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 structure of coronal loops

With the advent of space telescopes, coronal magnetic loops, both within and outside active regions, are being observed with renewed interest. This paper is an attempt to outline some general physical considerations pertinent to such loops, as a prelude to more sophisticated modelling. For example, a loop that is stretched (or possibly twisted) too much may be subject to a thermal instability that cools its core to a new equilibrium below 10⁵ K. Also a simple consequence of hydrostatic balance along an equilibrium loop is that, under some circumstances, the density inside a cool loop can be comparable with that outside, despite the much smaller scale height. Finally, when the equilibrium loop density is less than the ambient density, several small scale magnetohydrodynamic instabilities are sometimes efficient enough to generate a circulation that tends to equalize the densities.     

The thermal statics of coronal loops

The thermal statics of constant pressure coronal loops is discussed, with particular emphasis on non-equilibrium and scaling relations. An analytical solution showing explicitly the occurrence of non-equilibrium in radiation dominated loops is presented. In addition, the general scaling law for hot loops is given. However, in view of the uncertainties in the coronal heating function and the observational determined loop parameters, it is suggested that scaling laws are currently of limited value.     

The structure of coronal magnetic loops

We present here a model, based on observations, for the magnetic-field equilibrium of a cool coronal loop. The pressure structure, taken from the Harvard/Skylab EUV data, is used to modify the usual force-free-field form in quasi-cylindrical symmetry. The resulting field, which has the same direction but different strength, is calculated and its variation displayed. Finally, localized interchange stability is evaluated and discussed, as the first step in a subsequent complete magnetohydrodynamic-stability analysis.     

The structure of coronal magnetic loops

We present the second part of a complete theory for the plasma and field structure of a cool coronal arch, corresponding to those observed in the EUV from Skylab. The global magneto-hydrodynamic (MHD) stability of a previously described equilibrium-loop model is evaluated, and compared with that of an unmodified ambient force-free field. The influence of the photospheric boundary condition is also evaluated, producing a specification of stability limits which depend on the relative field and plasma pressures and scale widths. The resulting restrictions on the allowable field configuration of a coronal loop are then compared with observed values. The implications of this general method for deducing small-scale coronal magnetic-field structure from the measured plasma profile of an emissive feature are also described.     

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.     

Models of dynamic coronal loops

This paper reviews the basic ideas underlying one-dimensional fluid dynamic models of coronal loops and presents some of their most recent applications. These models are an important theoretical support to explore the new scenario provided by the data of Yohkoh, SOHO, and TRACE, and are useful to interpret observations, when supplemented by appropriate spectral synthesis codes. Possible developments are also discussed.     

The solar-surface boundary conditions of coronal magnetic loops

The dynamical properties of a realisticthermal-structure interface between a coronal loop and the chromosphere/photosphere are investigated by numerical simulations using acoustic and Alfvénic excitations. These properties are relevant to the end conditions seen by coronal MHD perturbations (e.g., waves or instabilities), in the absence of much slower energetics effects. Analytic studies of coronal-loop hydromagnetics have often made simplifying assumptions about the boundary conditions at the loop base in order to make their calculations tractable. However, in the presence of a transition region and chromosphere with rapidly varying plasma conditions, it is not clear how valid these heuristic assumptions are. In this study, we find that the discontinuous fluid-density model approximately represents the reflection/ transmission scaling with respect to varying transition-region density and temperature (i.e., dynamic impedance) ratios, although it does not quantitatively predict the chromospheric response to wave-like coronal activity. This disagreement is partially due to the finite width of the corona-to-photosphere transition.     

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.     

Plasma flow around coronal loops

Current dissipation models of coronal loop heating are studied. Turbulent current dissipation is shown to lead to a time dependent process because of an enormous mass motion induced in the current layer. A stationary heating process involves only ohmic heating, which requires a large current layer. To insure MHD stability, the loop must be composed of many elements with the oppositely directed currents. A stationary current dissipation process induces the plasma motion across the magnetic field into the loop and down the loop with the speeds v 10⁴ cm s^–1 and v 10⁴ cm s^–1, respectively. The pressure of the loop is also estimated to be proportional to the current density: p/J=6.3 × 10^-8dyn/statamp.     

Non-equilibrium ionization in coronal loops

The ionization conditions in coronal loops are investigated in the temperature range 2 × 10⁵–2 × 10⁶K, assuming velocity, density and temperature distributions computed for a siphon model of a pure hydrogen plasma. Use is made of the set of the carbon ions as an example of the general behaviour of the ions characteristic of that temperature range. It is found that the deviation from equilibrium ionization is large for subsonic-supersonic flow if the density is less than 5 × 10⁹cm^–-3, with the exception of the lower part of the first leg of very cool loops (T 2 × 10 K). With this exception cooler loops, given their larger density drop along the axis, show deviations from ionization equilibrium more easily than hotter ones, in spite of their lower flow velocity. We conclude that the possibility of a non-equilibrium state must be taken into account when deducing from measurements of line intensities the temperature of loops in which a flow may occur.Now at Institute for Plasma Research, Stanford University, as an E.S.A. Fellow.     

Thermal equilibria of coronal magnetic loops

Equations of thermal equilibrium along coronal loops with footpoint temperatures of 2 × 10⁴ K are solved. Three fundamentally different categories of solution are found, namely hot loops with summit temperatures above about 4 × 10⁵ K, cool loops which are cooler than 8 × 10⁴ K along their whole length and hot-cool loops which have summit temperatures around 2 × 10⁴ K but much hotter parts at intermediate points between the summit and the footpoints. Hot loops correspond to the hot corona of the Sun. The cool loops are of relevance for fibrils, for the cool cores observed by Foukal and also for active-region prominences where the magnetic field is directed mainly along the prominence. Quiescent prominences consist of many cool threads inclined to the prominence axis, and each thread may be modelled as a hot-cool loop. In addition, it is possible for warm loops at intermediate summit temperatures (8 × 10⁴K to 4 × 10⁵ K) to exist, but the observed differential emission measure suggests that most of the plasma in the solar atmosphere is in either the hot phase or the cool phase. Thermal catastrophe may occur when the length or pressure of a loop is so small that the hot solution ceases to exist and there are only cool loop solutions. Many loops can be superimposed to form a coronal arcade which contains loops of several different types.     

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.     

Radial oscillations of current-carrying coronal loops

The radial oscillations of coaxial magnetic flux tubes with an azimuthal field in the shell modeling current-carrying coronal loops are studied in the cool plasma approximation. Since the concept of current-carrying coronal loops provides a theoretical basis for studying simple loop flares, finding their parameters by means of coronal seismology is a topical problem of modern solar physics. The dispersion equation for radial oscillations is derived and the dispersion curves are constructed. Oscillations with arbitrarily long periods are shown to exist at the fundamental radial mode.     

The resonant excitation of transverse oscillations in coronal loops

We suggest a way of self-consistently solving the problem of the excitation and rapid damping of coronal loop oscillations observed from the TRACE (Transition Region and Coronal Explorer) satellite. Oscillations are excited on the dispersion branch of fast magnetoacoustic waves, which propagate mainly across the magnetic field. The rapid damping of the observed oscillations is governed by the dispersion spreading of the pulse of these waves that was produced, for example, by a solar flare. The fundamental oscillation period is close to the period of the fundamental mode. Dissipative processes attributable to the nonideality of the plasma and the coronal-loop footpoints play no fundamental role.     

Slow-mode oscillations of large-scale coronal loops

On several occasions, repetitive X-ray brightenings, sometimes accompanied by mass injections into adjacent loops, appeared quasi-periodically with mean periods close to 20 minutes. In all cases when X-ray images were available, the sites of these brightenings were in active regions which were associated with large-scale coronat loops of length (2 – 3) × 10⁵ km. Therefore, the primary source of these long-periodic pulsations might be slow-mode oscillations in these large-scale loops. Free MHD oscillations, proposed earlier by Roberts, Edwin, and Benz (1984), may fit the observed data.     

Kink oscillations of asymmetric coronal loops

The free oscillations of coronal loops with a constant density and a variable magnetic field changing according to parabolic laws are investigated. Using our developed method, we derive the wave equations with constant coefficients that describe the kink oscillations of symmetric and asymmetric magnetic flux tubes. For such models, we obtain analytical expressions for the oscillation spectra and amplitudes as well as the magnitudes and directions of the displacements of the extrema of the fundamental and first modes relative to their values for homogeneous tubes. For the first mode of an asymmetric loop, we have determined the dependence of the coordinate displacement for the internal node on the ratios of the magnetic field strengths in its asymmetric parts and the ratio of the amplitudes at the extremum points.     

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.     

Current confinement in solar coronal loops

In this paper solar coronal loops are regarded as regions of localized current flows. The main purpose is to investigate the consequences of current confinement rather than to produce a model. The physical and observational basis for this assumption are presented as well as the connection with previous studies on loop structure. A proper choice of the current profile allows us to treat quantitatively the equilibrium structure of the loops and their MHD and resistive stability properties. Regions of absolute stability against ideal kink modes are found. Explicit growth rates for the tearing-mode instability are computed. The possible relevance of other resistive effects is also discussed and the crucial importance of the small-scale geometry of the magnetic field outlined.