Damping of Longitudinal Magneto–Acoustic Oscillations in Slowly Varying Coronal Plasma

We investigate the propagation of MHD waves in a magnetised plasma in a weakly stratified atmosphere, representative of hot coronal loops. In most earlier studies, a time-independent equilibrium was considered. Here we abandon this restriction and allow the equilibrium to develop as a function of time. In particular, the background plasma is assumed to be cooling due to thermal conduction. The cooling is assumed to occur on a time scale greater than the characteristic travel times of the perturbations. We investigate the influence of cooling of the background plasma on the properties of magneto–acoustic waves. The MHD equations are reduced to a 1D system modelling magneto–acoustic modes propagating along a dynamically cooling coronal loop. A time-dependent dispersion relation that describes the propagation of the magneto–acoustic waves is derived using the WKB theory. An analytic solution for the time-dependent amplitude of waves is obtained, and the method of characteristics is used to find an approximate analytical solution. Numerical calculations of the analytically derived solutions are obtained to give further insight into the behaviour of the MHD waves in a system with a variable, time-dependent background. The results show that there is a strong damping of MHD waves and the damping also appears to be independent of the position along the loop. Studies of MHD wave behaviour in a time-dependent backgrounds seem to be a fundamental and very important next step in the development of MHD wave theory that is applicable to a wide range of situations in solar physics.     

Sausage Mode Propagation in a Thick Magnetic Flux Tube

The aim of this paper is to model the propagation of slow magnetohydrodynamic (MHD) sausage waves in a thick expanding magnetic flux tube in the context of the quiescent (VAL-C) solar atmosphere. The propagation of these waves is found to be described by the Klein–Gordon equation. Using the governing MHD equations and the VAL-C atmosphere model we study the variation of the cut-off frequency along and across the magnetic tube guiding the waves. Due to the radial variation of the cut-off frequency the flux tubes act as low frequency filters for the waves.     

Present and Future Observing Trends in Atmospheric Magnetoseismology

With modern imaging and spectral instruments observing in the visible, EUV, X-ray, and radio wavelengths, the detection of oscillations in the solar outer atmosphere has become a routine event. These oscillations are considered to be the signatures of a wave phenomenon and are generally interpreted in terms of magnetohydrodynamic (MHD) waves. With multiwavelength observations from ground- and space-based instruments, it has been possible to detect waves in a number of different wavelengths simultaneously and, consequently, to study their propagation properties. Observed MHD waves propagating from the lower solar atmosphere into the higher regions of the magnetized corona have the potential to provide excellent insight into the physical processes at work at the coupling point between these different regions of the Sun. High-resolution wave observations combined with forward MHD modeling can give an unprecedented insight into the connectivity of the magnetized solar atmosphere, which further provides us with a realistic chance to reconstruct the structure of the magnetic field in the solar atmosphere. This type of solar exploration has been termed atmospheric magnetoseismology. In this review we will summarize some new trends in the observational study of waves and oscillations, discussing their origin and their propagation through the atmosphere. In particular, we will focus on waves and oscillations in open magnetic structures (e.g., solar plumes) and closed magnetic structures (e.g., loops and prominences), where there have been a number of observational highlights in the past few years. Furthermore, we will address observations of waves in filament fibrils allied with a better characterization of their propagating and damping properties, the detection of prominence oscillations in UV lines, and the renewed interest in large-amplitude, quickly attenuated, prominence oscillations, caused by flare or explosive phenomena.     

Reflection Properties of Gravito-MHD Waves in an Inhomogeneous Horizontal Magnetic Field

We derive the dispersion equation for gravito-magnetohydrodynamical (MHD) waves in an isothermal, gravitationally stratified plasma with a horizontal inhomogeneous magnetic field. Sound and Alfvén speeds are constant. Under these conditions, it is possible to derive analytically the equations for gravito-MHD waves. The high values of the viscous and magnetic Reynolds numbers in the solar atmosphere imply that the dissipative terms in the MHD equations are negligible, except in layers around the positions where the frequency of the MHD wave equals the local Alfvén or slow wave frequency. Outside these layers the MHD waves are accurately described by the equations of ideal MHD. We consider waves that propagate energy upward in the atmosphere. For the plane boundary, z=0, between two isothermal plasma regions with horizontal but different magnetic fields, we discuss the boundary conditions and derive the equations for the reflection and transmission coefficients. In the simpler case of a gravitationally stratified plasma without magnetic field, these coefficients describe the reflection and transmission properties of gravito-acoustic waves.     

Linear and non-linear MHD wave propagation in steady-state magnetic cylinders

The propagation of linear and non-linear magnetohydrodynamic (MHD) waves in a straight homogeneous cylindrical magnetic flux tube embedded in a homogeneous magnetic environment is investigated. Both the tube and its environment are in steady state. Steady flows break the symmetry of forward (field-aligned) and backward (anti-parallel to magnetic field) propagating MHD wave modes because of the induced Doppler shifts. It is shown that strong enough flows change the sense of propagation of MHD waves. The flow also induces shifts in cut-off values and phase-speeds of the waves. Under photospheric conditions, if the flow is strong enough, the slow surface modes may disappear and the fast body modes may become present. The crossing of modes is also observed due to the presence of flows. The effect of steady-state background has to be considered particularly carefully when evaluating observation signatures of MHD waves for diagnostics in the solar atmosphere.     

The generation of MHD shock waves during the impulsive phase of the February 27, 1992 flare

In this paper a unique 2.3–4.2 GHz radio spectrum of the flare impulsive phase, showing fast positively drifting bursts superimposed on a slowly negatively drifting burst, is presented. Analyzing this radio spectrum it was found that the flare started somewhere near the transition region, where upward propagating MHD waves were generated during the whole impulsive phase. Moreover, it was found that behind a front of these ascending MHD waves the downward propagating electron beams, which bombarded dense layers of the solar atmosphere, were accelerated. It seems that, simultaneously with the increase of beam bombardment intensity, the intensity of MHD waves was increasing and thus the MHD shock wave generation and the electron beam acceleration and bombardment formed a self-consistently amplifying flare process. At higher coronal heights this process was followed by a type II radio burst, i.e. by the MHD flare shock. To verify this concept, the numerical modeling of the shock-wave generation and propagation in space from a flare site near the transition region up to 3 solar radii was made. Comparing the thermal and magnetic field disturbances, it was found that those of magnetic origin are more relevant in this case. Combining the results of interpretation and numerical simulation, a model of the February 27, 1992 flare is suggested and new aspects of this model are discussed.     

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.     

Non-linear damping of visco-resistive Alfvén waves in solar spicules

Interaction of Alfvén waves with plasma inhomogeneities generates phase mixing which can lead to dissipate Alfvén waves and to heat the solar plasma. Here we study the dissipation of Alfvén waves by phase mixing due to viscosity and resistivity variations with height. We also consider nonlinear magnetohydrodynamic (MHD) equations in our theoretical model. Non-linear terms of MHD equations include perturbed velocity, magnetic field, and density. To investigate the damping of Alfvén waves in a stratified atmosphere of solar spicules, we solve the non-linear MHD equations in the x–z plane. Our simulations show that the damping is enhanced due to viscosity and resistivity gradients. Moreover, energy variations is influenced due to nonlinear terms in MHD equations.     

Propagation of MHD body and surface waves in magnetically structured regions of the solar atmosphere

The fact that magnetically structured regions exist in the solar atmosphere has been known for a number of years. It has been suggested that different kinds of magnetohydrodynamic (MHD) waves can be efficiently damped in these regions and that the dissipated wave energy may be responsible for the observed enhancement in radiative losses. From a theoretical point of view, an important task would be to investigate the propagation and dissipation of MHD waves in these highly structured regions of the solar atmosphere. In this paper, we study the behavior of MHD body and surface waves in a medium with either a single or double (slab) magnetic interface by use of a nonlinear, two-dimensional, time-dependent, ideal MHD numerical model constructed on the basis of a Lagrangean grid and semi-implicit scheme. The processes of wave confinement and wave energy leakage are discussed in detail. It is shown that the obtained results depend strongly on the type of perturbations imposed on the interface or slab and on the plasma parameter, . The relevance of the obtained results to the heating problem of the upper parts of the solar atmosphere is also discussed.     

A Simulation Study of the Magnetic Reconnections Between Closed Loops and Open Magnetic Fields Driven by Horizontal Flows on Solar Surface

Numerous observational events in the solar atmosphere (e.g., solar ?ares and jets) are attributed to the energy conversion due to magnetic reconnec- tions. Magnetic reconnections are also involved in a new scenario of solar wind origin to play a crucial role in opening the closed magnetic loop and releasing its mass into the open magnetic funnel. In this scenario, the closed magnetic loop moves towards the supergranular boundary by the supergranular convection, and collides with the open magnetic funnel there to trigger the magnetic reconnec- tion between each other. This work aims at studying the occurrence and effect of magnetic reconnection in this scenario in detail. The magnetohydrodynamic (MHD) numerical simulation is an important approach to investigate the mag- netic reconnection process in the solar atmosphere. A two-dimensional MHD numerical model has been established, and in combination with the strati?ed temperature and density distributions in the solar atmosphere, the numerical simulation on the process of magnetic reconnection of the closed magnetic loops driven by the horizontal ?ows with the open magnetic ?elds has been performed on the scale of supergranulation. Based on a quantitative analysis of the simula- tion result, it is suggested that the process of magnetic reconnection can really realize the mass release of closed magnetic loops, and further supply to the new open magnetic structures to produce upward mass ?ows. Our results provide a basis for the further modeling of solar wind origin.     

Three-minute oscillations in sunspots: Seismology of sunspot atmospheres

We show that no eigenmodes of sunspot oscillations with periods of ～ 3 min or shorter exist. A complex spectrum of the 3-min oscillations arises, because the sunspot atmosphere is a multiband filter for slow MHD waves. To ascertain why the filter transmission bands appear, we have investigated the propagation of waves through a sunspot atmosphere using both multilayered isothermal model atmospheres and various empirical model atmospheres. It turns out that there are several different mechanisms responsible for the appearance of transmission bands in the atmospheric filter for slow waves. The filter lowest-frequency transmission band arises from the effect of a Fabry-Perot interference filter at the resonance frequency of the temperature plateau. The frequency of this band is always lower than the cutoff frequency of the temperature minimum. The next (in frequency) transmission band appears at the cutoff frequency. The higher-frequency transmission bands result from the antireflection of the atmosphere, an effect well-known in optics and acoustics. The nonlinearity of the 3-min oscillations observed in the upper chromosphere and transition region has only an indirect effect on the properties of the filter, increasing its transmission in most bands due to a decrease in the amplitude of the wave reflected from the upper atmosphere caused by nonlinear wave absorption. Knowledge of the formation mechanisms for the 3-min oscillation spectrum has allowed us to suggest a technique for estimating the parameters of sunspot atmospheres from the 3-min oscillation spectrum, i.e., to lay the foundations for the seismology of sunspot atmospheres.     

Hypersonic beam driven by high-energy particles in extragalactic radio sources

A mechanism is proposed for the formation of collimated beams in radio galaxies. The collimated flows which are non-thermally driven by high energy particles and magneto-hydrodynamic (MHD) waves are presented. The galactic nucleus surrounded by a cool gas is investigated. The cool gas accretes onto the nucleus and the accretion matter can confine the wave zone around the nucleus in which the high energy particles are completely locked to the MHD waves. When a quasi-radial magnetic field is embedded in the accretion flow, the MHD wave packets are collimated into the direction of symmetry axis of the galactic nuclear disc. The fluid around the nucleus is considered to be accelerated and heated by the MHD waves and ejected along the axis.A complete set of hydrodynamic equations which contain the energy transfers of high energy particles and MHD waves is presented. One-dimensional flows which are in pressure equilibrium with the surrounding accretion matter are calculated. When the energy density of the MHD waves is higher than that of the thermal energy, the fluid flow is strongly collimated in a narrow beam. When the MHD waves are strongly damped by the resistivity of the fluid at the great distance from the galactic centre, the collimated beam broadly reexpands. On the basis of the collimated beams driven by high energy particles, the radio morphology of the double radio sources is discussed.     

Dual pulsations in solar radio bursts at short centimeter wavelengths

It is well known that the oscillating MHD waves drive periodic variations in the magnetic field. But how the MHD waves can be triggered in the flaring loops is not yet well known. It seems to us that this problem should be connected with the physical processes occurring in the flare loop during a solar flare. A peculiar solar flare event at 04:00–04:51 UT on May 23, 1990 was observed simultaneously with time resolutions 1 s and 10 ms by Nanjing University Observatory and Beijing Normal University Observatory, which are about 1000 km apart, at 3.2 cm and 2 cm wavelengths, respectively. Two kinds of pulsations with quasi-periods 1.5 s and 40 s were found in radio bursts at the two short centimeter waves; however, the shorter quasi-periodic pulsations were superimposed upon the longer ones. From the data analysis of the above-mentioned quasi-periodic pulsations and associated phenomena in radio and soft X-ray emissions during this flare event published in Solar Geophysical Data (SGD), the authors suggest that the sudden increase in plasma pressure inside (or underlying) the flare kernel due to the upward moving chromospheric evaporated gas, which is caused by the explosive collision heating of strong non-thermal electrons injected downwards from the microwave burst source, plays the important role of triggering agents for MHD oscillations (fast magneto-acoustic mode and Alfvén mode) of the flare loop. These physical processes occurring in the flare loop during the impulsive phase of the solar flare may be used to account for the origin and observational characteristics of quasi-periodic pulsations in solar radio bursts at the two short centimeter wavelengths during the flare event of 1990 May 23. In addition, the average physical parameters N, T, B inside or underlying the flare kernel can be also evaluated.     

Linear MHD Wave Propagation in Time-Dependent Flux Tube

The propagation of magnetohydrodynamic (MHD) waves is an area that has been thoroughly studied for idealised static and steady state magnetised plasma systems applied to numerous solar structures. By applying the generalisation of a temporally varying background density to an open magnetic flux tube, mimicking the observed slow evolution of such waveguides in the solar atmosphere, further investigations into the propagation of both fast and slow MHD waves can take place. The assumption of a zero-beta plasma (no gas pressure) was applied in Williamson and Erdélyi (Solar Phys. 2013, doi: 10.1007/s11207-013-0366-9 , Paper I) is now relaxed for further analysis here. Firstly, the introduction of a finite thermal pressure to the magnetic flux tube equilibrium modifies the existence of fast MHD waves which are directly comparable to their counterparts found in Paper I. Further, as a direct consequence of the non-zero kinetic plasma pressure, a slow MHD wave now exists, and is investigated. Analysis of the slow wave shows that, similar to the fast MHD wave, wave amplitude amplification takes place in time and height. The evolution of the wave amplitude is determined here analytically. We conclude that for a temporally slowly decreasing background density both propagating magnetosonic wave modes are amplified for over-dense magnetic flux tubes. This information can be very practical and useful for future solar magneto-seismology applications in the study of the amplitude and frequency properties of MHD waveguides, e.g. for diagnostic purposes, present in the solar atmosphere.     

Numerical simulations for MHD coronal seismology

Magnetohydrodynamic（MHD） processes are important for the transfer of energy over large scales in plasmas and so are essential to understanding most forms of dynamical activity in the solar atmosphere. The introduction of transverse structuring into models for the corona modifies the behavior of MHD waves through processes such as dispersion and mode coupling. Exploiting our understanding of MHD waves with the diagnostic tool of coronal seismology relies upon the development of sufficiently detailed models to account for all the features in observations. The development of realistic models appropriate for highly structured and dynamical plasmas is often beyond the domain of simple mathematical analysis and so numerical methods are employed. This paper reviews recent numerical results for seismology of the solar corona using MHD.     

Coronal loop heating by discrete Alfvén waves

We have modeled the solar coronal active loop heating by discrete Alfvén waves. Discrete Alfvén waves (DAW) are a new class of Alfvén waves which can be described by the two-fluid model with finite ion-cyclotron frequency, or the MHD model with plasma current along the magnetic field line as shown by Appert, Vaclavik, and Villar (1984). We have modeled the coronal loop as a semi-toroidal plasma with the major toroidal radius much larger than the plasma radius. We have shown that the absorption of discrete Alfvén waves by the plasma through viscosity can account for at least 30% of the coronal heating rate density of 10^–4 J m^–3 s^–1.     

MHD oscillations in the solar spike radiation

In this paper the observed 1.4–1.6 s quasi-periodic oscillations in the spike radiation of the microwave outburst of 1981 May 16 are analysed in teras of MHD waves. We point out that the fast magnetoacoustic waves (“sausage” mode) propagating inside and outside a loop can modulate the magnetic field and the pitch angle distribution of the electron beams in the source region. The growth rate of electron-cyclotron-maser instability is then affected to give rise to the quasi-periodic oscillations. Quantitative estimates of relevant physical parameters are given.     

Magnetohydrodynamic behaviour of a finite amplitude disturbance in the solar magnetic field

In this paper, dynamic processes in the solar atmosphere are studied numerically from a complete set of MHD equations. Dynamic evolution of the non-linear magnetic field is produced by the finite amplitude of the azimuthai magnetic field at the base of the flux tube of the solar atmosphere. It is assumed that the initial configuration of the magnetic field is a force-free and potential field, the magnetic field is disturbed at the base, the plasma is driven and a part of the magnetic energy is transformed into the kinetic energy of the plasma.The compressed flow of the plasma has the features of fast MHD waves. The computation results give quantitatively the non-linear evolution of strong magnetic fields. These results could be used in an explanation of coronal transients, surge, spray and eruptive prominence events in the solar atmosphere, as well as in a modelling of plasma behaviour in high-β structure experiments in the laboratory.     

Waves and Oscillations in the Corona – (Invited Review)

It has long been suggested on theoretical grounds that MHD waves must occur in the solar corona, and have important implications for coronal physics. An unequivocal identification of such waves has however proved elusive, though a number of events were consistent with an interpretation in terms of MHD waves. Recent detailed observations of waves in events observed by SOHO and TRACE removes that uncertainty, and raises the importance of MHD waves in the corona to a higher level. Here we review theoretical aspects of how MHD waves and oscillations may occur in a coronal medium. Detailed observations of waves and oscillations in coronal loops, plumes and prominences make feasible the development of coronal seismology, whereby parameters of the coronal plasma (notably the Alfvén speed and through this the magnetic field strength) may be determined from properties of the oscillations. MHD fast waves are refracted by regions of low Alfvén speed and slow waves are closely field-guided, making regions of dense coronal plasma (such as coronal loops and plumes) natural wave guides for MHD waves. There are analogies with sound waves in ocean layers and with elastic waves in the Earth's crust. Recent observations also indicate that coronal oscillations are damped. We consider the various ways this may be brought about, and its implications for coronal heating.     

Long-lived Coronal Loop Profiles from TRACE

An initial study of long-lived loops observed with TRACE (Lenz et al., 1999) shows that they have no significant temperature stratification and that they are denser than the classic loop model predicts. Models that agree better with the observations include a loop consisting of a bundle of filaments at different temperatures and a loop with momentum input by MHD waves. Some implications for coronal heating models and mechanisms are discussed.