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.     

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 stability of magnetohydrostatic atmospheres

Using the analysis of Schindler et al. (1983) and Hood (1983), the local stability of Zweibel (1981) is extended to include the previously neglected magnetic tension terms. A particular magnetohydrostatic field is shown to be stable if the parameter, 2H, is small enough. This is in contradiction to the previous local analysis.     

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.     

On the possibility of constructing a radiative sunspot model in magnetohydrostatic equilibrium

It is currently believed that it is impossible to construct a radiative sunspot model in magnetohydrostatic equilibrium unless magnetic fields below the surface are excessively large (> 100 kG). This belief is based on results obtained using the mixing length theory of convection. We wish to point out that by using a different theory of convection, due to Öpik (1950), it is possible to compute a radiative sunspot model in which the field becomes no greater than 9000 G. By applying two boundary conditions, (i) depth of spot equals depth of convection zone, (ii) magnetic field has zero gradient at the base of the spot, we show that a radiative spot has a unique effective temperature for a given Wilson depression, . For = 650 km, we find T _e = 3800K ; for = 150 km, T _e = 3950K. According to our model, spots having T _e cooler than these values should not exist.     

The excitation equilibrium of coronal ions

The relative populations of levels of highly ionized Fe, Ni and Ca ions have been calculated for physical conditions appropriate to the solar corona. The results are presented in the form of tables. Line intensity ratios in the EUV and visible that are sensitive to electron density are discussed and compared with observations.     

A magnetohydrodynamic theory of coronal loop transients

A magnetohydrodynamic theory is presented for coronal loop transients. It is shown that the heliocentrifugal motion of a transient loop, as exhibited by the translational displacement of the axis of the loop, is driven by the magnetohydrodynamic buoyancy force exerted by the ambient medium. Self-induced hydromagnetic force, which includes the magnetic force produced by the internally driven current and the thermal force produced by the pressure imbalance between the internal and external gas pressures, causes the peripheral expansion of the loop, as exhibited by the lateral broadening and longitudinal stretching. This contention is substantiated by an analysis based on a model structure for a coronal loop.Besides accounting for the acceleration and expansion of a transient loop, this magnetohydrodynamic theory also provides an explanation for the initial ejection of a coronal loop from stationary equilibrium. Magnetic unwinding in consequence of abrupt magnetic activities at the solar surface will cause the periphery of a stationary coronal loop to expand. The increase in volume will enhance the magnetohydrodynamic buyoyancy force to exceed the gravitational force. Once a coronal loop is ejected from the solar surface, it will be continually accelerated and undergo expansion. Eventually a transient loop will blend with the ambient solar wind. This is also indicated by the theory presented in this paper.     

Energy equilibrium in a coronal magnetic loop

Temperature distribution in the cylindrically symmetric coronal magnetic loop, (i) with constant pressure and (ii) with the pressure varying along the radial distance, of the (a) hotter apex and (b) cooler apex than base is investigated analytically by considering the equilibrium between the heat conduction and radiation loss. If the temperature of the loop does not lie within one of the specified temperature ranges, then the distribution is calculated numerically.The effect of the inclusion of heating due to an external source is studied and found that it increases the length of the loop. On the basis of the observed phenomenon, that the magnetic field varies along the loop, the temperature distribution in the loop is investigated for the loop-geometries proposed by Antiochos and Sturrock (1976). It is concluded that for the larger compression in the area of cross section, the height of the loop decreases.Present investigation shows that no loop with equal apex and base temperatures can exist, but a small variation between the two temperatures supports the existence of the loop, which can be observed in nature.     

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.     

The influence of line-tying on coronal perturbations in a gravitationally stratified equilibrium

We study the influence of gravitational stratification of the solar atmosphere on the stability of coronal magnetic structures. In particular we question whether the (presumably stabilizing) influence of the anchoring of the magnetic field lines in the solar photosphere (line-tying) can be adequately modelled by either rigid wall or flow-through boundary conditions on the coronal perturbations, as is commonly done. Using the ideal MHD model without gravitational effects,inertial line-tying alone cannot lead to afull stabilization, as marginal stability cannot be crossed by including only the rapid density increase at the photospheric interface.We demonstrate, using the (localized) ballooning ordering, that when gravity and the corresponding intrinsically stable stratification of the photosphere is included, the points of marginal stability are no longer independent of the density. The sharp increase in density and associated decrease in pressure scale height at the solar surface leads to a stabilizing effect, which may result in a full transition from unstable to stable modes. Gravitational effects imply that rigid wall conditions represent photospheric field line anchoring better than flow-through conditions for determining the stability or modes of oscillation of a coronal equilibrium. Applying rigid wall conditions gives good approximations for frequencies that are much larger than photospheric time scales when the plasma is stable, and growth rates when the plasma is unstable. At the same time we show however that near marginal stability, even when gravity is included, rigid wall conditions are still violated.     

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.     

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.     

The temperature-density structure of coronal loops in hydrostatic equilibrium

The temperature and density structure are computed for a comprehensive set of coronal loops that are in hydrostatic and thermal equilibrium. The effect of gravity is to produce significant deviations from the usual uniform-pressure scaling law (T(pL) ^1/3) when the loops are taller than a scale height. For thermally isolated loops it lowers the pressure throughout the loop, which in turn lowers the density significantly and also the temperature slightly; this modifies the above scaling law considerably. For more general loops, where the base conductive flux does not vanish, gravity lowers the summit pressure and so makes the radiation decrease by more than the heating. This in turn raises the temperature above its uniform pressure value for loops of moderate length but lowers it for longer loops. A divergence in loop cross-section increases the summit temperature by typically a factor of 2, and decreases the density, while an increase in loop height (for constant loop length) changes the temperature very little but can halve the density.One feature of the results is a lack of equilibrium when the loop pressure becomes too large. This may explain the presence of cool cores in loops which originally had temperatures below 2 × 10⁶ K. Loops hotter than 2 × 10⁶ K are not expected to develop cool cores because the pressure necessary to produce non-equilibrium is larger than observed.     

The stability and uniqueness of coronal loops

A hydrodynamic model of high resolution is used to examine the stability of coronal loops to finite amplitude perturbations. The loop is heated by means of a low-amplitude energy input and its subsequent dynamic relaxation is followed.Firstly, the initial atmosphere is generated by solving the time independent form of the hydrodynamic equations. It is shown that the loop structure depends critically on the balance between the radiative losses and the quiescent heating at the base of the transition zone, i.e. on the concavity of the temperature profile in this region. This result already anticipates the need for high spatial resolution across the model transition zone.The dynamic evolution of the loop is then investigated for two classes of lower boundary conditions. In one case the chromospheric temperature is fixed throughout the simulation; in the other the low chromosphere is represented by a rigid insulating barrier. In both cases the loop is found to be stable: The loop is also unique to the extent that it relaxes to a state which is physically indistinguishable from its initial configuration. It is pointed out however, that a loop whose chromosphere is only marginally stable can evolve dynamically away from the initial static configuration.Finally, the observational consequences of the analysis are discussed. The differential emission measure profile is found to change its form as the loop cools, firstly, through an evaporative phase in which the coronal density increases; secondly, through a quasi-steady relaxation in which the enhanced coronal density gradually drains away to the chromosphere. This behaviour represents a possible observational test of the model.     

Loss of equilibrium in coronal loops

The loss of equilibrium in coronal magnetic field structures is a possible source of energy for coronal heating and solar flares. We investigate whether such a loss of equilibrium occurs when a coronal loop is progressively twisted by photospheric motions. In studies of 2-D cylindrical equilibria, long loops have been found to be of constant cross-sectional area along most of their length, with axial variations being confined to narrow boundary layers. We use this information to develop a 1-D line-tied model, for a 2-D coronal loop. We specify the twist in terms of the azimuthal field and more physically, in terms of the photospheric footpoint displacement. In the former case we find a loss of equilibrium, but not in the latter. We also examine a twisted loop with a non-zero plasma pressure. The loss of equilibrium is only found at high-plasma . It is conjectured that such high- can occur in flare loops and prior to a prominence eruption. However, when the plasma evolves adiabatically, there is no loss of equilibrium.     

The structure and stability of coronal magnetic fields

The high degree of symmetry often assumed in studies of the structure and stability of coronal magnetic field configurations is restrictive and can yield misleading results. We have therefore developed fully three-dimensional numerical methods for constructing force-free equilibria and for examining their stability properties, which make no assumptions about symmetry. A test of the stability analysis has been performed by applying it to the Gold-Hoyle twisted flux tube, which is known to be kink-unstable if the helical field makes more than about one turn between the line-tying end-plates. Our preliminary result is that the critical number of turns is about 1.1, in good agreement with the previous best estimate. However, we find that the growth rate, which has not been discussed previously, is orders of magnitude smaller than expected, even when the flux tube is twisted far beyond the stability limit.     

The thermal stability of coronal loops numerical simulations

We have studied the radiative stability of thermally isolated coronal loops with free-flow boundary conditions by nonlinear numerical simulation. We first establish a chromosphere-to-corona loop equilibrium (including the option of a deep chromosphere) by following the nonlinear evolution from an initial isothermal state with rigid boundaries. We then change the end conditions, to allow free flow and to fix the temperature, and investigate the response to non-isobaric perturbations. Within a family of loops of the same pressure, we find long hot loops to be stable and short cool loops to be unstable to the thermal chromosphericexpansion mode. The stable cases remain so, even when long chromospheric ends and/or gravity are added. In those cases which are unstable, we follow the subsequent nonlinear evolution which exhibits swelling of the chromosphere until the entire loop becomes cool and dense.     

The structure and stability of coronal magnetic fields

Solar Physics - The high degree of symmetry often assumed in studies of the structure and stability of coronal magnetic field configurations is restrictive and can yield misleading results. We have...     

The equilibrium and stability of magnetopolytropes

The theory of the oscillations of axisymmetric gaseous configurations with a prevalent magnetic field is presented. The virial tensor method is used to obtain the nine second harmonic modes of oscillations of the system. It is found that out of the nine modes, three are neutral, four are non-radial, and two are coupled. For the Prendergast spherical model it is found that one of the coupled modes is radial and the other non-radial. Both the radial and the non-radial modes obtained in this case agree with the corresponding formulae obtained byChandrasekhar andLimber (1954) andWoltjer (1962).The equilibrium structure of gaseous polytropes with toroidal magnetic fields is also investigated in detail for values of the polytropic indexn=1, 1.5, 2, 3 and 3.5. For this model the components of the moment of intertia and potential energy tensors together with the non-zero components of the supermatrix potential are obtained. The final results in terms of the effect of weak toroidal magnetic fields on the characteristic frequencies of distorted polytropes are presented in the form of tables.