Effects of Field-Aligned Flows on Standing Kink and Sausage Modes Supported by Coronal Loops

Fundamental standing modes and their overtones play an important role in coronal seismology. We examine the effects of a significant field-aligned flow on standing modes that are supported by coronal loops, which are modeled here as cold magnetic slabs. Of particular interest are the period ratios of the fundamental to its (n?1)th overtone [P ₁/nP _n ] for kink and sausage modes, and the threshold half-width-to-length ratio for sausage modes. For standing kink modes, the flow significantly reduces P ₁/nP _n in general, the effect being particularly strong for higher n and weaker density contrast [ $\rho_{0}/\rho_{\rm e}$ ] between loops and their surroundings. That said, even when $\rho_{0}/\rho_{\rm e}$ approaches infinity, this effect is still substantial, reducing the minimal P ₁/nP _n by up to 13.7?% (24.5?%) for n=2 (n=4) relative to the static case, when the Alfvén Mach number [M _A] reaches 0.8, where M _A measures the loop flow speed in units of the internal Alfvén speed. Although it is not negligible for standing sausage modes, the flow effect in reducing P ₁/nP _n is not as strong. However, the threshold half-width-to-length ratio is considerably higher in the flowing case than in its static counterpart. For $\rho_{0}/\rho_{\rm e}$ in the range [9,1024] and M _A in the range [0,0.5], an exhaustive parameter study yields that this threshold is well fitted by $(d/L)_{\rm cutoff, fit} = \frac{1}{2}\sqrt{\frac{1}{\rho_{0}/\rho_{\rm e}-1}} \exp (3.7 M_{\mathrm{A}}^{2} )$ , which involves the two parameters in a simple way. This allows one to analytically constrain the combination $(\rho_{0}/\rho_{\rm e}, M_{\mathrm {A}})$ for a loop with a known width-to-length ratio when a standing sausage oscillation is identified. It also allows one to examine the idea of partial sausage modes in more detail, and the flow is found to significantly reduce the spatial extent where partial modes are allowed.     

Non-linear waves in type-1 comet tails

The non-linear stabilization of Kelvin-Helmholtz instability occurring when disturbance amplitudes become finite is studied. A comet tail is considered as a plasma cylinder with free boundary immersed in the solar wind plasma. Due to Kelvin-Helmholtz instability of kink modes the type-1 comet tail is shown to take stable shape. However the sausage-like modes with wavelength of cylinder circumference cannot be stabilized due to finite amplitudes. The Kelvin-Helmholtz instability of sausage modes leads to the comet tail (or individual ray) losing its regular structure and breaking into individual clouds with the characteristic scale of the order of the cylinder diameter.     

A Manifestation of Negative Energy Waves in the Solar Atmosphere

Magnetosonic modes of magnetic structures of the solar atmosphere in the presence of inhomogeneous steady flows are considered. It is shown that, when the speed of the steady flow exceeds the phase speed of one of the modes, the mode has negative energy, and can be subject to an over-stability due to the negative energy wave instabilities. It is shown that registered steady flows in the solar atmosphere, with speeds below the threshold of the Kelvin–Helmholtz instability, can provide the existence of the magnetosonic negative energy wave phenomena. In particular, in isolated photospheric magnetic flux tubes, there are kink surface modes with negative energy, produced by the external granulation downflows. Dissipative instability of these modes due to finite thermal conductivity and explosive instability due to nonlinear coupling of these modes with Alfvén waves are discussed. For coronal loops, it is found that only very high-speed flows (>300 km s^-1) can produce negative energy slow body modes. In solar wind flow structures, both slow and fast body modes have negative energy and are unstable.     

Linear and non-linear MHD wave propagation in steady-state magnetic cylinders

The propagation of linear and non-linear magnetohydrodynamic (MHD) waves in a straight homogeneous cylindrical magnetic flux tube embedded in a homogeneous magnetic environment is investigated. Both the tube and its environment are in steady state. Steady flows break the symmetry of forward (field-aligned) and backward (anti-parallel to magnetic field) propagating MHD wave modes because of the induced Doppler shifts. It is shown that strong enough flows change the sense of propagation of MHD waves. The flow also induces shifts in cut-off values and phase-speeds of the waves. Under photospheric conditions, if the flow is strong enough, the slow surface modes may disappear and the fast body modes may become present. The crossing of modes is also observed due to the presence of flows. The effect of steady-state background has to be considered particularly carefully when evaluating observation signatures of MHD waves for diagnostics in the solar atmosphere.     

A mechanism for the formation of plasmoids and kink waves in the heliospheric current sheet

Satellite observations of the heliospheric current sheet indicate that the plasma flow velocity is low at the center of the current sheet and high on the two sides of current sheet. In this paper, we investigate the growth rates and eigenmodes of the sausage, kind, and tearing instabilities in the heliospheric current sheet with the observed sheared flow. These instabilities may lead to the formation of the plasmoids and kink waves in the solar wind. The results show that both the sausage and kink modes can be excited in the heliospheric current sheet with a growth time 0.05–5 day. Therefore, these modes can grow during the transit of the solar wind from the Sun to the Earth. The sausage mode grows faster than the kink mode for < 1.5, while the streaming kink instability has a higher growth rate for > 1.5. Here is the ratio between the plasma and magnetic pressures away from the current layer. If a finite resistivity is considered, the streaming sausage mode evolves into the streaming tearing mode with the formation of magnetic islands. We suggest that some of the magnetic clouds and plasmoids observed in the solar wind may be associated with the streaming sausage instability. Furthermore, it is found that a large-scale kink wave may develop in the region with a radial distance greater than 0.5–1.5 AU.Also at Department of Earth and Space Science, University of Science and Technology of China, Hefei Anhui 230029, China.     

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.     

Numerical simulations of driven MHD waves in coronal loops

The solar corona, modeled by a low-, resistive plasma slab, sustains MHD wave propagations due to footpoint motions in the photosphere. Simple test cases are undertaken to verify the code. Uniform, smooth and steep density, magnetic profile and driver are considered. The numerical simulations presented here focus on the evolution and properties of the Alfvén, fast and slow waves in coronal loops. The plasma responds to the footpoint motion by kink or sausage waves depending on the amount of shear in the magnetic field. The larger twist in the magnetic field of the loop introduces more fast-wave trapping and destroys initially developed sausage-like wave modes. The transition from sausage to kink waves does not depend much on the steep or smooth profile. The slow waves develop more complex fine structures, thus accounting for several local extrema in the perturbed velocity profiles in the loop. Appearance of the remnants of the ideal singularities characteristic of ideal plasma is the prominent feature of this study. The Alfvén wave which produces remnants of the ideal x ^–1 singularity, reminiscent of Alfvén resonance at the loop edges, becomes less pronounced for larger twist. Larger shear in the magnetic field makes the development of pseudo-singularity less prominent in case of a steep profile than that in case of a smooth profile. The twist also causes heating at the edges, associated with the resonance and the phase mixing of the Alfvén and slow waves, to slowly shift to layers inside the slab corresponding to peaks in the magnetic field strength. In addition, increasing the twist leads to a higher heating rate of the loop. Remnants of the ideal log ¦x¦ singularity are observed for fast waves for larger twist. For slow waves they are absent when the plasma experiences large twist in a short time. The steep profiles do not favour the creation of pseudo-singularities as easily as in the smooth case.     

Resonant interactions of modes in coronal magnetic flux tubes

Nonlinear resonant interactions of different kinds of fast magnetosonic (FMS) waves trapped in the inhomogeneity of a low- plasma density, stretched along a magnetic field (as, for example, in coronal loops) are investigated. A set of equations describing the amplitudes of interactive modes is derived for an arbitrary density profile. The quantitative characteristics of such interactions are found. The decay instability of the wave with highest frequency is possible in the system. If amplitudes of interactive modes have close values, the long-period temporal and spatial oscillations are in the system.For a quantitative illustration, the parabolic approximation of the transverse density profile has been chosen. Dispersion relations of FMS waves trapped in a low- plasma slab with a parabolic transverse density profile are found. The transverse structure of the waves in this case can be expressed through Hermitian polynomials. The interaction of kink and sausage waves is investigated. The sausage wave, with a sufficiently large amplitude, may be unstable with respect to the decay into two kink waves, in particular. The spatial scale of a standing wave structure and the time spectrum of radiation are formed due to the nonlinear interactions of loop modes which contain information about the parameters of the plasma slab.     

Numerical simulations of fast MHD waves in a coronal plasma

A method is presented for the numerical study of the temporal evolution of nonlinear periodic waves in solar coronal loops which are approximated by smoothed slabs of enhanced gas density embedded within a uniform magnetic field. This method uses a fast Fourier transform technique to calculate spatial derivatives and a modified Euler algorithm for the time scheme for solving cold magnetohydrodynamic equations that govern nonlinear perturbations. The numerical results show that nonlinearity can play a significant role, leading to wave breaking of the kink wave and slab demolition for the sausage one. The kink periodic wave adjusts better to the smoothed slab than the sausage wave.     

OSCILLATIONS IN PROMINENCE FINE-STRUCTURES

Oscillatory spectra of solar quiescent prominences highlight the importance of incorporating the effect of prominence fine-structure in the theory of prominence oscillations. We determine the magnetohydrodynamic modes of oscillation of an elementary, zero- model of a prominence fibril, arguing that the fast body kink modes, namely, the string and the internal magnetic Love modes, produce the observed short periodicities in prominence fine-structures. Estimates for the periods of these modes are presented: the modes are subject to testing in future high-resolution observations.     

The Effect of Flows on Transverse Oscillations of Coronal Loops

In this paper we study kink oscillations of coronal loops in the presence of flows. Using the thin-tube approximation we derive the general governing equation for kink oscillations of a loop with the density varying along the loop in the presence of flows. This equation remains valid even when the density and flow are time dependent. The derived equation is then used to study the effect of flows on eigenfrequencies of kink oscillations of coronal loops. The implication of the obtained results on coronal seismology is discussed.     

Dynamical processes in flux tabes and their role in chromospheric heating

We model the dynamical interaction between magnetic flux tubes and granules in the solar photosphere which leads to the excitation of transverse (kink) and longitudinal (sausage) tube waves. The investigation is motivated by the interpretation of network oscillations in terms of flux tube waves. The calculations show that for magnetic field strengths typical of the network, the energy flux in transverse waves is higher than in longitudinal waves by an order of magnitude. But for weaker fields, such as those that might be found in internetwork regions, the energy fluxes in the two modes are comparable. Using observations of footpoint motions, the energy flux in transverse waves is calculated and the implications for chromospheric heating are pointed out.     

3D Numerical Simulations of <Emphasis Type="Italic">f</Emphasis>-Mode Propagation Through Magnetic Flux Tubes

Three-dimensional numerical simulations have been used to study the scattering of a surface-gravity wave packet by vertical magnetic-flux tubes, with radii from 200 km to 3 Mm, embedded in stratified polytropic atmosphere. The scattered wave has been found to consist primarily of m=0 (axisymmetric) and m=1 modes. The ratio of the amplitude of these two modes was found to be strongly dependent on the radius of the flux tube. The kink mode is the dominant mode excited in tubes with a small radius, while the sausage mode is dominant for large tubes. Simulations of this type provide a simple, efficient, and robust way to start to understand the seismic signature of flux tubes, which have recently begun to be observed.     

Nonlinear Resonant Excitation of Fast Sausage Waves in Current-Carrying Coronal Loops

We consider a model of a coronal loop that is a cylindrical magnetic tube with two surface electric currents. Its principal sausage mode has no cut-off in the long-wavelength limit. For typical coronal conditions, the period of the mode is between one and a few minutes. The sausage mode of flaring loops could cause long-period pulsations observed in microwave and hard X-ray ranges. There are other examples of coronal oscillations: long-period pulsations of active-region quiet loops in the soft X-ray emission are observed. We assume that these can also be caused by sausage waves. The question arises of how the sausage waves are generated in quiet loops. We assume that they can be generated by torsional oscillations. This process can be described in the framework of the nonlinear three-wave interaction formalism. The periods of interacting torsional waves are similar to the periods of torsional oscillations observed in the solar atmosphere. The timescale of the sausage-wave excitation is not much longer than the periods of interacting waves, so that the sausage wave is excited before torsional waves are damped.     

Magnetosonic waves in structured atmospheres with steady flows

The magnetosonic modes of magnetic plasma structures in the solar atmosphere are considered taking into account steady flows of plasma in the internal and external media and using a slab geometry. The investigation brings nearer the theory of magnetosonic waveguides, in such structures as coronal loops and photospheric flux tubes, to realistic conditions of the solar atmosphere. The general dispersion relation for the magnetosonic modes of a magnetic slab in magnetic surroundings is derived, allowing for field-aligned steady flows in either region. It is shown that flows change both qualitatively and quantitatively the characteristics of magnetosonic modes. The flow may lead to the appearance of a new type of trapped mode, namelybackward waves. These waves are the usual slab modes propagating in the direction opposite to the internal flow, but advected with the flow. The disappearance of some modes due to the flow is also demonstrated.The results are applied to coronal and photospheric magnetic structures. In coronal loops, the appearance of backward slow body waves or the disappearance of slow body waves, depending upon the direction of propagation, is possible if the flow speed exceeds the internal sound speed ( 300 km s^–1). In photospheric tubes, the disappearance of fast surface and slow body waves may be caused by an external downdraught of about 3 km s^–1.     

Magneto-Acoustic Waves in Compressible Magnetically Twisted Flux Tubes

The oscillatory modes of a magnetically twisted compressible flux tube embedded in a compressible magnetic environment are investigated in cylindrical geometry. Solutions to the governing equations to linear wave perturbations are derived in terms of Whittaker’s functions. A general dispersion equation is obtained in terms of Kummer’s functions for the approximation of weak and uniform internal twist, which is a good initial working model for flux tubes in solar applications. The sausage, kink and fluting modes are examined by means of the derived exact dispersion equation. The solutions of this general dispersion equation are found numerically under plasma conditions representative of the solar photosphere and corona. Solutions for the phase speed of the allowed eigenmodes are obtained for a range of wavenumbers and varying magnetic twist. Our results generalise previous classical and widely applied studies of MHD waves and oscillations in magnetic loops without a magnetic twist. Potential applications to solar magneto-seismology are discussed.     

The Effect of Twisted Magnetic Field on the Resonant Absorption of MHD Waves in Coronal Loops

The standing quasi-modes in a cylindrical incompressible flux tube with magnetic twist that undergoes a radial density structuring is considered in ideal magnetohydrodynamics (MHD). The radial structuring is assumed to be a linearly varying density profile. Using the relevant connection formulae, the dispersion relation for the MHD waves is derived and solved numerically to obtain both the frequencies and damping rates of the fundamental and first-overtone modes of both the kink (m=1) and fluting (m=2,3) waves. It was found that a magnetic twist will increase the frequencies, damping rates and the ratio of the oscillation frequency to the damping rate of these modes. The period ratio P ₁/P ₂ of the fundamental and its first-overtone surface waves for kink (m=1) and fluting (m=2,3) modes is lower than two (the value for an untwisted loop) in the presence of twisted magnetic field. For the kink modes, particularly, the magnetic twists B _φ /B _z =0.0065 and 0.0255 can achieve deviations from two of the same order of magnitude as in the observations. Furthermore, for the fundamental kink body waves, the frequency band width increases with increasing magnetic twist.     

Sausage MHD Waves in Incompressible Flux Tubes with Twisted Magnetic Fields

Twisted magnetic flux tubes are of considerable interest because of their natural occurrence from the Sun’s interior, throughout the solar atmosphere and interplanetary space up to a wide range of applicabilities to astrophysical plasmas. The aim of the present work is to obtain analytically a dispersion equation of linear wave propagation in twisted incompressible cylindrical magnetic waveguides and find appropriate solutions for surface, body and pseudobody sausage modes (i.e. m = 0) of a twisted magnetic flux tube embedded in an incompressible but also magnetically twisted plasma. Asymptotic solutions are derived in long- and short-wavelength approximations. General solutions of the dispersion equation for intermediate wavelengths are obtained numerically. We found, that in case of a constant, but non-zero azimuthal component of the equilibrium magnetic field outside the flux tube the index ν of Bessel functions in the dispersion relation is not integer any more in general. This gives rise to a rich mode-structure of degenerated magneto-acoustic waves in solar flux tubes. In a particular case of a uniform magnetic twist the total pressure is found to be constant across the boundary of the flux tube. Finally, the effect of magnetic twist on oscillation periods is estimated under solar atmospheric conditions. It was found that a magnetic twist will increase, in general, the periods of waves approximately by a few percent when compared to their untwisted counterparts.     

Model of polarization state of compressional surface MHD waves in the low-altitude cusp

Examination of thermal plasma data obtained by low-altitude satellite measurements indicates that the intersection of the cusp in the dayside magnetosphere with the topside ionosphere creates a distinct plasma geometry at low altitudes. This region consists of one or two plasma discontinuities with steep boundaries. As a result of the plasma structuring in the cusp which commonly takes place in the winter hemisphere, the propagation of compressional surface MHD waves is supported. This point is illustrated by an analysis of the polarization state of compressional surface MHD waves propagating along a plasma layer with thickness a and ambient magnetic field B₀ parallel to the interfaces. The results obtained are applicable to the case of a single interface, which is derived in the limit a → ∞. In the general case the polarization of the compressional surface MHD waves in the plane transverse to the magnetic field B₀ is elliptical. This feature of the polarization state of the compressional surface modes does not follow from the former analysis by Edwin and Roberts (1982, Solar Phys. 76, 239) for a magnetic slab, because the disturbance components parallel to the interfaces and perpendicular to the magnetic field B₀ have not been examined. Although the absence of these components does not prove to be essential for deriving the exact dispersion equation for arbitrary wave directions of the surface modes, they must be included when considering polarization states. The surface mode polarization in the plasma layer changes its sense three times: at interfaces X = 0 and X = a and in the middle plane X = a/2. For the symmetrical (sausage) mode the wave disturbance component b_n transverse (normal) to the interfaces becomes zero in the middle plane; for the asymmetrical (kink) mode, the component b_t parallel to the interfaces and transverse to the ambient magnetic field is zeroed in the same plane. For a moving observer such as a satellite the polarization patterns which might be recorded change, depending on the velocity of the observer and the angles at which the layered cusp is traversed. An essential feature in the polarization of the compressional surface MHD modes is the presence of jumps in the magnetic disturbance component b_t at the interfaces. These jumps disappear only for propagation along the ambient magnetic field. In this particular case the component b_t vanishes and then the surface modes are undistinguishable from the body modes.