Short-Wave Oscillations of Coronal Magnetic Arcades

We investigate MHD waves in potential and force-free magnetic arcades describing bipolar active regions. The eikonal method allows us to study analytically the short waves, which are divided into Alfvén and magnetosonic waves. The eigen-modes of magnetic arcades are formed as a result of their reflection at the photosphere. The Alfvén mode oscillations of a certain frequency take place on magnetic surfaces. The fast-mode oscillations also take place on some surfaces but they are not magnetic surfaces. Both the Alfvén and fast-mode eigen-frequencies change continuously from one such surface to another. Each oscillation surface has a discrete set of eigen-frequencies.     

Three-Dimensional MHD Wave Propagation and Conversion to Alfvén Waves near the Solar Surface. I. Direct Numerical Solution

The efficacy of fast?–?slow MHD mode conversion in the surface layers of sunspots has been demonstrated over recent years using a number of modelling techniques, including ray theory, perturbation theory, differential eigensystem analysis, and direct numerical simulation. These show that significant energy may be transferred between the fast and slow modes in the neighbourhood of the equipartition layer where the Alfvén and sound speeds coincide. However, most of the models so far have been two dimensional. In three dimensions the Alfvén wave may couple to the magnetoacoustic waves with important implications for energy loss from helioseismic modes and for oscillations in the atmosphere above the spot. In this paper, we carry out a numerical “scattering experiment,” placing an acoustic driver 4 Mm below the solar surface and monitoring the acoustic and Alfvénic wave energy flux high in an isothermal atmosphere placed above it. These calculations indeed show that energy conversion to upward travelling Alfvén waves can be substantial, in many cases exceeding loss to slow (acoustic) waves. Typically, at penumbral magnetic field strengths, the strongest Alfvén fluxes are produced when the field is inclined 30°?–?40° from the vertical, with the vertical plane of wave propagation offset from the vertical plane containing field lines by some 60°?–?80°.     

3D MHD Coronal Oscillations about a Magnetic Null Point: Application of WKB Theory

This paper is a demonstration of how the WKB approximation can be used to help solve the linearised 3D MHD equations. Using Charpit’s method and a Runge?–?Kutta numerical scheme, we have demonstrated this technique for a potential 3D magnetic null point, B=[x,ε y,?(ε+1)z]. Under our cold-plasma assumption, we have considered two types of wave propagation: fast magnetoacoustic and Alfvén waves. We find that the fast magnetoacoustic wave experiences refraction towards the magnetic null point and that the effect of this refraction depends upon the Alfvén speed profile. The wave and thus the wave energy accumulate at the null point. We have found that current buildup is exponential and the exponent is dependent upon ε. Thus, for the fast wave there is preferential heating at the null point. For the Alfvén wave, we find that the wave propagates along the field lines. For an Alfvén wave generated along the fan plane, the wave accumulates along the spine. For an Alfvén wave generated across the spine, the value of ε determines where the wave accumulation will occur: fan plane (ε=1), along the x-axis (0<ε<1) or along the y-axis (ε>1). We have shown analytically that currents build up exponentially, leading to preferential heating in these areas. The work described here highlights the importance of understanding the magnetic topology of the coronal magnetic field for the location of wave heating.     

Quasi-Alfvén oscillations of magnetic surfaces

A system of equations has been derived for the modes of free oscillation of the magnetosphere when it is regarded as an adiabatic magnetic dipole trap filled with cold inhomogeneous plasma. The limiting case of infinite longitudinal conductivity corresponding to the assumption that the electric field is orthogonal to the geomagnetic field has been studied. The boundary at the ionosphere is supposed to be perfectly conducting.The eigenmode spectrum has been found to have discrete and continuous components. The eigenmodes of the discrete component correspond to quasi-magnetosonic modes and the eigenmodes of the continuous component to quasi-Alfvén modes.Assuming the magnetosphere to be axisymmetric, a general expression in the form of a Frobenius series has been derived for quasi-Alfvén oscillations of magnetic shells near resonant magnetic surfaces.     

Reply

A previous study of non-axisymmetric eigenmodes in a simple model plasmasphere is extended to a case in which the Alfvén speed varies across the magnetic field lines. An approximate ordinary differential equation is derived, and this contains a logarithmic singularity closely analagous to a result obtain by Tamao (1965). The singularity is found to lie close to the resonant lines of force for a hypothetical torsional oscillation. Some valid eigenfrequencies are stated for a region with appropriate further boundaries.     

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.     

Radiation of magnetoacoustic waves in the interstellar medium by physical variable stars

Radially pulsating stars are shown to radiate fast and slow magnetoacoustic waves into the interstellar gas. No Alfvén waves are excited, because the oscillations are radially symmetric. Calculations were performed for the following two limiting cases: hot, weakly magnetized interstellar plasma and cold plasma with a strong magnetic field. In these limiting cases, pulsating stars excite mostly fast magnetoacoustic waves, while the excitation of slow magnetoacoustic waves is weak. Magnetogasdynamic fields of density, velocity, and magnetic-field perturbations in the interstellar medium were found. Relations were derived to calculate the radiated power and its estimates are given for various conditions in the medium. It is shown that radially stratified wave structures with wavelengths from 1 AU to several tenths of a parsec must exist in the vicinity of pulsating stars.     

On Ponderomotive Effects Induced by Alfvén Waves in Inhomogeneous 2.5D MHD Plasmas

Where spatial gradients in the amplitude of an Alfvén wave are non-zero, a nonlinear magnetic-pressure gradient acts upon the medium (commonly referred to as the ponderomotive force). We investigate the nature of such a force in inhomogeneous 2.5D MHD plasmas by analysing source terms in the nonlinear wave equations for the general case of inhomogeneous B and ρ, and consider supporting nonlinear numerical simulations. Our equations indicate that there are two distinct classes of ponderomotive effect induced by Alfvén waves in general 2.5D MHD, each with both a longitudinal and transverse manifestation. i) Geometric effects: Gradients in the pulse geometry relative to the background magnetic field cause the wave to sustain cospatial disturbances, the longitudinal and transverse daughter disturbances – where we report on the transverse disturbance for the first time. ii) ?(c _A) effects: Where a pulse propagates through an inhomogeneous region (where the gradients in the Alfvén-speed profile c _A are non-zero), the nonlinear magnetic-pressure gradient acts to accelerate the plasma. Transverse gradients (phase mixing regions) excite independently propagating fast magnetoacoustic waves (generalising the result of Nakariakov, Roberts, and Murawski (Solar Phys. 175, 93, 1997)) and longitudinal gradients (longitudinally dispersive regions) perturb along the field (thus creating static disturbances in β=0, and slow waves in β≠0). We additionally demonstrate that mode conversion due the nonlinear Lorentz force is a one-way process, and does not act as a mechanism to nonlinearly generate Alfvén waves due to propagating magnetoacoustic waves. We conclude that these ponderomotive effects are induced by an Alfvén wave propagating in any MHD medium, and have the potential to have significant consequences on the dynamics of energy transport and aspects of dissipation provided the system is sufficiently nonlinear and inhomogeneous.     

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.     

A two-fluid modeling of kinetic Alfvén wave turbulence

We present an analytical model to explore the magnetic field turbulent spectrum by coupled high-frequency kinetic Alfvén wave (KAW) and slow mode of Alfvén wave (AW). The spectrum is computed as a realization of energy cascades from larger to smaller scales for a specific case of solar wind plasma at 1 AU. A two-fluid technique is implemented for the derivation of model equations leading two wave modes. These coupled, nonlinear equations are solved numerically. The nonlinearity in the system arises due to nonlinear ponderomotive force, which is believed to be responsible for the wave localization and magnetic islands formation. The numerical results show that the magnetic islands grow with time and attain a quasi-steady state after the modulation instability is saturated. The magnetic field spectrum and associated spectral indices are computed near the time of saturation of instability. The simulated spectrum in dispersion region follows a power-law with an index of ?2.5. The steeper spectrum could be attributed as energy transfer from larger to smaller scales and helps to study turbulence in solar wind. The magnetic field spectrum and spectral index show a good agreement with the observation of solar wind turbulent spectra.     

Stellar wind models with Alfvén waves

It is known that stellar winds from late type stars are of mixed thermal and magnetic origin. The stellar wind model presented in this work uses the hydrodynamic equations of mass and momentum conservation and closes the system of equations with a detailed energy equation. Both momentum and energy equations have terms due to the effects of Alfvén waves. A smooth transition between the two regimes for Alfvén wave propagation, the undamped and the damped modes, is achieved by considering the geometrical mean of both wave amplitudes. It will be shown that the initial push on the plasma is provided by the mechanical heating input, and that further out the Alfvén waves take over energetically.     

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.     

Attenuation of small-amplitude oscillations in a prominence–corona model with a transverse magnetic field

Observations show that small-amplitude prominence oscillations are usually damped after a few periods. This phenomenon has been theoretically investigated in terms of non-ideal magnetoacoustic waves, non-adiabatic effects being the best candidates to explain the damping in the case of slow modes. We study the attenuation of non-adiabatic magnetoacoustic waves in a slab prominence embedded in the coronal medium. We assume an equilibrium configuration with a transverse magnetic field to the slab axis and investigate wave damping by thermal conduction and radiative losses. The magnetohydrodynamic equations are considered in their linearised form and terms representing thermal conduction, radiation and heating are included in the energy equation. The differential equations that govern linear slow and fast modes are numerically solved to obtain the complex oscillatory frequency and the corresponding eigenfunctions. We find that coronal thermal conduction and radiative losses from the prominence plasma reveal as the most relevant damping mechanisms. Both mechanisms govern together the attenuation of hybrid modes, whereas prominence radiation is responsible for the damping of internal modes and coronal conduction essentially dominates the attenuation of external modes. In addition, the energy transfer between the prominence and the corona caused by thermal conduction has a noticeable effect on the wave stability, radiative losses from the prominence plasma being of paramount importance for the thermal stability of fast modes. We conclude that slow modes are efficiently damped, with damping times compatible with observations. On the contrary, fast modes are less attenuated by non-adiabatic effects and their damping times are several orders of magnitude larger than those observed. The presence of the corona causes a decrease of the damping times with respect to those of an isolated prominence slab, but its effect is still insufficient to obtain damping times of the order of the period in the case of fast modes.     

The effect of the lower ionosphere on hydromagnetic modes in the plasmasphere

Dimensionless resonant frequencies of hydromagnetic modes have been calculated for a simple model plasmasphere including a lower ionosphere. Results for the Alfvén mode are broadly consistent with those obtained by Hughes and Southwood [1976]. It is further concluded that the lower ionosphere, despite its strong damping effect for part of the day, does not provide much dissipative coupling between adjacent magnetic field shells in the Alfvén mode. The fast mode is found to be only slightly damped for horizontal wavelengths of global extent.     

Alfvén node-free vibrations of white dwarf in the model of solid star with toroidal magnetic field

In the context of white dwarf asteroseismology, we investigate the vibrational properties of a non-convective solid star with an axisymmetric purely toroidal intrinsic magnetic field of two different shapes. Focus is laid on the regime of node-free global Lorentz-force-driven vibrations about the symmetry axis at which material displacements have one and the same form as those for nodeless spheroidal and torsional vibrations restored by Hooke’s force of elastic shear stresses. Particular attention is given to the even-parity poloidal Alfvén modes whose frequency spectra are computed in analytic form, showing how the purely toroidal magnetic fields completely buried beneath the star surface can manifest itself in seismic vibrations of non-magnetic white dwarfs. The spectral formulae obtained are discussed in juxtaposition with those for Alfvén modes in the solid star model with the poloidal, homogeneous internal and dipolar external, magnetic field whose inferences are relevant to Alfvén vibrations in magnetic white dwarfs.     

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).     

Torsional oscillations of magnetized relativistic stars

Strong magnetic fields in relativistic stars can be a cause of crust fracturing, resulting in the excitation of global torsional oscillations. Such oscillations could become observable in gravitational waves or in high-energy radiation, thus becoming a tool for probing the equation of state of relativistic stars. As the eigenfrequency of torsional oscillation modes is affected by the presence of a strong magnetic field, we study torsional modes in magnetized relativistic stars. We derive the linearized perturbation equations that govern torsional oscillations coupled to the oscillations of a magnetic field, when variations in the metric are neglected (Cowling approximation). The oscillations are described by a single two-dimensional wave equation, which can be solved as a boundary-value problem to obtain eigenfrequencies. We find that, in the non-magnetized case, typical oscillation periods of the fundamental     torsional modes can be nearly a factor of 2 larger for relativistic stars than previously computed in the Newtonian limit. For magnetized stars, we show that the influence of the magnetic field is highly dependent on the assumed magnetic field configuration, and simple estimates obtained previously in the literature cannot be used for identifying normal modes observationally.     

The Alfvén wave parallel electric field in non-uniform space plasmas

We suggest a two-step mechanism for the generation of the parallel electric field at the Alfvén wave. At the first step, the coupling with the compressional mode due to the magnetic field non-uniformity and finite plasma pressure provides the parallel magnetic field of Alfvén wave. At the second step, the compressional mode acquires the parallel electric field due to coupling with the electrostatic mode as required by the quasi-neutrality condition in kinetics. The parallel electric field acquired by the Alfvén mode is considerably larger than that due to the single-step coupling between the Alfvén and electrostatic modes in kinetics.     

Cutoff frequency and the reflection of vertically propagating magnetoacoustic gravity waves in the solar atmosphere

Based on a plane-parallel isothermal model solar atmosphere stratified in the field of gravity, we investigate the main patterns of vertical propagation of magnetoacoustic gravity waves (MAGWs) in the approximation of a horizontal potential magnetic field. We have established that the cutoff frequency for MAGWs below which they cannot propagate does not depend on the magnetic field strength and is equal to that for acoustic gravity waves, the Lamb frequency. The cutoff frequency is shown to be unaffected by the linear interaction between counterpropagating MAGWs that results from a nonuniform height distribution of the Alfvén velocity and that causes the reflection of propagating waves at relatively large heights.     

Behaviour of Magnetoacoustic Waves in the Neighbourhood of a Two-Dimensional Null Point: Initially Cylindrically Symmetric Perturbations

The propagation of magnetoacoustic waves in the neighbourhood of a 2D null point is investigated for both β=0 and β ≠ 0 plasmas. Previous work has shown that the Alfvén speed, here v _A ∝r, plays a vital role in such systems and so a natural choice is to switch to polar coordinates. For β=0 plasma, we derive an analytical solution for the behaviour of the fast magnetoacoustic wave in terms of the Klein–Gordon equation. We also solve the system with a semi-analytical WKB approximation which shows that the β=0 wave focuses on the null and contracts around it but, due to exponential decay, never reaches the null in a finite time. For the β ≠ 0 plasma, we solve the system numerically and find the behaviour to be similar to that of the β=0 system at large radii, but completely different close to the null. We show that for an initially cylindrically-symmetric fast magnetoacoustic wave perturbation, there is a decrease in wave speed along the separatrices and so the perturbation starts to take on a quasi-diamond shape; with the corners located along the separatrices. This is due to the growth in pressure gradients that reach a maximum along the separatrices, which in turn reduces the acceleration of the fast wave along the separatrices leading to a deformation of the wave morphology.