Plasma waves caused by transient heat conduction in a coronal loop and anomalous resistivity

Numerical simulation is made of the transient heat conduction during local heating in a model coronal magnetic loop with an axial electric current. It is assumed that a segment near the top of the normal coronal loop is heated to above 10⁷ K by a sufficiently small heat input as compared with the total flare energy. A hump appears in the velocity distribution of electrons moving down the temperature gradient with speeds slightly below the thermal one. Consequently, electron plasma waves are excited. The high intensity of the waves persists in the upper region of the loop for more than a second until the termination of the simulation. The energy density of the plasma waves normalized with respect to thermal density is 10^–3.5 at maximum. A theoretical estimate gives an anomalous resistivity 5 orders of magnitude greater than an initial value. Based on the above result, we propose a model for impulsive loop flares.     

Steady heat conduction in coronal loop unstable against plasma instability

The Fokker-Planck equation is numerically solved to study the electron velocity distribution under steady heat conduction with an applied axial electric current in a model coronal loop.If the loop temperature is so high that the electron mean-free path is longer than the local temperature scale height along the loop, a velocity hump appears at about the local thermal electron velocity. The hump is attributed to cooler electrons moving up the temperature gradient to compensate for the runaway electrons moving down the gradient. If the ratio between the mean free path and temperature scale height is greater than about 2, negative absorption for the plasma waves can appear (waves grow). This effect is enhanced by the presence of axial electric current in the half of the coronal loop in which the electrons carrying the current are drifting up the temperature gradient. Thus, the plasma instability may occur in the coronal elementary magnetic flux tubes. Although the present paper is limited to show the critical condition and linear growth rate of the instability, the following scenarios may be inferred.If the flux tubes change from marginally stable to unstable against the plasma instability, due to an increase in the loop temperature, anomalous resistivity may suddenly appear because of the growth of plasma waves. Then a high axial electric field is induced that may accelerate particles. This could be the onset of impulsive loop flares.For a low electric current, if the loop temperature is sufficiently high to give the negative absorption for the plasma waves in a large part of the coronal loop, steady plasma turbulence may originate. This could be a source for the type I radio noise storm.     

A model of impulsive loop flares revisited

A critical examination of the components of the recent impulsive loop flare model of Takakura is made. It is found that his analysis of the stability of the electron distribution resulting from anomalous heat conduction is in error and electron plasma waves would not be excited. Rather, in the regions where the electron/proton temperature ratioT _e/T _i 10, electrostatic ion-cyclotron waves would be excited and in the regions whereT _e 10, ion-acoustic waves would be excited. Ratios ofT _e/T _i 10 occur only in the late time development behind the conduction fronts. Since the anomalous resistivity due to electrostatic ion-cyclotron waves is fortuitously about 70% of the one used by Takakura, the general development will follow closely the one calculated by him. Because the anomalous resistivity due to ion-acoustic waves is about 95 times the one used by Takakura, the development in the parts of the loop whereT _e/T _i 10 for late times would be altered considerably.Also Guest Worker at NOAA Space Environment Laboratory, Boulder, Colorado, U.S.A.     

On the heat conduction in a high-temperature plasma in solar flares

We have developed three types of mathematical models to describe the mechanisms of plasma heating in the corona by intense heat fluxes from a super-hot (T _e ? 10⁸ K) reconnecting current layer in connection with the problem of energy transport in solar flares. We show that the heat fluxes calculated within the framework of self-similar solutions using Fourier’s classical law exceed considerably the real energy fluxes known from present-day multi-wavelength observations of flares. This is because the conditions for the applicability of ordinary heat conduction due to Coulomb collisions of thermal plasma electrons are violated. Introducing anomalous heat conduction due to the interaction of thermal runaway electrons with ion-acoustic turbulence does not give a simple solution of the problem, because it produces unstable temperature profiles. Themodels incorporating the effect of collisional heat flux relaxation describe better the heat transport in flares than Fourier’s law and anomalous heat conduction.     

The structure of high-temperature solar flare plasma in non-thermal flare models

We solve the energy equation for the high-temperature (coronal) component of flare plasma for two models of energy input: (i) direct collisional heating by a beam of suprathermal electrons, and (ii) ohmic heating by the beam-neutralizing reverse current. We discuss the regimes where each case is applicable, and solve for the differential emission measure distribution of the coronal plasma in each case. Scaling laws between loop temperatures and injected electron fluxes are derived for both models; these are testable observationally through coordinated soft X-ray and hard X-ray observations, thus providing a method of discriminating between the two cases. We also readdress the question of the energetic importance of a return current which is below the instability threshold for generation of ion-acoustic plasma turbulence. We find that unless the ambient coronal density is very low ( 10⁹ cm ^–3), collisional heating will always dominate there, in agreement with the findings of previous authors. However, in the chromosphere/corona transition region, the relatively low temperature and correspondingly high plasma resistivity imply that reverse current ohmic heating can predominate the flare energetics, by up to an order of magnitude.Presidential Young Investigator.     

First phase heating and particle acceleration during solar flares by fast tearing modes

We develop a simple, but physically consistent, model of heating and particle acceleration by fast tearing modes, for modeling compact loop flares or erupting prominences. It is shown that there is a slow preheating, over many e -foldings of the instability, after which a rapid heating takes place in approximately one e-folding. The role of anomalous resistivity excited by the induced electric field during tearing is discussed, and how both thermal conduction and plasma expansion may play a role in cooling. Estimates for the total number of thermal and non-thermal electrons generated by one fast tearing mode are given, and it is argued that collisional tearing modes give rise to a primarily thermal plasma.     

Observations of steady anomalous magnetic heating in thin current sheets

A faint steadily emitting loop-like structure has been observed by HXIS in its low energy channels (3.5–8.0 keV) on November 5/6, 1980. These HXIS observations have permitted us to follow the thermal evolution of this loop for a period of about 15 hr and from this study we conclude that only a fraction of 0.1% of the volume of the loop is steadily heated at the rather large rate of 0.6 erg cm^-3 s^-1. We interpret this heating as the dissipation of magnetic fields in thin current sheets and we find that the dissipation with classical resistivity is very unlikely, while ion-kinetic tearing, as proposed by Galeev et al. (1981), suits the observations very well. The enhancement of the resistivity over the classical resistivity then turns out to be a factor 4 × 10⁴. Dissipation in extremely thin sheets via the ion-acoustic instability (Duijveman et al., 1981) cannot be completely excluded when the cross-field heat conductivity is anomalously enhanced by a factor 400.We identify the source of the X-ray emission in this paper with the H filament in the same region. The hot X-ray emitting plasma and the cool plasma radiating in H are thermally separated by the strong magnetic field.The main conclusion of the paper is that for the first time direct evidence is found for the steady dissipation of coronal magnetic fields via enhanced resistivity in thin current sheets.     

Coronal loop heating by Alfvén waves

The excitation and dissipation of global and surface Alfvén waves and their conversion into kinetic Alfvén waves have been analyzed for solar coronal loops using a cylindrical model of a magnetized plasma. Also the optimal conditions for coronal loop heating regimes with density of dissipated power 10³ erg cm^–3 s^–1 by the new scheme named combined Alfvén wave resonance are found. Combined Alfvén wave heating regime appears when the global Alfvén wave is immersed into the Alfvén continuum with the condition of not-so-sharp distribution of axial current.Instituto de Matemática, Universidade Federal Fluminense, Niterói, RJ, Brazil     

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.     

Low frequency turbulence in the solar corona and fundamental radiation of type III solar radio burst

On the basis of the previous numerical simulations, a new mechanism for the emission of the fundamental radio waves of solar radio type III bursts is presented. This hypothesis is to attribute the fundamental radio emission to the coalescence of the plasma waves with the low frequency turbulence, whistler or ion acoustic waves, pre-existing on the way of the electron beam which excite the plasma waves.It is estimated that ion acoustic waves could be occasionally unstable in the solar corona due to that drifting bi-Maxwellian distribution of electrons as observed in the solar wind, which is probably caused by collision-less heat conduction.It is also suggested that the reduced damping of the ion acoustic waves in such a distorted electron distribution in the corona may decrease the threshold electric current to cause the anomalous resistivity to be the onset of the solar flares.     

Space charge sheath in plasma-neutral gas interaction

A space charge sheath is found to be formed whenever a high-velocity magnetized plasma stream penetrates a gas cloud. The sheath is always located at the head of the plasma stream, and its thickness is very small compared to the length of the plasma stream. Soon after the sheath is formed it quickly slows down to the Alfvén critical velocity. The plasma behind the sheath continues to move at higher velocity until the whole plasma stream is retarded to the critical velocity. In the interaction at gas density 10¹⁹ m^–3 the sheaths are observed to be accompanied by a single loop of current with current density of 10⁵ Å m^–2. Maximum potential in the sheath ranges between 50 and 200 V.Presently available models for the sheath may explain the initiation of the sheath formation. Physical processes like heating of the electrons and ionization of the gas cloud which come into play at a later stage of the interaction are not included in these models. These processes considerably alter the potential structure in the sheath region. A schematic model of the observed sheath is presented here.Experiments reveal a threshold value of the magnetic field for plasma retardation to occur. This seems to correspond to the threshold condition for excitation of the modified two-stream instability which can lead to the electron heating. The observed current are found sufficient to account for the plasma retardation at a gas density of 10¹⁷ m^–3.     

Non-thermal processes in large solar flares

We analyze particle acceleration processes in large solar flares, using observations of the August, 1972, series of large events. The energetic particle populations are estimated from the hard X-ray and γ-ray emission, and from direct interplanetary particle observations. The collisional energy losses of these particles are computed as a function of height, assuming that the particles are accelerated high in the solar atmosphere and then precipitate down into denser layers. We compare the computed energy input with the flare energy output in radiation, heating, and mass ejection, and find for large proton event flares that:

1. The ～10–10² keV electrons accelerated during the flash phase constitute the bulk of the total flare energy.
2. The flare can be divided into two regions depending on whether the electron energy input goes into radiation or explosive heating. The computed energy input to the radiative quasi-equilibrium region agrees with the observed flare energy output in optical, UV, and EUV radiation.
3. The electron energy input to the explosive heating region can produce evaporation of the upper chromosphere needed to form the soft X-ray flare plasma.
4. Very intense energetic electron fluxes can provide the energy and mass for interplanetary shock wave by heating the atmospheric gas to energies sufficient to escape the solar gravitational and magnetic fields. The threshold for shock formation appears to be ～10³¹ ergs total energy in >20 keV electrons, and all of the shock energy can be supplied by electrons if their spectrum extends down to 5–10 keV.
5. High energy protons are accelerated later than the 10–10² keV electrons and most of them escape to the interplanetary medium. The energetic protons are not a significant contributor to the energization of flare phenomena. The observations are consistent with shock-wave acceleration of the protons and other nuclei, and also of electrons to relativistic energies.
6. The flare white-light continuum emission is consistent with a model of free-bound transitions in a plasma with strong non-thermal ionization produced in the lower solar chromosphere by energetic electrons. The white-light continuum is inconsistent with models of photospheric heating by the energetic particles. A threshold energy of ～5×10³⁰ ergs in >20 keV electrons is required for detectable white-light emission.

The highly efficient electron energization required in these flares suggests that the flare mechanism consists of rapid dissipation of chromospheric and coronal field-aligned or sheet currents, due to the onset of current-driven Buneman anomalous resistivity. Large proton flares then result when the energy input from accelerated electrons is sufficient to form a shock wave.     

Simulation for electron acceleration by DC electric field in the presence of ion sound waves and associated hard X-ray emission

Time-dependent Fokker-Planck equation was numerically solved to demonstrate the dynamics of electrons in a uniform coronal loop with an applied axial DC electric field in the presence of ion-sound waves. This electric field is attributed to an anomalous resistivity due to the ion-sound turbulence caused by an initially given critical current density.The electron momentum distribution becomes a steady state in the whole turbulent region in a short time for which some electrons can be accelerated to the maximum electric potential K _c. The steady energy distribution of electrons flowing out the end of the turbulent region has a very hard power-law-like spectrum with an index of about 0.75. The associated hard X-rays from a thick target also show a hard spectrum with a photon spectral index of 1.3. In order for to be much greater as observed in impulsive X-ray bursts, it is required that the source is a sum of many elementary loops with a power-law-like distribution in K _c with an index = – + 2.5.     

Steady models for the hard X-ray loops in the solar corona

In some gradual hard X-ray bursts with high intensity, hard X-ray source (15–40 keV) is steadily located in the corona along with softer X-ray source (5–10 keV).Two stationary models, high density and high temperature models, are proposed to solve the difficult problem of confinement of hot (or nonthermal) plasma in the direction of the magnetic field along the loops in the corona. In both models, an essential point is that the effective X-ray source is composed of fine dense filamentary loops imbeded in a larger rarefied coronal loop, and the electron number density in the filaments is so high as 10¹¹–10¹² cm^-3. If the density is so high heat conduction can be as reasonably small as of the order of 10²⁷ erg s ^-1 for the given emission measures of observed X-rays, since the required cross-sectional area is small and also classical conduction is valid. Collisional confinement of thermal tail, and nonthermal electrons if any, up to 50–60 keV in the filaments is also possible, so that the hard X-ray images can be loop like structure instead of double source (foot points).High density model is applicable to the coronal filamentary loops with temperature T _m < 5 × 10⁷ K at the loop summit. The heat flow from the summit downwards is lost almost completely by the radiation from the loop during the conduction to the foot points. A continuous energy release is assumed near the summit to maintain the stationary temperature T _m, and pressure balance is maintained along the loop. In this model, the number density at the summit is given by n _m - 10⁶ T _m ² /s_m, where s _m is the length of the loop from the summit to the foot point, and the distribution of temperature and density along the loop are given by T = T _m(s/s_m)^1/3 and n = n _m(s/s_m)^-1/3, respectively.High temperature model is applicable to the filamentary loops with higher temperature up to about 10^8.5 K and comparatively lower number density as 10¹¹ cm^-3 for the requirement of magnetic confinement of the hot plasma in radial direction. The radiation from the loop is negligibly small in this model so that the heat flux is nearly conserved down to the foot points. In this case, temperature gradient is smaller than that of the high density model, depending on the tapering of the magnetic bottle.In both models, the differential emission measure is maximum at the highest temperature T _m and the brightness distribution along the loop shows a maximum around the summit of the loop if some magnetic tapering is taken into account.     

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.     

Impulsive phase heating by uni-directional current systems in solar flares

The thermal interpretation of solar flare impulsive phase hard X-ray emission requires rapid heating of a substantial coronal volume to very high temperatures. In this study we investigate the possibility of producing such heating by current dissipation, driven by a tearing instability associated with a single uni-directional current system. Earlier research is synthesized by coupling the energy equation, including loss terms previously neglected, with an equation describing the evolution of the growing electric field. The resistivity due to the excitation of ion-cyclotron and ion-acoustic waves is computed by assuming marginal stability.It is found, for the fast tearing mode, that for initial growth rates _f 0.3 s^-1 (corresponding to a current channel width l 3 × 10⁵ cm), the electron heating is offset by convective losses, resulting in a very slow temperature rise. Furthermore, hard X-ray emitting temperatures (2 × 10⁸ K) are never realized. For the larger growth rates corresponding to smaller current channel widths, heating from 10⁷ to 10⁸ K can be achieved in a few seconds. However, in this regime the maximum volume that can be heated is only of order 10²⁰ cm³, some three to five orders of magnitude less than the volume of heated material that is inferred from hard X-ray emission measures. These results suggest that in the case of the fast tearing mode a more complicated geometry involving multiple small-scale, oppositely-directed, current channels may be necessary to achieve the required heating.     

Evolution of a flaring loop after injection of energetic electrons

For the November 5, 1980 flare it is investigated how the plasma in a large flaring loop responds to the injection of energetic electrons. Observations are compared with the results of a one-dimensional numerical simulation. For the simulation it is assumed that at the time the injection is started, the plasma is in an equilibrium state with a constant pressure along the loop and conductive heating compensated by radiative losses. Especially important for the evolution of the impulsively heated plasma is the penetration depth of the fast electrons compared to the depth of the transition layer. Both parameters are known from the observations. The injected energy is 2.6 × 10¹¹ ergs cm ^?2 in 30 s (as derived from the hard X-ray observations) and computations show that the high temperature plasma of the loop responds to it with upward motions of about 50 km s^?1, i.e. with velocities much smaller than the ion sound speed (≈ 500km s^?1). The heating of the plasma due to the absorption of beam energy can be understood using a constant density approximation. After the heating phase the plasma returns in about 5 min to its initial state by conductive cooling. The downward conducted energy is radiated away in the transition zone. The numerical simulation shows that impulsive heating by non-thermal electrons only does not explain the observed large increase in the density of the loop during the flare. It is therefore required that continuous energy and/or mass input occur after the impulsive phase.     

The effects of Alfvén waves on heating plasma in post-flare loops

In investigating the effects of collision Alfvén waves on the heating of a cool-type solar loop, like the post-flare loop, models are proposed, and the distributions of ion or electron density, temperature, pressure, and wave energy density are simulated. We assumed the magnetic field strength in the loop is about 100 G and found that Alfvén waves can propagate through the whole loop, that is to say, the decay length of collision Alfvén waves which we consider can reach to the height or length of the loop. Thus, the Alfvén wave heating is a considerable heating mechanism in cool loops. And we also found that the variations of density, pressure, and wave energy density are more significant than those of the temperature. In the whole loop, the temperature is of the order of 10⁴ K. In comparison with other parameters, the temperature can be considered as homogeneous; hence, the heat conductive flux in the simulations is omitted.     

Towards the circuit theory of solar flares

It has been shown that the main problems of the circuit theory of solar flares - unlikely huge current growth time and the origin of the current interruption - have been resolved considering the case of magnetic loop emergence and the correct application of Ohm's law. The generalized Ohm's law for solar flares is obtained. The conditions for flare energy release are as follows: large current value, > 10¹¹ A, nonsteady-state character of the process, and the existence of a neutral component in a flare plasma. As an example, the coalescence of a flare loop and a filament is considered. It has been shown that the current dissipation has increased drastically as compared with that in a completely ionized plasma. The current dissipation provides effective Joule heating of the plasma and particle acceleration in a solar flare. The ion-atom collisions play the decisive role in the energy release process. As a result the flare loop resistance can grow by 8–10 orders of magnitude. For this we do not need the anomalous resistivity driven by small-scale plasma turbulence. The energy release emerging from the upper part of a flare loop stimulates powerful energy release from the chromospheric level.     

Evolution of electron and proton temperatures in a flaring loop

We have investigated numerically how a temperature difference between electrons and protons is produced in a flaring loop by adopting a one-fluid, two-temperature model instead of a single-temperature model. We have treated a case in which flare energy is released in the form of heating of electrons located in the top part of the loop.In this case, a large temperature difference (T _e/T _p 10) appears in the corona in the energy-input phase of the flare. When the material evaporated from the chromosphere fills the corona, the temperature difference in the loop begins to shrink rapidly from below. Eventually, in the loop apex, the proton temperature exceeds the electron temperature mainly due to cooling of the electrons by conduction down the loop and heating of the protons by compression of the ascending material. In the late phase of the flare (t 15 min from the flare onset), the temperature difference becomes less than 2% of the mean temperature of electrons and protons at every point in the loop.