The effects of density stratification on standing fast body oscillations in coronal loops and dissipation

The standing magnetohydrodynamic (MHD) quasi-linear modes in a zero-β cylindrical magnetic flux tube that undergoes a longitudinal density stratification and radial density structuring are considered. The radial structuring is assumed to have a step-like density profile. The dispersion relation for the fast MHD body waves is derived and solved numerically to obtain the frequencies of the fundamental, first-overtone and second-overtone   k = 1, 2, 3  modes of both kink  ( m = 1)  and fluting  ( m = 2)  waves, where k and m are the longitudinal and azimuthal mode numbers, respectively. Damping rates due to both viscous and resistive dissipations in the presence of the density stratification are derived and solved numerically for the first three modes of both kink and fluting waves.     

Energy, angular momentum and wave action associated with density waves in a rotating magnetized gas disc

Both fast and slow magnetohydrodynamic (MHD) density waves propagating in a thin rotating magnetized gas disc are investigated. In the tight-winding or WKBJ regime, the radial variation of MHD density-wave amplitude during wave propagation is governed by the conservation of wave action surface density which travels at a relevant radial group speed C _g. The wave energy surface density and the wave angular momentum surface density are related to by = and = m respectively, where is the angular frequency in an inertial frame of reference and the integer m , proportional to the azimuthal wavenumber, corresponds to the number of spiral arms. Consequently, both wave energy and angular momentum are conserved for spiral MHD density waves. For both fast and slow MHD density waves, net wave energy and angular momentum are carried outward or inward for trailing or leading spirals, respectively. The wave angular momentum flux contains separate contributions from gravity torque, advective transport and magnetic torque. While the gravity torque plays an important role, the latter two can be of comparable magnitudes to the former. Similar to the role of gravity torque, the part of MHD wave angular momentum flux by magnetic torque (in the case of either fast or slow MHD density waves) propagates outward or inward for trailing or leading spirals, respectively. From the perspective of global energetics in a magnetized gas sheet in rotation, trailing spiral structures of MHD density waves are preferred over leading ones. With proper qualifications, the generation and maintenance as well as transport properties of MHD density waves in magnetized spiral galaxies are discussed.     

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.     

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.     

MHD waves in the solar north polar coronal hole

The effects, hitherto not treated, of the temperature and the number density gradients, both in the parallel and the perpendicular direction to the magnetic field, of O VI ions, on the MHD wave propagation characteristics in the solar North Polar Coronal Hole are investigated. We investigate the magnetosonic wave propagation in a resistive MHD regime where only the thermal conduction is taken into account. Heat conduction across the magnetic field is treated in a non‐classical approach wherein the heat is assumed to be conducted by the plasma waves emitted by ions and absorbed at a distance from the source by other ions. Anisotropic temperature and the number density distributions of O VI ions revealed the chaotic nature of MHD standing wave, especially near the plume/interplume lane borders. Attenuation length scales of the fast mode is shown not to be smoothly varying function of the radial distance from the Sun (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Structures in a class of magnetized scale-free discs

We construct analytically stationary global configurations for both aligned and logarithmic spiral coplanar magnetohydrodynamics (MHD) perturbations in an axisymmetric background MHD disc with a power-law surface mass density  Σ₀∝ r ^−α  , a coplanar azimuthal magnetic field   B ₀∝ r ^−γ  , a consistent self-gravity and a power-law rotation curve   v ₀∝ r ^−β  , where v ₀ is the linear azimuthal gas rotation speed. The barotropic equation of state  Π∝Σ ⁿ   is adopted for both MHD background equilibrium and coplanar MHD perturbations where Π is the vertically integrated pressure and n is the barotropic index. For a scale-free background MHD equilibrium, a relation exists among  α, β, γ  and n such that only one parameter (e.g. β) is independent. For a linear axisymmetric stability analysis, we provide global criteria in various parameter regimes. For non-axisymmetric aligned and logarithmic spiral cases, two branches of perturbation modes (i.e. fast and slow MHD density waves) can be derived once β is specified. To complement the magnetized singular isothermal disc analysis of Lou, we extend the analysis to a wider range of  −1/4 < β < 1/2  . As an illustrative example, we discuss specifically the  β= 1/4  case when the background magnetic field is force-free. Angular momentum conservation for coplanar MHD perturbations and other relevant aspects of our approach are discussed.     

Damped Oscillations of Coronal Loops

A mechanism of damped oscillations of a coronal loop is investigated. The loop is treated as a thin toroidal flux rope with two stationary photospheric footpoints, carrying both toroidal and poloidal currents. The forces and the flux-rope dynamics are described within the framework of ideal magnetohydrodynamics (MHD). The main features of the theory are the following: i) Oscillatory motions are determined by the Lorentz force that acts on curved current-carrying plasma structures and ii) damping is caused by drag that provides the momentum coupling between the flux rope and the ambient coronal plasma. The oscillation is restricted to the vertical plane of the flux rope. The initial equilibrium flux rope is set into oscillation by a pulse of upflow of the ambient plasma. The theory is applied to two events of oscillating loops observed by the Transition Region and Coronal Explorer (TRACE). It is shown that the Lorentz force and drag with a reasonable value of the coupling coefficient (c _d ) and without anomalous dissipation are able to accurately account for the observed damped oscillations. The analysis shows that the variations in the observed intensity can be explained by the minor radial expansion and contraction. For the two events, the values of the drag coefficient consistent with the observed damping times are in the range c _d ≈2 – 5, with specific values being dependent on parameters such as the loop density, ambient magnetic field, and the loop geometry. This range is consistent with a previous MHD simulation study and with values used to reproduce the observed trajectories of coronal mass ejections (CMEs).     

Magneto‐seismology of the solar atmosphere

Magnetohydrodynamic (MHD) waves in solar coronal loops, which were previously only predicted by theory have actually been detected with space‐borne instruments. These observations have given an important and novel tool to measure fundamental parameters in the magnetically embedded solar corona. This paper will illustrate how information about the magnetic and density structure along coronal loops can be determined by measuring the frequency or amplitude profiles of standing fast kink mode oscillations. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Compressible magnetohydrodynamic turbulence: mode coupling, scaling relations, anisotropy, viscosity-damped regime and astrophysical implications

We present numerical simulations and explore scalings and anisotropy of compressible magnetohydrodynamic (MHD) turbulence. Our study covers both gas-pressure-dominated (high β) and magnetic-pressure-dominated (low β) plasmas at different Mach numbers. In addition, we present results for super-Alfvénic turbulence and discuss in what way it is similar to sub-Alfvénic turbulence. We describe a technique of separating different magnetohydrodynamic modes (slow, fast and Alfvén) and apply it to our simulations. We show that, for both high- and low-β cases, Alfvén and slow modes reveal a Kolmogorov   k ^−5/3  spectrum and scale-dependent Goldreich–Sridhar anisotropy, while fast modes exhibit a   k ^−3/2  spectrum and isotropy. We discuss the statistics of density fluctuations arising from MHD turbulence in different regimes. Our findings entail numerous astrophysical implications ranging from cosmic ray propagation to gamma ray bursts and star formation. In particular, we show that the rapid decay of turbulence reported by earlier researchers is not related to compressibility and mode coupling in MHD turbulence. In addition, we show that magnetic field enhancements and density enhancements are marginally correlated. Addressing the density structure of partially ionized interstellar gas on astronomical-unit scales, we show that the viscosity-damped regime of MHD turbulence that we reported earlier for incompressible flows persists for compressible turbulence and therefore may provide an explanation for these mysterious structures.     

Numerical simulations for MHD coronal seismology

Magnetohydrodynamic（MHD） processes are important for the transfer of energy over large scales in plasmas and so are essential to understanding most forms of dynamical activity in the solar atmosphere. The introduction of transverse structuring into models for the corona modifies the behavior of MHD waves through processes such as dispersion and mode coupling. Exploiting our understanding of MHD waves with the diagnostic tool of coronal seismology relies upon the development of sufficiently detailed models to account for all the features in observations. The development of realistic models appropriate for highly structured and dynamical plasmas is often beyond the domain of simple mathematical analysis and so numerical methods are employed. This paper reviews recent numerical results for seismology of the solar corona using MHD.     

A dynamo model for axisymmetric and non-axisymmetric solar magnetic fields

More and more observations are showing a relatively weak, but persistent, non-axisymmetric magnetic field co-existing with the dominant axisymmetric field on the Sun. Its existence indicates that the non-axisymmetric magnetic field plays an important role in the origin of solar activity. A linear non-axisymmetric  α²– Ω  dynamo model is derived to explore the characteristics of the axisymmetric  ( m = 0)  and the first non-axisymmetric  ( m = 1)  modes and to provide a theoretical basis with which to explain the 'active longitude', 'flip-flop' and other non-axisymmetric phenomena. The model consists of an updated solar internal differential rotation, a turbulent diffusivity varying with depth, and an α-effect working at the tachocline in a rotating spherical system. The difference between the  α²–Ω  and the  α–Ω  models and the conditions that favour the non-axisymmetric modes under solar-like parameters are also presented.     

Wave excitation in three-dimensional discs by external potential

We study the excitation of density and bending waves and the associated angular momentum transfer in gaseous discs with finite thickness by a rotating external potential. The disc is assumed to be isothermal in the vertical direction and has no self-gravity. The disc perturbations are decomposed into different modes, each characterized by the azimuthal index m and the vertical index n , which specifies the nodal number of the density perturbation along the disc normal direction. The   n = 0  modes correspond to the two-dimensional density waves previously studied by Goldreich & Tremaine and others. In a three-dimensional disc, waves can be excited at both Lindblad resonances (LRs; for modes with   n = 0, 1, 2, …  ) and vertical resonances (VRs; for the   n ≥ 1  modes only). The torque on the disc is positive for waves excited at outer Lindblad/vertical resonances and negative at inner Lindblad/vertical resonances. While the   n = 0  modes are evanescent around corotation, the   n ≥ 1  modes can propagate into the corotation region where they are damped and deposit their angular momenta. We have derived analytical expressions for the amplitudes of different wave modes excited at LRs and/or VRs and the resulting torques on the disc. It is found that for   n ≥ 1  , angular momentum transfer through VRs is much more efficient than LRs. This implies that in some situations (e.g. a circumstellar disc perturbed by a planet in an inclined orbit), VRs may be an important channel of angular momentum transfer between the disc and the external potential. We have also derived new formulae for the angular momentum deposition at corotation and studied wave excitations at disc boundaries.     

Analytical theory of tachocline transport at mid latitudes

We provide a theory of magnetic diffusion, momentum transport, and mixing in the solar tachocline by considering magnetohydrodynamics (MHD) turbulence on a β plane subject to a large scale shear (provided by the latitudinal differential rotation). In the strong magnetic field regime, we find that the turbulent viscosity and diffusivity are reduced by magnetic fields only, similarly to the two-dimensional MHD case (without Rossby waves). In the weak magnetic field regime, we find a crossover scale (L_R) from a Alfvén dominated regime (on small scales) to a Rossby dominated regime (on large scales). For parameter values typical of the tachocline, L_R is larger than the solar radius so that Rossby waves are unlikely to play an important role in the transport of magnetic field and angular momentum. This is mainly due to the enhancement of magnetic back-reaction by shearing which efficiently generates small scales, thus strong currents. (© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Observation of multiple sausage oscillations in cool post-flare loop

Using simultaneous high spatial (1.3 arcsec) and temporal (5 and 10 s) resolution Hα observations from the 15 cm Solar Tower Telescope at Aryabhatta Research Institute of Observational Sciences (ARIES), we study the oscillations in the relative intensity to explore the possibility of sausage oscillations in the chromospheric cool post-flare loop. We use the standard wavelet tool, and find the oscillation period of ≈587 s near the loop apex, and ≈349 s near the footpoint. We suggest that the oscillations represent the fundamental and the first harmonics of the fast-sausage waves in the cool post-flare loop. Based on the period ratio   P ₁/ P ₂∼1.68  , we estimate the density scaleheight in the loop as ∼17 Mm. This value is much higher than the equilibrium scaleheight corresponding to Hα temperature, which probably indicates that the cool post-flare loop is not in hydrostatic equilibrium. Seismologically estimated Alfvén speed outside the loop is  ∼300–330  km s⁻¹  . The observation of multiple oscillations may play a crucial role in understanding the dynamics of lower solar atmosphere, complementing such oscillations already reported in the upper solar atmosphere (e.g. hot flaring loops).     

Rapid damping of the oscillations of coronal loops with an azimuthal magnetic field

We consider the MHD oscillations of an inhomogeneous coronal loop that consists of a dense cord surrounded by a shell. The magnetic field is longitudinal in the cord and has only an azimuthal component in the shell. The parameters of the loop are chosen to be such that there are no resonances; i.e., the resonance points are cut off. This choice is dictated by the formulated problem of considering the influence of the radiation of MHD waves into the surrounding space on the loop oscillations, thereby ruling out the possibility of resonant energy absorption. The wave radiation efficiency is high and allows low oscillation Q-factors, which are equal in order of magnitude to their observed values, to be obtained.     

A New Model for Propagating Parts of EIT Waves: A Current Shell in a CME

EIT waves are observed in EUV as bright fronts. Some of these bright fronts propagate across the solar disk. EIT waves are all associated with a flare and a CME and are commonly interpreted as fast-mode magnetosonic waves. Propagating EIT waves could also be the direct signature of the gradual opening of magnetic field lines during a CME. We quantitatively addressed this alternative interpretation. Using two independent 3D MHD codes, we performed nondimensional numerical simulations of a slowly rotating magnetic bipole, which progressively result in the formation of a twisted magnetic flux tube and its fast expansion, as during a CME. We analyse the origins, the development, and the observability in EUV of the narrow electric currents sheets that appear in the simulations. Both codes give similar results, which we confront with two well-known SOHO/EIT observations of propagating EIT waves (7 April and 12 May 1997), by scaling the vertical magnetic field components of the simulated bipole to the line of sight magnetic field observed by SOHO/MDI and the sign of helicity to the orientation of the soft X-ray sigmoids observed by Yohkoh/SXT. A large-scale and narrow current shell appears around the twisted flux tube in the dynamic phase of its expansion. This current shell is formed by the return currents of the system, which separate the twisted flux tube from the surrounding fields. It intensifies as the flux tube accelerates and it is co-spatial with weak plasma compression. The current density integrated over the altitude has the shape of an ellipse, which expands and rotates when viewed from above, reproducing the generic properties of propagating EIT waves. The timing, orientation, and location of bright and faint patches observed in the two EIT waves are remarkably well reproduced. We conjecture that propagating EIT waves are the observational signature of Joule heating in electric current shells, which separate expanding flux tubes from their surrounding fields during CMEs or plasma compression inside this current shell. We also conjecture that the bright edges of halo CMEs show the plasma compression in these current shells.     

The heating of the solar corona

The heating of the solar corona has been a fundamental astrophysical issue for over sixty years. Over the last decade in particular, space-based solar observatories (Yohkoh, SOHO and TRACE) have revealed the complex and often subtle magnetic-field and plasma interactions throughout the solar atmosphere in unprecedented detail. It is now established that any energy release mechanism is magnetic in origin - the challenge posed is to determine what specific heat input is dominating in a given coronal feature throughout the solar cycle. This review outlines a range of possible magnetohydrodynamic (MHD) coronal heating theories, including MHD wave dissipation and MHD reconnection as well as the accumulating observational evidence for quasi-periodic oscillations and small-scale energy bursts occurring in the corona. Also, we describe current attempts to interpret plasma temperature, density and velocity diagnostics in the light of specific localised energy release. The progress in these investigations expected from future solar missions (Solar-B, STEREO, SDO and Solar Orbiter) is also assessed.Received: 6 February 2003, Published online: 14 November 2003 Correspondence to: R. W. Walsh     

A New Look at Mode Conversion in a Stratified Isothermal Atmosphere

Recent numerical investigations of wave propagation near coronal magnetic null points (McLaughlin and Hood: Astron. Astrophys. 459, 641, 2006) have indicated how a fast MHD wave partially converts into a slow MHD wave as the disturbance passes from a low-β plasma to a high-β plasma. This is a complex process and a clear understanding of the conversion mechanism requires the detailed investigation of a simpler model. An investigation of mode conversion in a stratified, isothermal atmosphere with a uniform, vertical magnetic field is carried out, both numerically and analytically. In contrast to previous investigations of upward-propagating waves (Zhugzhda and Dzhalilov: Astron. Astrophys. 112, 16, 1982a; Cally: Astrophys. J. 548, 473, 2001), this paper studies the downward propagation of waves from a low-β to high-β environment. A simple expression for the amplitude of the transmitted wave is compared with the numerical solution.     

Superbubbles in magnetized interstellar media: blowout or confinement?

The importance of the interstellar magnetic field is studied in relation to the evolution of superbubbles with a three-dimensional (3D) numerical magnetohydrodynamical (MHD) simulation. A superbubble is a large supernova remnant driven by sequential supernova explosions in an OB association. Its evolution is affected by the density stratification in the galactic disc. After the superbubble size reaches 2–3 times the density scaleheight, it expands preferentially in the z -direction, until finally it can punch out a hole in the gas disc (blowout). On the other hand, the magnetic field running parallel to the galactic disc has the effect of preventing it from expanding in the direction perpendicular to the field. The density stratification and the magnetic fields have completely opposite effects on the evolution of the superbubble. We present results of a 3D MHD simulation in which both effects are included. As a result, it is concluded that when the magnetic field has a much larger scaleheight than the density, even for a model in which the bubble would blow out from the disc if the magnetic field were absent, a magnetic field with a strength of 5 μG can confine the bubble in | z |≲300 pc for ≃ 20 Myr (confinement). In a model in which the field strength decreases in the halo as B  ∝ ρ^1/2, the superbubble eventually blows out like a model with B  = 0 even if the magnetic field in the mid-plane is as strong as B  = 5 μG.     

Hall equilibrium of thin Keplerian discs embedded in mixed poloidal and toroidal magnetic fields

Axisymmetric steady-state weakly ionized Hall–magnetohydrodynamic (MHD) Keplerian thin discs are investigated by using asymptotic expansions in the small disc aspect ratio ε. The model incorporates the azimuthal and poloidal components of the magnetic fields in the leading order in ε. The disc structure is described by an appropriate Grad–Shafranov equation for the poloidal flux function ψ that involves two arbitrary functions of ψ for the toroidal and poloidal currents. The flux function is symmetric about the mid-plane and satisfies certain boundary conditions at the near-horizontal disc edges. The boundary conditions model the combined effect of the primordial as well as the dipole-like magnetic fields. An analytical solution for the Hall equilibrium is achieved by further expanding the relevant equations in an additional small parameter δ that is inversely proportional to the Hall parameter. It is thus found that the Hall equilibrium discs fall into two types: Keplerian discs with (i) small  ( R _d∼δ⁰)  and (ii) large  ( R _d≳δ^{− k} , k > 0)  radius of the disc. The numerical examples that are presented demonstrate the richness and great variety of magnetic and density configurations that may be achieved under the Hall–MHD equilibrium.