The stability of small flux tubes

We have investigated the influence of stationary velocity fields, twists of the field lines and changes of gas pressure within flux tubes on the interchange instability of magnetic flux tubes. A small flux tube is found to be stable. All three factors mentioned above can stabilize tubes with all fluxes. We estimate that, for the solar case, a change of gas pressure in flux tubes plays an important role in stabilizing magnetic flux tube.     

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.     

The kink instability in infinite cylindrical flux tubes: Eigenvalues for power-law twist profiles

We develop simple accurate methods of calculating ideal MHD instability eigenvalues for infinitely-long cylindrical tubes, with twist functionT(r)=B /rB _z . A complete theoretical treatment is presented for force-free magnetic equilibria with arbitraryT(r), and detailed semi-analytic results for the kink instability are given for the particular case of a power-law twistT(r)=r , where the index is non-negative. Our results show that the most rapidly growing and energetic instabilities occur in the Gold-Hoyle =0 field, with the instability progressively weakening with increasing . However, the maximum force eigenvalue is always small, so that even in the Gold-Hoyle case (where =O(10^–2) in dimensionless units) only a small proportion of the available magnetic energy can be released in the linear phase. Our results also confirm that the linear pinch (=) is remarkably weak (=O(10^–3)) yet relatively resistant to line-tying! It is shown that the weakness of the force eigenvalue implies that the influence of uniform gas pressure on stability is negligible. Implications for the energy-release mechanism in solar flares are discussed.     

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.     

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.     

The size dependence of contrasts and numbers of small magnetic flux tubes in an active region

Intensity contrasts and number densities of bright points, knots and pores ranging in size between 0.15 and 4 are studied using high resolution pictures in Mg b₁ of a young active region. On the average, the contrast in the wing of the line increases very strongly with decreasing size, while the continuum contrast increases more slowly. The ratio of contrast in the line to contrast in the continuum increases rapidly with decreasing size. The possibility is explored of using this contrast ratio as an indicator of size. The distribution of the contrast ratio in a part of the active region is used in this way to derive a size distribution of facular points. The resulting distribution has a limited accuracy, but is free from systematic distortion due to selection effects. Validity checks on the method are presented. We measure the size distribution of the pores in the same area, and combine the result with that for the facular points. The combined distribution shows that the surface area covered by magnetic elements with diameter has a maximum near = 0.8. It increases roughly proportional to for < 0.3 and falls off as for > 1.5. It is inferred that elements with 0.5 < < 1.6, which show no conspicuous contrast in the line wing or in the continuum, occupy as much area as the pores, and twice as much as the facular points. We suggest that the changing appearance of a facular area with increasing height of formation reflects at least as much the increasing weight of the small elements in the contrast as a real change in intrinsic properties (such as the diameter) of individual elements. A spatial resolution better than 0.1 may be needed to resolve the individual elements in plages and the chromospheric network.The observed variation of continuum contrast of facular points with size agrees with predictions based on magnetostatic flux tube models if a field strength of about 2000 G is assumed.     

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.     

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.     

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.     

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 interactions of modes in coronal magnetic flux tubes

Nonlinear resonant interactions of different kinds of fast magnetosonic (FMS) waves trapped in the inhomogeneity of a low- plasma density, stretched along a magnetic field (as, for example, in coronal loops) are investigated. A set of equations describing the amplitudes of interactive modes is derived for an arbitrary density profile. The quantitative characteristics of such interactions are found. The decay instability of the wave with highest frequency is possible in the system. If amplitudes of interactive modes have close values, the long-period temporal and spatial oscillations are in the system.For a quantitative illustration, the parabolic approximation of the transverse density profile has been chosen. Dispersion relations of FMS waves trapped in a low- plasma slab with a parabolic transverse density profile are found. The transverse structure of the waves in this case can be expressed through Hermitian polynomials. The interaction of kink and sausage waves is investigated. The sausage wave, with a sufficiently large amplitude, may be unstable with respect to the decay into two kink waves, in particular. The spatial scale of a standing wave structure and the time spectrum of radiation are formed due to the nonlinear interactions of loop modes which contain information about the parameters of the plasma slab.     

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.     

Converging and diverging convection around axisymmetric magnetic flux tubes

A numerical model of idealized sunspots and pores is presented, where axisymmetric cylindrical domains are used with aspect ratios (radius versus depth) up to 4. The model contains a compressible plasma with density and temperature gradients simulating the upper layer of the Sun's convection zone. Non-linear magnetohydrodynamic equations are solved numerically and time-dependent solutions are obtained where the magnetic field is pushed to the centre of the domain by convection cells. This central magnetic flux bundle is maintained by an inner convection cell, situated next to it and with a flow such that there is an inflow at the top of the numerical domain towards the flux bundle. For aspect ratio 4, a large inner cell persists in time, but for lower aspect ratios it becomes highly time dependent. For aspect ratios 2 and 3 this inner convection cell is smaller, tends to be situated towards the top of the domain next to the flux bundle, and appears and disappears with time. When it is gone, the neighbouring cell (with an opposite sense of rotation, i.e. outflow at the top) pulls the magnetic field away from the central axis. As this happens a new inner cell forms with an inflow which pushes the magnetic field towards the centre. This suggests that to maintain their form, both pores and sunspots need a neighbouring convection cell with inflow at the top towards the magnetic flux bundle. This convection cell does not have to be at the top of the convection zone and could be underneath the penumbral structure around sunspots. For an aspect ratio of 1, there is not enough space in the numerical domain for magnetic flux and convection to separate. In this case the solution oscillates between two steady states: two dominant convection cells threaded by magnetic field and one dominant cell that pushes magnetic flux towards the central axis.     

Sunspot proper motions and behaviour of rising magnetic flux tubes

Greenwich data for proper motions of the components of sunspot groups are analysed. We have found that only 4% of the 408 examined sunspot groups do not show systematic proper motions of the components at the beginning of their life. Interpreting proper motions as the results of ascending sub-photospheric magnetic flux tubes, information on characteristics of the tubes have been deduced. The influence of proper motions on the evaluation of the sunspot rotation rates is discussed.