Strong and Weak Damping of Slow MHD Standing Waves in Hot Coronal Loops

Slow magnetohydrodynamic (MHD) standing wave oscillations in hot coronal loops for both strong (i.e. τ_d/P∼ 1) and weak (i.e. τ_d/P≥ 2) damping are investigated taking account of viscosity, thermal conductivity and optically thin radiation. The individual effect of the dissipative terms is not sufficient to explain the observed damping. However, the combined effect of these dissipative terms is sufficient to explain the observed strong damping, as well as weak damping seen by SUMER. We find that, the ratio of decay time (τ_d) and period (P) of wave, i.e., τ_d/P (which defines the modes of damping, whether it is strong or weak) is density dependent. By varying density from 10⁸ to 10¹⁰ cm⁻³ at a fixed temperature in the temperature range 6 – 10 MK, observed by SUMER, we get two sets of damping: one for which τ _d/P∼ 1 corresponds to strong damping that occurs at lower density and another that occurs at higher density for which τ_d/P ≥ 2 corresponds to weak damping. Contrary to strong-damped oscillations, the effect of optically thin radiation provides some additional dissipation apart from thermal conductivity and viscosity in weak-damped oscillations. We have, therefore, derived a resultant dispersion relation including the effect of optically thin radiation. Solutions of this dispersion relation illustrate how damping time varies with physical parameters of loops in both strong and weak damping cases.     

Slow-Mode Oscillations and Damping of Hot Solar Coronal Loops

The effect of temperature inhomogeneity on the periods, their ratios (fundamental versus first overtone), and the damping times of the standing slow modes in gravitationally stratified solar coronal loops are studied. The effects of optically thin radiation, compressive viscosity, and thermal conduction are considered. The linearized one-dimensional magnetohydrodynamic (MHD) equations (under low-?? condition) were reduced to a fourth-order ordinary differential equation for the perturbed velocity. The numerical results indicate that the periods of nonisothermal loops (i.e., temperature increases from the loop base to apex) are smaller compared to those of isothermal loops. In the presence of radiation, viscosity, and thermal conduction, an increase in the temperature gradient is followed by a monotonic decrease in the periods (compared with the isothermal case), while the period ratio turns out to be a sensitive function of the temperature gradient and the loop lengths. We verify that radiative dissipation is not a main cooling mechanism in both isothermal and nonisothermal hot coronal loops and has a small effect on the periods. Thermal conduction and compressive viscosity are primary mechanisms in the damping of slow modes of the hot coronal loops. The periods and damping times in the presence of compressive viscosity and/or thermal conduction dissipation are consistent with the observed data in specific cases. By tuning the dissipation parameters, the periods and the damping times could be made consistent with the observations in more general cases.     

Morphology and spatial distribution of XUV and X-ray emissions in an active region observed from Skylab

We studied the morphology and spatial distribution of loops in an active region, using coordinated observations obtained with both the S082A XUV spectroheliograph and the S056 grazingincidence X-ray telescope on Skylab. The active region loops in the temperature range 5 × 10⁵ –3 × 10⁶ K fall basically into two distinctive groups: the hot loops with temperatures 2–3 × 10⁶ K as observed in coronal lines and X-rays, and the relatively cool loops with temperature 5 × 10⁵ –1 × 10⁶ K as observed in transition-zone lines (Ne vii, Mg ix). The brightest hot coronal loops in the active region are mostly low-lying, compact, closely-packed, and show greater stability than the transition-zone loops, which are fewer in number, large, and slender. The observed aspect ratio of the hot coronal loops is in the range of 0.1 and 0.2, which are almost two orders of magnitude larger than those for the Ne vii loops. Brief discussion of the MHD stability of the loops in terms of the aspect ratio is presented.     

The XUV structure of solar active regions

XUV spectroheliograms of 2 active regions are studied. The images are due to lines emitted at temperatures between 8 x 10⁴ K and 2 x 10⁶ K and thus are indicative of transition region and coronal structures. The hot coronal lines are formed solely in loop structures which connect regions of opposite photospheric magnetic polarity but are not observed over sunspots. Transition region lines are emitted in plages overlying regions of intense photospheric magnetic field and in loops or loop-segments connecting such regions. The hot coronal loops are supported hydrostatically while only some of the transition zone loops are. The coronal and transition zone loops are distinctly separated and are not coaxial. A comparison of direct measurements of electron densities using density sensitive line ratios with indirect measurements using emission measures and path lengths shows the existence of fine structures of less than a second of arc in transition region loops. From a similar analysis, hot coronal loops do not have any fine structure below about 2 seconds of arc.     

Nonlinearly selected frequencies in coronal loops

A nonlinear process for the resonant generation of low-frequency fast magnetosonic kink waves in coronal loops is discussed. The efficiency of the process is strongly enhanced due to the existence of a nonlinearly selected frequency produced by a constant frequency difference in the dispersion curves in the short wavelength limit. The kink wave with the selected frequency interacts with high-frequency kink and sausage waves. The efficiency of such interaction does not require coherence in the interactive waves. In a loop of width 2 × 10³ km, field strength 50 G and number density 5 × 10¹⁵ m^–3, the nonlinearly selected frequency is of order 46 mHz (period 21.8 s), but this may range through 11 mHz to 184 mHz (periods 86.5 s to 5.4 s) for typical coronal conditions.     

Thermal equilibria of isobaric coronal magnetic arcades

A coronal magnetic arcade can be thought of as consisting of an assembly of coronal loops. By solving equations of isobaric thermal equilibrium along each loop and assuming a base temperature of 2 × 10⁴ K, the thermal structure of the arcade can be found. The possible thermal equilibria can be shown to depend on two parameters L ^* p ^* and h ^*/p ^* representing the ratios of cooling (radiation) to condu and heating to cooling, respectively. Arcades can contain four types of loops: hot loops with summits hotter than 400000 K; cool loops at temperatures less than 80000 K along their lengths; hot-cool loops with cool summits and cool footpoints but hotter intermediate portions; and warm loops, cooler than 80000 K along most of their lengths but with summits as hot as 400000 K. Two possibilities for coronal heating are considered, namely a heating that is independent of magnetic field and a heating that is proportional to the square of the local magnetic field. When the arcade is sheared the thermal structure of the arcade may change, leading in some cases to non-equilibrium or in other cases to the formation of a cool core.     

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.     

Heating of coronal loops by fast mode MHD waves

A possible mechanism for the formation and heating of coronal loops through the propagation and damping of fast mode waves is proposed and studied in detail. Loop-like field structures are represented by a dipole field with the point dipole at a given distance below the solar surface. The density of the medium is determined by hydrostatic equilibrium along the field lines in an isothermal atmosphere. The fast mode waves propagating outward from the coronal base are refracted into regions with a low Alfvén speed and suffer collisionless damping when the gas pressure becomes comparable to the magnetic pressure. The propagation and damping of these waves are studied for three different cases: a uniform density at the coronal base, a density depletion within a given flux tube, and a density enhancement within a given flux tube. The fast mode waves are found to be important in the formation and heating of the loops if the wave energy flux density is of the order 10⁵ ergs cm^-2 s^-1 at the coronal base.The National Center for Atmospheric Research is sponsored by the National Science Foundation.     

Acoustic damping of fast kink oscillations of coronal loops

The damping of fast kink oscillations of solar coronal loops attributable to the radiation of MHD waves into the surroundings is considered in the thin-tube approximation. The oscillation damping decrement is calculated both by using a new energy method and by solving the dispersion equation for magnetic-tube eigenmodes. The two approaches are in good agreement under appropriate assumptions. The damping is negligible if MHD waves are radiated perpendicular to the magnetic field. The low Q factor of the loop oscillations in active regions found with the TRACE space telescope is associated with the generation of running waves that propagate along magnetic field lines.     

The coronal and transition region temperature structure of a solar active region

Using measurements of EUV and X-ray spectral lines we derive the differential emission measure vs electron temperature T from the transition region to the corona of an active region (10⁵ T <5 × 10⁶ K). The total emission measure and radiative losses are of order 3 × 10⁴⁸ cm^–3 and 4 × 10²⁶ ergss^–1 respectively. The emission measure at T > 10⁶ K (i.e. that mainly responsible for the X-ray emission) is about 75% of the total. We also examine the use of Mg x 625 Å as an indicator of coronal electron density. A set of theoretical energy balance models of coronal loops in which the loop divergence is a variable parameter is presented and compared with the observations. Particular attention is given to the limitations inherent in any such comparison.     

Spatial and temporal variations of solar coronal loops

Skylab EUV observations of an active region near the solar limb were analyzed. Both cool (T < 10⁶ K) and hot (T > 10⁶ K) loops were observed in this region. For the hot loops the observed intensity variations were small, typically a few percent over a period of 30 min. The cool loops exhibited stronger variations, sometimes appearing and disappearing in 5 to 10 min. Most of the cool material observed in the loops appeared to be caused by the downward flow of coronal rain and by the upward ejection of chromospheric material in surges. The frequent EUV brightenings observed near the loop footpoints appear to have been produced by both in situ transient energy releases (e.g. subflares) and the infall/impact of coronal rain. The physical conditions in the loops (temperatures, densities, radiative and conducting cooling rates, cooling times) were determined. The mean energy required to balance the radiative and conductive cooling of the hot loops is approximately 3 × 10^–3 erg cm^–3 s^–1. One coronal heating mechanism that can account for the observed behavior of the EUV emission from McMath region 12634 is heating by the dissipation of fast mode MHD waves.     

Characteristics of transverse oscillations in a coronal loop arcade

TRACE observations from 15 April 2001 of transverse oscillations in coronal loops of a post-flare loop arcade are investigated. They are considered to be standing fast kink oscillations. Oscillation signatures such as displacement amplitude, period, phase and damping time are deduced from 9 loops as a function of distance along the loop length. Multiple oscillation modes are found with different amplitude profile along the loop length, suggesting the presence of a second harmonic. The damping times are consistent with the hypothesis of phase mixing and resonant absorption, although there is a clear bias towards longer damping times compared with previous studies. The coronal magnetic field strength and coronal shear viscosity in the loop arcade are derived.     

Energy balance in a magnetically confined coronal structure observed by OSO-7

A model of a coronal region of enhanced Fexv and Fexvi emission is developed and its energy balance is examined using extreme ultraviolet observations from OSO-7 together with calculations of possible force-free coronal magnetic field configurations. The coronal emissions overlying the photospheric boundary between regions of opposite magnetic polarity are found to be associated with generally non-potential (current-carrying) magnetic fields in the forms of arches with footpoints in regions of opposite polarity. The orientation of these arches relative to the neutral line changes with degree of ionization of the emitting ion (which we infer from our limb observations to be a function of height) and may be evidence of differing electric currents along various field lines. The appearance of a coronal arch, seen side-on, can conveniently be represented by a parabola and a detailed analysis (Appendix) shows this to be a realistic approximation that should be generally useful in analyzing two-dimensional pictures of coronal structures. Applying this analysis to the most prominent coronal region observed in the radiations of Fexv and Fexvi, we find a maximum in the electron temperature, T _e , of 2.6 × 10⁶K at the top of arches whose heights are 20000–40000 km and whose footpoints are separated by ≈ 100000 km. A temperature gradient of ▽T _e ≈5 × 10^-5K cm^-1 is found in this coronal structure. Radiative losses are typically fifteen times greater than conductive losses and the energy deposition required to maintain the coronal feature is nearly uniformly distributed along its length.     

Characteristics of transverse oscillations in a coronal loop arcade

TRACE observations from 15 April 2001 of transverse oscillations in coronal loops of a post-flare loop arcade are investigated. They are considered to be standing fast kink oscillations. Oscillation signatures such as displacement amplitude, period, phase and damping time are deduced from 9 loops as a function of distance along the loop length. Multiple oscillation modes are found with different amplitude profile along the loop length, suggesting the presence of a second harmonic. The damping times are consistent with the hypothesis of phase mixing and resonant absorption, although there is a clear bias towards longer damping times compared with previous studies. The coronal magnetic field strength and coronal shear viscosity in the loop arcade are derived.     

Active Regions Observed in Extreme Ultraviolet Light by the Coronal Diagnostic Spectrometer on Soho

We present observations of five active regions made by the Coronal Diagnostic Spectrometer (CDS) on the Solar and Heliospheric Observatory (SOHO). CDS observes the Sun in the extreme ultraviolet range 150–780 Å. Examples of active region loops seen in spectral lines emitted at various temperatures are shown. Several classes of loops are identified: those that are seen in all temperatures up to 2 x 10⁶ K; loops seen at 10⁶ K but not reaching 1.6 x 10⁶ K; those at temperatures 2– 4 x 10^-5 K and occasionally at 6 x 10^-5 K but not reaching 10⁶ K. An increasing loop size with temperature and the relationship between the cool and hot structures is discussed. CDS observations reveal the existence of loops and other unresolved structures in active regions, at temperatures between 1.5– 4 x 10^-5 K, which do not have counterparts in lines emitted above 8 x 10^-5 K. Bright compact sources only seen in the transition region lines are investigated. These sources can have lifetimes of up to several days and are located in the vicinity of sunspots. We study the variability of active region sources on time scales from 30 sec to several days. We find oscillatory behaviour of Hei and Ov line intensities in an active region on time scales of 5–10 min.     

The temperature structure and pressure balance of magnetic loops in active regions

EUV observations show many active region loops in lines formed at temperatures between 10⁴K and 2×l0⁶K. The brightest loops are associated with flux tubes leading to the umbrae of sunspots. It is shown that the high visibility of certain loops in transition region lines is due principallly to a sharp radial decrease of temperature to chromospheric values toward the loop axis. The plasma density of these cool loops is not significantly greater than in the hot gas immediately surrounding it. Consequently, the internal gas pressure of the cool material is clearly lower. The hot material immediately surrounding the cool loops is generally denser than the external corona by a factor 3–4. When the active region is examined in coronal lines, this hot high pressure plasma shows up as loops that are generally parallel to the cool loops but significantly displaced laterally. In general the loop phenomenon in an active region is the result of temperature variations by two orders of magnitude and density variations of around a factor five between adjacent flux tubes in the corona.     

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 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 growth of filaments by the condensation of coronal arches

A model of filament formation based on the condensation of coronal arches is described. The condensation results from initiating the radiative instability within an arch by superimposing a transient energy supply upon the steady state heating mechanism. The transient energy supply increases the density within the arch so that when it is removed the radiative losses are sufficient to lead to cooling below the minimum in the power loss curve.Times from the initial formation of the condensation to its temperature stabilization as a cool filament have been calculated for various initial conditions. They lie in the range 10⁴ to 10⁵ s with the majority of the time spent above a temperature of 1 × 10⁶ K.Under the assumption that the condensation of a single arch forms an element of the filament, a complete filament requires the condensation of an arcade of loops. Using experimentally derived parameters, filament densities of 10¹¹ to 10¹² cm^–3 can be obtained.     

Temporal variations of loop structures in the solar atmosphere

Unique timelapse sequences of Skylab/ATM spectroheliograms reveal the following characteristics of normal (i.e. non-flare) loop structures in the solar atmosphere:

1. At the 0.5 × 10⁶ K temperature of Ne vii, emission is concentrated into individual spiky structures that project 10⁴–10⁵ km from their magnetic footpoints and live on the order of 30 min.
2. At the 1.0 × 10⁶ K temperature of Mg ix, the individual spikes are more diffuse, and have greater lengths and longer lifetimes (～ 1.5 hr) than their 0.5 × 10⁶ K counterparts. Perhaps for this reason, more 1.0 × 10⁶ K loops are visible than 0.5 × 10⁶ K loops at any given time.
3. At the 2.0 × 10⁶ K temperature of Fe xv, emission is confined to a number of relatively diffuse and irregularly shaped features whose collective patterns define closed field volumes in and between active regions. Although the individual features evolve on a time scale of roughly 6 hr, their collective patterns last for several days or more. Unlike the 0.5 × 10⁶ K features, the 2.0 × 10⁶ K features never form as a linear extension along an apparent magnetic field line, but seem to brighten and fade in place.

These results place severe constraints on theoretical models of coronal heating and mass flow.