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Astron. Astrophys. 318, 621-630 (1997)

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1. Introduction

The heating of the solar corona is an important problem of solar physics. It remains unsolved despite considerable efforts to propose efficient heating mechanisms. It is commonly admitted that the basic source of free energy is the photospheric plasma flow associated with convection. However, depending on the way this mechanical energy is supplied to the corona, the heating theories can be classified as either wave or current dissipation theories.

Wave heating is based on the generation of Alfvèn waves by the photospheric motions with a time scale of the order of the Alfvèn transit time scale. These waves propagate and may be subsequently dissipated within the corona by either phase-mixing (Heyvaerts & Priest 1983) or resonant absorption (Ionson 1978; Hollweg 1987; Einaudi & Mok 1987). Although the dissipation coefficients are extremely small, wave dissipation can be an efficient mechanism because large gradients develop during these processes. Presently, the wave heating theory is under intense investigation (Halberstadt & Goedbloed 1993-1995; Califano et al. 1992).

Although the heating by Alfvèn waves is a viable mechanism for open magnetic field region (coronal holes) and quiet closed field region, electric current dissipation mechanisms including magnetic reconnection are likely to dominate in active regions.

In current dissipation theory, the continuous twisting or braiding of magnetic flux tubes by photospheric motions on a slow time scale (compared with the Alfvèn transit time) is considered. This can drive fine scale current systems and subsequently rapid energy dissipation (Heyvaerts 1992). Parker (1983) proposed that the coronal magnetic structure becomes increasingly more complex due to braiding of magnetic field lines by complex photospheric boundary motions, and finally reaches an highly dissipative state. An approximate statistical description of the effect of magnetic braiding on the coronal electric system has been developed by Van Ballegooijen (1986). He has shown that the electric current develops a spectrum that reaches finer and finer scales as time goes on. Other authors, following Chiuderi (1980), have argued that a gradient length short enough for dissipation could result from MHD resistive instabilities developing in stressed coronal magnetic structures. Heyvaerts and Priest (1984, 1992), considering that initial instabilities should develop into a turbulent regime that would lead to dissipation, have presented global descriptions for coronal heating produced by a turbulent solar corona.

This paper deals with a particular aspect of these problems, namely, the development of the kink instability in line-tied coronal loops. The ideal MHD kink instability is one of the routes by which fine-scale current structures may form in the solar coronal medium. Although the dissipation coefficients are extremely small, dissipative effects would then rapidly destroy the almost singular current layers or filaments that are expected to form as a result of this instability, eventually leading to heating of the medium. In a recent paper, Longcope et al. (1994) have shown that (genuine) current sheets cannot develop in the ideal evolution of the line-tied coalescence instability. This result is consistent with recent numerical results of an MHD simulation on the ideal kink instability in line-tied cylindrical loop by Baty & Heyvaerts (1996). Therefore, more work remains to be done to describe in detail the mechanism by which kink instabilities could produce extremely fine-scale current gradients in line-tied cylindrical loops.

Coronal loops are magnetic flux tubes anchored in the photosphere. Because of the large density and small resistivity of the photospheric fluid, the magnetic field lines can be considered as "frozen" in the photospheric flow, which twists magnetic flux tube, i.e. induces currents in them, injecting magnetic energy into the loops. However, magnetic energy storage cannot proceed further if the loop becomes unstable on a time scale that is shorter than that of photospheric motions. Cylindrical line tied loops have been found to be ideally unstable when a critical twist is exceeded (Raadu 1972; Hood & Priest 1979; Einaudi & Van Hoven 1983). The twist is the change of the azimutal angle of a point following a given field line from one photospheric boundary to the other. Long and thin coronal loops are considered, so that, to a first approximation, toroidal curvature effects can be neglected in this study. The critical twist value depends on the aspect ratio and on the equilibrium configuration, the aspect ratio being the ratio of the loop length to its radius. The linear structure of the kink is fairly well established (Velli et al. 1990a; Mikic et al. 1990; Baty & Heyvaerts 1996). Although its structure bears some resemblance with the internal kink mode in cylindrical periodic geometry (representative of magnetic confinement devices for nuclear fusion, such as tokamaks and pinchs), the line-tying conditions suppress resonant singular surfaces. In addition, the mode has a ballooning character with a maximum amplitude at the axial midplane. However, few results are available on the non linear evolution of the mode. Strauss & Otani (1988) have performed numerical calculations in cylindrical geometry using a set of reduced MHD equations. Current sheets have been shown to form as in the periodic case, but no discussion of the circumstances under which this happens has been presented, and a non-zero resistivity was included to smooth the solution. Craig & Sneyd (1990) have also addressed this issue, using incompressible Lagrangian numerical calculations for the uniform twist Gold-Hoyle cylindrical line-tied flux tube; and they found that current sheets mainly appear near the tube ends. Finally, no current sheets formed in the line-tied cases treated by Mikic et al. (1990), because the loop was too short. In a recent paper, we clarified the situation by highlighting the mechanism by which fine-scale current structures may form (Baty & Heyvaerts 1996). Using fully three dimensional non linear MHD simulations in line-tied cylindrical loops, we considered two distinct initially unstable equilibria, the uniform-twist force-free Gold-Hoyle profile (GH) and a non force-free field with variable and localized twist profile (LT). For these two equilibria, the initial magnetic equilibrium has been shown to reach a bifurcated ideal MHD equilibrium. For the GH case, the magnetic structure obtained exhibits only non linear deformation of the initial equilibrium without formation of a fine scale magnetic structure. This results from the somewhat "pathological" character of the GH equilibrium, the twist of which having a constant value as a function of the radius (i.e. there is no shear). However, a current concentration extending all along the loop length and taking the form of an helical ribbon of intense current has been obtained for the LT case. This current concentration is non singular in the sense that it appears as a negative current spike with finite amplitude and non zero thickness. This contrasts with periodic configurations where a singular current sheet forms in the ideal limit. In the case of the loop studied by Baty & Heyvaerts, the thickness of the current layer has been found to be two orders of magnitude smaller than the loop radius. As pointed out by Mikic et al. (1990), the length of the loop probably plays an important role in determining the characteristics of the current layer. Line-tying effects should become less important for longer loop lengths, and a singular current sheet should form in the limit of infinite length. Recently, similar current concentrations have been obtained during the development of the sausage instability in coronal loops (Longbottom et al. 1996).

The aim of this paper is to investigate the dependence on the initial equilibrium parameters of the characteristics of the ideal current layer generated by the kink instability in line-tied solar loops. It is of particular interest to study how the thickness and the amplitude of the current concentration scales with the aspect ratio of the loop. We assume resistive effects to become important only in the non linear development, when the ideal kink mode has brought the initial equilibrium to a bifurcated kinked configuration. We plan to study the effect of resistivity in a future work. We do not deal here with resistive kink or tearing modes (Velli et al. 1990b; Otani & Strauss 1988; Mok & Van Hoven 1982) for which resistivity has an important effect on stability limits and on the dynamics of the linear phases. Numerical computations of ideally unstable kink modes have been carried out using the full 3D cylindrical non linear evolution MHD code, SCYL (Baty & Heyvaerts, 1996).

This paper is organised as follows. The physical model is presented in Sect. 2, where the numerical procedure is also briefly described. In Sect. 3, we present the initial unstable equilibria used as starting point of the simulations, and we give a brief study of the stability and of the linear phase of the kink mode. Results on the non linear development of the kink instability are reported in Sect. 4. A simple model is also provided to interpret the scaling of the characteristics of the current layer with the aspect ratio of the loop. Finally, in Sect. 5, we discuss the results with a particular attention on the consequences on the heating of the medium.

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© European Southern Observatory (ESO) 1997

Online publication: July 8, 1998