Nonthermal Velocities in the Solar Transition Zone and Corona

Nonthermal velocities are presented for spectral lines covering the temperature range 10 4–10 6 K, measured from high-spectral-resolution data for several solar features observed at the limb by the high resolution telescope and spectrograph (HRTS), including a coronal hole, quiescent regions and several small-scale active regions. These results are compared with predictions based on acoustic waves and heating via Alfvén waves. It is likely that more than one mechanism is operating simultaneously, in particular, resonant Alfvén wave heating, which is very sensitive to background plasma motions.     

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 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.     

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.     

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.     

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.     

Resonant Absorption of Alfvén Waves in Steady Coronal Loops

The effect of equilibrium flow on linear Alfvén resonances in coronal loops is studied in the compressible viscous MHD model. By means of a finite element code, the full set of linearised driven MHD equations are solved for a one-dimensional equilibrium model in which the equilibrium quantities depend only on the radial coordinate. Computations of resonant absorption of Alfvén waves for two classes of coronal loop models show that the efficiency of the process of resonant absorption strongly depends on both the equilibrium parameters and the characteristics of the resonant wave. We find that a steady equilibrium shear flow can also significantly influence the resonant absorption of Alfvén waves in coronal magnetic flux tubes. The presence of an equilibrium flow may therefore be important for resonant Alfvén waves and coronal heating. A parametric analysis also shows that the resonant absorption can be strongly enhanced by the equilibrium flow, even up to total dissipation of the incoming wave.     

Enhanced emission of Alfvén waves from sunspots during proton flares

Thomas (1978) has shown that, if Alfvén waves exist in a sunspot umbra, they are normally reflected so strongly by the temperature minimum as to be essentially undetectable in the upper solar atmosphere. However, it is known that in many proton flares, chromospheric emission overlies the umbra of a sunspot, indicating that the transition region (TR) between chromosphere and corona in the umbral flux tube has moved down to lower altitudes. As a result of this lowering, umbral Alfvén waves have readier access to the corona: the coronal leakage depends exponentially on the altitude of the TR. We find that the Alfvén wave flux which leaks out of the umbra into the corona can exceed 10⁷ ergs cm^-2 s^-1. A flux of this magnitude is expected to dissipate rapidly in the corona, thereby contributing to a positive feedback loop which ensures prolonged (1 hr) leakage of the umbral Alfvén waves into the corona. We propose that these Alfvén waves may contribute significantly to prolonged energization of proton flares in which umbral coverage occurs.     

Alfvén waves and the sunspot phenomenon

Parker's explanation of the sunspot phenomenon in terms of the enhanced emission of Alfvén waves (solar vulcanology) is shown to be compatible with observation only if 90% of the waves propagate downwards. Further difficulties arise if the region of cooling by Alfvén wave generation is restricted to a depth of 2 Mm. However, it is shown that, if Alfvén wave generation is included in a recent model proposed by Meyer, Schmidt, Weiss and Wilson, these difficulties may be resolved. The problem of the sharp umbra and penumbra boundaries is discussed and it is shown that features of this combined model are relevant to the flare phenomenon.     

Coronal loop heating by Alfvén waves

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

Propagation speeds and acoustic damping of waves in magnetic flux tubes

Propagation speeds are derived for the wave modes of a thin magnetic tube in an otherwise homogeneous magnetized or unmagnetized fluid. These results generalize results obtained by previous authors. There are three types of wave, a (torsional) Alfvén wave and two waves which are specific for the thin tube. These are named the longitudinal and transversal tube waves, according to their polarization properties. They can be camped by radiating an MHD or acoustic wave into the surroundings of the tube. Conditions for occurrence of this acoustic damping, and the damping rates, are derived. The behavior of the waves in the solar convection zone and corona is discussed.     

Nonlinear dissipation of Alfvén waves

Alfvén waves are generated easily in many cosmic plasmas, but they possess no linear damping mechanism since they are not compressive. The most prominent nonlinear damping occurs when one Alfvén wave decays into another plus a slow magnetosonic wave, or two Alfvén waves combine into one fast magnetosonic wave; the resulting magnetosonic waves can then be dissipated. The nonlinear coupling rates are presented, with special emphasis on the astrophysically important case of sound speed Alfvén speed. Streaming cosmic rays generate Alfvén waves moving in the direction of streaming, but they reabsorb the backward moving waves then produced by wave decay. The possible steady states for this system of cosmic rays and Alfvén waves turn out to be highly restricted.Supported by NSF grant GP-15218.     

A Two-Dimensional Alfvén-Wave-Driven Solar Wind Model

This paper presents a two-dimensional, Alfvén-wave-driven solar wind model, in which the wave energy is assumed to cascade from the low-frequency Alfvén waves to high-frequency ion cyclotron waves and to be transferred to the solar wind protons by cyclotron resonance at the Kolmogorov rate. A typical structure in the meridional plane consisting of a coronal streamer near the Sun, a fast wind in high latitudes, and a slow wind across the heliospheric current sheet, is found. The fast wind obtained in the polar region is essentially similar to that derived by previous one-dimensional flow-tube models, and its density profile in the vicinity of the Sun roughly matches relevant observations. The proton conditions at 1 AU are also consistent with observations for both the fast and slow winds. The Alfvén waves appear in the fast- and slow-wind regions simultaneously and have comparable amplitudes, which agrees with Helios observations. The acceleration and heating of the solar wind by the Alfvén waves are found to occur mainly in the near-Sun region. It is demonstrated in terms of one-dimensional calculations that the distinct properties of the fast and slow winds are mainly attributed to different geometries of the flow tubes associated with the two sorts of winds. In addition, the 2-D and 1-D simulations give essentially the same results for both the fast and the slow winds.     

Turbulence Heating for Protons in High-Speed Solar Wind

In high-speed solar wind, propagating Alfvén waves can be transferred into fast magnetosonic waves. When both the magnetic field strength and Alfvén wave velocity approach zero, fast magnetosonic waves will be transferred into ion-acoustic waves. As the phase velocity of ion-acoustic waves is slightly greater than the thermal velocity of protons, the turbulence energy of ion-acoustic waves can largely be absorbed by protons and can cause the mean temperature of protons to be greater than that of electrons by stochastic turbulence heating of ion-acoustic waves for protons.     

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.     

Solar Wind and Chromospheric Network

A physical model of the transition region, including upflow of the plasma in magnetic field funnels that are open to the overlying corona, is presented. A numerical study of the effects of Alfvén waves on the heating and acceleration of the nascent solar wind originating in the chromospheric network is carried out within the framework of a two-fluid model for the plasma. It is shown that waves with reasonable amplitudes can, through their pressure gradient together with the thermal pressure gradient, cause a substantial initial acceleration of the wind (on scales of a few Mm) to locally supersonic flows in the rapidly expanding magnetic field trunks of the transition region network. The concurrent proton heating is due to the energy supplied by cyclotron damping of the high-frequency Alfvén waves, which are assumed to be created through small-scale magnetic activity. The wave energy flux of the model is given as a condition at the upper chromosphere boundary, located above the thin layer where the first ionization of hydrogen takes place.Among the new numerical results are the following: Alfvén waves with an assumed f ^-1 power spectrum in the frequency range from 1 to ⁴ Hz, and with an integrated mean amplitude ranging between 25 and 75 km s⁴, can produce very fast acceleration and also heating through wave dissipation. This can heat the lower corona to a temperature of 5× 10⁵ K at a height of h=12,000 km, starting from 5× 10⁴ K at h=3000 km. The resulting thermal and wave pressure gradients can accelerate the wind to speeds of up to 150 km s^-1 at h=12,000 km, starting from 20 km s^-1 at h=3000 km in a rapidly diverging flux tube. Thus the nascent solar wind becomes supersonic at heights well below the classical Parker-Type sonic point. This is a consequence of the fact that any large wave-energy flux, if it is to be conducted through the expanding funnel to the corona, implies the building-up of an associated wave-pressure gradient. Because of the diverging field geometry, this might lead to a strong initial acceleration of the flow. There is a multiplicity of solutions, depending mainly on the coronal pressure. Here we discuss two new (as compared with a static transition region model) possibilities, namely that either the flow remains supersonic or slows down abruptly by shock formation, which then yields substantial coronal heating up to the canonical 10⁶ K for the proton temperature.     

A new numerical method for simulating the solar wind Alfvén waves: Development of the Vlasov-MHD model

A novel scheme of plasma simulation particularly suited for computing the one-dimensional nonlinear evolution of parallel propagating solar wind Alfvén waves is presented. The scheme is based on the Vlasov and the MHD models, for solving the longitudinal and the transverse components, respectively. As long as the nonlinearity is not very large (so that the longitudinal and transverse components are well separated), our Vlasov-MHD model can correctly describe evolution of finite amplitude parallel Alfvén waves, which are typical in the solar wind, both in the linear and nonlinear stages. The present model can be applied to discussions of phenomena where the parallel Alfvén waves play major roles, for example, the solar coronal heating and solar wind acceleration by the Alfvén waves propagating from the photosphere.     

Solar coronal heating by high-frequency dispersive Alfvén waves

It is shown that high-frequency dispersive kinetic Alfvén waves can cause significant electron heating in the solar corona. The heating is produced by collisionless electron Landau dissipation of the parallel electron current associated with high-frequency dispersive kinetic Alfvén waves, which have a parallel electric field.     

Impulsively generated MHD waves and their detectability in solar coronal loops

Impulsively generated magnetohydrodynamic waves in solar coronal loops, with arbitrary plasma , are studied numerically by a flux-corrected transport algorithm. Numerical results show that the total reflection which occurs in the region of low Alfvén speed leads to trapped fast kink magnetosonic waves. These waves propagate along the slab and exhibit periodic, quasi-periodic, and decay phases. As a consequence of the difference in wave propagation speeds, the time signatures of the slow magnetosonic waves are delayed in time in comparison to the time signatures of the fast magnetosonic and Alfvén waves. An interaction between the waves can generate a longer lasting and complex quasi-periodic phase of the fast wave. We discuss also the observational detectability of such MHD waves in optical, radio, and soft X-ray wavelenghts.     

On the dissipation rates for Alfvén waves in the solar transition region

The present paper concerns Alfvén waves, in a resistive and viscous atmosphere, under a steep temperature gradient (Section 1). The dissipative Alfvén wave equation is deduced assuming uniform vertical background magnetic field, and allowing for arbitrary profiles of Alfvén speed, and viscous and resistive diffusivities as functions of altitude (Section 2). A three-parameter family of temperature profiles, allowing for independent choice of initial and asymptotic temperature, and of initial temperature gradient, is used to re-write the wave equation, with the temperature as the independent variable, instead of altitude (Section 3). It is shown that, for the conditions prevailing in the solar transition region between the chromosphere and corona, two approximations of the dissipative wave equations may be considered, the simplest leading to solution in terms of Gaussian hypergeometric functions (Section 4). The exact analytical solution allows calculation of the (i) velocity and (ii) magnetic field perturbations, (iii) kinetic, (iv) magnetic and (v) total energy density, (vi) energy flux, (vii) rate-of-strain and (viii) electric current, and (ix) viscous, (x) resistive and (xi) total rate of dissipation (Section 5). These are plotted versus temperature, across the transition region from the chromosphere to the corona, for the quiet and active Sun (Section 6). The feasibility of heating of the transition region by dissipation of Alfvén waves is discussed (Section 7), by comparing empirical heating rates, with theoretical values for a range of physical conditions, including initial velocity perturbations 5 to 15 km s ^–1, background magnetic field 12 to 120 G, wave periods 60 to 300 s, thickness of the transition region 100 to 300 km, resistive and anomalous diffusivities to 100 and viscous and turbulent diffusivities to 100 . The conclusion is that dissipation of Alfvén waves is not an effective heating mechanism for the transition region and corona, although it may be for the chromosphere (see Campos and Mendes, 1995, and references therein).