Resistive ballooning modes in line-tied coronal fields

The stability of coronal arcades to localized resistive interchange modes in the ballooning ordering, including photospheric line tying, is investigated. It is found that the anchoring of magnetic footpoints in the photosphere is not sufficient to stabilise ballooning modes, once resistivity is taken into account. All configurations with a pressure profile decreasing from the arcade axis at some point are unstable, a purely growing mode being excited. The dependence of the growth rate on the parameter Rm ^–1 k ² , where is the resistivity and k the wavenumber in a direction perpendicular to the equilibrium field, can be described by a power law with varying index: for small values of k ² and an ideally stable field one finds Rm ^–1. As k ² is increased or marginal stability is approached one finds Rm^–1/3. T implications of these localised instabilities to the temporal evolution and overall energy balance of arcade structures in the solar corona is discussed.     

Resistive ballooning line-tied boundary conditions

Ideal and resistive ballooning modes are investigated for different ratios of a two-layer stratified density region representing a model for the photospheric/coronal boundary. Construction of the ballooning equations using a WKB approach is justified by comparison between the values of the growth rate obtained using Hain-Lüst and ballooning equations together with a WKB integral relation. Different values of the density ratio, radius, and resistivity are considered. Sausage-type and kink-type instabilities are found. One of these, depending on the value of r remained unstable for large density ratios. The other instability tended to marginal stability as the density ratio was increased, and allowed parallel and perpendicular flows across the boundary. This is contrary to the predictions of both the rigid-wall and flow-through conditions.     

The effects of parallel and perpendicular viscosity on resistive ballooning modes in line-tied coronal magnetic fields

The study of resistive ballooning instabilities in line-tied coronal magnetic fields is extended by including viscosity in the stability analysis. The equations that govern the resistive ballooning instabilities are derived and the effects of parallel and perpendicular viscosity are included using Braginskii's stress tensor. Numerical solutions to these equations are obtained under the rigid wall boundary conditions for arcades with cylindrically-symmetric magnetic fields. It is found that viscosity has a stabilizing effect on the resistive ballooning instabilities with perpendicular viscosity being more important by far than parallel viscosity. The strong stabilizing effect of perpendicular viscosity can lead to complete stabilization for realistic values of the equilibrium quantities.Research Assistant at the Belgian Fund for Scientific Research.     

Resistive tearing in line-tied magnetic fields: Slab geometry

The resistive tearing-mode instability of a current carrying plasma sheet is investigated including the stabilising photospheric line-tying boundary conditions. This end condition prohibits a single Fourier mode and so requires a series expansion in harmonics of the fundamental sheet excitation. Equilibria in which there exist field lines that do not connect to the photosphere are unstable provided the ratio of the sheet length to characteristic transverse scale is smaller than a critical value that depends on the equilibrium profile. Line-tying has a strong stabilising effect on the fundamental periodic mode. That tearing mode harmonic which develops close to the instability threshold, leads to a configuration with one X point and one 0 point. Its linear growth rate follows the usual constant- scaling with resistivity S ^-3/5, where S is the magnetic Reynolds number.     

Simple tests for the ideal MHD stability of line-tied coronal magnetic fields

Methods for investigating the stability of line-tied, cylindrically-symmetric magnetic fields are presented. The energy method is used and the perturbed potential energy integral is manipulated to produce simple tests that predict either stability to general coronal disturbances or instability to localized modes, both satisfying photospheric line-tying. Using these tests the maximum amount of magnetic energy, that can be stored in the coronal magnetic field prior to an instability, can be estimated. The tests are applied to four different classes of equilibria and results are obtained for both arcade and loop geometries.     

Bounds on the ideal MHD stability of line-tied 2-D coronal magnetic fields

A tractable method for investigating the linear stability of line-tied 2-D coronal magnetic fields is introduced. It is based on the Bernstein et al. (1958) energy principle and can be applied to non-isothermal equilibria with gravity, having a translational invariance. The perturbed potential energy integral is manipulated to produce either necessary conditions for stability to localized modes or sufficient conditions for stability to global modes. Each condition only requires the solution of a set of ordinary differential equations, integrated along the magnetic field lines. The tests are employed to two different classes of equilibria. A linear force-free field is shown to be completely stable, regardless of the shear. The role of pressure gradients, footpoint displacements, line-tying and stratification on an isothermal magneto-hydrostatic equilibrium is assessed.     

Resistive instabilities in coronal conditions

Resistive instabilities in a context referring to the solar corona are rigorously investigated. Various equilibrium configurations are considered, differing, among other things, by their behaviour with respect to fast, ideal instabilities. The computations presented cover in a unified scheme all known regimes of resistive modes and allow one to determine the fastest timescale over which resistivity can play a role. Comparisons with previous work as well as possible extensions are presented.     

The shape of twisted,line-tied coronal loops

The magnetostatic equilibrium of a coronal loop in response to slow twisting of the photospheric footpoints is investigated. A numerical code is used to solve the full non-linear 2-D axisymmetric problem, extending earlier linearised models which assume weak twist and large aspect ratio. It is found that often the core of the loop tends to contract into a region of strong longitudinal field while the outer part expands. It is shown that, away from the photospheric footpoints, the equilibrium is very well approximated by a straight 1-D cylindrical model. This idea is used to develop a simple method for prescribing the footpoint angular displacement and calculating the equilibrium.     

The ideal magnetohydrodynamic stability of a line-tied coronal magnetohydrostatic equilibrium

An energy method is used to determine a condition for local instability of field lines in magnetohydrostatic equilibrium which are rooted in the photosphere. The particular equilibrium studied is isothermal and two-dimensional and may model a coronal arcade of loops where variations along the axis of the arcade are weak enough to be ignorable. If line tying conditions are modelled by perturbations that vanish on the photosphere, then, when the field is unsheared, the condition for stability is necessary and sufficient. However, when the axial field component is non-zero, so that the field is sheared, the stability condition is only sufficient.It is found that when < 0.34 the equilibrium is stable. When = 0.34 a magnetic neutral line appears at the photosphere and it is marginally stable. When > 0.34 a magnetic island is present and all the field lines inside the island are unstable as well as some beyond it. As increases, the size of the island and the extent of unstable field lines increase. The effect of the instability is likely to be to create small-scale filamentation in the solar corona and to enhance the global transport coefficients.     

Resistive tearing mode in coronal neutral sheets

The apparent stability of coronal neutral sheets with respect to the resistive tearing mode has been attributed by previous authors to the influence of a weak normal component of the confining magnetic field. To check this hypothesis a normal mode analysis is performed applying rigorously singular perturbation technique. Allowance is made for a value of the normal component which is large measured in the appropriate units deduced from the dynamics of the one-dimensional tearing mode. The structure of the eigenmodes is completely changed: the singular layer decays into a broad band of filaments with antiparallel flow directions and spatial oscillations in the perturbed current density appear. Surprisingly, the growth rate is not changed. If parameters for a typical neutral sheet in the middle corona (0.5 solar radii) are inserted, the result is that no stabilization by a normal component occurs, if the value of the growth time predicted by the one-dimensional theory is far shorter than ten minutes - independent of the values assumed for the width of the neutral sheet or the resistivity.     

The evolution of line-tied coronal arcades including a converging footpoint motion

It has been demonstrated in the past that single, two-dimensional coronal arcades are very unlikely driven unstable by a simple shear of the photospheric footpoints of the magnetic field lines. By means of two-dimensional, time-dependent MHD simulations, we present evidence that a resistive instability can result if in addition to the footpoint shear a slow motion of the footpoints towards the photospheric neutral line is included. Unlike the model recently proposed by van Ballegooijen and Martens (1989), the photospheric footpoint velocity in our model is nonsingular and the shear dominates everywhere. Starting from a planar potential field geometry for the arcade, we find that after some time a current sheet is formed which is unstable with respect to the tearing instability. The time of its onset scales with the logarithm of the magnetic diffusivity assumed in our calculation. In its nonlinear phase, a quasi-stationary situation arises in the vicinity of the x-line with an almost constant reconnection rate. The height of the x-line above the photosphere and the distance of the separatrix footpoints remain almost constant in this phase, while the helical flux tube, formed above the neutral line, continuously grows in size.     

Modelling coronal magnetic fields

The hairy ball model of coronal magnetic fields has a spherical source surface separating potential and radial magnetic fields. In the present model the source surface is chosen such that the wind speed equals the Alfvénic speed at selected points on the source surface. Results have been obtained for a dipole base field and an isothermal corona.Proceedings of the 14th ESLAB Symposium on Physics of Solar Variations, 16–19 September 1980, Scheveningen, The Netherlands.     

Electric fields in coronal magnetic loops

We analyze spectra taken with the 40 cm coronograph at Sacramento Peak Observatory, for evidence of Stark effect on Balmer lines formed in coronal magnetic structures. Several spectra taken near the apex of a bright post-flare loop prominence show significant broadening from H₁₀ to the limit of Balmer line visibility in these spectra, at about H₂₀ The most likely interpretation of the increasing width is Stark broadening, although unresolved blends of Balmer emissions with metallic lines could also contribute to the trend. Less significant broadening is seen in 3 other post-flare loops, and the data from 5 other active coronal condensations observed in this study show no broadening tendency at all, over this range of Balmer number. The trend clearly observed in one post-flare loop requires an ion density of n _i ? 2 × 10¹² cm^?3, if it is to be explained entirely as Stark effect caused by pressure broadening. But mean electron densities measured directly from the Thomson scattering at λ3875 in the same SPO spectra, yield n _e ? 3?7 × 10¹⁰ cm^?3 for the same condensations observed within that loop. Comparison of this evidence from electron scattering, with densities derived from emission measures and line-intensity ratios, argues against a volume filling factor small enough to reconcile the values of n _i and n _e derived in this study. This discrepancy leads us to suggest that the Stark effect observed in these loops, and possibly also in flares, could be caused by macroscopic electric fields, rather than by pressure broadening. The electric field required to explain the Stark broadening in the brightest post-flare loop observed here is approximately 170 V cm^?1. We suggest an origin for such an electric field and discuss its implications for coronal plasma dynamics.     

Thermal condensations in coronal magnetic fields

Conditions under which cool condensations can form in the solar corona are investigated using the powerful phase plane method to analyse the energy and hydrostatic balance equations. The importance of the phase plane approach is that the conclusions deduced are not sensitive to the actual choice of boundary conditions adopted which only determine the actual contour. The importance of heating variations and area divergence are studied as well as the influence of gravity for their effect on the formation of cool condensations. The cool temperature at which optically thin radiation and heating balance is important and the links with other cool solutions are mentioned.     

Magnetic fields in a coronal condensation

Using the potential field model, we have calculated the magnetic structure of the coronal condensation, which appeared on the east limb of the solar disk at the 22 September 1968 eclipse. The comparison between calculated magnetic field geometry and observed contour of the coronal condensation shows general agreement. From this analysis, we see that the magnetic field extending to the lower coronal region is approximately the potential field during the steady period of an active region. The observed contour of the coronal condensation may be considered as the projection of the magnetic tubes of the active region on the plane perpendicular to the line of sight. The matter flows along the magnetic field lines.     

Stability of line-tied 1-d coronal loops: Significance of an extended Suydam criterion

The stability of coronal magnetic loops is investigated with the influence of the dense photosphere (line-tying) included. The stability method, based on the Finite Fourier Series method developed by Einaudi and Van Hoven (1981, 1983), is applied to two different equilibria and the approximate critical conditions for the onset of different azimuthal instabilities are investigated. It is shown that, for nearly force-free loops, the extended Suydam criterion, obtained by De Bruyne and Hood (1989) for localized modes, predicts the existence of a global kink instability when a localized mode is just destabilized. For loops with substantial gas pressure gradients it is the localized modes that are destabilized first of all and the extended Suydam criterion gives the necessary and sufficient conditions for an instability. In this latter case, the instability threshold for the kink mode is quite close to the localized mode threshold. Finally, it is shown that the growth times of the instabilities are comparable to the Aflvén travel times along the loop when the extended Suydam criterion is violated.     

Periodic oscillations found in coronal velocity fields

By making line profile analyses of the Fe xiv 5303 coronal line, temporal variations of three fundamental quantities, the line intensity, the FWHM, and the Doppler velocity, have been investigated for an active region. The power spectrum shows that the line-of-sight Doppler velocity fluctuated periodically at two locations with a period of nearly 300 seconds, while no periodic oscillations were found in line intensity. As to the FWHM, some evidence of periodic fluctuations are recognized but in a less distinct manner.The Sacramento Peak Observatory is operated by the Associated Universities for Research in Astronomy Inc. by contact with the National Science Foundation.     

The evolution of coronal magnetic fields

Slow photospheric motions can produce flow speeds in the corona which are fast enough to violate quasi-static evolution. Therefore, high-speed flows observed in the corona are not necessarily due to a loss of equilibrium or stability. In this letter we present an example where the flow speed increases indefinitely with, height, while the coronal magnetic energy increases quadratically with time.     

XUV observations of coronal magnetic fields

Spectroheliograms obtained with the Naval Research Laboratory's Extreme Ultraviolet Spectrograph (S082A) on Skylab are compared with Kitt Peak National Observatory magnetograms. A principal result is the characteristic reconnection of flux from an emerging bipolar magnetic region to previously existing flux in its vicinity. Examples of the disappearance of magnetic flux from the solar atmosphere are also shown. The results of a particularly simple, potential field calculation are shown for comparison with the Skylab observations.     

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.