Tunneling and interference of Alfvén waves

The propagation and interference of Alfvén waves in magnetic regions is studied. A multilayer approximation of the standard models of the solar atmosphere is used. In each layer, there is a linear law of temperature variation and a power law of Alfvén velocity variation. The analytical solutions of a wave equation are stitched at the layer boundaries. The low-frequency Alfvén waves (P > 1 s) are able to transfer the energy from sunspots into the corona by tunneling only. The chromosphere is not a resonance filter for the Alfvén waves. The interference and resonance of Alfvén waves are found to be important to wave propagation through the magnetic coronal arches. The transmission coefficient of Alfvén waves into the corona increases sharply on the resonance frequences. To take into account the wave absorption in the corona, a method of equivalent schemes is developed. The heating of a coronal arch by Alfvén waves is discussed.     

Alfvén-magnetosonic waves interaction in the solar corona

The nonlinear propagation of the Alfvén and magnetosonic waves in the solar corona is investigated in terms of model equations. Due to viscous effects taken into account the propagation of the fast wave itself is governed by Burgers type equations possessing both expansion and compression shock solutions. Numerical simulations show that both parallely and perpendicularly propagating fast waves can steepen into shocks if their amplitudes are in excess of some sizeable fraction of the Alfvén velocity. However, if the magnetic field changes linearly in the perpendicular direction, then formation of perpendicular shocks can be hindered. The Alfvén waves exhibit a tendency to drive both the slow and fast magnetosonic waves whose propagation is described by linearized Boussinesq type equations with ponderomotive terms due to the Alfvén wave. The limits of the slow and fast waves are investigated.     

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.     

On the role of hydromagnetic waves in the corona and the base of the solar wind

Hydromagnetic waves are of interest for heating the corona or coronal loops and for accelerating the solar wind. This paper enumerates some of the limitations that must be considered before hydromagnetic waves are taken seriously. In the lowest part of the corona, waves interact so that a significant fraction of the coronal wave flux should have periods as 10 s. If the problem of interest determines either a flux of wave energy or a dissipation rate, the distance that each wave mode can travel can be specified, and for at least one mode it must be consistent with the size and location of the region where the waves are to act. Heating of coronal loops observed by X-rays can be explained if the strength of the magnetic field along the loop lies within a rather narrow range and if the wave period is sufficiently short. In general, Alfvén waves travel furthest and reach high into the corona and into the solar wind. The radial variation of the magnetic field is the most important parameter determining where the waves are dissipated. Heating of coronal helmets by Alfvén waves is probable.The National Center for Atmospheric Research is sponsored by the National Science Foundation.     

Magnetoacoustic-gravity surface waves

The linearized theory for the parallel propagation of magnetoacoustic-gravity surface waves is developed for an interface of a horizontal magnetic field above a field-free medium. The media either side of the interface are taken to be isothermal. The dispersion relation is obtained for the case of a constant Alfvén speed. In the absence of gravity the interface may support one or two surface modes, determined by the relative temperatures and magnetism of the two media. The effect of gravity on the modes is examined and dispersion diagrams and eigenfunctions are given. In the usual - k _x diagnostic diagram, the domain of evanescence is shown to be divided into two distinct regions determining whether a given mode will have a decaying or growing vertical velocity component. In the absence of a magnetic field the transcendental dispersion relation may be rewritten as a polynomial. This polynomial possesses two acceptable solutions only one of which may exist in any given circumstances (depending on the ratio of the densities). If the gas density within the field exceeds that in the field-free medium, then the f-mode may propagate. The f-mode exists in a restricted band of horizontal wavenumber and only when the field-free medium is warmer than the magnetic atmosphere. An analytical form for the wave speed of the f-mode is obtained for small values of the Alfvén speed. It is shown that the f-mode is related to the fast magnetoacoustic surface wave, merging into that mode at short wavelengths.     

Alfvén waves in the solar atmosphere

The nonlinear propagation of Alfvén waves on open solar magnetic flux tubes is considered. The flux tubes are taken to be vertical and axisymmetric, and they are initially untwisted. The Alfvén waves are time-dependent axisymmetric twists. Their propagation into the chromosphere and corona is investigated by solving numerically a set of nonlinear time-dependent equations, which couple the Alfvén waves into motions parallel to the initial magnetic field (motion in the third coordinate direction is artificially suppressed). The principal conclusions are: (1) Alfvén waves can steepen into fast shocks in the chromosphere. These shocks can pass through the transition region into the corona, and heat the corona. (2) As the fast shocks pass through the transition region, they produce large-velocity pulses in the direction transverse to B _o. The pulses typically have amplitudes of 60 km s^–1 or so and durations of a few tens of seconds. Such features may have been observed, suggesting that the corona is in fact heated by fast shocks. (3) Alfvén waves exhibit a strong tendency to drive upward flows, with many of the properties of spicules. Spicules, and the observed corrugated nature of the transition region, may therefore be by-products of magnetic heating of the corona. (4) It is qualitatively suggested that Alfvén waves may heat the upper chromosphere indirectly by exerting time-dependent forces on the plasma, rather than by directly depositing heat into the plasma.     

Mode Conversion of Solar p Modes in non-Vertical Magnetic Fields – i. two-Dimensional Model

Sunspots absorb incident p modes. The responsible mechanism is uncertain. One possibility is mode conversion to slow magnetoacoustic–gravity waves. In vertical field mode conversion can adequately explain the observed f-mode absorption, but is too inefficient to explain the absorption of p modes. In this investigation we calculate the efficiency of fast-to-slow magnetoacoustic–gravity wave conversion in non-vertical field. We assume two-dimensional propagation where the Alfvén waves decouple. It is found that resultant p-mode absorption is significantly enhanced for moderate inclinations at higher frequencies, whereas for p modes at lower frequencies, and the f mode in general, there is no useful enhancement. However, the enhancement is insufficient to explain the observed p-mode absorption by sunspots. Paper II considers the efficiency of mode conversion in non-vertical field with three-dimensional propagation, where fast and slow magnetoacoustic–gravity waves and Alfvén waves are coupled.     

Alfvén waves in sunspots

The propagation of Alfvén waves in a simple model of a sunspot is considered. The vertical structure near the center of the umbra is modelled realistically, but the horizontal structure is not considered. The full wave equation is solved, without recourse to the WKB approximation. Only wave propagation in the vicinity of the central field line in an axially symmetric spot is examined, and it is assumed that this field line is open. By taking wave reflections into account, we find that the observations of non-thermal motions near the temperature minimum (Beckers, 1976) and in the corona (Beckers and Schneeberger, 1977) are both consistent with an upward-propagating Alfvénic energy flux density of a few times 10⁷ erg cm^–2 s^–1. This flux density is too small to cool the sunspot, but it is large enough to supply the energy requirements of the transition region and corona above a sunspot. This conclusion depends on the assumptions that the observed motions are indeed Alfvénic with periods near 180 s.The National Center for Atmospheric Research is sponsored by the National Science Foundation.     

Mode Conversion of Solar <Emphasis Type="Italic">p</Emphasis>-Modes in Non-Vertical Magnetic Fields

Sunspots absorb and scatter incident f- and p-modes. Until recently, the responsible absorption mechanism was uncertain. The most promising explanation appears to be mode conversion to slow magnetoacoustic-gravity waves, which carry energy down the magnetic field lines into the interior. In vertical magnetic field, mode conversion can adequately explain the observed f-mode absorption, but is too inefficient to account for the absorption of p-modes. In the first paper of the present series we calculated the efficiency of fast-to-slow magnetoacoustic-gravity wave conversion in uniform non-vertical magnetic fields. We assumed two-dimensional propagation, where the Alfvén waves decouple. In comparison to vertical field, it was found that mode conversion is significantly enhanced in moderately inclined fields, especially at higher frequencies. Using those results, Cally, Crouch, and Braun showed that the resultant p-mode absorption produced by simple sunspot models with non-vertical magnetic fields is ample to explain the observations. In this paper, we further examine mode conversion in non-vertical magnetic fields. In particular, we consider three-dimensional propagation, where the fast and slow magnetoacoustic-gravity waves and the Alfvén waves are coupled. Broadly speaking, the p-mode damping rates are not substantially different to the two-dimensional case. However, we do find that the Alfvén waves can remove similar quantities of energy to the slow MAG 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.     

Perturbations of a twisted solar coronal loop: The relation between surface waves and instabilities

The spectrum of propagating waves and instabilities on a current-carrying, zero gas pressure, twisted magnetic flux loop is analysed for several models of the magnetic field structure. A surface wave mode of the fast Alfvén wave is found to exist, with damping of the wave when Alfvén resonance absorption occurs. If the loop is surrounded by a uniform, purely axial magnetic field, then the surface wave is always stable. If the loop is surrounded by a nonuniform field which is continuous with the loop's field, then the surface wave may connect to the unstable external kink mode.     

Gravitational instability of partially-ionized plasma in an oblique magnetic field

The effects of Hall currents, finite conductivity, and collision with neutrals have been studied on the gravitational instability of a partially-ionized plasma. It is assumed that plasma is permeated by an oblique magnetic field. The dispersion relation has been obtained and numerical calculations have been performed to obtain the dependence of the growth rate of the gravitationally unstable mode on the various physical parameters involved. It is found that Jeans's criterion remains unchanged in the presence of Hall currents, finite conductivity, and collisions. The Hall currents, finite conductivity, and collisions have destabilizing influence on the unstable mode of wave propagation of a gravitational instability of partially-ionized plasma.     

The nature of the sunspot phenomenon

The properties of Alfvén waves in a vertical column of field are pointed out as a guide in treating the complicated problem of overstability. There are internal Alfvén waves of arbitrary form propagating along the magnetic field, without disturbing the fluid outside the column. There are also surface waves which involve the fluid both inside and outside the column of field. The surface waves propagate at a speed less than the Alfvén speed.Convective forces couple the internal and external fluid motions. If the forces are not too strong, the identity of the modes, as internal waves or surface waves, is maintained. The surface waves are unstable and, we suggest, may contribute to some of the activity of a sunspot. We suggest that the internal Alfvén wave modes are of more central interest for producing the basic sunspot phenomena. They represent the degenerate case, and their form is worked out in some detail. The overstable Alfvén wave modes peak sharply near the outer edge of the field, and do not strongly disturb the fluid outside. We suggest that this effect contributes to the sharp edge of the sunspot umbra.Recent observations by Giovanelli show intense wave activity originating inside the edge of the umbra. We tentatively identify the activity with the peak in the overstable modes within the umbra.This work was supported in part by the National Aeronautics and Space Administration under Grant NGL 14-001-001.     

Nonlinear wave propagation along a magnetic flux tube

The weakly nonlinear wave propagation of a slow sausage surface wave traveling along a magnetized slab with a thin nonuniform boundary layer is considered. The ideal incompressible MHD equations are used and the nonlinearities are assumed to be due to second harmonic generation. A nonlinear dispersion relation and the related nonlinear Schrödinger equation is derived. The existence of a continuous thin interface leads to sharply peaked field amplitudes due to resonant interaction with local Alfvén waves. It is shown that the nonlinear effects from processes within the thin layer are much more important than those from the main slab. Furthermore, the nonlinear interaction with local Alfvén waves yields a nonlinear damping rate of the wave that is much larger than the linear damping rate when the transition layer is sufficiently thin.     

Solar atmospheric heating

The solar atmosphere is heated by a flux of mechanical waves propagating in one or more of the modes: acoustic, Alfvén and gravitational.The acoustic theory is compared with observational data and found inadequate. First, the theory is based quantitatively on the Böhm-Vitense convection zone model, and large-scale convective motions (supergranulation) and magnetic fields (unipolar regions) show that convection has another form. On the other hand, when granular motions are invoked the energy flux is too small. Second, atmospheric heating is localized in faculae, and enhanced acoustic flux beneath these regions is no longer explicable. Finally, the short periods of 10–30 s invoked recently appear inexplicable. Objections to the gravitational wave heating process are given briefly.Previous objections to Alfvén waves as an energy source followed from the belief that fields were generally uniform and of strength 50 G, now known to be incorrect. Models of Alfvén wave generation are based on (i) granule eddy motions, (ii) overstable oscillations in subsurface flux tubes and sunspot flux ropes, and (iii) supergranule motions, both horizontal and vertical.The first provides waves which propagate along thin flux tubes oscillating as taut wires in a compressible fluid; they may explain mottles, fibrils and other small emission features. The second may explain the enormous dissipation in spot groups, including flares. The third has been invoked earlier to explain spicules, and may have effects in the solar wind.     

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.     

Solar p-modes oscillations and heating of the corona

The properties of the resonator are considered for fast magnetoacoustic waves. It is shown that tunnel penetration of waves from the resonator leads either to heating of the medium in the Alfvén resonance vicinity (if the inclination angle of the magnetic field is smaller than the critical angle), or to excitation of Alfvén waves at the Alfvén resonance (if the inclination angle is larger than the critical angle). This suggests that non-radiative heating of the corona can be due to solar p-mode oscillations.     

Magnetoacoustic surface waves at a single interface

The occurrence of magnetoacoustic surface waves at a single magnetic interface one side of which is field-free is explored for the case of non-parallel propagation. Phase-speeds and penetration depths of the waves are investigated for various Alfvén speeds, sound speeds and angles of propagation to the applied field. Both slow and fast magnetoacoustic surface waves can exist depending on the values of sound speeds and propagation angle. The fast waves penetrate more than the slow waves.The parallel propagation of fast and slow magnetoacoustic surface waves on a magnetic-magnetic interface is investigated. The slow surface wave is unable to propagate below a critical sound speed. In a low -plasma, only the fast mode exists (0 0).     

Effects of plasma rotation on alfvén solitons in pulsar magnetospheres

Nonlinear Alfvén wave in a hot rotating and strongly magnetized electron-positron plasma is considered. Using relativistic two fluid equations, the dispersion relation for Alfvén wave in the rotating plasma is obtained. Large amplitude Alfvén solitons are found to exist in the rotating pulsar plasma. Rotational effects on solitons are discussed.     

Filamentation instability of Alfvén waves

The filamentation instability of finite amplitude left-hand circularly polarized Alfvén waves has been investigated taking into account the second-order density and magnetic field perturbations that are created by the Alfvén wave pressure. The minimum scale length and time over which the filamentation occurs, are found. Our results are applied to Alfvén waves that should scatter cosmic rays in the interstellar medium (ISM).