsi, matérialiser leur position. Cependant, la mesure de $A'(0, P)$ reste très délicate.

Devant cette indétermination, une autre condition aux limites peut être utilisée: celle de la densité surfacique de masse observée. Cela conduit à un problème non linéaire car on impose le produit $Bx.Bz$ à la limite de la protubérance: il est résolu par itérations, en partant d'une configuration obtenue en imposant le champ vertical dans la protubérance (cela pour éviter les discontinuités dans le champ produites lorsque l'on impose le champ horizontal). Deux voies sont possibles: on peut imposer une densité ne dépendant que de la hauteur $z$, ou que de la ligne de champ. La première méthode est applicable aux observations: un exemple est donné à la figure 17. La courbe 1 a été obtenue en imposant le champ horizontal dans la protubérance; puis, par itérations, on impose progressivement une autre fonction de masse: courbe 2. Cette fonction de masse peut être ensuite amplifiée par un facteur multiplicatif: courbe 3 à 5, le calcul s'effectuant toujours par itérations successives. La deuxième méthode permet, en revanche, d'étudier la formation quasi-statique d'une protubérance: la matière dense est ajoutée au sommet des arches, ce qui déforme progressivement la configuration initiale. Ces méthodes sont prometteuses pour l'interprétation des données observationnelles: elles permettent de construire la configuration magnétique à partir des observations de champ magnétique. Une généralisation à trois dimensions est à l'étude. Le champ magnétique sans force et linéaire peut se mettre sous la forme générale:

$$B = \nabla \wedge (\nabla \wedge (\phi \mathbf{e}_z) + \alpha \cdot \phi \cdot \mathbf{e}_z)$$

où la fonction $\phi(x, y, z)$ vérifie $\Delta\phi + \alpha^2\phi = 0$ comme dans le problème à deux dimensions. Cependant, le manque de conditions aux limites observationnelles, déjà important à deux dimensions, est encore plus gênant ici.
<figure><img src="image_84.png" /><figcaption>**Figure 18** : Schématisation de la formation des pieds des protubérances par instabilité de Rayleigh-Taylor.</figcaption></figure>

Le plasma dense du filament est supporté par un champ magnétique uniforme. Nakagawa et Malville (1969).

<figure><img src="image_85.png" /><figcaption>**Figure 19** : Schématisation de la configuration magnétique tridimensionnelle Normale pour une protubérance.</figcaption></figure>

Les lignes de champ entre les pieds sont fortement cisaillées et étirées. Du plasma froid permet de les visualiser comme des structures fines formées de longues arches. Le champ au voisinage des pieds est formé de diverses arches peu inclinées sur la direction du filament. Elles donnent un aspect de rosaces aux pieds des protubérances lorsqu'elles sont vues de dessus avec une bonne résolution spatiale. Poland et al (1987).

## 2.2 Un modèle tridimensionnel de protubérances solaires: article A9

Les pieds des protubérances, régions où le plasma froid de la protubérance rejoint la chromosphère, sont énigmatiques aussi bien du point de vue des observations que des modèles. Trop bas au limbe solaire, ils ne peuvent être observés car la lumière photosphérique est diffusée par l'atmosphère terrestre: le rayon du disque du coronographe doit être supérieur a celui du disque solaire d'environ $10^4$ km pour éliminer cette lumière parasite. Nakagawa et Malville (1969) ont modélisé la formation des pieds par une instabilité de Rayleigh-Taylor d'un plasma dense supportée par un champ magnétique: figure 18. L'ordre de grandeur de l'espacement des pieds observé est obtenu pour des valeurs raisonables des paramètres. Milne et al (1979) ont décrit un modèle unidimensionnel montrant le couplage entre les aspects magnétique et thermique de l'équilibre. Lorsque le $\beta$ du plasma devient supérieur à (ou de l'ordre de) l'unité, le champ ne peut soutenir le plasma, ce qu'ils interprètent comme la formation de pieds. Cependant, cela est contraire aux observations: on observe que les pieds des protubérances se forment en premier, sans que de la matière dense ne chute depuis de hautes altitudes. En fait, les pieds doivent être reliés à la configuration bidimensionnelle du champ magnétique de la photosphère: il est concentré dans le pourtour des cellules de mésogranulation, de taille moyenne identique à la distance moyenne entre les pieds d'un filament.

Cet article (A9) décrit la configuration magnétique coronale résultant d'un champ photosphérique concentré en tubes de flux, répartis périodiquement (figures 2 de cet article). Le champ magnétique est supposé être sans force:

$$ j \wedge B = 0 \\ \text{où } \text{rot}B = \alpha.B $$

Seuls, des champs linéaires sont considérés ($\alpha$ = constante): cela permet d'avoir un modèle analytique où le champ est développé sur une base de fonctions. Ce modèle prévoit la variation de l'intensité et de la direction du champ magnétique à basse altitude: le champ est moins incliné sur le plan de la protubérance et plus intense près des concentrations de champ photosphérique. Cette modulation, le long de l'axe de la protubérance, disparaît avec l'altitude, en accord avec les mesures effectuées à une altitude supérieure
à environ $10^4$ km au-dessus de la photosphère: Leroy et al (1984). La configuration magnétique obtenue est similaire à celle présentée par Poland et al (1987): la figure 19 représente le schéma possible de la configuration magnétique Normale déduite de l'apparence visuelle des protubérances solaires. La présence de matière froide soutenue par le champ magnétique est modélisée dans cet article (A9) par des lignes de courant. Cela conduit à la présence de pieds, bien localisés spatialement (figures 7 et 9 de l'article A9), même pour des champs photosphériques peu concentrés (figures 3 et 9a de l'article A9). Ces pieds sont localisés à la frontière des cellules convectives, c'est-à-dire au-dessus des concentrations de champ photosphérique pour une configuration Normale. En revanche, dans la configuration Inverse, ils sont placés entre les concentrations de champ.

## 2.3 Structure fine dans les filaments solaires: observations et stabilité thermique: article A2

L'observation d'un filament quiescent à une haute résolution spatiale (< 0.5") permet d'étudier la structure fine vue en projection sur le disque solaire. Les vitesses observées étant bien inférieures aux vitesses sonore et d'Alfven, nous présentons un modèle statique. La structure thermique est simplifiée par un damier à 2 dimensions, formé de régions froides ($T \approx 10^4$ K) et chaudes ($T \approx 10^6$ K): figure 1 de l'article. Le champ magnétique est supposé uniforme, et seule la stabilité thermique du plasma est étudiée. Les flux de conduction thermique, parallèle et orthogonale au champ magnétique, sont estimés par différences finies. L'équilibre périodique n'est trouvé possible que dans un domaine relativement étroit du chauffage et de la pression (environ 2 décades). Les temps d'instabilité sont caractéristiques de la fonction de rayonnement: de quelques minutes à quelques heures. Le plasma chaud (instable) est stabilisé par le plasma froid si la taille des cellules est suffisamment faible: typiquement, si elle est inférieure à $5.10^7$ m parallèlement au champ magnétique ou inférieure à $10^4$ m orthogonalement. Ces valeurs encadrent la taille typique des structures fines: $10^6$ m. Il est cependant à noter que l'équation d'équilibre thermique est fortement non linéaire, ce qui conduit, par une résolution complète de l'équation, à des tailles de la région froide beaucoup plus faibles que celles de la région chaude. Les tailles indiquées ci-dessus sont donc caractéristiques de la dimension de la structure chaude.

On en conclut que seule, la conduction parallèle au champ magnétique peut produire des structures froides de l'ordre de $10^6$ m.

## 2.4 Discussion de l'équilibre de la configuration Inverse

### 2.4.1 Equilibre

Dans le modèle de configuration Inverse défini par Kuperus et Raadu (1973), figure 5, la protubérance est soutenue par l'ancrage des lignes de champ dans la photosphère, car la force de Lorentz du champ magnétique créé par la distribution photosphérique sur les courants de la protubérance est dirigée vers le bas. L'équilibre vertical d'une ligne de courant d'intensité I s'écrit alors:

$$ -\lambda g + IB_x + R(h) I^2 = 0 \\ \begin{array}{l} \text{gravité} \\ \text{force} \\ \text{de} \\ \text{Laplace} \end{array} \qquad \begin{array}{l} \text{Répulsion de l'image:} \\ \text{modélisation de l'ancrage} \\ \text{des lignes de champ dans} \\ \text{la photosphère} \end{array} $$

avec $\lambda$: densité massique de la protubérance,

g: gravité solaire,

Bx: composante horizontale du champ coronal au niveau de la ligne de courant ($Bx<0$ ici),

h: hauteur de la protubérance au-dessus de la photosphère, et $R(h) = \mu_0/4\pi h$ pour un champ coronal potentiel.

**Remarque:** les conventions sont ici $I > 0$ pour une configuration Normale, et $I < 0$ pour une configuration Inverse, mais sont différentes d'un auteur à l'autre!

Pour une configuration Normale, I est positif, et l'équilibre est obtenu approximativement par l'égalité des deux premiers termes. Pour une configuration Inverse, I est négatif, et le dernier terme doit égaler les deux premiers. Les différents paramètres étant fixés, cela conduit à une intensité au moins 10 fois plus intense que dans le cas précédent (voir l'article A4).
### 2.4.2 Problème dans la modélisation de la configuration Inverse

Les modèles de configuration Inverse antérieurs à 1988 développent l'idée initiale de Kuperus et Raadu (1974). Cependant, le support de la protubérance par la répulsion de la protubérance sous la photosphère pose de nombreux problèmes. On peut les résumer en trois points:

- les hauteurs d'équilibre sont beaucoup plus faibles que pour la configuration Normale, pour la même valeur des paramètres; cela est contraire aux observations car les grandes protubérances quiescentes sont de configuration Inverse. Des paramètres très différents sont donc nécessaires pour chaque configuration, en particulier pour l'intensité I du filament;

- le mécanisme pouvant former un courant $I < 0$ n'est pas connu; un candidat hypothétique est la reconnexion des lignes de champ forcée par des mouvements photosphériques convergents;

- une nappe de courant ne peut être entièrement soutenue dans un champ dipolaire fermé.

Anzer (1984) a montré l'importance du dernier point pour des champs potentiels, figure 7, mais cela se généralise à des champs sans force. Dans une configuration Inverse, le support ne provient que de la répulsion des courants images, de ceux de la protubérance, par rapport au plan photosphérique. Dans l'équilibre de la structure interne, intervient aussi l'interaction entre les courants de la protubérance. L'équilibre à l'altitude z, où circule le courant surfacique $i(z)$, dans l'hypothèse d'un champ coronal potentiel s'écrit alors:

$$-\mu_0 g + i(z) \cdot B_z + i(z) \cdot \int_0^\infty \frac{\mu_0 \cdot i(z) \cdot dz'}{2\pi (z' - z)} + i(z) \cdot \int_0^\infty \frac{\mu_0 \cdot i(z) \cdot dz'}{2\pi (z' + z)} = 0$$

Le troisième terme est dû à l'attraction des courants de la protubérance. Si l'on se place près du sommet, z est supérieure à z' dans la majeure partie des intégrales, les trois premiers termes sont donc négatifs. De plus les deux derniers termes peuvent se regrouper en une seule intégrale qui est négative, si z est proche du haut de la protubérance et si la densité de courant n'a pas de singularité. Le haut de la protubérance ne peut donc être en équilibre, et la structure se contracte, sans parvenir à trouver un équilibre. Cela provient de la décroissance de la force d'interaction (entre deux courants) avec la distance: l'attraction des courants de la protubérance est plus importante que la répulsion des courants images, au sommet de la protubérance. Cet effet est amplifié dans un champ sans force linéaire car les courants volumiques

écrivent la force d'interaction entre deux courants: Amary et Aly (1988). Le support par la répulsion de l'image n'est donc pas suffisant.

### 2.4.3 Résumé des travaux

On peut avoir un support à la partie supérieure de la protubérance par un gradient de la pression magnétique associé à la composante By du champ. Cependant, la variation de l'intensité du champ nécessaire est importante et cette composante By doit décroître avec la hauteur de la protubérance, ce qui est contraire aux observations. Il existe, actuellement, deux modèles possibles : le support dans une région quadrupolaire: figure 6c, par exemple, et la formation dans un champ torsadé: Priest et al (1988). Ce dernier modèle sera discuté plus en détail dans le chapitre C portant sur l'évolution des protubérances. La première solution est actuellement étudiée en suivant l'évolution de deux régions bipolaires qui convergent l'une vers l'autre. Une nappe de courant se forme alors au contact des deux régions bipolaires, et un processus de formation avec une reconnexion lente des lignes de champ peut alors avoir lieu. Le processus de formation est analogue à celui proposé par Kuperus et Raadu (1974): figures 5.

## 3 Conclusion

Les observations nous montrent des objets étonnants, qui se forment rapidement (quelques heures, à quelques jours), qui évoluent et changent d'apparence continuellement, et qui donnent une éruption soudaine en quelques dizaines de minutes, pour se reformer ensuite à la même position. La structure spatiale est aussi complexe: structures fines et pieds n'ont pas fini de nous intriguer!

Les progrès effectués dans cette thèse, sur la structure magnétique, peuvent se résumer en quatre points principaux. D'abord, l'ensemble des solutions, pour des conditions aux limites observationnelles données (dans la photosphère et dans la protubérance), a été exploré. Ensuite, la composante du champ magnétique parallèle à la protubérance a été introduite dans les modèles; les observations (chapitre A) ont montré son importance pour la formation et la stabilité des protubérances. En troisième lieu, un modèle tri-dimensionnel a été construit, ce qui permet de prévoir la configuration du
<figure><img src="image_86.png" /><figcaption>Figure 20 : Schéma de l'aspect des protubérances quiescentes.</figcaption></figure>

(a) Une protubérance jeune, avec peu d'arches. Les pieds sont formés en premier et la structure s'allonge progressivement en formant de nouvelles arches.

(b) Une protubérance bien développée. Deux systèmes de structures fines sont visibles: l'un formant les arches, l'autre donnant les structures quasi-verticales.

(c) Une protubérance ancienne. La structure fine est formée d'un ensemble de filets froids très torsadés.

Rompolt (1987).

champ magnétique dans les pieds des protubérances (celle-là est encore inconnue d'après les observations). Enfin, l'étude systématique des propriétés des configurations Normales et Inverses a été réalisée. On ne ne peut cependant pas encore confronter ces configurations aux observations car beaucoup de paramètres sont mal connus. De nombreux progrès sont attendus grâce à l'observation des champs vectoriels avec le télescope Thémis. L'étude des oscillations dans les protubérances pourra aussi fournir de nombreuses informations sur la distribution du champ magnétique et la densité des protubérances.

La structure magnétique et thermique interne est modélisée séparément de la structure externe du champ magnétique, et, de plus, chaque modèle ne considère que quelques aspects du problème. Une protubérance est un milieu complexe où le plasma interagit à la fois avec le champ magnétique et le rayonnement. Le problème majeur est de comprendre le mécanisme de stabilisation de la température, vers 7000 K, dans la protubérance, puis de relier cette région à la zone de transition dominée par la conduction thermique. Un problème analogue est présent au niveau de la chromosphère. L'interaction plasma-champ magnétique me semble secondaire, car le β du plasma étant faible, le champ magnétique impose sa géométrie au plasma. Il sera ensuite nécessaire de coupler ce modèle interne à la structure magnétique externe, si possible dans un modèle tridimensionnel. Ce dernier aspect est très important car il donne le lien avec la convection, et donc avec l'ensemble de la dynamo solaire. Un modèle d'évolution quasi-stationnaire, en liaison avec les rouleaux de convection est à construire. De même, si quelques modélisations des structures fines ont été développées, celles-ci gardent en grande partie leurs mystères. Pourquoi ont-elles un aspect si rapidement changeant? De façon générale, quelques pièces du puzzle ont certainement été réunies, mais peut-on réellement estimer le nombre de pièces manquantes? Une schématisation, même simple, de l'apparence des protubérances laisse rêveur: figure 20.

A-t-on des aspects différents de l'évolution d'un même objet, ou les trois classes (protubérances quiescentes de haute et de basse altitude, et protubérance de plage, mais on peut en définir beaucoup plus: D'Azambuja et D'Azambuja, 1948) sont-elles des entités différentes? Pour ce faire, il faut comprendre comment se forment et évoluent les protubérances solaires.

# 4 Publications
# - A5: The magnetic field around quiescent solar prominences computed from observational boundary conditions

P. Démoulin<fnref n="1" />, J.M. Malherbe<fnref n="1" />, and E.R. Priest<fnref n="2" />

Received March 8, accepted August 17, 1988

**Summary.** A generalization of Anzer's model (1972) for quiescent prominence support is presented. The coronal field is assumed to be current-free except inside the prominence where currents support dense material against gravity. The prominence is taken to be an infinitely thin current sheet of finite extent along the vertical axis. The hypothesis of two-dimensional fields allows us to use complex functions to solve the mixed boundary problem which is defined by the observed vertical field in the photosphere and the horizontal magnetic field in the prominence. These boundary conditions are not sufficient to determine a unique solution for the magnetic field. We are able to decrease the indeterminacy by physical considerations and present some models for Normal (N) and Inverse (I) configurations. In both cases it is possible to find field configurations which can support the finite prominence against gravity.

**Key words:** solar prominences - magnetic field

## 1. Introduction

Prominences are thin condensed sheets of cold material ($T \approx 8000$ K) suspended in the hot corona ($T \approx 10^6$ K). As the pressure is almost the same as the coronal one, the prominence density is typically one hundred times greater than the coronal density and so this cold material needs to be supported against gravity by magnetic fields.

Available observational data are the electron density and the temperature of the cool sheet, the horizontal magnetic field in the prominence from limb observations and the magnetic flux pattern in the photosphere from disc observations. All these quantities have been reviewed by Hiroyama (1985).

Theoretical models simplify the global structure of prominences, in general by excluding the periodic feet of prominences and modelling the prominence by a straight line of current or by a thin current sheet. Two basic geometries have been proposed by Kippenhahn and Schlüter (1957) and Kuperus and Raadu (1974): they are respectively of the Normal and Inverse type. In the N configuration, the prominence field has the same direction as the potential field created by the photospheric pattern. In the I configuration, the field has the opposite direction and so it needs, for a current free model, an X point, below the prominence,

which separates the prominence field region from low-lying closed field lines.

Many authors have tried to compute the coronal field from photospheric data. The following have assumed a potential field in the corona: Schmidt (1964), Adams and Pneumann (1976), Sakurai (1982). These calculations cannot give realistic fields near prominences since there must be current to support them. Other works supposed that the field is a linear force-free field (e.g. Nakagawa and Raadu, 1972). These models were more realistic than previous ones, but they cannot use the measured magnetic field in the prominence from limb observations. Moreover as they only use the photospheric magnetic field pattern, they must impose more or less realistic boundaries conditions in the corona at the computation box edges. (Such boundary conditions are implicit if they use Fourier transforms, assuming a periodic magnetic field.) The choice of boundary conditions allows a certain quantity of magnetic flux in the box which changes greatly with different kinds of boundary conditions. Then we must have a box greater than the prominence height to avoid these effects, but it is not evident that we can assume that the field is a constant-α force-free field everywhere. In particular, there are large concentrations of magnetic fields in the photosphere like sunspots or pores which are more complex than a constant-α force-free field. We conclude that, even if such numerical linear force-free field codes might be useful to understand the field behaviour near a prominence, the solutions are not necessary realistic.

The aim of this paper is to use observational magnetic field data, to understand the field topology near a prominence. It involves a mixed boundary problem: we know the vertical field in the photosphere and the horizontal one in the prominence. This work has been started by Anzer (1972) who modelled the prominence as an infinitely thin current sheet stretching a distance H along the vertical axis from the photosphere. For simplicity the field is assumed potential in the corona and is two-dimensional. The field is computed by using complex functions. The advantage of this method is that the observational field in the photosphere and in the prominence are automatically included. One limitation is that we cannot impose the density behaviour in the prominence: it is a result of the computation, and Anzer found a downward force rather than supporting force in the low part of the prominence in all the cases. One possibility to avoid this is to move the base of the prominence above the photosphere. Then the prominence is assumed to be present only in the interval $[P, Q]$ of the vertical axis. The generalization of Anzer's work to

<footnote n="1">¹ Observatoire de Paris, Section de Meudon, DASOP (UA 326), F-92195 Meudon Principal Cedex, France</footnote>

<footnote n="2">² University of St. Andrews, Department of Applied Mathematics, St. Andrews KY 16 9SS, Scotland, UK</footnote>

<footnote>Send offprint requests to : P. Démoulin</footnote>
this case needs the introduction of a free parameter: the flux between the photosphere and the prominence base as discussed at the end of page 333 of Anzer's paper. This parameter could be adjusted to give a "good field behaviour". To avoid this, we preferred to compute directly the magnetic field, as discussed in Sect. 4.

Another way to build a prominence magnetic field with complex functions, and so potential fields, has been investigated by Malherbe and Priest (1983). They proposed two-dimensional analytical magnetic configurations for both *I* and *N* configurations. The advantages of this method are simplicity, and the production of a great number of models with the two-dimensional topology in both *I* and *N* configurations, and with a positive support in the whole prominence. The disadvantages are that both magnetic fields and prominence densities cannot be imposed.

We want here to generalize the paper of Anzer to more general configurations, in particular to *I* configurations, with a finite current sheet in the segment [*P*, *Q*] of the vertical axis. This mixed boundary problem with photospheric and prominence observational fields is described in Sect. 2 and is solved using complex variables in Sect. 3. It leads to an indeterminacy as the data are not sufficient to ensure the uniqueness of the solution. We limited it using physical considerations and discuss the relevant liberty in Sect. 4.

## 2. Description of the model

### 2.1. Assumptions

We make several assumptions in our model, as follows.

#### 2.1.1. Two-dimensional geometry

The spatial coordinates are described by a cartesian coordinate system. The prominence is supposed to be in a vertical plane with *x*-axis horizontal and perpendicular to this plane, *z*-axis along the axis of the prominence and *y*-axis in the vertical direction (Fig. 1).

Although a prominence appears on the sun as a periodic structure falling down to the chromosphere at points named "feet", we assume here that the global magnetic field can be reasonably well approximate by a two-dimensional model: we assume that all quantities are independent of the *y* coordinate (An analytical model for prominence feet is presented in Démoulin and Priest, 1988, in relation with the supergranular chromospheric pattern).

Observations do not show significantly different magnetic fluxes both sides of prominences, so that we suppose the magnetic field to be symmetric with respect to the prominence plane.

#### 2.1.2. Field structure

The field is taken to be current-free (potential) throughout the corona, (except in the prominence). It satisfies:

$$ \text{rot } \mathbf{B} = 0 \quad \text{and } \text{div } \mathbf{B} = 0 $$

which gives in two dimensions:

$$ \frac{\partial B_x}{\partial y} = \frac{\partial B_y}{\partial x} \quad \text{and} \quad \frac{\partial B_x}{\partial x} + \frac{\partial B_y}{\partial y} = 0 $$

<figure><img src="image_87.png" /></figure>

<figure><img src="image_88.png" /><figcaption>Fig. 1. a The 2D geometry of the computations. The prominence is modelled by a current sheet between *P* and *Q* on the vertical axis. We impose the horizontal field ($B_x$) at the prominence location and the vertical field ($B_y$) at the photospheric level. b The transformation, from the plane to the half plane using the conform transformation $\xi \to \xi^2 = \xi^4$</figcaption></figure>

This means that $B_y + iB_x$ is an analytic function of the variable $\xi = x + iy$. The field can also be represented by one of the two following potentials $\varphi$ or $\psi$,

$$ \text{with } \mathbf{B} = -\text{grad} \quad \mathbf{B} = \text{rot}(\psi \mathbf{e} z) $$

The function $A(x, y) = \psi + i\varphi$ is analytic and $B(x, y) = -dA(x, y)/d\xi$

The field ($B_x$, $B_y$) can be described either by $B(\xi)$ or $A(\xi)$. Anzer (1972) has chosen the second possibility, and here we prefer the first one for the reason described in the introduction: there is no extra free parameter, such as the flux below the prominence. The unknown quantity is only the function $G(\xi)$ (see below).

As the field is potential and two dimensional, the component $B_x(x, y)$ needs to be a constant. Since it does not affect the mass balance we forget it in the analysis below.

#### 2.1.3. Prominence shape and current

The prominence is represented by an infinitely thin current sheet in the interval [*P*, *Q*] of the vertical axis. If *d* is the thickness of the prominence we define the surface density $\lambda$ and the surface current *J* by taking the limit $d \to 0$:

$$ \lambda = \lim_{d \to 0} \int_{-d/2}^{d/2} e \, dx $$

$$ J = \lim_{d \to 0} \int_{-d/2}^{d/2} j_z \, dx $$

and the integration or rot $\mathbf{B} = \mu_{0}j$ from $x = -d/2$ to $x = d/2$ gives, at the limit $d \to 0$

$$ B_y(0^+, y) - B_y(0^-, y) = \mu_0 J $$

if we neglect the contribution of $\partial B_x/\partial y$ (see Sect. 2.2.2).

From the symmetry condition $B_y(0^+, y) = -B_y(0^-, y)$, we find:

$$ J = \frac{2}{\mu_0} B_y(0^+, y) \quad (1) $$

#### 2.1.4. Static model

The model is magnetostatic since the time-scale for quiescent filament evolution (except in eruptive phases) is much greater than the sound or Alfvén travel-time of the structure (days to months compared to a few minutes).

The filament is in equilibrium between gravity and magnetic forces.

As we must fix the vertical extension of the prominence and the horizontal field in the prominence, these calculations cannot be extended to an evolution model. The aim is just to determine the magnetic configuration near a prominence, and we use ideal MHD. The prominence is infinitively thin and so we do not introduce the thermal coupling of the filament to the photosphere or the corona along field lines.

### 2.2. Mass support

#### 2.2.1. Force balance in a prominence

With the assumptions of Sect. 2.1 prominence material is subject to four forces:

- the solar gravity,

- the external Laplace force between the prominence current and the magnetic field,

- the internal repulsion force between the current of the prominence,

- the external support force by the repulsion of an image prominence current below the photosphere.

The last force is the line tying of the magnetic fields in the dense photosphere. It was first introduced by Kuperus and Raadu (1974).

The balance between all these forces gives the surface mass densities of the prominence when we know the magnetic fields $B_x$ and $B_y$ at the prominence location.

#### 2.2.2. Total magnetic field support

The *x* component of the Lorentz force $\mathbf{j} \wedge \mathbf{B}$ is even in *x* and then the horizontal equilibrium is satisfied for the assumed symmetrical prominence.

The *y* component of the Lorentz force is:

$$ f_y = \frac{B_x}{\mu_0} \frac{\partial B_y}{\partial x} - \frac{1}{2\mu_0} \frac{\partial B_x^2}{\partial y} \quad \text{Both terms are odd in } x $$

The first term is the magnetic tension and the second the magnetic pressure. We integrate this force over the prominence width from $x = -d/2$ to $x = +d/2$, and the magnetic force on a small segment *dy* of the prominence becomes:

$$ dF_B = \frac{d}{\mu_0} \int_{-d/2}^{d/2} \left( B_x \frac{\partial B_y}{\partial x} - \frac{\partial B_x^2}{2\partial y} \right) dx $$

We assume that the horizontal magnetic field $B_x$ does not change greatly across the prominence, then:

$$ dF_B = \left[ 2B_x(0, y) \cdot B_y\left(\frac{d}{2}, y\right) - \frac{1}{2} \frac{\partial B_x^2(0, y)}{\partial y} \cdot d \right] \frac{dy}{\mu_0} \quad (2) $$

using the symmetry condition

$$ B_y\left(\frac{d}{2}, y\right) = -B_y\left(-\frac{d}{2}, y\right) $$

Observations by Leroy et al. (1983), (1984) provide for quiescent prominences:

$$ B_p = \sqrt{B_x^2 + B_z^2} \approx 8G \quad 0 = \arcsin\left(\frac{B_x}{B_p}\right) \sim 10^0 \text{ to } 25^0 \text{ so } B_x \sim 1 \text{ to } 4G $$

$$ \text{and } \frac{\partial B_p}{\partial y} \le 0.5 \text{ G/10 Mm} $$

So the ratio between the two terms of Eq. 2 is:

$$ R = \frac{\frac{\partial B_x}{\partial y} \cdot d}{2 \cdot B_y\left(\frac{d}{2}, y\right)} \le \frac{1}{16 \cdot B_y\left(\frac{d}{2}, y\right)} $$

with a typical size *d* = 5 Mm

$B_y$ cannot be observed. For $B_y \ge B_x = B_p \sin 0$, the magnetic pressure term is negligible compared to the magnetic tension term. The magnetic pressure term is only important for very low values (typically 0.1G) which gives an unrealistic and very low prominence mass. This term provides also a downward force and so does not contribute to support of the prominence. Hence, for all these reasons, this term should play a secondary role in realistic prominences and we set it to zero, assuming constant $B_x$ in the prominence. (We neglect the vertical gradient found by Leroy et al.). The slowly increasing horizontal field component, observed in quiescent prominences, is in fact not important either for mass balance, or for field line topology as shown by the present computations.

#### 2.2.3. Current forces

The repulsive force exerted on a part *dy* of the prominence due to the image prominence current below the photosphere is:

$$ dF_{RI} = \left( \int_{P}^{Q} \frac{\mu_0 \cdot J(0, y_1) dy_1}{2\pi(y_1 + y_2)} \right) \cdot J(0, y) dy $$

and, using (1) we get:

$$ dF_{RI} = \frac{2}{\pi \cdot \mu_0} \left( \int_{P}^{Q} \frac{B_y(0, y_1) \cdot dy_1}{y + y_1} \right) \cdot B_y(0, y) \cdot dy \quad (3) $$

We must also consider the action of currents exerted by the other parts of the prominence (which is attractive) on element *dy*:

$$ dF_{AP} = \frac{2}{\pi \cdot \mu_0} \left( \int_{P}^{Q} \frac{B_y(0, y_1) dy_1}{-y + y_1} \right) \cdot B_y(0, y) dy \quad (4) $$
This integral is a principal part whose convergence for $y = P$ or $y = Q$ is assured by $\lim_{x \to 0} B_y(0, y) = 0$ and $\lim_{x \to 0} B_y(0, y) = 0$. (The current density goes to zero at the ends of the prominence and so the vertical magnetic field is a continuous function at $y = P$ and $y = Q$.) This integral is positive in the low part and negative in the upper part of the prominence: the upper currents exert an attraction, and so a support against gravity, on the low part of the prominence. But as it is an internal force in the prominence itself, this term has no global effect on the total mass support and the integral

$$ \int_P^Q B_y(0, y) \left( \int_P^Q \frac{B_y(0, y_1) dy_1}{-y + y_1} \right) dy \text{ vanishes.} $$

These forces, given by (3) and (4), have not to be added to other forces for the mass balance as it is included in (2): the photospheric pattern as well as the prominence current, and the image current creates a magnetic field, and the data of the horizontal field $B_x$ in the prominence includes this effects. Even if $df_{RI}$ and $df_{AP}$ have no effect in the force balance, it is interesting to consider how great is the repulsion support and the "self pinch" effect in the prominence. A large value of this last term means that the prominence will evolve certainly to a current line configuration, which is not observed. So configurations with big currents (or $B_y$) in prominences lead to such difficulties. This self pinch effect due to large currents has been analyzed by Anzer and Priest (1984) using complex variables.

### 2.2.4. Force balance

The vertical equilibrium force gives the surface mass density as a function of the magnetic field on the vertical axis:

$$ \dot{\lambda}g = \frac{2}{\mu_0} B_x(0, y) \cdot B_y(0, y) \quad (5) $$

for a width $d$ which is taken infinitely small. We normalize quantities to $B_0$ and $L$ defined by.

$$ B_x = B'_x B_0 \qquad B_y = B'_y B_0 \qquad y = y' \cdot L; \quad (5) \text{ becomes:} $$

$$ \dot{\lambda} = \frac{2 \cdot B_0^2}{\mu_0 g} B'_y(0, y') \cdot B'_x(0, y'). $$

The relative contribution to the density of image repulsion ($\epsilon = +1$) and attraction of prominence currents ($\epsilon = -1$) are:

$$ \frac{\dot{\lambda}_e}{\dot{\lambda}} = \frac{1}{\pi \cdot B'_x(0, y')} \cdot \int_P^Q \frac{B'_y(0, y') dy_1'}{\epsilon y_1' + y_1'} $$

For smooth field behaviour in the prominence, the surfacic mass density is of the order of $\dot{\lambda}_0 = 2B^2/(\mu_0 g)$ which gives with $B_0 = 3G$; $\dot{\lambda}_0 \approx 7 \cdot 10^{-4} \text{ kg m}^{-2}$.

This is in agreement with surface densities deduced from observations of electron density in quiescent prominences: with $n_e \approx 10^{11}$ electrons per cm$^3$ an ionization degree of 0.1 and a width $d = 5$ Mm, we find $\dot{\lambda} \approx 10^{-4} \text{ kg m}^{-2}$. This order of magnitude is also consistent with the recent observations of Bommier et al. (1986) suggesting that $n_e \approx 10^{10}$ electrons per cm$^3$ and an ionization degree of 1.

We compare now the magnitude of the different supporting forces. If we assume that the current density is almost constant in the prominence we have:

$$ \dot{\lambda}_{e=1} = \frac{1}{\pi \cdot B'_x(0, y')} \cdot \int_P^Q \frac{B'_y(0, y_1') \cdot dy_1'}{y' + y_1'} \approx \frac{B'_y}{B'_x \pi} \log_e \left( \frac{y' + Q'}{y' + P'} \right) \\ \le \frac{B'_y}{\pi} \log_e \left( \frac{P' + Q'}{2P'} \right) $$

$$ \text{for } Q' \approx 3P', \text{ for example, } \log_e \left( \frac{P' + Q'}{2P'} \right) \approx 0.3 \\ \text{and } \frac{\dot{\lambda}_{e=1}}{\dot{\lambda}} \approx \frac{B'_y}{10 \cdot B'_x} $$

It means that the repulsion by the prominence image below the photosphere provides an efficient support mechanism only when $B'_y > 10 B'_x$, which implies that the magnetic field changes its direction rapidly from the horizontal one in the center of the prominence to an almost vertical one at the edges. The peculiar behaviour would be enhanced by a powerful "self pinch" effect as discussed before. (In the equilibrium equation, the effects of the currents in the prominence itself are more important than those of the image currents below the photosphere, since they are closer). For N configurations, the current (or $B_y(0, y)$) does not need to be large: both image and self current effects are negligible. For I configurations, the "potential photospheric pattern" magnetic force is downward and there are two ways of supporting the prominence. In the first case, the repulsion force due to image currents is large, and so $B_y(d/2, y) \gg B_x(d/2, y)$, with the difficulties discussed above. The second case has support by a horizontal field which does not connect directly the photosphere: this model needs a quadrupolar region (or open magnetic field lines) or helical configuration as proposed by Pneumann (1983). From an observational view, Tang (1986) shows that quiescent prominences are formed more likely on neutral line between bipolar regions than on neutral lines inside bipolar regions. As quiescent prominences are, mostly of the I type, it means that we may investigate quadrupolar configurations for this case.

## 3. Mathematical formulation for the half plane $y \ge 0$

### 3.1. The Dirichlet problem

If we known an analytical function $f(\xi_c)$ along a closed boundary C, we know it everywhere in the surface delimited by this contour and the solution is given by

$$ f(\xi) = \frac{1}{2i\pi} \int_{\phi} \frac{f(\xi_c)}{\xi_c - \xi} d\xi_c \quad (6) $$

We can apply this concept to the region between the real axis $Ox$ and a semi-circle centered at $(0, 0)$ and of radius $R$. If we suppose that the solution $f(\xi = re^{i\theta})$ has a finite limit, $f(\infty)$, when $r$ increases to infinity, the integration on the half circle has the finite limit $i \cdot f(\infty)$. $\pi$ and (6) may be expressed as an integral along the $x$ axis:

$$ f(\xi) = \frac{1}{2i\pi} \int_{-\infty}^{+\infty} \frac{f(x) dx}{x - \xi} + \frac{f(\infty)}{2} \quad (7) $$

Using the same technique with $\bar{\xi}$, instead of $\xi$, and taking the conjugate of (7), we get:

$$ 0 = -\frac{1}{2i\pi} \int_{-\infty}^{+\infty} \frac{\bar{f}(x) dx'}{x - \bar{\xi}} + \frac{\bar{f}(\infty)}{2} $$

Adding this equation from (7) we can now choose to prescribe the imaginary part of $f(x)$ along $x$ axis.

$$ f(\xi) = \frac{1}{\pi} \int_{-\infty}^{+\infty} \frac{\operatorname{Im}(f(x)) dx}{x - \bar{\xi}} + \operatorname{Re}(f(\infty)) \quad (8) $$

### 3.2. The mixed boundary-value problem for the half plane

In the prominence magnetic field we want to impose as observed quantities $B_y$ in the photosphere and $B_x$ in the prominence so we have to solve a mixed boundary value problem for the analytical function $B(\xi) = B_y + iB_x$. The Dirichlet problem may be solved either for the real or the imaginary part of the function $f(\xi)$ specified on the boundary, but not for mixed conditions. Muskhelishvili (1952) gives the solution of this problem for the half plane, introducing another analytical function $g(\xi)$ which converges at infinite distance, and is purely real or purely imaginary along the boundary (namely the real axis), so that (8) could be rewritten by changing $f(\xi)$ to $f(\xi) \cdot g(\xi)$ as:

$$ f(\xi) \cdot g(\xi) = \frac{1}{2i\pi} \int_{-\infty}^{+\infty} \frac{g(x) \cdot f(x) - \bar{g}(x) \cdot \bar{f}(x)}{x - \xi} dx + \frac{f(\infty) \cdot g(\infty) + \bar{f}(\infty) \cdot \bar{g}(\infty)}{2} $$

$$ \text{so: } f(\xi) \cdot g(\xi) = \frac{1}{\pi} \int_{g \text{ Real}} \frac{\operatorname{Re}(g(x)) \cdot \operatorname{Im}(\bar{f}(x))}{x - \xi} dx + \frac{1}{\pi} \int_{g \text{ imaginary}} \frac{\operatorname{Im}(g(x)) \cdot \operatorname{Re}(\bar{f}(x))}{x - \xi} dx + \frac{f(\infty) \cdot g(\infty) + \bar{f}(\infty) \cdot \bar{g}(\infty)}{2} \quad (9) $$

### 3.3. Application to solar prominences

The geometry of the boundary conditions is shown in Fig. (1a). The boundary condition on the imaginary (vertical) axis outside the segment $[P, Q]$ is not imposed by magnetic field observations, but as we have no prominence here, so no current, and as the field has to be chosen symmetric with respect to the vertical axis, the vertical field $B_y$ vanishes in this region. In order to use (9) we have to transform the positive part of the imaginary axis into the negative part of the real axis. This transformation, from the half-plane to the plane, can be realized using the conformal transformation $\xi \to \xi^2 = \xi'$. When doing this transformation, we pass from Fig. 1a, to the Fig. 1b, and become able to use (9) with the variable $\xi'$. The inverse transformation $\xi = \sqrt{\xi'}$ gives the solution for solar prominences (mixed boundary-value problem for the quarter plane):

$$ B(\xi) \cdot G(\xi) = \frac{2}{\pi} \int_P^Q \frac{\operatorname{Re}(G(it)) \cdot B_x(t) \cdot t \cdot dt}{t^2 + \xi^2} + \frac{2}{\pi} \int_0^\infty \frac{\operatorname{Im}(G(t)) \cdot B_y(t) \cdot t \cdot dt}{t^2 - \xi^2} - B_x(\infty) I_m G(\infty) \quad (10) $$

where the function $G(\xi)$ is:

- real on the interval $y \in [P, Q]$,

- imaginary on the positive part of the $x$-axis and on the positive part of the $y$-axis except in $[P, Q]$,

- has a finite value $g(\infty)$ at infinity,

We can notice that the conditions on the function $G(\xi)$ are very slack and we have to restrict this choice further using physical considerations.

There is also an indeterminacy on the field so that we can add a function $G'(\xi)$ to the computed magnetic field without changing the boundary conditions. This indeterminacy is infinite since the constraints on $G'(\xi)$ are the same as for $G(\xi)$.

This is not in contradiction with the Dirichlet problem of Sect. 3.1. which specified that the solution for $f(\xi)$ is unique: here the solution for $B(\xi) \cdot G(\xi)$ is unique when we have specified its imaginary part on the whole boundary (positive $x$ and $y$ axis and finite value at infinity). The fact that we have to find a good function $G(\xi)$ leads to an indeterminacy for $B(\xi)$. We propose a way to have only one solution by determining $G(\xi)$ at the end of the next section but this requires observational data that are not available at this time: we need both magnetic field components $B_x$ and $B_y$ on the entire positive $x$ and $y$ axes. As it will not be possible to measure all these quantities still for a long time (a vector magnetograph operating both on the disk and at the limb is necessary), we discuss, in the next Sect. (4), with the actually available data, a way to reduce the indeterminacy on $G(\xi)$ and so on $B(\xi)$.

### 3.4. Behaviour of the integrals

#### 3.4.1. Magnetic field computation in the prominence

The magnetic field is given by (10), which we rewrite as:

$$ B(iy) = \frac{2}{\pi \cdot G(iy)} \left( - \int_P^Q I_y(y, t) dt + \int_0^\infty I_x(y, t) dt \right) \quad (11) \\ \text{(taking } B_c(\infty), \operatorname{Im}(G(\infty)) = 0 \text{)} $$

For $y$ between $P$ and $Q$, $G(\xi)$ is real. The second integral, along the $x$-axis, has no singularity and is real, and so contributes only to the vertical field at the edge of the prominence. The first integral, on the $y$-axis, is singular and is not real. The real part of $I_y$ has two peaks of opposite sign which grow to infinity when $|t-y|$ decreases to zero on both sides of the point at height $y$, and $R_c(\int_Q^P I_y dt)$ is a principal part which contributes to the vertical field at the edge of the prominence. The imaginary part of $I_y$ has the limit of a Dirac distribution when $x$ decreases to zero, so that $\operatorname{Im}(\int_P^Q I_y \cdot dt)$ gives the right contribution to set the horizontal magnetic field in the prominence to its boundary value.

In Eq. (11) only the first integral has a contribution for $B_x$ in the interval $[P, Q]$, and this equation gives $B_x$ imposed by the boundary condition. But both terms in Eq. (11) contribute to create the vertical field at the edges of the prominence. As the sign of $B_x$ is imposed, the sign of $B_y$ tells if the prominence can be supported. This condition of support depends on the contribution of both the photospheric and prominence boundary conditions in general.

#### 3.4.2. Magnetic field on the $y$-axis outside the prominence

Equation (11) gives, by construction, a zero vertical field on the $y$-axis outside the prominence as the Dirac distribution, (in the limit as $x$ decreases to zero), is outside the interval $[P, Q]$ of integration. The horizontal field $B_x$ has a contribution from both the photospheric and the prominence boundary conditions. For a physical function $G(\xi)$ (for which $1/G(\xi)$ is zero at $\xi = ip$ and $\xi = iQ$) both $B_x$ and $B_y$ are continuous along the entire $y$-axis
(with a current density which decreases to zero at both ends P and Q of the prominence).

### 3.4.3. Magnetic field on the x-axis

As on the vertical axis, (10) gives the imposed vertical field along the x-axis. The horizontal field is given by:

$$B_x(x, 0) = \frac{2/\pi}{\operatorname{Im}(G(x))} \left[ \int_P^Q \frac{\operatorname{Re}(G(it)) \cdot B_x(t) \cdot t \cdot dt}{t^2 + x^2} - \int_0^\infty \frac{\operatorname{Im}(G(t) \cdot B_y(t) \cdot t \cdot dt)}{t^2 - x^2} \right] \quad (12)$$

The second integral is singular at $t = x$.

## 4. Reduction of the multiplicity of the solution

### 4.1. Limit at infinity

From (10) the asymptotic behaviour of $B(\xi)$ when $|\xi|$ increases to infinity is

$$\lim_{|\xi| \to x} B(\xi) \propto \xi^2 \cdot G(\xi) \quad \text{if } B_x(x) \cdot \operatorname{Im}(G(x)) = 0 \quad (13)$$

We can impose that $\lim_{|\xi| \to x} B(\xi) = \text{constant}$ as physically acceptable. But it is not really necessary as we assume a current-free magnetic field. This hypothesis must be valid only in a finite region, typically smaller than the coronal gravity scale-height. The region considered also must be quiet. It means that the magnetic field computed by this analytical mean cannot be extended to large distances. An observational support of this is the slow increase with height of the horizontal magnetic field in high quiescent prominences.

So we can consider both functions $G(\xi)$ giving a decreasing or increasing magnetic field at infinity, but we restrict ourselves to only fields which decrease to zero or are constant at infinity.

### 4.2. Constraints at the end of the current sheet

One particular solution which satisfies the condition that $G(\xi)$ be real for $y$ between $P$ and $Q$ and $G(\xi)$ be imaginary on the other parts of the axis is:

$$G_i(\xi) = i \sqrt{(\xi^2 + P^2)} \sqrt{(\xi^2 + Q^2)}$$

The determination of the square root is the classical one (the positive part of the real axis is the line of the cut such that if $z = \operatorname{Re}i^0$, $\sqrt{z} = \sqrt{\operatorname{Re}i^0/2}$ for $y \ge 0$). From (10) we have $B(\xi)$ $\propto 1/\sqrt{(\xi^2 + P^2)\sqrt{(\xi^2 + Q^2)}}$ and a singularity at points $\xi = iP$ and $\xi = iQ$ since the integrals in (10) do not go to zero at this point. It means that there is an infinite density of current at the ends of the current sheet. Such behaviour is not physically acceptable and we must choose instead a function like:

$$G_2(\xi) = \frac{i}{\sqrt{(\xi^2 + P^2) \cdot \sqrt{(\xi^2 + Q^2)}}} \quad (14)$$

It gives a current density which goes to zero at the ends of the current sheet, so the vertical magnetic field $B_y$ is continuous at these points.

The solutions for $B(\xi)$ are still not unique since we can multiply $G_2(\xi)$ by any analytical function real on both axis. So we rewrite $G(\xi)$ as:

$G(\xi) = G_2(\xi) \cdot F(\xi)$ and rewrite (10) as:

$$B(\xi) = \frac{-2i}{\pi} \frac{\sqrt{(\xi^2 + P^2)} \cdot \sqrt{(\xi^2 + Q^2)}}{F(\xi)} \times \left[ \int_0^\xi \frac{B_y(t) \cdot F(t) \cdot t \cdot dt}{\sqrt{t^2 + P^2} \cdot \sqrt{t^2 + Q^2} \cdot (t^2 - \xi^2)} + \int_P^Q \frac{B_x(t) \cdot F(it) \cdot t \cdot dt}{\sqrt{t^2 - P^2} \sqrt{Q^2 - t^2} (t^2 + \xi^2)} + i B_x(\infty) \cdot \frac{\sqrt{(\xi^2 + P^2)} \cdot \sqrt{(\xi^2 + Q^2)}}{F(\xi)} \right] \quad (15)$$

With $F(\xi)$ analytic with the constraint that $F(\xi)$ be real on both axes.

### 4.3. Discussion

As $G(\xi)$ is chosen real on the interval $[P, Q]$ and imaginary on other parts of the axis, it means that a magnetic field $B_G(\xi) = G(\xi)$ has only a vertical field in the prominence and a horizontal one along the rest of the axis. Then, if we add to the solution $B(\xi)$ of (10) an arbitrary function $G(\xi)$ we do not change the imposed boundary values at the prominence and at the photosphere. This fact is also shown in (10) by the presence of the term $-B_x(\infty) \cdot \operatorname{Im}[g(\infty)]$. Clearly adding $G(\xi)$ to some solution affects only the unknown part of the magnetic field on both axes. To remove this ambiguity we need to know $B_x$ and $B_y$ in the photosphere, and also inside, below and above the prominence.

### 4.4. Uniqueness of the solution

If we know $B(\xi)$ on both axes, we can use (15), with the hypothesis that $\lim_{|\xi| \to \infty} B(\xi) \cdot G(\xi) = 0$, to determine $F(\xi)$ on the axes $0x$ and $0y$.

$$\text{On } 0x: B_x(x, 0) \cdot F(x) = -\frac{2}{\pi} \sqrt{x^2 + P^2} \sqrt{x^2 + Q^2} \times \left[ \int_0^\infty \frac{B_y(t) \cdot F(t) \cdot t \cdot dt}{\sqrt{t^2 + P^2} \cdot \sqrt{t^2 + Q^2} (t^2 - x^2)} + \int_P^Q \frac{B_x(t) \cdot F(t) \cdot t \cdot dt}{\sqrt{t^2 - P^2} \sqrt{t^2 + Q^2} (t^2 + y^2)} \right]$$

$$\text{On } 0y: B_y(0, y) \cdot F(y) = +\frac{2}{\pi} \sqrt{y^2 - P^2} \sqrt{Q^2 - y^2} \times \left[ \int_0^\infty \frac{B_y(t) \cdot F(t) \cdot t \cdot dt}{\sqrt{t^2 + P^2} \cdot \sqrt{t^2 + Q^2} (t^2 + x^2)} + \int_P^Q \frac{B_x(t)}{\sqrt{t^2 - P^2} \sqrt{t^2 + Q^2} (t^2 - y^2)} \right]$$

These are integral equations which determine the real function $F(x)$ on $0x$ and $F(iy)$ on $[P, Q]$. They are linear for the unknown function $F$, but the difficulty comes from the singularity of the integrals at points $t = P$, $t = Q$, $t = x$ and $t = y$, so a careful discretisation needed.

After the use of (15) in the rest of the y-axis, we can determine directly the function $F(iy)$ on the entire y-axis when we know the horizontal magnetic field below and above the prominence.

The function $F(\xi)$ is then computed everywhere using the solution of the Dirichlet problem (Sect. 3.1.): then the magnetic field results from (15).

Observations with the future vector magnetograph Themis (1991) will provide the magnetic vector in the photosphere and in the prominence. But it will not give the boundary conditions along the whole y-axis. The field below and above the prominence remains unobservable, and even the observations in the prominence will not give the field at the prominence corona interface that we need in fact here for the resolution. The determination of $B_y(iy)$, and so the current in the prominence, could only be measured at hotter temperatures, if we assume that the prominence is a cool sheet surrounded by a hotter plasma.

Another way is to use the measurement of electron density to compute surface density, and then $B_y$ with the force balance. But it leads to great uncertainty since the degree of ionization and the width of the prominence are not well known.

In fact, the horizontal field below and above the prominence remains unknown, and so the determination is not complete. Here we prefer to use only the prominence horizontal field, and the vertical photospheric field, which are well known, as shown in Fig. 1.

## 5. Results

### 5.1. Boundary conditions and choice of $F(\xi)$

#### 5.1.1. Magnetic boundary conditions

We used similar distributions of the magnetic field as Anzer (1972). The base flux was averaged to 4 G as plotted in Fig. 2a, with a decrease at 100 Mm from the origin. The prominence horizontal field was taken constant: $B_x(0, y) = 8G$ in the prominence.

<figure><img src="image_89.png" /><figcaption>Fig. 2a and b. The boundary conditions adopted in this article: a the vertical field $B_y(x, 0)$ on the photosphere; b the horizontal field in the prominence. The horizontal field in the prominence is multiplied by $-\beta$ in the computations</figcaption></figure>

In the computations Fig. 2b (a small increase with height, as observed, has no effect on the general topology of the field lines). But, from an analysis of the linear polarisation of $D_3$ and $H_\beta$ lines using the Hanle effect, Leroy et al. (1983, 1984) have measured the orientation of the magnetic field in prominences. They found that the field is almost parallel to the prominence axis: it means that the field of 8G used by Anzer may be too large, and only the component along the x-axis must be used to support the prominence. So the field in the prominence is reduced in our paper by a factor $\beta = \sin(\theta)$. As $|\theta| \sim 10^6$ to $25^6$ ($\beta \approx 0.2, 0.4$) we must use values nearer 3G for $B_x(0, y)$ rather than Anzer's values.

Another important result from Leroy et al. is that the majority of quiescent filaments have a magnetic field oppositely directed to the potential case. It means that the Inverse configuration may be the most common support for quiescent prominences. These last correspond to $\beta < 0$.

The extension of the prominence in the vertical direction is taken from $P = 20$ Mm to $Q = 40$ Mm. (This choice is not critical). In particular we can take $P = 0$ (as in Anzer's paper) without changing the force support. (In the simple example described below, the magnetic force has the same sign everywhere in the prominence: this behaviour is different from Anzer's computation where the upper part of the prominence is supported but not the lower part).

#### 5.1.2. Choice of $F(\xi)$

First we limit our study with $B_x(\infty)$. $\operatorname{Im}[g(\infty)] = 0$ in (10). The effect of a non zero value is discussed in Sect. 5.4. We investigate two cases for $F(\xi)$: $F(\xi) = 1$ and $F(\xi) = \xi^2$. In the first case $\lim_{|\xi| \to \infty} B(\xi) = \text{constant}$ (horizontal field), and in the second $\lim_{|\xi| \to \infty} B(\xi) \propto 1/\xi^2$ (closed dipolar field).

Alternatives to this last case can also be investigated using functions $F(\xi)$ such as $\xi^2 + h^2$. It gives a pole at $y = ih$ or $x = h$, and the field has the behaviour of a dipole at infinity. We can also impose at these points a zero magnetic field behaving like $1/(\xi^2 \pm h^2)$ on the axis, but the field grows like $\xi^2$ at infinity. Such variations to the initial models give more complicated configurations, and we have no observational data to justify them so that we present here only the two basic configurations:

$$1: F(\xi) = 1$$

$$2: F(\xi) = \xi^2$$

The numerical integration is made by a routine which handles integrands with singularities at a finite number of points. We chose the NAG routine DO1ALF. The algorithm is described in De Doncker (1978).

#### 5.2. Case $F(\xi) = 1$

##### 5.2.1. Analytical case

The integrations in (10) can be done analytically in the case when $B_y(x, 0) = B_x^0$ on $0x$

and $B_x(0, y) = B_y^0$ on $0y$ in the interval $[P, Q]$

where $B_x^0$ and $B_y^0$ are arbitrary constants. The integration leads to:

$$B_x - iB_y(x, y) = \frac{1}{\pi} B_x^0 \cdot \ln \left( \frac{A(0)}{A(-1/\xi^2)} \right) + B_x^0$$
$$ 
\begin{array}{l}
\text{with } A(u) = a(u) + \sqrt{a^2(u) - b^2} \\
\\
a(u) = u(\xi^2 + P^2)(\xi^2 + Q^2) + \xi^2 + \frac{P^2 + Q^2}{2} \\
b = 1/2(Q^2 - P^2).
\end{array}
\qquad (16)
$$

The integral on the photospheric boundary gives a complicated behaviour, but the integral on the prominence boundary gives only a constant field: $B_x^0$.

### 5.2.2. Potential case

The case without any prominence has $P=Q=0$ and the magnetic field horizontal everywhere on the positive vertical axis. There is no contribution from $B_x$ in this case, and (16) gives an infinite field since $B_y$ is set to a constant on all the horizontal axis. A numerical computation is needed with a more realistic field, as in Fig. 2a, with $\lim_{x \to 0} B_y(x, 0) = 0$ and $\lim_{x \to \infty} B(x, 0) = 0$. The field is represented in Fig. 3a: the photospheric field has an importance only in the low part of the box, with a gap near the origin since the field vanishes there. The upper part of the figure is filled with a field coming from photospheric regions far from the

origin, and with the opposite polarity. The figure is part of a quadrupolar configuration.

### 5.2.3. Normal configuration: $B_x(0, y) < 0, \beta > 0$.

The analytical solution (16) gives singular field lines near the origin and at infinity. This could be avoided by integrating the second integral of (10) only from $h_1 > 0$ to a finite limit $h_2$. Then:

$$ B_x - iB_y(x, y) = B_x^0 + \frac{1}{\pi} B_y^0 \ln \left( \frac{A \left( \frac{1}{h_2^2 - \xi^2} \right)}{A \left( \frac{1}{h_1^2 - \xi^2} \right)} \right) $$

The behaviour of the first integral of Eq. (10) is simple in this case: it is a constant: $B_x^0$. But the behaviour of the second term is complicated and so we discuss here only the general behaviour directly by using (10) for this term.

The vertical field at the edges of the prominence is created only by the photospheric pattern:

$$ B_y(0, y) \approx \frac{2 \cdot B_y^0}{\text{Re}(G_2(i_y))\pi} \left( \int_0^x \frac{\text{Im}(G_2(t) \cdot t \cdot dt)}{t^2 + y^2} \right) \qquad (17) $$

<figure><img src="image_90.png" /><figcaption>Fig. 3a and b. The magnetic configuration obtained in the case $G(\xi) = G_2(\xi)$ ($F(\xi) = 1$ in Eq. (15)) for: a the potential configuration; b the I configuration with $\beta = 0.2$ ($B_x = +1.6$ G in the prominence). The I configuration has a neutral point below the prominence ($y \approx 10$ Mm). The N configuration is not plotted as there is no support with this function $F(\xi)$</figcaption></figure>

which is of same sign as $B_y^0$ since $\text{Im}(G_2(\xi)) > 0$ on $0x$ and $\text{Re}(G_2(iy)) > 0$ for $y$ between P and Q.

The horizontal field outside the prominence on the y-axis is:

$$ B_x(0, y) = B_x^0 - \frac{2 \cdot B_y^0}{\text{Im}(G_2(iy)) \cdot \pi} \left( \int_0^y \frac{\text{Im}(G_2(t)) \cdot t \cdot dt}{t^2 + y^2} \right) \qquad (18) $$

Again, the second term has the same sign as $-B_x^0$ below the prominence and as $B_y$ above the prominence.

By convention we take the photospheric field $B_y(x, 0) > 0$. For the N case (17) gives a positive vertical magnetic field at the edge of the prominence. Then $B_x(0, y) \cdot B_y(0, y) < 0$ for every cases.

So, the function $G_2(\xi)$, or $F(\xi) = 1$ does not provide a good magnetic field for a prominence.

### 5.2.4. Inverse configuration

In this case $\beta < 0$, $B_x(0, y) > 0$ and the prominence is supported by the horizontal magnetic field. The surface mass density has the regular behaviour shown in Fig. 4b. The surface density is an order of magnitude lower than $\lambda_0$ computed in Sect. 2.2.4, as the vertical field is much lower: $B_y(0, y) \approx 1$ Gauss: Fig. 4a. We have here a model of a low density prominence. To increase the mass supported we need to increase the average magnetic base field: the mass supported is proportional to $B_y^0$.

Figure 4b shows that the density contribution from the repulsion image is negligible, as the base magnetic field $B_y$ is too low compared to $B_x$: Eq. (17). The “self pinch” effect is however most important: curve 3 of Fig. 4b.

Figure 3b gives the field topology in the case $\beta = -0.2$. The configuration possesses an X-point below the prominence. Its height $h$ is given by $B_x(0, h) = 0$ in Eq. (18). The height $h$ increases with increasing photospheric field $B_y^0$ to the limit $y = P$ for large $B_y^0$. This X-point disappears for very large prominence magnetic fields. In the case $B_y^0 \approx 4G$, it disappears for $\beta \approx -0.8$.

These results show that, for the case when $F(\xi) = 1$, we cannot find support for Normal configuration, but there is support for an Inverse configuration in all the cases. The field has no peculiar behaviour at any points, and it is limited at infinity by horizontal magnetic field. The repulsion by the image prominence current is always negligible when $|\beta| > 0.1$.

### 5.3. Case $F(\xi) = \xi^2$

As before we compute the numerical solution with the fields of Figs. 2.

#### 5.3.1. Potential case

$P=Q=0$. The magnetic configuration is given in Fig. 5a. It is similar to Anzer's case with a magnetic strength which decreases like $1/|\xi|^2$ at infinity, and with a singularity at the origin like $1/\xi^2$. This potential field is created by a bipolar region with finite dimension. The behaviour near the point $x = 100$ Mm $y = 0$ is artificial since we fix $B_y(x, 0) = 0$ at larger distances.

#### 5.3.2. Normal configuration

In the case $F(\xi) = \xi^2$ both integrals in (10) give a contribution to the vertical field near the prominence. We compute first separately

this contribution with the boundary field of Figs. 2. Figure 6a gives the field on the vertical axis when we impose $B_y(x, 0) = 0$ on the photosphere, and 6b gives the same when we impose $B_x(0, y) = 0$ at the edge of the prominence.

The normal configuration is built by adding these two computations with the first one multiplied by $\beta > 0$. The contribution of the first case (“field prominence”) is more important for the vertical field in the prominence than the second, for $\beta \approx 1$, which gives a downward force. To have a N support we may decrease the prominence boundary field and the support exists for $\beta \le 0.3$. Figure 6c gives the magnetic field on the vertical axis for $\beta = 0.2$: the vertical field in the prominence is low, but could support a thin prominence. Again, if we want to increase the mass supported we may increase the photospheric field.

The magnetic field lines are represented in Fig. 5b for the case $\beta = 0.2$. The prominence is supported by an arcade closing the opposite polarity of the photosphere. The configuration below the prominence is complicated as there is a pole at the origin.

<figure><img src="image_91.png" /><figcaption>Fig. 4a. The magnetic field along the y-axis: Bx is shown by the solid curve, By by the dashed curve, for an I configuration with $\beta = -0.2$. The choice $F(\xi) = 1$ gives no divergence of the magnetic field, but the limit of the field at infinity is a constant</figcaption></figure>

<figure><img src="image_92.png" /><figcaption>Fig. 4b. The mass density deduced from: 1: support by the total horizontal magnetic field of the prominence, 2: support by the repulsion image, 3: support by the interaction between current in the prominence</figcaption></figure>
<figure><img src="image_93.png" /><figcaption>Fig. 5a and b. The same as Fig. 3 for the case $F(\xi) = \xi^2$, a is the potential case and b is the N configuration for $\beta = 0.2$. The potential case is the same as Anzer's but the N configuration is different with support at all heights in the prominence. The behaviour near the origin comes from a singularity of the solution in this point</figcaption></figure>

This pole imposes its own behaviour near the origin and gives a neutral point on the y-axis at an altitude $y \approx 1$ Mm.

### 5.3.3. Inverse configuration

From Fig. 6a and 6b we see that in the *I* case ($\beta < 0$, $B_x(0, y) > 0$) the support from the horizontal magnetic field is not possible as $B_x(0, y) \cdot B_y(0, y)$ is always negative. A *I* configuration could not be achieved in this case.

For the case $F(\xi) = \xi^2$ we can support a N prominence for $\beta < 0.3$. The field decreases like $1/\xi^2$ at infinity, but presents a pole at the origin. This could be avoided by setting the photosphere at a greater height, but below the prominence with new boundary conditions on the photosphere, qualitatively not different from the previous one, and so we have a model without a singularity. This could be achieved more systematically by using the technique described in Sect. 4.4. as the singular behaviour of $1/F(\xi)$ near the origin will disappear.

### 5.4. Discussion with $B_x(\infty)G(\infty) \neq 0$

As discussed in Sect. 4.3; we can add an arbitrary function $G(\xi)$ to a solution $B(\xi)$, provided that $G(\xi)$ is analytic on the quarter

plane $x > 0$, $y > 0$ and is purely real on the interval $y \in [P, Q]$ and imaginary in other parts of the axis. This leads to a great freedom and in all cases we can support a prominence for both N and *I* configurations by choosing the right sign for $G(iy)$ in the interval $[P, Q]$. All the configurations described in Malherbe and Priest (1983) can be found by this procedure, but we cannot choose between these models as observational data are missing. In this paragraph, we limit the discussion to non-zero values of $B_x(\infty)$ in Eq. (10). If $F(\xi) = 1$, $\mathrm{Im}[G(\infty)] = 0$ and there is no additional contribution to Sect. 5.2. If $F(\xi) = \xi^2$, $\mathrm{Im}[G(\infty)] = 1$ and we add $B_a(\xi) = -B_x(\infty)/(\xi^2 - G_z(\xi))$ to the results of Sect. 5.3. This function is plotted in Fig. 7a. The plot shows that the field defined by this function has no vertical field on the x-axis and no horizontal field on the y-axis in the prominence. The singularity of the function at the origin is seen clearly. To demonstrate the possibilities of this method we used a large value of $\beta$: $|\beta| = 0.5$, and then by adding $B_a(\xi)$ with the right sign we can support both N and *I* configurations by a horizontal field in the prominence. $B_x(\infty)$ is chosen to be 10 Gauss, and then the field obtained has, at infinity, the limit of a horizontal field of 10 Gauss. The field on the y-axis is plotted in Figs. 8a and 8b for, respectively, the N and *I* configurations.

The horizontal field on the y-axis varies monotonically with a neutral point below the prominence in both cases. The imposed

<figure><img src="image_94.png" /></figure>

<figure><img src="image_95.png" /></figure>

<figure><img src="image_96.png" /><figcaption>Fig. 6a-c. The magnetic field component on the y-axis as in Fig. 4a, a is for a zero base flux and b for a zero horizontal field in the prominence. The sum of these two fields with $\beta < 0.3$ gives an N configuration supported by the horizontal field as $B_x$. $B_y$ is positive in the prominence as in the plot of c with $\beta = 0.2$. The subtraction of a and b gives an *I* configuration not supported</figcaption></figure>

constant horizontal field in the prominence could be changed to an increasing function with height, then the horizontal field on the vertical axis changes regularly with height with no particularity in the prominence. In both cases the horizontal field supports the prominence, with a typical mass density as discussed above.

The field line topologies are plotted in Figs. 7b and 7c. The general behaviour is not so different: the field is almost horizontal at large altitudes and the field lines are concentrated at very low heights below the prominence. (The *I* case gives an island below the prominence, but it is an artefact due to the low resolution computation).

## 6. Conclusion

From the vertical field observed in the photosphere on the solar disc and from the horizontal field observed in prominences at the limb, we have built a method of computing the magnetic field in the neighbourhood of prominences. The method is an extension of Anzer's: the computations are made in two dimensions with complex functions. We choose to work with the complex magnetic field, rather than with the potential, as Anzer did since it is easier to define the problem for a finite current sheet at a fixed height (*P*) above the photosphere. The solution of the problem is not unique because it requires the definition of an arbitrary function $G(\xi)$ and we investigate different models of prominences lying at a finite height above the photosphere (in Anzer's case, prominences go down to the surface). We try to choose functions $G(\xi)$ using the constraint that the limit of the field at infinity must be finite. We consider also only finite surface current densities in the prominence, which gives a continuous magnetic field in the prominence and in the neighbourhood. We compute different solutions of this problem for the Normal and Inverse configurations. We find surface mass densities comparable with the observed electron density in prominences for typical photospheric fields of 4G.

If we impose the magnetic field to decrease at infinity like $1/\xi^2$, we must introduce a pole somewhere. It could be in the prominence, or on the positive y-axis, which is not physically acceptable, or along the photosphere. We choose the *lase* case. A way to avoid this difficulty is to put the photosphere at a positive height: the boundary conditions are changed, but the solution has no singularity above the photosphere. The computation of these new boundary conditions could be done by the method proposed in Sect. 4.4, which solves also the problem of the interminacy of the solution, but the observed necessary boundary conditions are not available at this time. We need knowledge of the vector fields in the photosphere and the prominence, but the horizontal magnetic field below and above the prominence will remain unknown, and so might be treated as free parameters for the determination of the prominence field environment. Another extension of this work consists in treating the problem by allowing currents in the corona. This step could be done with a linear force free-field (Démoulin and Raadu 1989). The problem is treated using finite differences and the boundary conditions are fixed on a box (some conditions as in the present paper, plus free floating conditions on the other boundaries). The degree freedom of this computation comes from the boundary condition on the two sides of the box, the magnetic field below the prominence and the shear $\alpha$ of the field. Different topological field configurations could be found by varying these conditions. The problem
<figure><img src="image_97.png" /><figcaption>Fig. 7. a The magnetic field $B_\xi(\xi) = -B_x(x)/(G_2(\xi)\xi^2)$ which field has no horizontal field in the prominence and no vertical field on the photosphere, and so it could be added to the field computed with the boundaries of Fig. 2, to give an another solution. $B_y$ choosing $B_x(x)$ we can make both N and I models with a support for the prominence. b shows the N configuration, with $\beta = 0.5$, and $B_x(\infty) = 10G$. Fig. 8b gives the same for the I configuration with $\beta = -0.5$ and $B_x(\infty) = +10G$.</figcaption></figure>

<figure><img src="image_98.png" /><figcaption>Fig. 8a and b. The field component on the y-axis of Figs. 7b and 7c. Both cases could be supported by a horizontal field for surface densities typical of those observed.</figcaption></figure>

could be generalized to three dimensions, but the problem of unknown boundary data is so important that it cannot yet be applied to solar prominences in a realistic way.

*Acknowledgements.* The authors thank S. Bordet for typing the manuscript.

References

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P. Démoulin¹, E.R. Priest², U. Anzer³

¹ Observatoire de Paris, Section de Meudon, DASOP (UA. 326)
F 92195 Meudon Principal Cedex, France

² Applied Mathematics Department. University of St. Andrews
KY 16 9SS St. Andrews - Scotland

³ Max-Planck-Institut fur Physik und Astrophysik
Institut fur Astrophysik D-8046 Garching F.R.G.

**Submitted to** : Astronomy and Astrophysics, the Main Journal
**Section** : The Sun

**Send offprint requests to** : P. Démoulin

**Key words** : Solar prominences, magnetic field, force-free field.

**Thesaurus codes** : 19.81.1 - 19.84.1
## SUMMARY

We propose to model the 3D structure of quiescent prominences by a superposition of two fields. A 3D force-free field with constant $\alpha$ is assumed to exist in the corona prior to the prominence formation. The prominence itself is represented by a line current which interacts with the coronal field. At present the linear force-free field of the current line is not included, and we neglect this effect in the force balance. A three-dimensional field is represented by analytical functions and concentration of the magnetic field at the chromospheric level by convection cells is taken into account. When the field created by the chromospheric pattern supports the prominence, the prominence feet are found to be located at supergranulae centres otherwise they are located at cells boundaries.

## 1. INTRODUCTION

At the solar limb, quiescent prominences consist of bridges of chromospheric -like material embedded in the hot corona. Their typical dimensions are 200 Mm for the length, 50 Mm for the height and 6 Mm for the width. Periodically the prominence connects the photosphere to form arches and columns like a bridge. A great number of such examples can be found in d'Azambuja and d'Azambuja (1948). The distance between the columns, or feet of the filament, is of the order of the prominence height and the supergranule diameter : 30 Mm. As the prominence density is 100 times greater than the coronal one, they can be supported by a magnetic field of typically 10G. The field is observed to be nearly aligned along the prominence axis : the angle between the magnetic field and the prominence axis, measured by Leroy et al (1983, 1984), is about 20°.

The observations on the disc show also feet as regions where the filament seems larger, which could be due to perspective effects. The material seems to be organized in long and parallel threads which connect feet. At the location of the feet the threads seem to fan in every direction like a rosace (Simon et al, 1986a). The suggestion that these threads could represent the direction of the magnetic field in quiescent prominences is controversial since the observed velocity of a few kms⁻¹ are much lower than the sound, free fall and Alfven speed, and so a support is needed. The regions of the feet are also special for the dynamics, since most of a quiescent prominence is seen on the disc to possess upward motions of a few kms⁻¹ (Mein 1977, Martres et al 1981), Malherbe et al 1981-1983a, Engvold 1986), the regions of feet are more likely to have downward motions Martin (1973), Simon et al (1986a) as for active filaments : Schmieder et al (1985).

The physical parameters of prominence feet are very poorly known. The measurement of the strength and the direction of the magnetic field has not been achieved with the Hanle effect since the photospheric scattered light entering the coronograph disturbs greatly the measurement of the polarisation degree. The density is also poorly known for the same reasons.

Little theoretical work has been attempted on the filament feet. Since most of the models are in two dimensions and assume invariance along the filament axis. Nakagawa and Malville (1969) try to give an explanation for feet, by studying the stability of a cool plasma supported by a magnetic field. The
Rayleigh-Taylor instability is stabilised at low wavelength, and so eventually the dense material falls. They considered a uniform magnetic field ($B_0$) along the x axis, parallel to the horizontal interface that separates two uniform plasmas. Above, the plasma has the typical density of prominences, and below of the corona. For this simple equilibrium the dispersion equation becomes :

$$ \omega^2 = -g \cdot k \frac{\rho - \rho_c}{\rho - \rho_c} + \frac{2 B_0^2 k_x}{\mu_0 (\rho - \rho_c)} $$

(eg Priest, 1982)

The fastest growing mode for a disturbance along the field ($ky = kz = 0$)

is :

$$ k_F = \frac{(\rho_o^P - \rho_o^c) g \cdot \mu_0}{2 B_0^2} $$

With $\rho_o^P \approx 2 \cdot 10^{-11} \text{kg m}^{-3}$, $\rho_o^c \ll \rho_o^P$, $B_0 = 10G$ and $G = 274 \text{ms}^{-2}$ it gives a scale length of the order of 30 Mm, the same as the typical length between adjacent filament feet. Nakagawa and Malville have developed this idea to include thermal pressure, a simple energy equation and a shear. Their results are similar to the simple case described above. Sometimes a longer distance (~90 Mm) between structures is observed, which corresponds to a nearly perpendicular supporting field, while shorter distances (~ 15 Mm) correspond to fields nearly parallel (20° - 30°) to the filament axis. One possible criticism is that, in such a simple uniform magnetic field, a disturbance orthogonal to the equilibrium field is not stabilised at any wavelength, since no magnetic pressure or tension is produced by the perturbation. The stabilising effect of photospheric line tying may be incorporated in this case, just as Hood and Priest (1980b) have done for a magnetic arcade configuration.

Milne et al (1978) have built a one-dimensional quiescent prominence with the aim of coupling the magnetohydrostatic equilibrium with thermal equilibrium along the field lines. They found that an equilibrium state is impossible when the prominence plasma beta is greater than or of the order of unity. When the plasma beta is greater than this upper limit the dense material may fall down to the photosphere to form the prominence feet. The observed motions of a few kms⁻¹ are slower than the free-fall velocity : $\sqrt{g\Omega} \ge 20 \text{ kms}^{-1}$, and the downward motions in feet are not systematic. This leads to the conclusion that the magnetic field may support, at least partially, the dense material of the feet.

It could lead also to the picture that the prominence is formed at a great height in the corona by some thermal instability or by injection, and then falls back periodically to give the feet. Such a view, however, is inconsistent with observations, where it is seen that the feet are formed together with the filament body. In fact we can understand the existence of the feet by reformulating the conclusions of Milne et al (1978). At low heights the prominence material is so heavy that the dense plasma cannot be supported by the magnetic field. No cold and dense condensations are formed. At the location of the feet, the field needs to be greater, so the feet are likely to be near the supergranule boundaries, where most of the chromospheric flux is concentrated. This conclusion is supported by the rosace behaviour of a cool thread when we see a foot from the top : its location may be at the junction of several cells with divergent magnetic field lines. The body of a filament, seen from above, looks like long and parallel threads supported by nearly parallel field lines connecting two cell edges.

These conclusions are supported by the observations of Plocieniak and Rompolt (1972), who found that the feet tend to occur at the intersection of 3 or 4 cells. But this result is controversial because the supergranules are usually defined by chromospheric magnetograms, and, since the filament lies always along a neutral line of the vertical field, the supergranule cells are not well determined near the filament. In particular the right distance between feet could be obtained by locating the feet at the centres of supergranules, which sometimes seems to occur. For this case Plocieniak and Rompolt have suggested that some of this controversy may be due to the fact that sometimes feet occur at the centre of a cell, but do not extend down to the chromosphere. They refer to these features as "suspended legs". Martin (1986) has confirmed the location of the feet near the junction of several cells, since they are located in regions of convergent motions in the photosphere. She reports also that it is the location of converging fields of opposite polarity which are cancelling. This new feature must be confirmed in the future, since the observations also show the activation of quiescent filaments, and so one wonders if they also refer to the regions where the filament is rather quiet.

The object of the present paper is to describe the external magnetic field of a prominence in connection with the photospheric magnetic flux pattern. We want to model the feet, so a three-dimensional model is needed. The motion of the flux pattern near a quiescent filament is very slow, so we can use a
stationary model. The field in the corona should be approximately force-free and from the Taylor-hypothesis Heyvaerts and Priest (1984) have suggested that it would develop into a field with constant α. One criticism is that the coronal plasma could be very well described as perfectly conducting if we consider typical global scale lengths of a prominence, but not for much lower lengths of a few meters. Then reconnection may be important such low scales, but the field evolution may be globally slow as observed and a global description of the field may be adopted with a linear force-free field for the whole corona. One should also bear in mind that the condition α = const. can only be fulfilled over a finite region because fields which have constant α everywhere lead to an infinite total magnetic energy (Seehafer, 1978). One other difficulty is that the field is not force-free in the prominence as the Lorentz force must balance the gravitational one. We have treated this as a singularity in the force-free field, with infinitesimally small width, and we add a line current in a simple case, to model the prominence. This model and the hypothesis are described in Section 2. The linear force-free field configurations are analysed in Section 3, and then we add some current lines to study the location of the feet above the supergranule pattern, in Section 4. More concentrated photospheric flux patterns are analysed in Section 5.

## 2. DESCRIPTION OF THE MODEL

### 2.1. Behaviour Expected of the Magnetic Field Near a Prominence

The prominence is in a channel where the photospheric magnetic field changes polarity.

The prominence reaches periodically down to the photosphere, so we may expect that, in these regions, the magnetic field has a different configuration (although not measured so far).

The magnetic field is important near the boundaries of supergranule cells, and even if the field sprays out like a fan in the corona as modelled by Gabriel (1976), the concentration of field near the intersections of 3 or 4 cells may be related to the prominence feet.

The top of a prominence has nearly the same height above feet as between them. This suggests that the magnetic field becomes nearly uniform at a height, which is roughly of the order of the gravitational scale height of the corona.

The field in the prominence is nearly horizontal and inclined at about 20° to the prominence axis (Leroy et al, 1983, 1984). The strength is about 10G and the field seems to increase slowly with height: 0.5 G/10 Mm, although Athay et al (1983) did not confirm this increase and it only applies to very large prominences. This small increase is in fact not important, Démoulin et al. (1988b), and so the prominence field may be considered roughly constant with height.

The prominence has a very narrow width, so we can model it by a current sheet or a line current.

### 2.2. Hypothesis

a) Ideal MHD. The magnetic Reynolds number, $R_m = 2 \times 10^{-9} \cdot l \cdot v \cdot T^{3/2}$ for $v \approx 20$ kms$^{-1}$, $l \approx 10$ Mm, $T \approx 10^6$ is of the order of $10^{11}$ and the diffusivity of the plasma could be neglected when we consider the global structure of the prominence field (the slow evolution of the field and the heating may, however, depend greatly on the reconnection at small scales).

b) Plasma speeds are observed to be lower than the sound and Alfven speeds, so, as a first approximation, we can consider a static model. (A slow evolution through a series of static models may be a good representation of the field of a prominence).

c) For typical observed parameters, the plasma beta of the corona is lower than unity ($\beta \lesssim 0.1$), so the field is almost force-free.

d) At the location of the prominence gravity is important and so the field is not force-free. But as the prominence width is much smaller than the other dimensions, we can model the prominence as a singularity in the force-free field. For simplicity we use here current lines. (An extension would be to use a current sheet, but qualitatively the results are unlikely to change).

e) We assume a certain value for the current, and then deduce the mass density from the vertical equilibrium. We postulate that the mass needed to form the prominence can be supplied at this height by some mechanism (such as thermal instability or injection from the chromosphere).
f) We suppose that the observed chromospheric pattern of the magnetic field remains concentrated up to the chromosphere, where the magnetic field becomes to be force-free.

g) The prominence is assumed to be located in a vertical plane.

## 2.3. General solutions

A linear force-free field satisfies :

$$ \vec{\text{rot}} \vec{B} = \alpha \vec{B} \quad \text{and} \quad \text{div} \vec{B} = 0 \qquad (1) $$

with α a constant

It leads to the equation (if one uses Cartesian coordinates) :

$$ \vec{\Delta B} + \alpha^2 \vec{B} = 0 \qquad (2) $$

which we can resolve by using a Fourier series with components behaving like :

$$ B_0 e^{i(k_x x + k_y y + k_z z)} $$

Where Equation (2) gives $k_x^2 + k_y^2 + k_z^2 = \alpha^2$

The series is limited at large values of k by α. Such solutions would have spacial oscillations and a reversed field above the surface (z = 0). To avoid this we can take a dependence in the vertical direction behaving like $e^{-\ell z}$. It has the property of a regular decrease as expected for solar magnetic fields. Then the horizontal components can be expressed as functions of $B_z$ as follows :

$$ B_x = \frac{\alpha \frac{\partial B_z}{\partial y} - \rho \frac{\partial B_z}{\partial x}}{\alpha^2 + \rho^2} \qquad B_y = \frac{-(\rho \frac{\partial B_z}{\partial y} + \alpha \frac{\partial B_z}{\partial x})}{\alpha^2 + \rho^2} \qquad (3) $$

and $B_z$ satisfies :

$$ \frac{\partial^2 B_z}{\partial x^2} + \frac{\partial^2 B_z}{\partial y^2} + (\alpha^2 + \rho^2) B_z = 0 \qquad (4) $$

The solution of equation (4) could be found by a using Fourier series in the x,y plane. Then (3) gives the horizontal components of the magnetic field. The y-direction is chosen to be the axis of the prominence and the x direction is

normal to its plane. Since the prominence lies at the dividing line of opposite vertical magnetic fields, we chose the base flux to be odd in x. A prominence could be approximated by an infinite and periodic structure in the y-direction, with the same behaviour in the positive and in the negative directions. Then by choosing the origin of the y-axis, we can limit the Fourier series to even functions in y. The basic function is then :

$$ B_z = B^0 \cos(k_y y) \sin(k_x x) e^{-\ell z} \qquad (5) $$

and (3) gives :

$$ B_x = \frac{B^0}{k_x^2 + k_y^2} \left( -\alpha k \sin(k_x x) \sin(k_y y) - \rho k_x \cos(k_x x) \cos(k_y y) \right) e^{-\ell z} \qquad (6) $$

$$ B_y = \frac{B^0}{k_x^2 + k_y^2} \left( \rho k_y \sin(k_x x) \sin(k_y y) - \alpha k_x \cos(k_x x) \cos(k_y y) \right) e^{-\ell z} \qquad (7) $$

$$ \text{with } k_x^2 + k_y^2 = \alpha^2 + \ell^2 \qquad (8) $$

The development of such functions has been used by Nakagawa and Raadu (1972) to describe the coronal field above an active region, from the observed vertical magnetic field in the photosphere. Here we use a small number of harmonics to describe the magnetic field around a quiescent prominence. The periodicity of such solutions in the y-direction is well adapted to the periodic structure of a prominence. The solution is also periodic in x-direction orthogonal to the prominence axis, so we limit the region to the interval $|x| < L/2$ with the constraint that at the boundaries, the field is vertical, so that there is no flux link with neighbouring regions. This field configuration is sketched in Figure 1. The prominence feet are schematically drawn to connect periodically the photosphere (z = 0), and the prominence height is set to be approximately independent of the y-coordinate. The field is inclined at an angle $\varphi$ to the prominence axis. In the two-dimensional case $\varphi$ is the same along a field line (the line is coplanar) ; in the three-dimensional case $\varphi$ varies from the top to the bottom.
## 2.4. Photospheric Pattern

Photospheric magnetograms show that the magnetic flux is concentrated at the boundary of supergranulation cells. This pattern is irregular with a variety of cell sizes, but is stable in time. As an example, there are relatively few changes in the global behaviour in one day of the photospheric pattern observed by Martin (1986), even when there is newly emerging flux and the filament changes its location by 10 000 km. From these slow changes, we infer a static picture. To simplify the theoretical analysis we also simplify the irregular pattern to more symmetrical ones like those represented in Figure 2. The neutral line is assumed to be a straight line, with symmetry of opposite polarities on both sides.

Case $C_1$ has the prominence located above a row of convection cells. If $d_C$ is the size of the cells, then the periodicity $L_y$ of the model in the y-direction is $L_y = d_C$, and so the feet are equidistant with $d_C \approx 30 000$ km, which matches the average of the observations. In the x-direction the field concentrations are also separated by $d_C$.

In case $C_2$ the prominence is located over the cell boundaries. There is no flux concentration below the prominence since the field variation in the x-direction imposes a neutral line $\ell y = d_C$, but the nearest flux concentrations are twice as far away in the x direction as in the model $C_1$.

Scheme $C_3$ et $C_4$ lead to more realistic cells : they are hexagonal with flux concentration at the intersection of three cells. The scheme $C_3$ sets the prominence location above a row of convective cells. The periodicity in the y-direction is again $d_C$.

From a field pattern point of view scheme $C_3$ could be obtained by superposing two patterns like $C_1$ with $\ell y_1 = d_C$ and $\ell y_2 = d_C$. A location of the prominence as in case $C_4$ leads to an asymmetric field pattern, which may distort the prominence plane, and the feet are only 0.4. $d_C$ apart. Such field patterns do not match the observations.

From the different possibilities described above, we want to take a simple case but sufficiently general. Case $C_1$ to $C_3$ could be simplified in a single model. At low heights ($\ll d_C$) the prominence field is approximately described by the nearest photospheric cells, so we retain only one row of flux concentration on each side of the prominence. In scheme $C_3$, at low heights, we may choose the row distant from $d_C/2$ to the prominence (case $C_{3a}$), but at greater heights ($\sim$

$d_C$) we must choose the row distant from $d_C$ to the prominence : Case $C_{3b}$ (the higher harmonic of a force-free field rapidly decreases with height : equations (6) and (7)). In conclusion a crude approximation to the scheme $C_1$ to $C_3$ could be periodic rows of flux concentrations, periodically spaced at a distance $d_C$, with a distance $L/2$ to the prominence (Figure 1). $L/2$ is equal to $d_C/2$, $d_C$, $d_C/2$ and $d_C$ in schemes $C_1$, $C_2$, $C_{3a}$ and $C_{3b}$ respectively.

## 3. BASIC MODEL

### 3.1. Description

The Fourier analysis described in paragraph 2.3 is well suited to model the photospheric flux distribution of paragraph 2.4. From the photospheric pattern we can find the amplitude of the harmonics. Here we want to build an analytical model, and so we take only the lowest harmonics.

We choose :

$$ B_z(x,y,z) = B_0 \cdot [\sin(k_x x)e^{-\ell_0 z} + b_1 \sin(k_x x) \cos(k_y y)e^{-\ell_1 z}] \quad (9) $$

$$ \ell_0^2 = k_x^2 - \alpha^2 $$

$$ \ell_1^2 = k_x^2 + k_y^2 - \alpha^2 $$

$$ k_x = 2\pi/L_x = \pi/L \quad k_y = 2\pi/L_y \quad (L_y \ 30\ 000 \ km) $$

The flux concentrations near the point $|x| = L/2$, $y = nL_y$ with n an integer would be more concentrated if we add higher harmonics, but this simplest case shows the general behaviour. In most computations we take $L = d_C$ (which is the case for flux patterns $C_1$ or $C_{3a}$ of Figure 2).
## 3.2. Chromospheric Field

There is, in general, one unipolar region on either side of prominence, with no parasite polarity. So at the chromospheric level $z = 0$, we may impose that

$$|b_1| < 1,$$

The Figure 3 shows the example, with $b_1 = 0.8$, that we use in this section (only one period in each direction, is shown and the flux concentration is located at $|x| = \pm 15$ Mm). A constant $\alpha' = 3$ is used so as to give a magnetic field inclined to the oy axis at nearly 20°, as observed on the Sun ($\alpha' = \alpha L$). A ratio $L_y/L_x = 0.5$ is taken to give the same cell size in the x direction as in the y direction.

## 3.3. Two-Dimensional Arcades

This particular case is obtained by taking $k_y = 0$ in equations (5) to (8). The field lines are contained in a plane inclined at an angle $\varphi$ to the prominence axis (Figure 1).

$$\lg \varphi = \frac{B_x}{B_y} = \frac{\sqrt{k_x^2 - d^2}}{d}$$

The observations of Leroy et al (1983, 1984) show that the angle $\varphi$ has a low value in the prominence $\varphi < 30°$. It implies that the field configuration is far from current-free (or potential) and so a current must be allowed along field lines. This current is important, since $\alpha$ is near its maximum value ($k_x$).

The scale-height of the field is :

$$h = l^{-1} \approx L/\pi \cdot \operatorname{tg} \quad (k_x \approx \alpha)$$

so the field strength decreases by 2.7 at a height of 0.9 L for $\varphi \approx 20°$ and of 1.8L for $\varphi \approx 10°$. The vertical extension of a force-free field is lower than the horizontal scale, unless the field is greatly sheared.

## 3.4. Field in the Prominence Plane

The magnetic field on the plane $x = 0$ is :

$$B = \begin{cases} -B_0 \frac{l_0}{k_x} e^{-l_0 z} \left( 1 + l_1 \frac{k_x^2}{k_x^2 + k_y^2} \frac{l_1}{l_0} \cos(k_y y) e^{-(l_1 - l_0)z} \right) & (1) \\ -B_0 \frac{l_0}{k_x} e^{-l_0 z} \left( 1 + l_1 \frac{k_x^2}{k_x^2 + k_y^2} \frac{l_0}{l_0} \cos(k_y y) e^{-(l_1 - l_0)z} \right) & (2) \\ 0 & \end{cases}$$

The field has no vertical component at the location of the prominence.

### 3.4.1. Shear angle

The shear angle $\theta$ defined by $\operatorname{tg}\theta = B_y/B_x$ varies periodically along the y-axis. At the chromospheric level (Figure 3) the shear is less important near the origin of the y axis than at $|y| = 15$ Mm. So, near a chromospheric flux concentration, the field lines cross the prominence plane at a small angle. An approximate relation can be obtained when the field is nearly aligned with the y-axis and the first harmonic has a low contribution :

$$\varphi \approx \varphi_0 \left( 1 + \frac{k_x^2}{k_x^2 + k_y^2} \left( \frac{l_1}{l_0} - 1 \right) l_1 \cos(k_y y) e^{-(l_1 - l_0)z} \right) \quad (12)$$

where $\varphi = \theta + \frac{\pi}{2}$ is the field angle with the y-axis (Figure 1), and $\varphi_0$ is the angle for the two-dimensional arcade. The evolution of the angle $\varphi$ with y for different heights is also plotted in Figure 4a in the case of the base flux of Figure 3. Both equation (12) and Figure 4a show that the modulation with y of $\varphi$ decreases rapidly with height since it comes from the presence of the first harmonic. The scale height $h_1 = 1/l_1$, is of the order of 5 Mm, and the angle becomes nearly constant at heights greater than $2h_1 = 10$ Mm. The shear constant $\alpha'$ has been chosen to give the average value of $\varphi$ measured by Leroy et al (1983, 1984). At heights greater than 10 Mm, we find a nearly constant shear angle as observed. But Leroy et al could not observe below a height of 10 Mm since the scattered light of the photosphere by the atmosphere dominates. Also observations have not been done in the feet of prominences. This model gives a great variation of the shear angle along the prominence axis at low heights. The
localisation in the prominence feet is described in Section 4.

At low heights $\phi$ is negative at the centre of the supergranule cells, so the field has an inverse configuration from the chromospheric pattern, without a parasite polarity. Such configurations have also been found in a two-dimensional field (Démoulin et al 1988a). This appears only for a greatly sheared magnetic field at low heights with an important reversed second harmonic, but it can be obtained in a bipolar region. This leads to the presence of a dip in the magnetic arcade occurring very close to the feet.

### 3.4.2. Field Strength

Figure 3 shows the horizontal field at the chromospheric level. It is more stronger between the two flux concentrations. Figure 4b shows the variation with height. The amplitude of the oscillation along the y axis decreases rapidly with height. As for the shear angle, the field magnitude becomes nearly constant at heights greater than $2 L_1 \approx 10$ Mm which agrees with the observations of Leroy et al. Nevertheless the field decreases more rapidly with height than would be expected from observations (Figure 4c). Two ways can avoid the rapid field decrease of Figure 4c. One is to increase the shear. A scale height of 50 Mm results with $\phi \approx 100°$, but a field strength that is constant with height can only be obtained with $\phi$ of the order of a few degrees, which is not observed. The other way is to increase the size of the chromospheric scale. Prominences are seen to form between field polarity regions much larger than the size of a supergranule cell. Then a two-dimensional harmonic with a periodicity greater than $d_c$ along the x axis may for simplicity be added, and the scale height of the field is multiplied by the same factor. But the other harmonics, used here, must be less sheared, since the constant $\alpha'$ is limited by constant $k_x$ of the large-scale two-dimensional harmonic. Figure 4c is then transformed to more rapidly decreasing field, to which we add a nearly constant field. Then the global properties are not qualitatively changed, but the field decreases less rapidly at large heights.

Departures from the chromospheric flux of Figure 3 are computed in Figure 5. If $L_y/L_x$ is decreased, $k_y$ is increased for a given value $k_x$ and the scale height $h_1$ is decreased: equation (9). For a given value of the amplitude $b_1$, the horizontal field in the plane $x=0$ is also decreased (Equation (11)). Then the first harmonic decreases more rapidly with height. It explains the behaviour of

the curves in Figure 5a which are taken at the centre of a supergranule cell where the first harmonic is reversed from the zeroth order harmonic. In particular for great values of $k_y$, the field strength increases with height. It is similar to the behaviour described by Démoulin et al (1988c). The great magnetic pressure at large heights leads to the formation of dip in the force-free field. The field direction is reversed in this region, as shown in Figure 4a near $|y| = 15$ Mm. When the shear is increased, the scale height of the magnetic field increases. It is particularly important at large shears when $\alpha'$ balances $\sqrt{k_x^2 + k_y^2}$ in the constant $\ell_1$ (equation 9)). This occurs at the centre of the supergranule cells as shown in Figure 5b. But a slightly increasing field strength with height is only present for a great shear: $\alpha' \approx 3.1$, which leads to an angle of the order of 10 degrees at large heights (or for the lowest harmonic).

### 3.5. Field Line Length

The length of a field line connecting two points of the photosphere is an important parameter for the thermal equilibrium along the field line, under a balance between radiative cooling, coronal heating and parallel thermal conduction. It has been shown that nearly all the loop is filled with coronal material, so that a uniform temperature equilibrium seems a good approximation. At coronal temperatures, radiative cooling drives a thermal instability of the plasma since the radiation decreases as the temperature increases. Field (1965) has investigated the stability of such a uniform medium. At short lengths, greater than a few times 10 Mm (which depends on the pressure and the heating) the medium is unstable to linear perturbations. Oran et al (1982) have studied the non-linear development which leads to the formation of a cold condensation, like a prominence.

Further work is necessary to understand the possibility of the formation of prominences by thermal instability. At this stage it seems that we may refer to the work of Field for which the condensation only forms in sufficiently long loops. Nevertheless the problem is not yet well understood, and destabilizing the corona without destabilizing the transition region is difficult. It needs a heating which varies greatly with height and not only with pressure and temperature.
Whatever be the final result of the thermal stability, an important parameter is the loop length. The isocontours of field line length are plotted in Figure 6 for the base flux of Figure 3, except that Figure 6a is computed with $\alpha' = 1.5$. The dashed lines represent the isocontours of the distance of the footpoints (at the chromospheric level $z=0$) from the nearest concentration of flux. As the two feet of a field line are not at the same distance from a flux concentration, an average value is taken. We compute this since the pressure could be different from one field line to another one. The difference could come from below the plane $z=0$ where the magnetic field is concentrated by the convective cells. The region of concentrated field may have a lower pressure to achieve horizontal force balance. At the photospheric and chromospheric level, the plasma pressure decreases much more rapidly than the magnetic pressure, which produces a nearly force-free field in the corona. Nevertheless, the field line anchored in region where the plasma pressure is greater preserves this enhanced pressure at greater heights if the temperature distribution is the same in a horizontal plane. A pressure enhancement increases the radiative cooling and so favours thermal instability according to the work of Field.

At low shears, as in Figure 6a, a field line starting from a positive polarity connects the nearby negative polarity. Then, as the shear is smaller near the flux concentration, the length is shorter than that of a field line starting between concentrations of flux. This difference decreases with height as shown in Figure 6a, and at $y=0$, as the summit increases, so the footpoint is closer to the flux concentration. As the shear increases, so $\alpha'$ increases and only lower field lines can connect adjacent flux concentrations. As previously, they have lower lengths than those anchored between flux concentrations. At greater heights (at $y=0$) the field lines come from regions of lower field strength and so greater shear. The field line length then increases as seen in Figure 6b for a height of about 7 Mm. At greater heights the field lines again connect two flux concentrations and the length is more important at $y=0$ than at $|y| = 15$ Mm. As the constant $\alpha'$ increases, this effect becomes more important and the field line length on the plane $x=0$, at $y=0$, becomes lower, then greater, then lower than the length at $|y| = 15$ Mm and so on many times. However, as the isocontours in Figure 6 are equidistant, the relative value $\Delta\ell/\ell$, is much more important at low heights. The field line length could be lower by a factor of 0.5 near the chromospheric concentration than between them.

From the thermal stability of a coronal loop, we find that the lower loops near chromospheric concentration (and so up to supergranule boundary) are shorter and so more stable. The effect of the pressure variation from one field line to another gives the opposite effect, since the plasma pressure is smaller on lower field lines. Other important parameters for thermal stability are also missing, such as the heating pattern and the footpoint boundary conditions. It is within the scope of this paper to study this problem in more detail, but a more complete model may explain why the filament feet are formed at all and where they are located relative to the supergranule pattern.
# 4. LOCATION OF THE PROMINENCE FEET

## 4.1. Prominence Modelling

There are two main classes of field configuration that support prominences (Leroy et al, 1983-84). In the first configuration the horizontal magnetic field crosses the prominence in the same direction as a potential field created by the photospheric pattern. It has so-called Normal (N) polarity (Priest, 1988). The second configuration has an Inverse (I) field direction. Kippenhahn and Schluter (1957) were the first to propose an N configuration (KS) to support a prominence. They started from a potential symmetrical field and cut it to give a current sheet at the centre, where a prominence could exist. The Lorentz force from the magnetic field produced by the photospheric magnetic pattern and the prominence currents gives the support. Kuperus and Raadu (1974) proposed an Inverse configuration (KR) and a way to form a prominence in a current sheet. They included line tying of the field lines in the photosphere and modelled it by the repulsion of an image current below the photosphere. In fact, both these models could be extended to more complex configurations where both the field and current repulsion forces could support the prominence. In particular I configurations could be supported by the field created by the photospheric pattern, if the region is quadrupolar.

Prominences can be modelled by current sheets, or by current lines, with a singularity of mass. We have to add a line current I to the previous force-free field configurations in the plane above the neutral line. Amari and Aly (1987) have constructed such solution for a two-dimensional linear force-free field model. At present the generalisation to three dimensions is not obvious and the linear force-free field created by a current line warped in three dimensions is not known. In other way, the superposition of a linear force-free field (from photospheric boundary) with a potential field (from line current) given an flimsy as the total field have volumique forces. Then, let be the current line field be a linear force-free field with the same α of the photospheric field, the total field have not net force at each point except a singular Lorentz force at the location of the prominence. For the equilibrium of the current the only change, from the potential field line current, is the form of the repulsion term. Demoulin and Priest (1988a) show in two dimensions that the results are not qualitatively change in this case. Moreover it is difficult to improve the

repulsion term as it is sensitive to the boundary conditions. This conclusion extended to the three dimensional case, show that it is not important to resolved the problem of the force-free field of a current line. However it is important for the global field topology, but in this case we need to know the prominence current repartition. Such considerations are beyond the scope of this paper and we represent only the field created by the photospheric boundary.

For a given photospheric flux polarity, the prominence current could have two directions, and with a vertical magnetic field positive for x > 0 at the chromospheric level, one has an upward Lorentz force when the current is directed in the positive y-direction. This case has Normal polarity since the current creates a magnetic field in the same direction as the photospheric pattern. The negative direction for the current gives an inverse configuration which could be supported, in general, only by the effect of line tying in the photosphere. (There is an exception to this case, even in a bipolar region, since a force-free field could have a reversed direction at large shears and low heights and then could support a negative current (Démoulin et al.; 1988c).

We discuss here the two cases $I_p>0$ and $I_p<0$, called a positive current configuration PCC and NCC, respectively.

## 4.2. Line Current Equilibrium

The prominence modelled by a line current is in equilibrium between the gravity force ($\vec{f}_g = \lambda \vec{g}$), the Lorentz force ($\vec{f}_L = \vec{I} \wedge \vec{B}$) and the repulsion of the image current: $\vec{f}_L = \frac{\mu_0 I^2}{4\pi h} \vec{u}_z$

If the line current is located in a vertical plane, where the vertical magnetic field is zero, the Lorentz force components are:

$$ \vec{f}_L = \begin{cases} -I \cdot \text{By} \cdot \sin \delta & \text{I. By. sin } \delta \\ I \cdot \text{Bx} \cdot \sin \delta & \text{I. Bx. sin } \delta \\ -I \cdot \text{Bx} \cdot \cos \delta & \text{-I. Bx. cos } \delta \end{cases} \quad (13) $$

Where δ is the inclination of the line current to the horizontal plane.

The repulsion force has been approximated because the line current is not straight. To be more rigorous, the repulsion term would have to be expressed by an integral along the line current. Here the errors of the approximation are
such as to increase the current repulsion at the dips of the line current and decrease it at the summits. This is not a serious effect since boundary conditions could affect it more (Amari and Aly, 1986) and the result on the line current equilibrium would not change qualitatively (Demoulin et al, 1988a).

The total force $\vec{f}_g + \vec{f}_y + \vec{f}_R$ would not in general be zero since the Lorentz force is only vertical for a line current parallel to the y-axis, as implicitly assumed in two-dimensional models. We discuss this problem only for positive currents ($I_p$) because for negative currents, the repulsion term must dominate the Lorentz force, so that departures from the vertical of the Lorentz force are less important.

The lorentz force is normal to the line current in the Oyz plane. At a dip of the current line, the y-component of the lorentz force strengthens this region to increase the dip. Then the field lines are pressed together and the magnetic pressure and the dip line tying at the photospheric level will provide a feed-back to help find an equilibrium. We suppose the prominence feet are located at such a line current dip, then this horizontal force tries to decrease the width of the prominence feet. We can note that such a force is much smaller than the force of vertical support : $f_y / f_x = \tan \delta$ when $\delta$ is much lower than 45°.

The x-component of the Lorentz force is much greater by a factor $1/\tan \delta$ than the y-component ; i.e. a factor 3 for $\approx 20°$. Both Bx and By are even in y at the location of the prominence plane. Then the vertical equilibrium shows that $\delta$ is an odd function and so $f_y$ is also an odd function and periodic with a period $2.L_y$. The line current is periodically pulled first in the positive and then is the negative x direction. The average force is zero, but such a periodic warping of the prominence is too small to be not observed.

The magnetic pressure due to the field line tying in the photosphere with a highly sheared field could act very efficiently to impose a nearly planar geometry for the filament. It would deform the analytical force-free field configuration presented here in a way that would need to be calculated numerically. This horizontal component of the Lorentz force may also produce a slow reconnection of adjacent field lines. It decreases the shear, and so the energy of the system. Then new field lines are needed to support the filament, provided by the supergranule cells or giant convection rolls. This reconnection may be important only near the feet of prominences, where the horizontal components of the Lorentz force are important, and the close-back of the field

lines could be related to the downflow observed in filament feet. It may explain why such velocities are much lower than the sound, Alfven or free fall velocities, and also the rosette appearance of threads near a foot.

Clearly, more sophisticated models are needed to explain the observations more fully.

The vertical equilibrium is given by :

$$ \lambda g = B_x \cdot I_p \cdot \cos \delta + \frac{\mu_0 \cdot I_p}{4\pi h} \delta^2 $$

which can be written with dimensionless variables

$$ I'_{p} = I_{p} / \sqrt{\frac{\lambda \cdot g \cdot L}{\mu_{0}}} $$

$$ \delta' = \delta \cdot L, \quad h' = h / L $$

$$ \text{and} \quad \eta' = \sqrt{\frac{(B_x(z=0))^2 \cdot L}{\lambda \cdot g \cdot \mu_0}} \quad (14) $$

$1 = B' \cdot x \cdot I'_{p} \cos \delta \cdot \eta' - I'^2 / 4\pi h'$

The line current is also not in equilibrium, at the location of the line current dip, since we do not compute (15) completely but instead assume that $\cos \delta = 1$, which gives a good approximation when $\delta$ is low. The effect of $\cos \delta$ when $I_p$ is positive is to decrease the support of the magnetic field at the boundaries of the line current dip, which makes the width of the dip a little greater. When $I_p$ is negative, the force-free field give a downward force and the effect is opposite.

In the next sections 4.3 and 4.4, we study the equilibrium height of a line current in a linear force-free field of the basic case plotted in Figure 3.
### 4.3. Positive Current Configuration (PCC)

For a positive current $I_p$, the repulsion of the image current is negligible, except at low heights. Then for small ondulations of the current line, the equilibrium height is approximately given by the isocontours of $B'_x$. If we assume that the prominence mass, $\lambda$, as well as the current, $I$, are independent of $y$ then the line current moves down, where the field normal to the prominence plane is lower and forms a prominence foot. It is localised between concentrations of magnetic flux, near the centre of a supergranule cell (Figures 3 and 7a). When the line current dip increases to low heights, the repulsion between currents increases and stops the current line above the photospheric plane. It increases the size of the dip as shown for the lower line current of Figure 7a.

In Demoulin et al (1988a), we have studied the equilibrium of a line current in a two-dimensional linear force-free field. The KS configuration in this model has the same force-free field and current geometry as the case studied in this section. We have considered two harmonics, the first and the third one. It is found that when the third one is added reversal from the first one to give a field that slowly decreases or increases with height, so that three equilibrium height are possible for a given current $I_p$. The medium solution is found to be unstable (Figure 3 of Demoulin et al 1988a). Approximate conditions have been derived to determine three equilibrium positions, namely

$$ \eta'^2 B' (h')^2 > 4 \cdot R' (h') $$

$$ dB'(h') / dh' > 0 $$

Where $R'(h')$ is the repulsion term equal $1/4\pi h'$ here.

The first leads to a value $\eta'_{crit}$ so that for $\eta' > \eta'_{crit}$ there are three possible solutions when the second condition is approximately satisfied. This means that the third harmonics must be reversal and sufficiently great, but it can occur without parasit photospheric polarity at the base, and with a potential or a highly sheared force-free field.

The first harmonic in the $y$ direction of equation (11) plays here the role of the third one in the two dimensional analysis of Démoulin et al (1988a). Then

three equilibrium positions are possible if $\eta'$ is great enough ($\eta' > \eta'_{crit}$). Figure 7b shows such a case with $\eta' = 10$, and the current line dip of Figure 7a is greatly increased. The medium solution is unstable, while the upper and lower ones are stable, to two-dimensional linear disturbances. Nevertheless the upper one has a greater gravitational energy, and it may be unstable to nonlinear perturbations. The equilibrium position of a current line may then fall to low heights at the centre of supergranule cell.

Variation of the parameters gives results somewhat similar to those obtained with the two-dimensional model, even if the periodic length is shorter along the prominence axis than across. When the shear $\alpha'$ is increased, the region where the three equilibria are possible for a given $I'_p$ increases in size along the vertical. When a reversed two-dimensional harmonic is added, the region increases in size in both the horizontal and vertical directions. The increase of the period length ($L_y$) along $y$ decreases $k_y$, and so increase the scale height and the importance for $B_x$ and $B_y$ of the first harmonic.

In conclusion, when the prominence current is in the positive direction, the Lorentz force is upward, and the height of the line current decreases between the photospheric flux concentrations. There is also the existence of multiple height equilibria for a given current if the magnetic field is too strong (more precisely $\eta' > \eta'_{crit}$).

### 4.4. Negative Current Configuration (NCC)

Support is achieved here by the repulsion between the current and its image below the photosphere. It needs a current much greater, by a factor 10, than in the previous configuration. The Lorentz force is downward, and current line dips are then located where the horizontal field $B_x$ is greatest. In this configuration the prominence feet are located near the boundary cells. Figure 8a shows a typical case with $\eta' = 5$.

The two-dimensional equilibrium of a line current in the negative current configuration has been calculated in Demoulin et al (1988c) to have multiple equilibrium heights one needs, in contrast with the previous positive current configuration, a third harmonics with the same direction as the fundamental. Approximate conditions have been given to have three equilibriums positions, namely
$$ \frac{\eta'^2}{d} \frac{B'(h')}{\left(\frac{B'(h')}{R'(h')}\right)} \sim \frac{4}{\eta} \frac{R'(h')}{dh'} \sim 0 $$

The first is the same as in the PCC, so that $\eta' > \eta'_{crit}$, and the second needs the magnetic field to decrease more rapidly with height than the repulsion term $R'(h')$.

Again the first harmonic plays here qualitatively the role of the third one in the two-dimensional work. The field strength decreases more rapidly with height near the flux concentration, where the line current makes a dip. As in the PCC it increases the dip, and a narrow prominence foot can be formed. An example, with the base flux of Figure 3 is given in Figures 8a and 8b. A region with three solutions is found only at great heights, where the two-dimensional arcade dominates (Figure 4c). This result depends greatly on the form of the repulsion term $R'(h') = 1/4\pi h'$ here, which is the repulsion between the two currents. Amari and Aly have shown that this term is greatly changed at large heights with the field vertical at the boundary. This leads to a repulsion term which decreases at great heights like the first two-dimensional arcade, and the existence of three solutions for a given current disappears. Then as the possibility of three solutions depend on the form of the repulsion term and the boundary conditions, their existence in Figures 8 needs to be investigated more generally. In the next section we try to find other base flux distributions which have a solution at low heights with a more rapidly decreasing field.

The dips in the current line of Figures 8 are small. They do not increase with a greater $n$. Since at low height the repulsion term dominates the magnetic field, the solution of Equation (15) is $I'(h') \approx -4\pi h'$ which does not depend on $n$. When the shear is lower, the magnetic strength decreases more rapidly with height, and so the region where three solutions are possible increases in size to lower heights. The lower height is decreased from 30 Mm in Figure 8b with $\alpha' = 3$, to 7 Mm with $\alpha' = 2$. An increasing of the periodicity length along oy has the same effect, and setting $L_y/L_x = 1$ leads to a lower height of the order of 12 Mm.

In conclusion, when the prominence current is in the negative direction, the prominence feet are located near the supergranule boundaries. However with diffuse flux concentration, as shown in Figure 3, the foot points are not well marked.

## 5. FLUX CONCENTRATION IN THE CHROMOSPHERE

### 5.1. Chromospheric Flux Concentration

The magnetic field is concentrated, below the photosphere, by the convective cells. As the height increase in the solar atmosphere, so the plasma beta of the plasma decreases and the magnetic field dominates the plasma in the corona. Thus, the magnetic field fans out into the corona from the flux concentrations and the great concentration observed in photospheric magnetograms at the network decreases with height (Gabriel 1976). It is not known how important this effect is at the height where the magnetic field becomes force-free (we have called this region the upper chromosphere in the text). Here we wish to investigate the effect of a more concentrated base flux, with no parasite polarity.

### 5.2. Photospheric Flux Pattern

The description of Section 3 may be extended to include any number of harmonics so that Equation (5) becomes:

$$ B_z(x,y,z) = \sum_{n_x=1}^{\infty} \sum_{n_y=0}^{\infty} B_{n_x, n_y} \cdot \cos(k_y y) \sin(k_x x) e^{-\ell z} $$

$$ \text{with } \ell^2 = k_x^2 + k_y^2 - \alpha^2 $$

The field remains odd in x and even in y. To simplify the description of the base flux, we assume that $B_z(x,y,z)$ could be written as the product of functions of x and y.

$$ B_z(x,y,z) = \sum_{n_x=1}^{\infty} \sum_{n_y=0}^{\infty} B_{n_x, n_y} \sin(k_x x) \cos(k_y y) e^{-\ell z} $$

A periodic sum of Dirac functions is obtained in the y-direction when all the coefficients $B_y$ are set equal to unity. Here we limit the development to the fourth harmonic, with coefficient different from unity to avoid parasite polarity. Figure 9a shows one example where the flux is more concentrated both in the x and y direction than in Figure 3.
### 5.3. Positive Current Configuration (PCC)

The concentration of the base flux increases the region of low field strength and also increases the horizontal size of the line current dip well. One example is shown in Figure 9b with the base flux of Figure 9a. All the other parameters are the same as for Figure 7b. The region with three solutions has disappeared since the second condition is not satisfied. It comes from the second harmonic in the x-direction which is not reversed from the fundamental, and so the field decreases too rapidly with height even between two flux concentrations. If we concentrate the field only in the y direction ($B_x' = 1, B_y' = 1, B_y'' = 1.1, B_y''' = 0.9$ and $B_y'''' = 0.6$) then a region of three equilibrium heights is present, with a much greater extent than in Figure 7b.

The effect of concentrating the base flux in the PCC is to increase the horizontal size of the dip. The dip in the current line is important (three positions being possible) only when the base concentrations are pulled near the neutral line.

### 5.4. Negative Current Configurations (NCC)

The base flux concentration is more important in this case than in the previous one since it creates more deeper and narrower dips in the current lines. Two examples are shown on Figure 9c and 9d similar to Figures 8a and 8b except for the base flux. The flux concentration increase in the y-direction makes a more bigger dip in the current line of Figures 8a and 8b, but does not create a region with three solutions at low height since the field strength does not decrease more rapidly than the repulsion term. A two-dimensional harmonic in the same direction as the first one would help to build a more rapidly decreasing field with height and Figures 9c and 9d show the results of concentration in the two directions. Such results are similar to those obtained in the PCC of Figures 7a and 7b.

With a more concentrated base flux, we have obtained a large dip in the current line equilibrium. The prominence feet are located here at the boundary of two super granule cells. An intersection between three or four cells would be an even more favourable region for this type of feet since the two oppositely concentrated regions need to be close.

## 6. CONCLUSION

A three-dimensional linear force-free field configuration has been investigated to model the external field of a prominence. The prominence is modelled by a current line, and its vertical equilibrium in the force-free field is studied.

We have studied at first the force-free field configuration alone. It may represent well the magnetic field before the formation of a prominence, or in the presence of a low density prominence. We expand the field in a Fourier series. Only a few harmonics are necessary to give the qualitative behaviour of a prominence-like field. The chromospheric field pattern is represented by concentrations of vertical magnetic flux on each side of a polarity inversion line.

This represents the observed location where solar prominences are forming. A bipolar region is used, but the bridge appearance of prominences requires a variation in the V-direction. The force-free field resulting from these boundary conditions is built up from magnetic arcades which connect the two opposite polarities (there are no open field lines). In the prominence plane, the magnetic field is horizontal, stronger and less sheared near the chromospheric flux concentrations than between them. At a height greater than 10 Mm, the field becomes nearly uniform along the prominence axis, and important variations are localised at the prominence feet level. The field line lengths are less important near the base flux concentrations than between them, because of the smaller field shear. From a thermal instability point of view (which can form the cool prominence material) one needs a large length to trigger the instability. The condensation may start between the base flux concentrations, and as at the centre of supergranule cells. This is consistent with the observations which show that prominence formation start from the feet and continues to greater heights. In this model the feet are formed at the centre of supergranule cells.

To study the appearance of a well-formed but not too heavy prominence, we add a line of current and mass in the prominence plane. This current line is in vertical equilibrium between three forces: gravity, an interaction with the force-free field, and the repulsion of an image current below the photosphere, modelling the line tying of the field line in the photosphere. There are two possible directions for the current line. One where the force-free field
supports the current against gravity, called PCC in the text, and the second where the current has the opposite sense, with a downward force, called NCC in the text. In PCC, the line current falls down, where the field strength is lower, i.e. at the centre of the supergranule cells. In NCC, there is the opposite effect, so the dip is located at the cell boundary. In each case when a critical parameter is reached, three equilibrium positions are possible for the line current at the location of the dip in the line current. It has the effect of increasing the depth of the dip in each case. The effect of a more highly concentrated base flux is not qualitatively important, in the PCC, but is needed in the NCC to produce a significant dip.

When the line current is not horizontal, the Lorentz force from the horizontal force-free field is not strictly vertical, and a local equilibrium cannot be achieved because the two other forces are vertical. However, a global equilibrium of the line current is obtained, and the extra horizontal forces may deform the initial field and its reaction compensate them.

The three-dimensional force-free field with the current lines is a reasonable representation of the magnetic field around a solar prominence. The locations of the feet depend on the direction of the prominence current. The feet are located at the centre of the super granule cells in the positive case, and at the boundaries in the negative case. A comparison with observations is not directly possible since the two preceding cases are not directly related to the observed Normal and Inverse configurations. A more realistic model with a current sheet does not create a singular field as a line current does. With such a model, the field created by the prominence currents could be lower than that created by the photospheric pattern for a low density prominence.

In future, this work needs to be extended in several directions to be more realistic.

The field which results from the postulated line currents has to be modified in such a way that no Lorentz forces are generated in the corona outside the prominence. (A force-free field solution is needed for the field created by the current line). Also non-linear effects and the closure of the electric current system have to be included.

A relation with the motion of the convective cells may be important. We have to understand how the motions at the photospheric level could deform the magnetic field to form a magnetic configuration where a condensation could happen.

A slow reconnection, in relation with cells motion has to be understood, particularly how such a configuration can exist for such a long time.

Acknowledgements

The authors thank S. Bordet for typing the manuscript.
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FIGURE CAPTIONS

Figure 1 :

A schematic prominence showing the coordinate system used here. The hatching represents the bottom and top of the prominence. Periodically the prominence reaches the photospheric level in structures called feet. A coplanar magnetic arcade is shown dashed. $\varphi$ is the angle between the prominence axis Oy and the local horizontal magnetic field.

Figure 2 :

A schematic diagram of convective supergranule cells in the photosphere. The plasma motion represented by arrows in one cell concentrates the field at the cell boundaries. High concentration is obtained at the intersection of 3 or 4 cells, represented by circles. Possible prominence locations are indicated by a thick straight line for several symmetric positions. Figure 2a represents a rough approximation with square cells, and Figure 2b a more realistic situation with hexagonal cells. Both cases could be approximated roughly by the configuration $C_1$.

Figure 3 :

Field pattern at the chromospheric level ($z = 0$) for the basic model. The size of a supergranule cell is taken to be 30 Mm. The periodicity length along the x-direction is twice that along the y direction ($\ell y/\ell x = 0.5$), $\alpha' = 3.$, and the three-dimensional harmonic has an amplitude of $b_1 = 0.8$ times the two dimension one. The arrows represent the horizontal field, the lines are the isocontour of the vertical component, with continuous lines for positive field and dashed lines for negative ones. This case is used in Figures 4 to 8

Figure 4 :

The evolution of the magnetic field in the prominence plane ($x = 0$) for the basic case of Figure 3.

(a) and (b) represent the angle $\varphi$ of the field with the neutral line and the field strength above the neutral line for different heights (in Mega meters). At low heights, the field has an Inverse polarity since $\varphi$ is negative. The shear angle $\theta = \frac{\Pi}{2} - \varphi$ is lower near the photospheric flux

concentration, where the magnetic field is greater.

(c) represents the evolution of the magnetic strength with height ($z$) for different values of $y$. Near the flux concentration the field decreases rapidly, but between these concentrations, it is nearly constant at low heights.

Figure 5 :

Evolution of the magnetic field strength with height ($z$) midway between two photospheric flux concentration, $\ell y/\ell x$ (top) and $\alpha'$ (bottom). This figure is computed for the photospheric flux of the basic model (Figure 3). An increasing field magnitude with height needs both $\ell y/\ell x$ and $\alpha'$ to be large. The field is slowly decreasing at large heights when $\alpha'$ is close to $\pi$. $\alpha' = 3.1$ corresponds to an angle $\varphi$ of only $90^\circ$ for $z > 10$ Mm.

Figure 6 :

The isolength of the field line (continuous curves) in the basic case of Figure 3. The lines are labelled by the length (in Mm) of the field lines from one footpoint to the other. At low heights the field lines are shorter near the origin. The opposite effect is present at great heights.

The dashed lines represent the isocontours of the distance of the footpoints to the nearest flux concentration (since this distance is not the same for the two footpoints, an average is taken).

The upper case is for $\alpha' = 1.5$, and the isocontour of length are equidistant by $\Delta\ell = 9.7$ Mm, the isocontours of distance of $\Delta d = 5$ Mm; the lower case is for $\alpha' = 3$ and $\Delta\ell = 12$ Mm, $\Delta d = 5$ Mm

Figure 7 :

The field created by current line has not been added to the photospheric field pattern.

Dashed lines are the isocontours of the normal field ($Bx$). Continuous thick lines represent the equilibrium location of current lines in the force-free field for the basic model. Both figure are plotted with a positive current (a) with $\eta' = 5$. (b) with $\eta' = 10$. The later case shows one example where three positions are possible for the current line. The intermediate position is unstable, and the line current may fall to the lower equilibrium position where the gravitational energy is lower.
Figure 8 :

The same as for Figure 7 but with a negative current. (a) : $\eta' = 5$, (b) $\eta' = 10$. The location of the feet are different from the PCC : they are located near the flux concentrations, i.e. at the supergranule boundaries.

Figure 9 :

(a) Configurations with a more highly concentrated magnetic flux at the chromospheric level. The harmonic amplitudes are $B_x' = 1$, $B_y' = 0.5$, $B_z' = 1$, $B_x^2 = 1.1$, $B_y^3 = 0.9$, $B_z^4 = 0.6$, and the parameters $\alpha' = 3$, $L_y/L_x = 0.5$. There are no parasite polarities. (b) represents the PCC for this base flux, with $\eta' = 10$. Comparison with Figure 7b shows that the dip is larger, but that the possibility of three positions has disappeared since $B_x$ is positive and large enough. (c) (d) represent the NCC with $\eta' = 5$ and $\eta' = 10$, respectively. the dip is narrower and more important than with the less concentrated flux of Figures 8a and 8b.

<figure><img src="image_99.png" /><figcaption>Figure 1</figcaption></figure>

<figure><img src="image_100.png" /><figcaption>Figures 2</figcaption></figure>
<figure><img src="image_101.png" /><figcaption>Field angle on prominence axis</figcaption></figure>

<figure><img src="image_102.png" /><figcaption>Magnetic strength along prominence axis</figcaption></figure>

<figure><img src="image_103.png" /></figure>

<figure><img src="image_104.png" /></figure>
<figure><img src="image_105.png" /><figcaption>Horizontal magnetic strength along the vertical</figcaption></figure>

<figure><img src="image_106.png" /></figure>

<figure><img src="image_107.png" /></figure>

<figure><img src="image_108.png" /></figure>

<figure><img src="image_109.png" /></figure>

<figure><img src="image_110.png" /></figure>
<figure><img src="image_111.png" /><figcaption>Figure 7b</figcaption></figure>

<figure><img src="image_112.png" /><figcaption>Figure 8a</figcaption></figure>

<figure><img src="image_113.png" /><figcaption>Figure 8b</figcaption></figure>

<caption>Figures 8</caption>
<figure><img src="image_114.png" /></figure>

<figure><img src="image_115.png" /></figure>

<figure><img src="image_116.png" /></figure>

<figure><img src="image_117.png" /></figure>
# C1: THERMAL STABILITY ANALYSIS OF THE FINE STRUCTURE OF SOLAR PROMINENCES.

Pascal Démoulin
Jean - Marie Malherbe
Brigitte Schmieder

Observatoire de Paris - Section de Meudon
92 195 Meudon Cedex - France

and

Mickael A. Raadu

Royal Institute of technology
Department of plasma physics
S- 100 44 Stockholm - Sweden

## SUMMARY

We analyse the linear thermal stability of a 2D periodic structure (alternatively hot and cold) in a uniform magnetic field. The energy equation includes wave heating (assumed proportional to density), radiative cooling and both conduction parallel and orthogonal to magnetic lines. The equilibrium is perturbed at constant gas pressure. With parallel conduction only, it is found to be unstable when the length scale $l_{\parallel}$ is greater than 45 Mm. In that case, orthogonal conduction becomes important and stabilizes the structure when the length scale $l_{\perp}$ is smaller than 5 km. On the other hand, when $l_{\perp}$ is greater than 5 km, the thermal equilibrium is unstable, and the corresponding time scale is about $10^4$ s: this result may be compared to observations showing that the lifetime of the fine structure of solar prominences is about one hour; consequently, our computations suggest that the size of the unresolved threads could be of the order of 10 km only.

## FUNDAMENTAL EQUATIONS OF THERMAL EQUILIBRIUM.

We use the 2D "chessboard" of figure 1, which displays a periodic hot (T1) and cold (T2) structure in a uniform magnetic field (this could be the case in the central parts of prominences, see Leroy et al., 1983). We write equations for thermal equilibrium of hot and cold cells, as :

$$ 
\begin{cases}
\rho_1 h - \rho_1^2 Q(T_1) - 2 \frac{F_{\parallel}}{l_{\parallel}} - 2 \frac{F_{\perp}}{l_{\perp}} = 0 \\
\rho_2 h - \rho_2^2 Q(T_2) + 2 \frac{F_{\parallel}}{l_{\parallel}} + 2 \frac{F_{\perp}}{l_{\perp}} = 0
\end{cases}
$$

Where $F_{\parallel} = \frac{\rho_{o\parallel}}{3.5} \frac{T_1^{3.5} - T_2^{3.5}}{l_{\parallel}}$

and $F_{\perp} = - \frac{\rho_{o\perp}}{1.5} \frac{T_1^{-1.5} - T_2^{-1.5}}{l_{\perp}}$

are respectively the heat flux parallel and orthogonal to the magnetic field. $\rho_1$ and $\rho_2$ are the densities of respectively hot and cold cells. $k_{o\parallel}$ and $k_{o\perp}$ are conduction coefficients; $k_{o\perp}$ depends on the strength of the magnetic field B.
We assume that the gas pressure remains constant ($ρ_1 T_1 = ρ_2 T_2 = ρ T$ = constant and use the cooling function $Q(T)$ given byildner (1974). Unknown quantities are $h$, $T_1$, $T_2$. The equilibrium state ($h, T_2$), when $T_1$ is fixed, is given by the set of 2 equations above. Possible solutions are shown in figure 2 (top).

## ANALYSIS OF THERMAL STABILITY

We perturb the equilibrium at constant gas pressure $P$ and heating $h$ ($T→T+δT$). Equations are linearized assuming that $δT ∝ e^{βt}$, where $β$ is a growth rate. We assume also that there is no motion in a direction perpendicular to the magnetic field, so that $l_⊥$ = constant. Hence, mass conservation gives $1/ρ$ = constant. We get two solutions for $β$: the first one is always negative (stable solution), but the second one can either be positive (unstable) or negative (stable). It depends on the values of equilibrium parameters $1//$, $l_⊥$ and $T_1$. The magnetic field strength $B$ was kept constant (1 Gauss).

## RESULTS

Figure 2 (bottom) gives the growth rate as a function of $l_⊥$ and hot temperature $T_1$ (parallel conduction was neglected there). The thermal equilibrium is unstable when $β > 0$; this is always the case when $l_⊥$ is too large (> $10^6$ m). When $l_⊥$ is smaller than $10^5$ m, the equilibrium may be stable if $T_1$ does not exceed a critical value $T_{max}$ ($T_{max} = 10^6$ K for $l_⊥ = 10^4$ m).

Figure 3 gives $β$ as a function of $l_⊥$ and $1//$ for a 2 D model ($T_1$ was kept constant). It shows that, when $1// < 45$ Mm, the thermal equilibrium is stable; when $1// > 45$ Mm, it is unstable, unless $l_⊥ < 5$ km. When $l_⊥ > 5$ km, the time scale for instability is approximately equal to $10^4$ s and corresponds to the observed life time of the fine structure in solar prominences. This result suggests that the size of thin threads could be as small as 10 km. The temperature of hot cells ($T_1$) used in the computations was $10^6$ K.

<figure><img src="image_118.png" /></figure>

**figure 1** : the geometry of the model : the structure is periodic in $x$ direction (length scale $l_⊥$) and $y$ direction (length scale $1//$). The magnetic field $B$ is parallel to $y$. $z$ is the vertical axis. White areas are hot ($T_1$) and tenuous ($ρ_1$); dashed cells are cold ($T_2$ and dense ($ρ_2$)).

**figure 2** : next page

**top** : $\log Q(T) + (\text{orthogonal conduction})/ρ^2$ as a function of $\log(T)$ for different values of the length scale $l_⊥$ (1D calculation with $l_⊥ = 10^3$, $10^4$, $10^5$, $10^6$ and $10^7$ m). Equilibrium solutions are located at points A, B, C (intersection with the straight line $\log(T) + \log(\frac{\delta h}{P_m})$). When the heating becomes too large, the cold solution A does not exist any more.

**bottom** : the growth rate as a function of hot temperature $T_1$ (1 D calculation). The function $\operatorname{sgn}(β) / \log|β|$ is displayed for different values of $l_⊥$ (same as above).
<figure><img src="image_119.png" /></figure>

<figure><img src="image_120.png" /><figcaption>figure 2</figcaption></figure>

<figure><img src="image_121.png" /><figcaption>log l// (Mm)

figure 3 : the growth rate for the 2D calculation as a function of l// and 1⊥. Isocontours of sgn (β) log|β| are displayed. The dotted area is stable (β < 0) ; the white one (top) is unstable (β > 0). The dashed region (left and bottom) represents the domain where a cold equilibrium does not exist (see figure 2).</figcaption></figure>

REFERENCES

Dunn, R.B. : 1960, Thesis Ph.D, Harvard University

Engvold, O. : 1976, Solar Phys., **49**, 283

Hildner, E. : 1974, Solar Phys., **35**, 123

Leroy, J.L., Bommier V., Sahal S. : 1983, Astron. Astrophys., **83**, 135

Maltby, P. : 1976, Solar Phys., **46**, 149.
### C3:

Analyse de la stabilité des structures fines des protubérances solaires quiescentes.

P.Démoulin (Observatoire de Meudon)

L'observation des protubérances quiescentes au limbe solaire met en évidence de trés fines structures de moins d'une seconde d'arc. Ici on présente les observations Ha, en projection sur le disque solaire, d'une protubérance. Elles ont été obtenues grâce au spectrographe à double passage soustractif multicanal (DPSM) du Pic du Midi. Les clichés mettent en évidence des structures fines dans le corps du filament aussi bien sur la carte des intensités que sur celle des vitesses.

Beaucoup de modèles de protubérances ignorent cette structuration à petite échelle en supposant que le filament est homogène, comme le montrent les images prises avec une faible resolution. Il a été suggeré que les structures fines se forment dans une nappe de courant grâce à l'instabilité de 'tearing' couplée à l'instabilité thermique. Ici on envisage l'influence de la conduction parrallèle et orthogonale au champ magnétique sur la stabilité des structures fines. Les conditions d'équilibre et le taux de croissance des instabilités sont déterminés par une analyse linéaire de la stabilité.

La conduction parallèle stabilise des échelles de longueur de 10 000 à 100 000 km qui sont typiques, respectivement de la longueur des structures fines et de la hauteur d'un filament quiescent. La conduction orthogonale stabilise des échelles de longueur de l'ordre de 10 km avec une conductivité normale. Cette taille peut être amplifiée par une conduction anormale en milieu turbulent d'un ou deux ordre de grandeur.

Les observations actuelles montrent que les filaments sont constitués de filets de matière froide d'un diamètre de l'ordre de 1000 km. Cependant l'étude précédente permet de prévoir des tailles encore plus faibles, de l'ordre de 100 km, qui pourront être résolues avec Themis. On pourra alors mieux comprendre la physique des filaments quiescents qui semblent être un enchevêtrement de tubes chauds et froids, alors que les modèles actuels les décrivent comme des corps homogènes de plasma froid. La mesure du champ magnétique à trois dimensions permettra de positionner l'alignement de celui-ci sur les structures fines.
Astron. Astrophys. 183, 142-150 (1987)

ASTRONOMY
AND
ASTROPHYSICS

- A2:

# Fine structures in solar filaments

## I. Observations and thermal stability

P. Démoulin¹, M. A. Raadu², J. M. Malherbe¹, B. Schmieder¹

¹ Observatoire de Paris - Section de Meudon, DASOP, UA 326, F-92195 Meudon Principal Cedex, France

² Royal Institute of Technology, Department of Plasma Physics, S-10044 Stockholm, Sweden

Received January 30, accepted April 9, 1987

**Summary.** Limb observations of quiescent prominences show very fine structures of less than one arcsecond. Here Hα observations on the disk, made with the Multichannel Subtractive Double Pass (MSDP) spectrograph operating at the Observatoire du Pic du Midi are presented. They show long fine structures in the body of the filament, both in intensity and velocity maps. Many prominence models ignore such fine structures and suppose that the quiescent filament is a uniform body such as is seen in unresolved images. It has been suggested that in a current-sheet model of a prominence, fine structures may be produced as a consequence of the tearing mode coupled to thermal instability. Here we investigate the role of parallel and orthogonal conduction on the stability of a periodic fine structure. Equilibrium conditions are found and growth rates determined for linear perturbations.

**Key words:** solar prominences – thermal instability – fine structure

### 1. Introduction

Quiescent prominences are structures consisting of cool plasma ($\approx$7000 K) supported by magnetic field ($\approx$10G) in the hot solar corona ($\approx$10° K). Seen on the disk they are dark features which lie on the neutral line between regions of opposite magnetic polarity. Their typical dimensions are 2 10⁸ m long, 5 10⁷ m high and only 2 10⁶ m thick. The main body is a stable structure lasting for a few months.

Observations of quiescent prominences at the solar limb reveal a fine structure when the seeing is better than 0.5. Dunn (1960) described these structures: there are very fine almost vertical threads with a width of 3 10⁵ m, or less. (3 10⁵ m is the minimum size set by the instrumental technique, smaller details could not be resolved). Engvold (1976), however, reported that prominences are composed of knots which perhaps are the basic structures forming the threads, and are only seen under very good observing conditions. The fine structure is continuously evolving with a typical time scale of about 10⁵ s: (Orrall and Zirker, 1961), although Engvold (1976) has reported that threads have a life time of around one hour and that bright knots have an average lifetime of only 8 min.

Send offprint requests to: P. Demoulin

There is a controversy in the literature concerning the velocity. Observations at the solar limb, (Dunn, 1960; Menzel and Wolbach, 1960a) show that there is a downward motion between 10 to 20 km s⁻¹ in hedgerow prominences. Pettit (1932) found velocities of only 5 to 10 km s⁻¹ in a quiescent one, while observations on the solar disk show that there is an upward motion of a few km s⁻¹: (Mein, 1977; Martres et al., 1981; Malherbe et al., 1981–1983a). A discussion of the accuracy of velocity determinations may be found in Schmieder et al. (1984).

The downward motion can be explained by material falling in a helicoidal force-free field with a radius of 1000 km: (Malville, 1978). These velocities may also be explained by the slow falling down of the plasma braked by an electromotive force: (Mercier and Heyvaerts, 1977). However, the latter authors argue that the diffusion is insufficient to explain the observations. The upward motions have been modeled by Malherbe and Priest (1983b). The motion is a result of the evolution of the magnetic support induced by photospheric motions.

Almost all theoretical models regard quiescent prominences as a uniform body as observed with insufficient resolution. These are therefore not useful to describe the fine structures. Only a few models have been proposed to model the detailed structure of prominences. One of the most attractive models was constructed by Kuperus and Tandberg-Hanssen (1967), who described a way to create the fine structure by the combined action of thermal and tearing mode instabilities. The magnetic field between two regions of opposite polarity is opened by the "safety valve mechanism" of Parker (1963) and a neutral sheet is formed. The plasma is there strongly compressed and thermally isolated from conduction by islets of magnetic field formed by the tearing mode instability. Consequently, the hot plasma cools to form long thin threads with a diameter of typically 300 km. Recently a thorough investigation has been made of the effects of the thermal and tearing modes with a consistent simultaneous treatment of both instabilities (see for example Van Hoven et al., 1984; Tachi et al., 1985). A significant new result is that magnetic reconnection may occur even on the shorter radiative instability time scale (Steinolfson and Van Hoven, 1984). This means that small-scale magnetic structures can be formed simultaneously with thermal instability.

In this paper we will consider the role of thermal energy balance in a prominence model consisting of alternating slabs of cool and hot (coronal) matter. This leads to constraints on the dimension of the fine structure. Further constraints and estimates