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.     

Gravitational instability in isotropic MHD plasma waves

The effect of compressive viscosity, thermal conductivity and radiative heat-loss functions on the gravitational instability of infinitely extended homogeneous MHD plasma has been investigated. By taking in account these parameters we developed the six-order dispersion relation for magnetohydrodynamic (MHD) waves propagating in a homogeneous and isotropic plasma. The general dispersion relation has been developed from set of linearized basic equations and solved analytically to analyse the conditions of instability and instability of self-gravitating plasma embedded in a constant magnetic field. Our result shows that the presence of viscosity and thermal conductivity in a strong magnetic field substantially modifies the fundamental Jeans criterion of gravitational instability.     

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.     

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.     

Magnetohydrodynamic Waves in an Asymmetric Magnetic Slab

Analytical models of solar atmospheric magnetic structures have been crucial for our understanding of magnetohydrodynamic (MHD) wave behaviour and in the development of the field of solar magneto-seismology. Here, an analytical approach is used to derive the dispersion relation for MHD waves in a magnetic slab of homogeneous plasma enclosed on its two sides by non-magnetic, semi-infinite plasma with different densities and temperatures. This generalises the classic magnetic slab model, which is symmetric about the slab. The dispersion relation, unlike that governing a symmetric slab, cannot be decoupled into the well-known sausage and kink modes, i.e. the modes have mixed properties. The eigenmodes of an asymmetric magnetic slab are better labelled as quasi-sausage and quasi-kink modes. Given that the solar atmosphere is highly inhomogeneous, this has implications for MHD mode identification in a range of solar structures. A parametric analysis of how the mode properties (in particular the phase speed, eigenfrequencies, and amplitudes) vary in terms of the introduced asymmetry is conducted. In particular, avoided crossings occur between quasi-sausage and quasi-kink surface modes, allowing modes to adopt different properties for different parameters in the external region.     

Chromospheric and coronal heating due to the dissipation of fast magnetoacoustic waves

The dissipation of ducted, fast, magnetoacoustic waves by ion viscosity and electron heat conduction in a radiating, optically thin atmosphere has been re-examined and the results compared with two previously published, conflicting sets of results. In general, the dissipation length of the waves increases with magnetic field strength and decreases with increase in density, and is a few wavelengths for waves of periods of several seconds in the active corona. Oscillations with such periods have been observed in the corona, so waves could, given the right conditions, be dissipated there, the energy so released being a contributory factor to coronal heating.     

Magneto-gravity waves and the heating of the solar corona

It is generally believed that the heating of the solar corona is caused by waves originating in the photosphere and propagating into the corona where their energy is dissipated. The medium through which these waves propagate is in general permeated by magnetic fields complicating the behaviour of this propagation considerably. We have therefore analysed the wave motions in a plasma permeated by constant magnetic and gravitational fields. In general, three waves modes were found, which we called the + mode, –mode, and the Alfvén mode. Each mode was found to be strongly coupled to each of the three kinds of motion; acoustic, gravity, and hydromagnetic. However, the Alfvén mode was found to be separable from the dispersion relation, and therefore independent of compressibility and gravity. The local dispersion relation is derived and expressed in nondimensional form independent of the constants that describe a particular atmosphere. From the dispersion relation one can show that rising waves propagate either with a constant or a growing wave amplitude depending on the magnitudes and directions of the gravitational field, magnetic field, and the wave vector. The variation of the density with height is taken into account by a generalized W.K.B. method. Equations are found which give the height at which wave reflection occurs, giving the upper bound for possible wave propagation.Work supported by the National Aeronautics and Space Administration under Research Grant NGR-29-001-016.On leave of absence from the Desert Research Institute and Department of Physics, University of Nevada, Reno, Nevada, U.S.A.     

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.     

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.     

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.     

Cylindrical Hall – MHD Waves: A Nonlinear Solution

The exact nonlinear cylindrical solution for incompressible Hall – magnetohydrodynamic (HMHD) waves, including dissipation, essentially from electron – neutral collisions, is obtained in a uniformly rotating, weakly ionized plasma such as exists in photospheric flux tubes. The ω – k relation of the waves, called here Hall – MHD waves, demonstrates the dispersive nature of the waves, introduced by the Hall effect, at large axial and radial wavenumbers. The Hall – MHD waves are in general elliptically polarized. The partially ionized plasma supports lower frequency modes, lowered by the factor δ≡ratio of the ion mass density to the neutral particle mass density, as compared to the fully ionized plasma (δ=1). The relation between the velocity and the magnetic field fluctuations departs significantly from the equipartition found in Alfvén waves. These short-wavelength and arbitrarily large amplitude waves could contribute toward the heating of the solar atmosphere.     

The heating of the solar corona

The heating of the solar corona has been a fundamental astrophysical issue for over sixty years. Over the last decade in particular, space-based solar observatories (Yohkoh, SOHO and TRACE) have revealed the complex and often subtle magnetic-field and plasma interactions throughout the solar atmosphere in unprecedented detail. It is now established that any energy release mechanism is magnetic in origin - the challenge posed is to determine what specific heat input is dominating in a given coronal feature throughout the solar cycle. This review outlines a range of possible magnetohydrodynamic (MHD) coronal heating theories, including MHD wave dissipation and MHD reconnection as well as the accumulating observational evidence for quasi-periodic oscillations and small-scale energy bursts occurring in the corona. Also, we describe current attempts to interpret plasma temperature, density and velocity diagnostics in the light of specific localised energy release. The progress in these investigations expected from future solar missions (Solar-B, STEREO, SDO and Solar Orbiter) is also assessed.Received: 6 February 2003, Published online: 14 November 2003 Correspondence to: R. W. Walsh     

Effect of Flow on Resonant Absorption of Slow MHD Waves in Coronal Arcades

Resonant absorption of slow MHD waves is studied numerically by using the SGH method and is applied to a model of a coronal arcade in the presence of equilibrium plasma flows. The arcade is approximated by a 1D horizontal magnetic slab that is non-uniform along the vertical direction and which is surrounded by two homogeneous media. While propagating from the photosphere upwards into the corona, the magneto-acoustic waves can be resonantly absorbed in the inhomogeneous region of the arcade. Computational results show that the resonant absorption of the impinging waves strongly depends on the equilibrium model and on the characteristics of the driving wave. The results also indicate that the presence of an equilibrium plasma flow along the magnetic field of the arcade reduces the resonant absorption for the flow speed parameters considered.     

Dissipating waves in a hot,compressible, magnetic,structured atmosphere

The damping of ducted, fast, magnetohydrodynamic (MHD) waves by ion viscosity and electron heat conduction in a radiating, optically thin, warm, structured atmosphere has been evaluated. Dissipation is more effective in a warm plasma than in a cold one but, for waves ducted by solar coronal loops, dissipation is only efficient if the periods of the waves are shorter than a few tens of seconds and only if the background magnetic field is less than about 15 G. It appears that MHD waves of longer periods and in stronger magnetic fields will survive the dissipative mechanisms considered here and may be manifest as observable coronal oscillations.     

Mhd Study of Coronal Waves: A Numerical Approach

The solar corona, modelled by a low β, resistive plasma slab sustains MHD wave propagations due to footpoint motions in the photosphere. The density, magnetic profile and driver are considered to be neither very smooth nor very steep. The numerical simulation presents the evolution of MHD waves and the formation of current sheet. Steep gradients in slow wave at the slab edges which are signature of resonance layer where dissipation takes place are observed. Singularity is removed by the inclusion of finite resistivity. Dissipation takes place around the resonance layer where the perturbation develops large gradients. The width of the resonance layer is calculated. The thickness of the Alfvén resonance layer is more than that of the slow wave resonance layer. Attempt is made to distinguish between slow and Alfvén wave resonance layers. Fast waves develop into kink modes. As plasma evolves the current sheets which provide the heating at the edges gets distorted and fragment into two current sheets at each edge which in turn come closer when the twist is enhanced. This revised version was published online in July 2006 with corrections to the Cover Date.     

On the role of transition region on the Alfvén wave phase mixing in solar spicules

Alfvénic waves are thought to play an important role in coronal heating and solar wind acceleration. Here we investigate the dissipation of standing Alfvén waves due to phase mixing at the presence of steady flow and sheared magnetic field in the stratified atmosphere of solar spicules. The transition region between chromosphere and corona has also been considered. The initial flow is assumed to be directed along spicule axis, and the equilibrium magnetic field is taken 2-dimensional and divergence-free. It is determined that in contrast to propagating Alfvén waves, standing Alfvén waves dissipate in time rather than in space. Density gradients and sheared magnetic fields can enhance damping due to phase mixing. Damping times deduced from our numerical calculations are in good agreement with spicule lifetimes. Since spicules are short living and transient structures, such a fast dissipation mechanism is needed to transport their energy to the corona.     

MHD shock interactions in coronal structures

We consider the magnetohydrodynamic (MHD) interactions of solar coronal fast shock waves of flare and/or nonflare origin with the boundaries of coronal streamers and coronal holes. Boundaries are treated as MHD tangential discontinuities (TD). Different parameters of the observed corona are used in the investigation. The general case of the oblique interaction is studied.It is shown that a solar fast shock wave must be refracted usually as a fast shock wave inside the coronal streamer. For the special case of the velocity shear across TD, a slow shock wave is generated. On the contrary, the shock wave refracted inside the coronal hole is indeed a slow shock wave.The significance of different effects due to the interaction of fast and slow shock waves on the coronal magnetic field is noticed, especially at the time of a coronal mass ejection (CME). It is also shown, that an oblique fast MHD coronal shock wave may trigger an instability at the boundary of a streamer considered as a TD. It might have a relation with the observed process of abrupt disappearance of the streamer's boundary in the solar corona.On leave from the Academy of Sciences, Central Astronomical Observatory Pulkovo, 196140, St. Petersburg, Russia.     

Coronal heating by selective decay of MHD turbulence

We suggest that the solar corona is stationarily heated by Joule dissipation of magnetohydrodynamic (MHD) turbulence. This turbulence is continuously being fed by subphotospheric convective motions at very low wavenumbers where the plasma can be considered ideal. The cascade process due to the nonlinear interaction of the turbulent modes, effectively transfers energy to even shorter wavenumbers until Joule effect becomes dominant and converts the cascading magnetic energy into heat.Assuming that a stationary spectrum is established, we show that MHD turbulence naturally provides a way of enhancing the Joule dissipation of macroscopic magnetic stresses and a heating rate can be derived.     

Effect of rotation and self-gravity on the propagation of MHD waves

Magnetohydrodynamics waves and instabilities in rotating, self-gravitating, anisotropic and collision-less plasma were investigated. The general dispersion relation was obtained using standard mode analysis by constructing the linearized set of equations. The wave mode solutions and stability properties of the dispersion relations are discussed in the propagations transverse and parallel to the magnetic field. These special cases are discussed considering the axis of rotation to be in transverse and along the magnetic field. In the case of propagation transverse to the magnetic field with axis of rotation parallel to the magnetic field, we derived the dispersion relation modified by rotation and self-gravitation. In the case of propagation parallel to the magnetic field with axis of rotation perpendicular to the magnetic field, we obtained two separate modes affected by rotation and self-gravitation. This indicates that the Slow mode and fire hose instability are not affected by rotation. Numerical analysis was performed for oblique propagation to show the effect of rotation and self-gravitation. It is found that rotation has an effect of reducing the value of the phase speeds on the fast and Alfven wave modes, but self-gravitation affect only on the Slow modes, thereby reducing the phase speed compare to the ideal magneto hydrodynamic (MHD) case.     

Particle acceleration and nonthermal radiation in space plasmas

Acceleration processes for fast particles in astrophysical and space plasmas are reviewed with emphasis on stochastic acceleration by MHD turbulence and on acceleration by shock waves. Radiation processes in astrophysical and space plasmas are reviewed with emphasis on plasma emission from the solar corona and electron cyclotron maser emission from planets and stars.