Possible Role of mhd Waves in Heating the Solar Corona

Heating of the solar corona by MHD waves has been investigated. Taking account of dissipation mechanisms self-consistently, a new general dispersion relation has been derived for waves propagating in a homogeneous plasma. Solution of this sixth-order dispersion relation provides information on how the damping of both slow and fast mode waves depends upon the plasma density, temperature, field strength, and angle of propagation relative to the background magnetic field. Wave quantities with and without dissipation are presented. In particular, we consider one of the most important clues from space observations that viscosity of coronal plasma may be orders of magnitude different from its classical value in heating of the corona by MHD waves.     

Oscillatory Modes of a Prominence – PCTR – Corona Slab Model

Oscillations of magnetic structures in the solar corona have often been interpreted in terms of magnetohydrodynamic waves. We study the adiabatic magnetoacoustic modes of a prominence plasma slab with a uniform longitudinal magnetic field, surrounded by a prominence – corona transition region (PCTR) and a coronal medium. Considering linear small-amplitude oscillations, we deduce the dispersion relation for the magnetoacoustic slow and fast modes by assuming evanescentlike perturbations in the coronal medium. In the system without PCTR, a classification of the oscillatory modes according to the polarisation of their eigenfunctions is made to distinguish modes with fastlike or slowlike properties. Internal and external slow modes are governed by the prominence and coronal properties, respectively, and fast modes are mostly dominated by prominence conditions for the observed wavelengths. In addition, the inclusion of an isothermal PCTR does not substantially influence the mode frequencies, but new solutions (PCTR slow modes) are present.     

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.     

Effect of Variable Background on an Oscillating Hot Coronal Loop

We investigate the effect of a variable, i.e. time-dependent, background on the standing acoustic (i.e. longitudinal) modes generated in a hot coronal loop. A theoretical model of 1D geometry describing the coronal loop is applied. The background temperature is allowed to change as a function of time and undergoes an exponential decay with characteristic cooling times typical for coronal loops. The magnetic field is assumed to be uniform. Thermal conduction is assumed to be the dominant mechanism for damping hot coronal oscillations in the presence of a physically unspecified thermodynamic source that maintains the initial equilibrium. The influence of the rapidly cooling background plasma on the behaviour of standing acoustic (longitudinal) waves is investigated analytically. The temporally evolving dispersion relation and wave amplitude are derived by using the Wenzel–Kramers–Brillouin theory. An analytic solution for the time-dependent amplitude that describes the influence of thermal conduction on the standing longitudinal (acoustic) wave is obtained by exploiting the properties of Sturm–Liouville problems. Next, numerical evaluations further illustrate the behaviour of the standing acoustic waves in a system with a variable, time-dependent background. The results are applied to a number of detected loop oscillations. We find a remarkable agreement between the theoretical predictions and the observations. Despite the emergence of the cooling background plasma in the medium, thermal conduction is found to cause a strong damping for the slow standing magneto–acoustic waves in hot coronal loops in general. In addition to this, the increase in the value of thermal conductivity leads to a strong decay in the amplitude of the longitudinal standing slow MHD waves.     

Viscous Damping of Non-Adiabatic MHD Waves in an Unbounded Solar Coronal Plasma

The damping of MHD waves in solar coronal magnetic field is studied taking into account thermal conduction and compressive viscosity as dissipative mechanisms. We consider viscous homogeneous unbounded solar coronal plasma permeated by a uniform magnetic field. A general fifth-order dispersion relation for MHD waves has been derived and solved numerically for different solar coronal regimes. The dispersion relation results three wave modes: slow, fast, and thermal modes. Damping time and damping per periods for slow- and fast-mode waves determined from dispersion relation show that the slow-mode waves are heavily damped in comparison with fast-mode waves in prominences, prominence–corona transition regions (PCTR), and corona. In PCTRs and coronal active regions, wave instabilities appear for considered heating mechanisms. For same heating mechanisms in different prominences the behavior of damping time and damping per period changes significantly from small to large wavenumbers. In all PCTRs and corona, damping time always decreases linearly with increase in wavenumber indicate sharp damping of slow- and fast-mode waves.     

Influence of a uniform coronal magnetic field on solar p modes: coupling to slow resonant MHD waves

The influence of a constant coronal magnetic field on solar global oscillations is investigated for a simple planar equilibrium model. The model consists of an atmosphere with a constant horizontal magnetic field and a constant sound speed, on top of an adiabatic interior having a linear temperature profile. The focus is on the possible resonant coupling of global solar oscillation modes to local slow continuum modes of the atmosphere and the consequent damping of the global oscillations. In order to avoid Alfvén resonances, the analysis is restricted to propagation parallel to the coronal magnetic field. Parallel propagating oscillation modes in this equilibrium model have already been studied by Evans and Roberts (1990). However, they avoided the resonant coupling to slow continuum modes by a special choice of the temperature profile. The physical process of resonant absorption of the acoustic modes with frequency in the cusp continuum is mathematically completely described by the ideal MHD differential equations which for this particular equilibrium model reduce to the hypergeometric differential equation. The resonant layer is correctly dealt with in ideal MHD by a proper treatment of the logarithmical branch cut of the hypergeometric function. The result of the resonant coupling with cusp waves is twofold. The eigenfrequencies become complex and the real part of the frequency is shifted. The shift of the real part of the frequency is not negligible and within the limit of observational accuracy. This indicates that resonant interactions should definitely be taken into account when calculating the frequencies of the global solar oscillations.     

Oscillations of coronal loops and second pulsations of solar radio emission

The dispersion properties of the sausage eigenmodes of oscillations in a thin magnetic flux tube are numerically analyzed in terms of ideal magnetohydrodynamics (MHD). The period of the modes accompanied by the emission of MHD waves into the surrounding medium, which leads to acoustic damping of oscillations, is determined by the radius of the tube, not by its length. The dissipation of the sausage oscillations in comparatively high (?0.7R _⊙) and tenuous (?6 × 10⁸ cm^?3) coronal loops is considered. Their Q factor has bound found to be determined by the acoustic damping mechanism. The ratio of the plasma densities outside and inside the loop and the characteristic height of the emission source have been estimated by assuming the quasi-periodic pulsations of meter-wavelength radio emission to be related to the sausage oscillations.     

Radial oscillations of current-carrying coronal loops

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

Solar coronal heating by magnetosonic waves

Solar coronal heating by magnetohydrodynamic (MHD) waves is investigated. ultraviolet (UV) and X-ray emission lines of the corona show non-thermal broadenings. The wave rms velocities inferred from these observations are of the order of 25–60 km s⁻¹ . Assuming that these values are not negligible, we solved MHD equations in a quasi-linear approximation, by retaining the lowest order non-linear term in rms velocity. Plasma density distribution in the solar corona is assumed to be inhomogeneous. This plasma is also assumed to be permeated by dipole-like magnetic loops. Wave propagation is considered along the magnetic field lines. As dissipative processes, only the viscosity and parallel (to the local magnetic field lines) heat conduction are assumed to be important. Two wave modes emerged from the solution of the dispersion relation. The fast mode magneto-acoustic wave, if originated from the coronal base can propagate upwards into the corona and dissipate its mechanical energy as heat. The damping length-scale of the fast mode is of the order of 500 km. The wave energy flux associated with these waves turned out to be of the order of 2.5×10⁵ ergs cm⁻² s⁻¹ which is high enough to replace the energy lost by thermal conduction to the transition region and by optically thin coronal emission. The fast magneto-acoustic waves prove to be a likely candidate to heat the solar corona. The slow mode is absent, in other words cannot propagate in the solar corona.     

Influence of a chromospheric magnetic field on solar acoustic modes

This paper studies the effect of a magnetic atmosphere on the global solar acoustic oscillations in a simple Cartesian model. First, the influence of the ratio of the coronal and the photospheric temperature τ and the strength of the magnetic field at the base of the corona B_c on the oscillation modes is studied for a convection zone-corona model with a true discontinuity. The ratio τ seems to be an important parameter. Subsequently, the discontinuity is replaced by an intermediate chromospheric layer of thickness L and the effect of the thickness on the frequencies of the acoustic waves is studied. In addition, nonuniformity in the magnetic field, plasma density and temperature in the transition layer gives rise to continuous Alfvén and slow spectra. Modes with characteristic frequencies lying within the range of the continuum may resonantly couple to Alfvén and/or slow waves.     

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 Oscillations of Coronal Loops Including the Shell

We investigate the MHD waves in a double magnetic flux tube embedded in a uniform external magnetic field. The tube consists of a dense hot cylindrical cord surrounded by a co-axial shell. The plasma and the magnetic field are taken to be uniform inside the cord and also inside the shell. Two slow and two fast magnetosonic modes can exist in the thin double tube. The first slow mode is trapped by the cord, the other is trapped by the shell. The oscillations of the second mode have opposite phases inside the cord and shell. The speeds of the slow modes propagating along the tube are close to the tube speeds inside the cord and the shell. The behavior of the fast modes depends on the magnitude of Alfvén speed inside the shell. If it is less than the Alfvén speed inside the cord and in the environment, then the fast mode is trapped by the shell and the other may be trapped under the certain conditions. In the opposite case when the Alfvén speed in the shell is greater than those inside the cord and in the environment, then the fast mode is radiated by the tube and the other may also be radiated under certain conditions. The oscillation of the cord and the shell with opposite phases is the distinctive feature of the process. The proposed model allows to explain the basic phenomena connected to the coronal oscillations: i) the damping of oscillations stipulated in the double tube model by the radiative loss, ii) the presence of two different modes of perturbations propagating along the loop with close speeds, iii) the opposite phases of oscillations of modulated radio emission, coming from the near coronal sources having sharply different densities.     

Interfacing MHD Single Fluid and Kinetic Exospheric Solar Wind Models and Comparing Their Energetics

An exospheric kinetic solar wind model is interfaced with an observation-driven single-fluid magnetohydrodynamic (MHD) model. Initially, a photospheric magnetogram serves as observational input in the fluid approach to extrapolate the heliospheric magnetic field. Then semi-empirical coronal models are used for estimating the plasma characteristics up to a heliocentric distance of 0.1 AU. From there on, a full MHD model that computes the three-dimensional time-dependent evolution of the solar wind macroscopic variables up to the orbit of Earth is used. After interfacing the density and velocity at the inner MHD boundary, we compare our results with those of a kinetic exospheric solar wind model based on the assumption of Maxwell and Kappa velocity distribution functions for protons and electrons, respectively, as well as with in situ observations at 1 AU. This provides insight into more physically detailed processes, such as coronal heating and solar wind acceleration, which naturally arise from including suprathermal electrons in the model. We are interested in the profile of the solar wind speed and density at 1 AU, in characterizing the slow and fast source regions of the wind, and in comparing MHD with exospheric models in similar conditions. We calculate the energetics of both models from low to high heliocentric distances.     

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.     

On poloidal mode coupling in the continuous spectrum of 2D equilibria

The continuous spectrum of linear ideal MHD is determined analytically in 2D magnetostatic models for coronal loops and arcades by means of a perturbation expansion. Poloidal mode coupling, induced by non-circularity of the cross-sections of the magnetic surfaces and/or variation of the plasma density along the magnetic field lines, is shown to occur in first order. The coupling is most pronounced on and near rational surfaces for particular poloidal and toroidal mode numbers and produces gaps in the continuous spectrum of ideal MHD.     

Coronal MHD perturbation field generated by localized perturbers in a photospheric active region

We investigate the two-dimensional boundary value problem of the linear wave excitation in the solar corona by a pair of periodic perturbers localized at the photosphere. The physical properties of the corona allow us to consider a magnetic configuration such as a potential arcade which is in a magneto-hydrostatic equilibrium with the surrounding plasma. The model excludes the A1fvén mode and since the slow mode is absent in a potential arcade, the excited waves are then the fast magneto-acoustic modes. The characteristic magnetic field scale length is twice the scale height of the coronal plasma, assumed isothermal and the induced fluid motions are in the direction perpendicular to the magnetic field, in the cross-sectional plane of the arcade.A particular example of two localized perturbers in a photospheric active region is given, pointing out the effects produced in the perturbed fields by the variation of the different parameters involved. Such example shows that, while a modification of the linear size or the horizontal wave number of the perturbers does not affect significantly the propagating disturbances, the variation of the perturbers' frequency produces important effects. Such effects can be summarized by saying that low-frequency perturbers are able to disturb the whole coronal magnetic configuration above the active region while high-frequency perturbers only disturb the coronal region just above them, producing a channeling feature. Moreover, for high frequencies the perturbations grow withz all the way from the photosphere while for very low frequencies evanescent waves are dominant, although thanks to the influence of the medium their amplitude starts to grow from somez on.     

Generation of Intermediate Drift Bursts by Magnetohydrodynamic Waves in the Solar Corona

The plasma mechanism of radio emission generation in an inhomogeneous medium is investigated. In the model under study, the electron beam with loss-cone distribution generates upper-hybrid waves that, in turn, are transformed into radio emission. It is shown that the influence of the plasma density inhomogeneity limits the plasma waves’ intensity considerably due to variation in their wave vector. The results are used to interpret the intermediate drift (IMD) bursts. A model is proposed in which these bursts are reflections of propagating small-scale (with amplitudes of about 1% and sizes of hundreds of kilometers) magnetohydrodynamic (MHD) disturbances of magnetic tubes. It is shown that this model allows us to explain the spectral parameters of the bursts in question. At present, the lack of precise and independent data about the magnetic field does not allow us to decide definitively between the existing models (whistler or MHD waves) of the IMD bursts; nevertheless, if the proposed model is correct, it can be used to determine the characteristics of the coronal MHD waves.     

太阳多极活动区中的磁重联、剪切低磁弧内的强电流和冕穴结构

探讨了复杂磁结构上空日冕物理状态与磁剪切的关系．结果表明在强磁场的磁中性线上方磁剪切会引起具有强电流和较强等离子体压力的低磁弧．这可解释Ｙｏｈｋｏｈ 卫星的观测结果     

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.     

Coronal Seismology Using Transverse Oscillations of Non-planar Coronal Loops

We continue studying the robustness of coronal seismology. We concentrate on two seismological applications: the estimate of coronal scale height using the ratio of periods of the fundamental harmonic and first overtone of kink oscillations, and the estimate of magnetic-field magnitude using the fundamental harmonic. Our analysis is based on the model of non-planar coronal loops suggested by Ruderman and Scott (Astron. Astrophys. 529, A33, 2011), which was formulated using the linearized MHD equations. We show that the loop non-planarity does not affect the ratio of periods of the fundamental harmonic and first overtone, and thus it is unimportant for the estimates of the coronal scale height. We also show that the density variation along the loop and the loop non-planarity only weakly affect the estimates of the magnetic-field magnitude. Hence, using the simplest model of coronal loops, which is a straight homogeneous magnetic cylinder, provides sufficiently accurate estimates for the magnetic-field magnitude.