Resonant behaviour of MHD waves on magnetic flux tubes

The resonances that appear in the linear compressible MHD formulation of waves are studied for equilibrium states with flow. The conservation laws and the jump conditions across the resonance point are determined for 1D cylindrical plasmas. For equilibrium states with straight magnetic field lines and flow along the field lines the conserved quantity is the Eulerian perturbation of total pressure. Curvature of the magnetic field lines and/or velocity field lines leads to more complicated conservation laws. Rewritten in terms of the displacement components in the magnetic surfaces parallel and perpendicular to the magnetic field lines, the conservation laws simply state that the waves are dominated by the parallel motions for the modified slow resonance and by the perpendicular motions for the modified Alfvén resonance.The conservation laws and the jump conditions are then used for studying surface waves in cylindrical plasmas. These waves are characterized by resonances and have complex eigenfrequencies when the classic true discontinuity is replaced by a nonuniform layer. A thin non-uniform layer is considered here in an attempt to obtain analytical results. An important result related to earlier work by Hollweg et al. (1990) for incompressible planar plasmas is found for equilibrium states with straight magnetic field lines and straight velocity field lines. For these equilibrium states the incompressible and compressible surface waves have the same frequencies at least in the long wavelength limit and there is an exact correspondence with the planar case. As a consequence, the conclusions formulated by Hollweg et al. still hold for the straight cylindrical case. The effects of curvature are subsequently considered.     

Resonant behaviour of MHD waves on magnetic flux tubes

The absorption of solar five-min oscillations by sunspots is interpreted as the resonant absorption of sound waves by a magnetic cylinder. The absorption coefficient is calculated both analytically under certain simplifying assumptions, and numerically under more general conditions. The observed magnitude of the absorption coefficient, which is up to 0.5 or even more, can be explained for suitable ranges of parameters. Limitations in the present model are also discussed.     

Resonant behaviour of MHD waves on magnetic flux tubes

Resonant absorption of MHD waves on a nonuniform flux tube is investigated as a driven problem for a 1D cylindrical equilibrium. The variation of the fractional absorption is studied as a function of the frequency and its relation to the eigenvalue problem of the MHD radiating eigenmodes of the nonuniform flux tube is established. The optimal frequencies producing maximal fractional absorption are determined and the condition for total absorption is obtained. This condition defines an impedance matching and is fulfilled for an equilibrium that is fine tuned with respect to the incoming wave. The variation of the spatial wave solutions with respect to the frequency is explained as due to the variation of the real and imaginary parts of the dispersion relation of the MHD radiating eigenmodes with respect to the real driving frequency.     

The equilibrium of coronal flux tubes under toroidal forces

The properties of slender isolated flux tubes, taking into account curvature effects, were investigated by Parker (1975, 1979) and Spruit (1981), and many studies have been made concerning the equilibrium of slender flux tubes in the solar corona. In this paper we use a different approach considering the coronal loop as a part of a circular torus and studying the position of its top when the loop is in equilibrium under toroidal forces. Toroidal forces were considered by Shafranov (1966) for toroidal pinches and the equilibrium can be studied for different values of the toroidal current intensity and external magnetic field. The results show that it is possible to have a coronal flux tube in equilibrium without considering gravity and external magnetic field. Furthermore, the total twist of the flux tube and its variation with the toroidal intensity has been studied.     

Linear MHD stability of infinitely long coronal flux tubes

Twisted magnetic flux tubes are often used to model the filed in coronal loops, and much attention has been given to analysing their stability. Previous astrophysical studies have concentrated on establishing the existence of an instability or determining stability bounds, and little information seems available on the associated eigenvalues, which give crucial information on the energy released. This paper develops methods of determining eigenvalues for infinitely long flux tubes. The most striking feature of the results is that the eigenvalues are always small-of order 10^–2 (in dimensionless units) even for the fastest helical kink modes (m=1). The more localized higher-m modes have even smaller eigenvalues. A family of flux tubes with field line twist proportional tor is investigated, and it appears that the most energetic instabilities occur in the Gold-Hoyle tube with uniform twist (=0). Implications of these results are discussed.     

Catastrophe of coronal magnetic flux ropes in partially open magnetic fields

Using a 2.5-D, time-dependent ideal MHD model in Cartesian coordinates, a numerical study is carried out to find equilibrium solutions associated with a magnetic flux rope in the corona. The ambient magnetic field is partially open, consisting of a closed arcade in the center and an open field at the flank. The coronal magnetic flux rope is characterized by its magnetic properties, including the axial and annular magnetic fluxes and the magnetic helicity, and its geometrical features, including the height of the rope axis, the halfwidth of the rope and the length of the vertical current sheet below the rope. It is shown that for a given partially open ambient magnetic field, the dependence of the geometrical features on the magnetic properties displays a catastrophic behavior, namely, there exists a certain critical point, across which an infinitesimal enhancement of the magnetic parameters causes a finite jump of the geometrical parameters for the rope. The amplitude of the jump depends on the extent to which the ambient magnetic field in open, and approaches to zero when the ambient magnetic field becomes completely closed. The implication of such a catastrophe in solar active phenomena is briefly discussed.     

Topological stability of finite-length magnetic flux tubes

It has been suggested that the activity of cosmical magnetic fields is a consequence of a general topological nonequilibrium in the neighbourhood of magnetostatic equilibria. Evidence for this suggestion can be obtained from the Kolmogorov-Arnold-Moser theorem of classical mechanics, applied to the magnetic field line flow as a Hamiltonian system. A finite-length magnetic flux tube, however, always possesses two independent sets of flux surfaces - or, equivalently, the corresponding Hamiltonian system two independent first integrals - and is topologically stable if in the volume occupied by the tube there are no singular (null) points of the magnetic field and the normal field component does not change its sign on the end faces of the tube. Therefore, the concept of nonequilibrium due to flux surface destruction is not applicable to solar atmospheric loops with each end situated in the interior of one polarity of the photospheric normal field component. Further, it seems unlikely that the tearing-mode mechanism can play a role in such loops.     

The rise of twisted magnetic flux tubes

Sunspots are caused by the eruption of magnetic flux tubes through the solar photosphere: current theories of the internal magnetic field of the Sun suggest that such tubes must rise relatively unscathed from the base of the convection zone. In order to understand how the structure of the magnetic field within a buoyant flux tube affects its stability as it rises, we have considered the quasi-two-dimensional rise of isolated magnetic flux tubes through an adiabatically stratified atmosphere. The magnetic field is initially helical; we have investigated a range of initial field configurations, varying the distribution and strength of the twist of the field.     

Thermodynamical properties of unresolved magnetic flux tubes

We propose a diagnostic method, based on the observation of circular polarization signals in line pairs, to derive the thermodynamical properties of unresolved magnetic elements in the solar atmosphere. The concept of response function for the ratio of circular polarization signals in two lines is introduced and its main properties are analyzed. Some detailed calculations for suitably selected line pairs are presented.     

Leaky and non-leaky oscillations in magnetic flux tubes

An extensive analysis, both analytic and numerical, of waves in flux tubes imbedded in (possibly) magnetic surroundings is given. It is shown that any wave confined to the tube and its neighbourhood can be put into one of seven categories. Simple criteria for deciding the existence of each type in any particular case are derived. Many other (leaky) modes are found which excite waves in the external medium and thereby lose energy to the surroundings. A number of asymptotic analyses allow much information to be gained about these without the need for numerical solution of the complicated equations involved. Three particular cases, pertaining to photospheric flux tubes, H fibrils, and coronal loops, are considered in detail.     

Implications of rapid footpoint motions of photospheric flux tubes for coronal heating

Some recent observations at Pic-du-Midi (Mulleret al., 1992a) suggest that the photospheric footpoints of coronal magnetic field lines occasionally move rapidly with typical velocities of the order 3 km s^–1 for about 3 or 4 min. We argue that such occasional rapid footpoint motions could have a profound impact on the heating of the quiet corona. Qualitative estimates indicate that these occasional rapid motions can account for the entire energy flux needed to heat the quiet corona. We therefore carry out a mathematical analysis to study in detail the response of a vertical thin flux tube to photospheric footpoint motions in terms of a superposition of linear kink modes for an isothermal atmosphere. We find the resulting total energy that is asymptotically injected into an isothermal atmosphere (i.e., an atmosphere without any back reflection). By using typical parameter values for fast and slow footpoint motions, we show that, even if the footpoints spend only 2.5% of the time undergoing rapid motions, still these rapid motions could be more efficient in transporting energy to the corona than the slow motions that take place most of the time.     

Stability of magnetic flux tubes in close binary stars

Photometric and Doppler imaging observations of active binaries indicate the existence of starspots at preferred longitudes (position angles with respect to the companion star). We investigate the stability of magnetic flux tubes in the convection zone of close, fast‐rotating binary stars and explore whether the observed preferred longitudes could be caused by tidal forces and the deformation of the active star. We assume a synchronized binary system with spin axes perpendicular to the orbital plane and a rotation period of a few days. The tidal force and the deviation from spherical structure are considered in lowest‐order perturbation theory. The magnetic field is in the form of toroidal magnetic flux rings, which are stored in mechanical equilibrium within the stably stratified overshoot region beneath the convection zone until the field has grown sufficiently strong for the undulatory instability to initiate the formation of rising loops. Frequencies and geometry of stable as well as growth rates of unstable eigenmodes are determined by linear stability analysis. Particular consideration is given to the question whether the effects of tidal forces and perturbations of the stellar structure can force a rising flux loop to enter the convection zone at specific longitudes.     

Time-dependent interaction of granules with magnetic flux tubes

The time-dependent interaction of the granulation velocity field with a magnetic flux tube is investigated here. It is seen that when a magnetic field line is displaced normal to itself so as to simulate thebuffeting action of granules, a flow of gas is initiated along the field. By choosing a lateral velocity field which is consistent with observations of granules, it is found that the resulting gas motion is a downward flow with a velocity compatible with the observed downflow in isolated photospheric flux tubes. It is therefore proposed that the observed photospheric downflow is a manifestation of the interaction of granules with flux tubes.     

Twisting together magnetic flux tubes under helical symmetry

This paper investigates nonlinear interaction ofmagnetic flux tubes, as a possible cause of current-sheet formation. We focus attention on Gold–Hoyle tubes because of their simple analytic form, using a frictional magnetic relaxation code to find eventual equilibria. We assume that all fields possess helical symmetry so the problem becomes essentially two-dimensional and high resolution can be more easily achieved. When the tubes are not twisted together current sheets form only if the plasma pressure is zero. Twisting the tubes when the plasma pressure is small but finite results in curved current sheets and line currents. Current singularities are identified by performing calculations at increasing grid resolutions, and observing a regular increase in the maximum current.     

Exact static equilibrium of vertically oriented magnetic flux tubes

A method is prescribed for generating exact solutions of magnetostatic equilibrium describing a cylindrically symmetric magnetic flux tube oriented vertically in a stratified medium. Given the geometric shape of the field lines, compact formulae are presented for the direct calculation of all the possible distributions of pressure, density, temperature and magnetic field strength compatible with these field lines under the condition of static equlibrium. The plasma satisfies the ideal gas law and gravity is uniform in space. A particular solution is obtained by this method for a medium sized sunspot whose magnetic field obeys the similarity law of Schlüter and Temesváry (1958). With this solution, it is possible for the first time to illustrate explicitly the confinement of the magnetic field of the cool sunspot by the hotter external plasma in an exact relationship involving both magnetic pressure and field tension as well as the support of the weight of the plasma by pressure gradients. It is found that the cool region of the sunspot is not likely to extend much more than a few density scale heights below the photosphere. The sunspot field approaches being potential in the neighbourhood of the photosphere so that the Lorentz force exerting on the photosphere is less than what the magnetic pressure would suggest. This accounts for how the sunspot field can be confined in the photosphere where its magnetic pressure is often observed to even exceed the normal photospheric pressure. The energy mechanism operating in the sunspot and the question of mechanical stability are not treated in this paper.Normally at Lau Kuei Huat (Singapore) Private Limited, 55 Shipyard Road, Singapore 22, Singapore.     

Twisted magnetic flux tubes: Effect of small twist

The shape of a magnetic flux tube is investigated when photospheric motion causes small twist at the magnetic footpoints. Using a Fourier-Bessel series expansion, the previous results of Zweibel and Boozer (1985) and Steinolfson and Tajima (1987), when the twist is small, are substantiated. A twisting motion that is restricted to a finite region is investigated. Inside the twisted region, the tube contracts, but in the outer region the field remains straight, except for a slight expansion at the outside of the loop near the footpoints. The amount of twist depends on the radial position and can in fact be larger in the contracted region with the twist decreasing as the tube expands. An axial boundary-layer region is present, as noted by the above authors, through which the field adjusts to the line-tied magnetic footpoint positions. An analysis of the boundary layer shows that the thickness remains constant as the loop-length is increased with the result that the main part of the loop has constant cross-sectional area and can be described by cylindrically-symmetric fields. This new 1-D model predicts the main behaviour of the loop without the need to solve the more complicated 2-D problem directly. It is speculated that the boundary layers will remain even when the twist becomes large and a simple example is presented. A detailed parametric study of different twist profiles shows how the central part of the loop responds.Using the result that the majority of the loop can be described by a constant cross-sectional area, a model for a toroidal loop is presented that models coronal loops in a more realistic manner. The main result from this section is that the coronal loops can only remain in equilibrium if they are confined by an external magnetic field (possibly potential in nature) and not by a gas pressure unless additional physical effects are included.     

Axisymmetric and non-axisymmetric modulated MHD waves in magnetic flux tubes

Nonlinear modulated both axisymmetric and non-axisymmetric MHD wave propagation in magnetic flux tubes is studied. In the cylindrical coordinates, ordinary differential equation with cubic nonlinearity is derived. In both cases of symmetry, the equation has solitary solutions. Modulation stability of the solutions is studied. The results of the study show that the propagation of axisymmetric soliton causes rising of plasma temperature in peripheral regions of a magnetic flux tube. In the non-axisymmetric case, it gives also temperature rising effect. Results of theoretical study are examined on idealized model of chromospheric spicule.