Scaling Law for the Heating of Solar Coronal Loops

We report preliminary results from a series of numerical simulations of the reduced magnetohydrodynamic equations used to describe the dynamics of magnetic loops in active regions of the solar corona. A stationary velocity field is applied at the photospheric boundaries to imitate the driving action of granule motions. A turbulent stationary regime is reached, characterized by a broadband power spectrum Ek approximately k-3&solm0;2 and heating rate levels compatible with the energy requirements of active region loops. A dimensional analysis of the equations indicates that their solutions are determined by two dimensionless parameters: the Reynolds number and the ratio between the Alfvén time and the photospheric turnover time. From a series of simulations for different values of this ratio, we determine how the heating rate scales with the physical parameters of the problem, which might be useful for an observational test of this model.     

The coupling resonance of torsional alfven wave and fast wave in coronal loops

The method of Orthogonal Function Series Expansion (OFSE) is generalized and applied to the study of the evolution of the coupling of nondissipative torsional Alfven wave and fast wave in coronal loops. Using this method, the intrinsic angular frequency of the overall wave mode can be described mathematically and that of the Alfven waves along the magnetic lines in the coronal loop during the coupling of the Alfven and fast waves can be analyzed both theoretically and numerically. Also with this method, the relation between the coupling driven term and the Alfven wave resonance may be analyzed. Results of computation reveal the place of appearance of coupling resonance as well as the characteristics of the amplitudes of the Alfven and fast waves. As found by the calculations, if the footpoint driven angular frequency is not equal to the intrinsic angular frequency of the overall wave mode of the coronal loop and when a δ section appears at the place of coupled resonance, the radial gradient of the fast wave's amplitude is quite large. Sometimes it approximates to a discontinuity, and this is extremely favorable for the dissipation of the fast wave. If the footpoint driven angular frequency is equal to the intrinsic angular frequency of the overall wave mode and when a δ section occurs in the Alfven wave amplitude, abundant small-scale structures appear in the radial direction. Then the location of resonance approximately becomes a discontinuity, very favorable to the dissipation of the Alfven wave.     

Solar Flare Statistics with a One-Dimensional Mhd Model

The dynamics of dissipative events in coronal loops is modeled with the compressible MHD equations in one space dimension in slab geometry, with a forcing that mimics the footpoint motions. Using a set of numerical simulations, the statistics of strong velocity and magnetic field gradients that develop, leading to a bursty dissipation, are analyzed. Agreement with existing observations of X-ray solar flares obtains concerning the power-law distribution of the luminosity histogram, including when the dynamics are simplified to that of the Burgers equation; in that latter case, this allows for recasting the analysis in terms of avalanche-type models.     

The dynamics of driven magnetic reconnection in coronal arcades

The dynamics of interacting coronal loops and arcades have recently been highlighted by observations from theYohkoh satellite and may represent a viable mechanism for heating the solar corona. Here such an interaction is studied using two-dimensional resistive magnetohydrodynamic (MHD) simulations. Initial potential field structures evolve in response to imposed photospheric flows. In addition to the anticipated current sheet about theX-point separating the colliding flux systems, significant current layers are found to lie all the way along the separatrices that intersect at theX-point and divide the coronal magnetic field into topologically distinct regions. Shear flows across the separatrices are also observed. Both of these features are shown to be compatible with recent analytical studies of two-dimensional linear steady-state magnetic reconnection, even though the driven system that has been simulated is not strictly ‘open’ in the sense implied by steady-state calculations. The implications for future steady-state models are also discussed. The presence of the neutral point also brings into question any constant-density approximations that have previously been used for such quasi-steady coronal evolution models. This results from the intimate coupling between the neutral point and its separatrices communicated via the gas pressure. In terms of the detailed energetics during the arcade evolution, preliminary results reveal that on the order of 3% of the energy injected by the footpoint motions is lost purely through ohmic dissipation. We would therefore anticipate a local hot spot between the interacting flux systems, and a brightening distributed along the length of any separatrix field lines. Furthermore, as the resistivityη is reduced, the flux annihilation rate and the ohmic dissipation rate are found to scale independently ofη.     

Topological changes of the photospheric magnetic field inside active regions: A prelude to flares?

The detection of magnetic field variations as a signature of flaring activity is one of the main goals in solar physics. Past efforts gave apparently no unambiguous observations of systematic changes. In the present study, we discuss recent results from observations that scaling laws of turbulent current helicity inside a given flaring active region change in response to large flares in that active region. Such changes can be related to the evolution of current structures by a simple geometrical argument, which has been tested using high Reynolds number direct numerical simulations of the MHD equations. Interpretation of the observed data within this picture indicates that the change in scaling behavior of the current helicity seems to be associated with a topological reorganization of the footpoint of the magnetic field loops, namely with the dissipation of small scales structures in turbulent media.     

Implications of rapid footpoint motions of photospheric flux tubes for coronal heating

Some recent observations at Pic-du-Midi (Mulleret al., 1992a) suggest that the photospheric footpoints of coronal magnetic field lines occasionally move rapidly with typical velocities of the order 3 km s^–1 for about 3 or 4 min. We argue that such occasional rapid footpoint motions could have a profound impact on the heating of the quiet corona. Qualitative estimates indicate that these occasional rapid motions can account for the entire energy flux needed to heat the quiet corona. We therefore carry out a mathematical analysis to study in detail the response of a vertical thin flux tube to photospheric footpoint motions in terms of a superposition of linear kink modes for an isothermal atmosphere. We find the resulting total energy that is asymptotically injected into an isothermal atmosphere (i.e., an atmosphere without any back reflection). By using typical parameter values for fast and slow footpoint motions, we show that, even if the footpoints spend only 2.5% of the time undergoing rapid motions, still these rapid motions could be more efficient in transporting energy to the corona than the slow motions that take place most of the time.     

Coronal heating by magnetic reconnection

The theory of magnetic reconnection has advanced substantially over the past few years. There now exists a new generation of fast two-dimensional models known as almost-uniform reconnection and nonuniform reconnection, depending on the boundary conditions. Also, we are beginning to explore the uncharted region of three-dimensional reconnection, where regimes of “spine reconnection” and “fan reconnection” have been discovered. Furthermore, part of the coronal heating problem appears to have been solved with recent observational support for the Converging Flux Model in which heating is produced by coronal reconnection driven by footpoint motions.     

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.     

Numerical simulation of coronal heating by resonant absorption of Alfvén waves

The heating of coronal loops by resonant absorption of Alfvén waves is studied in compressible, resistive magnetohydrodynamics. The loops are approximated by straight cylindrical, axisymmetric plasma columns and the incident waves which excite the coronal loops are modelled by a periodic external driver. The stationary state of this system is determined with a numerical code based on the finite element method. Since the power spectrum of the incident waves is not well known, the intrinsic dissipation is computed. The intrinsic dissipation spectrum is independent of the external driver and reflects the intrinsic ability of the coronal loops to extract energy from incident waves by the mechanism of resonant absorption.The numerical results show that resonant absorption is very efficient for typical parameter values occurring in the loops of the solar corona. A considerable part of the energy supplied by the external driver, is actually dissipated Ohmically and converted into heat. The heating of the plasma is localized in a narrow resonant layer with a width proportional to ^1/3. The energy dissipation rate is almost independent of the resistivity for the relevant values of this parameter. The efficiency of the heating mechanism and the localization of the heating strongly depend on the frequency of the external driver. Resonant absorption is extremely efficient when the plasma is excited with a frequency near the frequency of a so-called collective mode.     

Global Simulations of Differentially Rotating Magnetized Disks: Formation of Low-beta Filaments and Structured Coronae

We examine the excitation of oscillations in the magnetic network of the Sun through the footpoint motion of photospheric magnetic flux tubes located in intergranular lanes. The motion is derived from a time series of high-resolution G-band and continuum filtergrams using an object-tracking technique. We model the response of the flux tube to the footpoint motion in terms of the Klein-Gordon equation, which is solved analytically as an initial value problem for transverse (kink) waves. We compute the wave energy flux in upward-propagating transverse waves. In general we find that the injection of energy into the chromosphere occurs in short-duration pulses, which would lead to a time variability in chromospheric emission that is incompatible with observations. Therefore, we consider the effects of turbulent convective flows on flux tubes in intergranular lanes. The turbulent flows are simulated by adding high-frequency motions (periods 5-50 s) with an amplitude of 1 km s(-1). The latter are simulated by adding random velocity fluctuations to the observationally determined velocities. In this case, we find that the energy flux is much less intermittent and can in principle carry adequate energy for chromospheric heating.     

What is Moss?

TRACE observations of active regions show a peculiar extreme ultraviolet (EUV) emission over certain plage areas. Termed `moss' for its spongy, low-lying, appearance, observations and modeling imply that the phenomenon is caused by thermal conduction from 3–5 MKcoronal loops overlying the plage: moss is the upper transition region emission of hot coronal loops. The spongy appearance is due to the presence of chromospheric jets or `spicules' interspersed with the EUV emission elements. High cadence TRACE observations show that the moss EUV elements interact with the chromospheric jets on 10 s time scales. The location of EUV emission in the moss does not correlate well to the locations of underlying magnetic elements in the chromosphere and photosphere, implying a complex magnetic topology for coronal loop footpoint regions. We summarize here the key observations leading to these conclusions and discuss new implications for understanding the structuring of the outer solar atmosphere. Supplementary material to this paper is available in electronic form at http://dx.doi.org/10.1023/A:1005286503963     

Siphon flows in coronal loops: I. Adiabatic flow

It is now known that the corona is filled with a multitude of loop-like structures. The likelihood of these loops being in static equilibrium is small and so this paper explores the possibility of steady isothermal or adiabatic flows, driven by a pressure difference between the loop feet. For a symmetric loop the flow becomes supersonic at the summit and is then retarded by a shock-wave at some point on the downflowing leg. The effect of adiabatic flow is to lower both pressure and temperature by at least a factor of two and so provide a possible explanation for the cool cores that are sometimes observed in coronal loops. Asymmetric loops, whose cross-sectional area increases or decreases in the flow direction, are found to possess a wide range of both subsonic and shocked flows. Converging loops have subsonic flows if the pressure difference between the footpoints is small, but shocked flows if the pressure difference is large enough. Diverging loops exhibit only shocked flows towards a low pressure footpoint, but can have either subsonic or shocked flow towards a high pressure footpoint. Flows in diverging loops can therefore be either accelerated or decelerated.     

Spectral and Spatial Variations of Flare Hard X-ray Footpoints

In a sample of strong RHESSI M-class flares we have made a study of the relationship between the `hardness' of the HXR spectrum and the intensity in the 30–50 keV energy range. In all events we find clear evidence for a `soft–hard–soft' pattern of correlation between hardness and flux, on time scales as short as 10 s. We investigate whether or not this pattern is intrinsic to the acceleration mechanism. The RHESSI images in this energy range are dominated by footpoint brightenings, and we have searched for a correlation between footpoint separation velocity and spectral hardness, to be compared qualitatively with theoretical flare models. We find quite systematic footpoint motions, and also note that episodes in which footpoint separation varies rapidly often correspond with episodes of significant change in the flare spectral index, though not as the simplest flare models would predict. We report also on one of our events, on 14 March 2002, which exhibits highly sheared HXR footpoint ribbons extending over a scale of 100 arc sec. For this flare we find a correlation between footpoint motion and hard X-ray flux. Supplementary material to this paper is available in electronic form at http://dx.doi.org/10.1023/A:1022479610710     

NONLINEAR MHD SIMULATIONS OF WAVE DISSIPATION IN FLUX TUBES

The phase mixing and resonant dissipation of Alfvén waves is studied in both the 'closed' magnetic loops and the 'open' coronal holes observed in the hot solar corona. The resulting energy transfer from large to small length scales contributes to the heating of these magnetic structures. The nonlinear simulations show that the periodically varying shear flows that occur in the resonant layers are unstable. In coronal holes, the phase mixing of running Alfvén waves is speeded up by the 'flaring out' of the magnetic field lines in the lower chromosphere.     

Acceleration and Storage of Energetic Electrons in Magnetic Loops in the Course of Electric Current Oscillations

A mechanism of electron acceleration and storage of energetic particles in solar and stellar coronal magnetic loops, based on oscillations of the electric current, is considered. The magnetic loop is presented as an electric circuit with the electric current generated by convective motions in the photosphere. Eigenoscillations of the electric current in a loop induce an electric field directed along the loop axis. It is shown that the sudden reductions that occur in the course of type IV continuum and pulsating type III observed in various frequency bands (25?–?180 MHz, 110?–?600 MHz, 0.7?–?3.0 GHz) in solar flares provide evidence for acceleration and storage of the energetic electrons in coronal magnetic loops. We estimate the energization rate and the energy of accelerated electrons and present examples of the storage of energetic electrons in loops in the course of flares on the Sun or on ultracool stars. We also discuss the efficiency of the suggested mechanism as compared with the electron acceleration during the five-minute photospheric oscillations and with the acceleration driven by the magnetic Rayleigh–Taylor instability.     

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.     

The rate of flux pile-up magnetic reconnection in reduced MHD

The rate of two-dimensional flux pile-up magnetic reconnection is known to be severely limited by gas pressure in a low-beta plasma of the solar corona. As earlier perturbational calculations indicated, however, the pressure limitation should be less restrictive for three-dimensional flux pile-up. In this paper the maximum rate of reconnection is calculated in the approximation of reduced magnetohydrodynamics (RMHD), which is valid in the solar coronal loops. The rate is calculated for finite-magnitude reconnecting fields in the case of a strong axial field in the loop. Gas pressure effects are ignored in RMHD but a similar limitation on the rate of magnetic merging exists. Nevertheless, the magnetic energy dissipation rate and the reconnection electric field can increase by several orders of magnitude as compared with strictly two-dimensional pile-up. Though this is still not enough to explain the most powerful solar flares, slow coronal transients with energy release rates of order 10²⁵– 10²⁶ erg s^–1and heating of quiet coronal loops are within the compass of the model.     

On the Stability of Colliding Flows: Radiative Shocks, ThinShells, and Supersonic Turbulence

High-resolution numerical simulations reveal the turbulent character of the interaction zone of colliding, radiative, hypersonic flows. As the shocked gas cools radiatively, the cooled matter is squeezed into thin, high density shells. The remaining kinetic energy causes supersonic turbulence within these shells, before it is finally dissipated by internal shocks and vortex cascades. The density is far from homogeneous. High density filaments and large voids coexist. Its mean value is significantly below the stationary value. Similarly, areas with supersonic velocities are found next to subsonic regions. The mean velocity is slightly below or above the sound speed. While quasi uniform flow motions are observed on smaller scales the large scale velocity distribution is isotropic. Part of the turbulent shell is occupied by relatively uniform flow-patches, resembling coherent structures. Astronomical implications of the turbulent interaction zone are multifarious. It probably drives the X-ray variability in colliding wind binaries as well as the surprising dust formation on orbital scales in some WR-binaries. It lets us understand the knotty appearance of wind-driven structures as planetary and WR-ring nebulae, symbiotics, supernova remnants, galactic supperbubbles. Also, WR and other radiatively driven, clumpy winds, advection dominated accretion, cooling flows and molecular cloud dynamics in star-forming regions may carry its stamp This revised version was published online in July 2006 with corrections to the Cover Date.     

VERY LARGE ARRAY OBSERVATIONS OF EVOLVING NOISE STORM SOURCES ON THE SUN

The presence of coronal magnetic fields connecting active regions is inferred from decimetric observations of solar noise storms with the Very Large Array (VLA) and from soft X-ray images taken by Yohkoh. Temporal changes in the noise storms appear to be correlated with some soft X-ray bursts detected by both Yohkoh and the GOES satellite. Combined analysis of the radio and X-ray data suggests a re-arrangement of the coronal magnetic field during the onset of impulsive noise storm burst emission. On one day during the combined VLA–Yohkoh–GOES observations, two widely-separated active regions appear to be connected by a faint trans-equatorial 91 cm source as well as two distinct soft X-ray loops. The two active regions show anti-correlated fluctuations in decimetric radio emission. On another day of combined VLA–Yohkoh observations, a series of 91 cm noise storm bursts are observed along the major axis of the associated noise storm continuum. Time sequences of Yohkoh soft X-ray images show a contraction of coronal loops prior to the onset of this series of bursts and a corresponding increase in the X-ray flux in the apparent footpoint of the overarching loop containing the noise storm. These observations imply that energy from a realignment of the magnetic field is being transferred, possibly by accelerated particles, along loops connecting separated active regions on the Sun.     

On the quality of resonant absorption as a coronal loop heating mechanism

The qualityQ of a resonance is defined as the ratio of the total energy contained in the system to the dissipation per driving cycle. Hence, a good quality resonance is one with little losses, i.e., little dissipation per driving cycle. However, for heating coronal plasmas by means of resonant absorption of waves, bad quality resonances are required. Here, the quality of the MHD resonances that occur when an inhomogeneous coronal loop is excited by incident waves is investigated for typical coronal loop parameter values in the frame work of linear, resistive MHD. It is shown that the resonances in coronal loops have bad quality and, hence, yield a lot of Ohmic heating per driving cycle compared to the total energy stored in the loop. As a consequence, the time scales of the heating process are relatively short and resonant absorption turns out to be a viable candidate for the heating of the magnetic loops observed in the solar corona.