Phase mixing of Alfvén waves propagating in non-reflective magnetic plasma configurations

The ability of phase mixing to provide efficient damping of Alfvén waves even in weakly dissipative plasmas made it a popular mechanism for explaining the solar coronal heating. Initially it was studied in the equilibrium configurations with the straight magnetic field lines and the Alfvén speed only varying in the direction perpendicular to the magnetic field. Later the analysis of the Alfvén wave phase mixing was extended in various directions. In particular it was studied in two-dimensional planar magnetic plasma equilibria. Analytical investigation was carried out under the assumption that the wavelength is much smaller than the characteristic scale of the background quantity variation. This assumption enabled using the Wentzel, Kramers, and Brillouin (WKB) method. When it is not satisfied the study was only carried out numerically. In general, even the wave propagation in a one-dimensional inhomogeneous equilibrium can be only studied numerically. However there is one important exception, so-called non-reflective equilibria. In these equilibria the wave equation with the variable phase speed reduces to the Klein-Gordon equation with constant coefficients. In this paper we apply the theory of non-reflective wave propagation to studying the Alfvén wave phase mixing in two-dimensional planar magnetic plasma equilibria. Using curvilinear coordinates we reduce the equation describing the Alfvén wave phase mixing to the equation that becomes a one-dimensional wave equation in the absence of dissipation. This equation is further reduced to the equation which is the one-dimensional Klein-Gordon equation in the absence of dissipation. Then we show that this equation has constant coefficients when a particular relation between the plasma density and magnetic field magnitude is satisfied. Using the derived Klein-Gordon-type equation we study the phase mixing in various non-reflective equilibria. We emphasise that our analysis is valid even when the wavelength is comparable with the characteristic scale of the background quantity variation. In particular, we study the Alfvén wave damping due to phase mixing in an equilibrium with constant plasma density and exponentially divergent magnetic field lines. We confirm the result previously obtained in the WKB approximation that there is enhanced Alfvén wave damping in this equilibrium with the damping length proportional to ln(Re), where Re is the Reynolds number. Our theoretical results are applied to heating of coronal plumes. We show that, in spite of enhanced damping, Alfvén waves with periods of the order of one minute can be efficiently damped in the lower corona, at the height about 200 Mm, only if the shear viscosity is increased by about 6 orders of magnitude in comparison with its value given by the classical plasma theory. We believe that such increase of the shear viscosity can be provided by the turbulence

Effect of siphon flow on resonant damping of kink oscillations in magnetic flux tubes

The effect of siphon flow on kink oscillations of magnetic flux tubes is studied in the thin tube and thin boundary layer (TTTB) approximation. The presence of a transitional layer results in oscillation damping due to resonance absorption. To calculate the damping rate we use the regular perturbation method with the ratio of transitional layer thickness to tube radius as a small parameter. We found a dependence of the ratio of decrement to the oscillation frequency, γ/ω1, on the ratio, χ, of flow velocity magnitude to the Alfvén speed in the tube core. The general theoretical results are applied to a particular case where the density radial dependence in the transitional layer is linear. We consider two models. In the first model, the radial dependence of the velocity amplitude is such that the resonance in the transitional layer occurs where the flow velocity is zero. In the second model, the flow velocity is non-zero in the whole transitional layer. In both cases, γ/ω1 is an increasing function of χ. In the first case, the presence of flow can lead to an increase in γ/ω1 by more than a factor of two. In the second model, we only carry out the calculation in the case where the plasma density inside the tube is much larger than the density of the surrounding plasma. In this model, the effect of flow is less pronounced than in the first model, and the presence of flow can increase γ/ω1 by a factor of 0.25 at most. We discuss the application of the obtained results to coronal and prominence seismology. We conclude that while for typical values of velocity in coronal loops the effect of flow is weak, it can be quite substantial in prominence seismology

Phase mixing of Alfven waves in axisymmetric non-reflective magnetic plasma configurations

We study damping of phase-mixed Alfvén waves propagating in non-reflective axisymmetric magnetic plasma configurations. We derive the general equation describing the attenuation of the Alfvén wave amplitude. Then we applied the general theory to a particular case with the exponentially divergent magnetic field lines. The condition that the configuration is non-reflective determines the variation of the plasma density along the magnetic field lines. The density profiles exponentially decreasing with the height are not among non-reflective density profiles. However, we managed to find non-reflective profiles that fairly well approximate exponentially decreasing density. We calculate the variation of the total wave energy flux with the height for various values of shear viscosity. We found that to have a substantial amount of wave energy dissipated at the lower corona, one needs to increase shear viscosity by seven orders of magnitude in comparison with the value given by the classical plasma theory. An important result that we obtained is that the efficiency of the wave damping strongly depends on the density variation with the height. The stronger the density decrease, the weaker the wave damping is. On the basis of this result, we suggested a physical explanation of the phenomenon of the enhanced wave damping in equilibrium configurations with exponentially diverging magnetic field lines

Resonant damping and instability of propagating kink waves in flowing and twisted magnetic flux tubes

We study the propagation and stability of kink waves in a twisted magnetic tube with the flow. The flow velocity is assumed to be parallel to the magnetic field, and the magnetic field lines are straight outside the tube. The density is constant inside and outside of the tube, and it monotonically decreases from its value inside the tube to that outside in the transitional or boundary layer. The flow speed and magnetic twist monotonically decrease in the transitional layer from their values inside the tube to zero outside. Using the thin tube and thin boundary layer (TTTB) approximation, we derived the dispersion equation determining the dependence of the wave frequency and decrement/increment on the wavenumber. When the kink wave frequency coincides with the local Alfvén frequency at a resonant surface inside the transitional layer, the kink wave is subjected to either resonant damping or resonant instability. We study the properties of kink waves in a particular unperturbed state where there is no flow and magnetic twist in the transitional layer. It is shown that in a tube with flow, the kink waves can propagate without damping for particular values of the flow speed. Kink waves propagating in the flow direction either damp or propagate without damping. Waves propagating in the opposite direction can either propagate without damping, or damp, or become unstable. The theoretical results are applied to the problem of excitation of kink waves in spicules and filaments in the solar atmosphere

Effect of transitional layer on frequency of kink oscillations

In this article we study the effect of a transitional layer in a magnetic tube on the real part of the frequencies of kink oscillations. In our analysis, we use the model of a straight magnetic tube with the density and cross-section radius varying along the loop, and the thin-tube thin-boundary (TTTB) approximation. First, we calculate the correction to the fundamental frequency and show that it is positive and of the order of the ratio of the transitional layer thickness to the loop radius ℓ. The increase in the fundamental frequency results in the decrease in the estimate of the magnetic-field magnitude. Then we study the effect of the transitional layer on the ratio between the fundamental frequency and the first overtone frequency that is used for estimating the atmospheric scale height. We show that the correction to the frequency ratio is of the order of ℓ2, and thus it can be neglected for moderate values of ℓ

Excitation of decayless kink oscillations by random motion

We study kink oscillations of a straight magnetic tube with a transitional region at its boundary. The tube is homogeneous in the axial direction. The plasma density monotonically decreases in the transitional region from its value inside the tube to that in the surrounding plasma. The plasma motion is described by the linear magnetohydrodynamic equations in the cold plasma approximation. We use the ideal equations inside the tube and in the surrounding plasma, but take viscosity into account in the transitional region. We also use the thin tube and thin transitional or boundary layer (TTTB) approximation. Kink oscillations are assumed to be driven by a driver at the tube footpoint. We derive the equation describing the displacement in the fundamental mode and overtones. We use this equation to study kink oscillations in both the case of harmonic and random driving. In the case of random driving, we assume that the driver is described by a stationary random function. The displacements in the fundamental mode and overtones are also described by stationary random functions. We derive the relation between the power spectra of the fundamental mode and all overtones and the power spectrum of the driver. We suggest a new method of obtaining information on the internal structure of coronal magnetic loops based on the shape of graphs of the power spectrum of the fundamental mode

Nonlinear generation of fluting perturbations by kink mode in a twisted magnetic tube

We study the excitation of fluting perturbations in a magnetic tube by an initially imposed kink mode. We use the ideal magnetohydrodynamic (MHD) equations in the cold-plasma approximation. We also use the thin-tube approximation and scale the dependent and independent variables accordingly. Then we assume that the dimensionless amplitude of the kink mode is small and use it as an expansion parameter in the regular perturbation method. We obtain the expression for the tube boundary perturbation in the second-order approximation. This perturbation is a superposition of sausage and fluting perturbations

Kink waves in twisted and expanding magnetic tubes

We study kink and fluting waves in expanding and twisted magnetic flux tubes. We use the thin-tube and zero-beta plasma approximations. The equilibrium magnetic field is force free with a constant proportionality coefficient between the electrical current and the magnetic field. We derive the equation governing the kink and fluting waves in a tube. Using this equation we study the propagation of kink waves in a particular case of a magnetic tube homogeneous in the axial direction. We show that while there is only one propagating kink wave with the phase speed equal to the kink speed in an untwisted tube, in a twisted tube there are two wave modes, accelerated and decelerated. The phase speed of the accelerated wave exceeds the kink speed, while the phase speed of the decelerated wave is less than the kink speed. We also show that the standing modes are defined by the same eigenvalue problem as that in the case of an untwisted tube. Hence, the frequencies of the standing-wave modes are not affected by the twist. This implies that the seismological results based on the observation of the standing-wave mode frequencies remain valid when the twist is taken into account. The only effect of twist is the variation of the direction of polarisation of the coronal magnetic-loop displacement along the loop. As a result, an apparent node can be detected near the loop apex if only one component of the loop displacement is observed. This can lead to an incorrect conclusion that the observed coronal loop kink oscillation was the first overtone, while in fact it was the fundamental mode

Stability of KdV solitons with respect to transverse perturbations: Absolute and convective instabilities

We study the stability of one-dimensional solitons propagating in an anisotropic medium. We derived the Kadomtsev-Petviashvili equation for nonlinear waves propagating in an anisotropic medium. By a proper variable substitution this equation reduces either to the KPI or to the KPII equation. In the former case solitons are unstable with respect to the normal modes of transverse perturbations, and in the latter they are stable. We only consider the case when the solitons are unstable. We formulated the linear stability problem. Using the Laplace–Fourier transform, we found the solution describing the evolution of an initial perturbation. Then, using Briggs' method we studied the absolute and convective instabilities. We found that a soliton is convectively unstable unless it propagates at an angle smaller then critical with respect to a critical direction defined by the condition that the group velocity is parallel to the phase velocity. The critical angle is proportional to the ratio of the dispersion length to the soliton width, which is a small parameter. The coefficient of proportionality is expressed in terms of the phase speed and its second derivative with respect to the angle between the propagation direction and the critical direction. As an example we consider the stability of solitons propagating in Hall plasmas

Ultra-high energy cosmic ray investigations by means of EAS muon density measurements

A new approach to investigations of ultra-high energy cosmic rays based on the ground-level measurements of the spectra of local density of EAS muons at various zenith angles is considered. Basic features of the local muon density phenomenology are illustrated using a simple semi-analytical model. It is shown that muon density spectra are sensitive to the spectrum slope, primary composition, and to the features of hadronic interaction. New experimental data on muon bundles at zenith angles from 30 degrees to horizon obtained with the coordinate detector DECOR are compared with CORSIKA-based simulations. It is found that measurements of muon density spectra in inclined EAS give possibility to study characteristics of primary cosmic ray flux in a very wide energy range from 10^15 to 10^19 eV.Comment: 7 pages, 7 figures. Presented at CRIS-2006, Catania, Italy, May 29 - June 2, 2006. Accepted for publication in Nucl. Phys. B (Proc. Suppl.