MHD waves in coronal loops with a shell

We consider a model of a coronal loop in the form of a cord surrounded by a coaxial shell. Two slow magnetosonic waves longitudinally propagate within a thin flux tube on the m=0 cylindrical mode with velocities close to the tube velocities in the cord and the shell. One wave propagates inside the cord, while the other propagates inside the shell. A peculiar feature of the second wave is that the plasma in the cord and the shell oscillates with opposite phases. There are two fast magnetosonic waves on each of the cylindrical modes with m>0. If the plasma density in the shell is lower than that in the surrounding corona, then one of the waves is radiated into the corona, which causes the loop oscillations to be damped, while the other wave is trapped by the cord, but can also be radiated out under certain conditions. If the plasma density in the shell is higher than that in the cord, then one of the waves is trapped by the shell, while the other wave can also be trapped by the shell under certain conditions. In the wave trapped by the shell and the wave radiated by the tube, the plasma in the cord and the shell oscillates with opposite phases.     

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.     

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.     

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.     

Numerical simulations of fast MHD waves in a coronal plasma

The temporal evolution of ducted waves under coronal conditions is studied in the framework of linearized low MHD by means of numerical simulations. Coronal loops are represented by smoothed slabs of enhanced gas density embedded within a uniform magnetic field. The simulations show that for a smoothed density profile there is an energy leakage from the slab, associated with the propagation of sausage and kink waves. Wave energy leakage in the kink wave is generally small, whereas the wave energy in sausage waves leaks more strongly for long wavelengths and smoother slabs.     

The effect of density stratification on the resonant absorption of MHD waves in coronal loops

The standing quasi-modes of the ideal magnetohydrodynamics (MHD) 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 be a linearly varying density profile. Using the relevant connection formulae of the resonant absorption, the dispersion relation for the fast MHD body waves is derived and solved numerically to obtain both the frequencies and damping rates of the fundamental and first-overtone,   k = 1, 2  , modes of both the kink  ( m = 1)  and fluting  ( m = 2)  waves, where k and m are the longitudinal and azimuthal mode numbers, respectively.     

MHD waves in coronal magnetic arcades

We analyze the eigenmodes of the solar coronal magnetic arcade that describes the magnetic field of a bipolar active region using the eikonal method for ideal magnetohydrodynamic equations. We write out the eikonal equations for Alfvèn and magnetoacoustic waves and derive the equations for the amplitudes of the zeroth approximation. We construct the wave fields for Alfvèn and fast magnetoacoustic modes and derive the expressions for the eigenfrequencies. We show that Alfvèn modes of a given frequency are near a number of magnetic surfaces, while fast magnetoacoustic eigenmodes are near nonmagnetic surfaces. A discrete set of eigenfrequencies that continuously change from one surface to another corresponds to each such surface.     

Energy spectrum and pitch angle distribution of particles reflected by MHD shock waves of fast mode

To study the macroscopic acceleration process for non-thermal particles at the front of MHD shock waves, two limiting treatments, namely the “adiabatic” and the “kink” treatments have been developed. They correspond to cases of (particle gyroradius)/(width of shock transition region) ? 1 and ? 1, respectively. The effects of the acceleration process on energy and pitch angle distributions of reflected particles are examined by using each of these treatments and results are compared. It is shown that these two treatments give almost the same energy and pitch angle distribution in the case of nearly-perpendicular shock waves. In the case of nearly-parallel shock waves, the pitch angle distributions differ significantly, there being reflected particles in the adiabatic loss cone when the kink treatment is employed, while the ranges of the energy distribution for these two treatments do not differ greatly. Analytic representation for the acceleration in the adiabatic treatment is given for the later usage.     

Acoustic damping of fast kink oscillations of coronal loops

The damping of fast kink oscillations of solar coronal loops attributable to the radiation of MHD waves into the surroundings is considered in the thin-tube approximation. The oscillation damping decrement is calculated both by using a new energy method and by solving the dispersion equation for magnetic-tube eigenmodes. The two approaches are in good agreement under appropriate assumptions. The damping is negligible if MHD waves are radiated perpendicular to the magnetic field. The low Q factor of the loop oscillations in active regions found with the TRACE space telescope is associated with the generation of running waves that propagate along magnetic field lines.     

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)     

WKB estimates for the onset of ideal MHD instabilities in solar coronal loops

A WKB approach, based on the method of Connor, Hastie, and Taylor (1979), is used to obtain simple estimates of the critical conditions for the onset of ideal MHD instabilities in line-tied solar coronal loops. The method is illustrated for the constant twist, Gold-Hoyle (1960) field, and the critical conditions are compared with previous and new numerical results. For the force-free case, the WKB estimate for the critical loop length reduces to . For the sufficiently non-force-free case the critical length can be expressed in the forml ₀ +l ₁/m. The results confirm the findings of De Bruyne and Hood (1992) that for force-free fields them = 1 mode is the first mode to become unstable but for the sufficiently strong non-force-free case this reverses with them → ∞ mode being excited first.     

Non-adiabatic discrete Alfvén waves in coronal loops and prominences

Discrete Alfvén waves in coronal loops and prominences are investigated in non-ideal magnetohydrodynamics. The non-ideal effects included are anisotropic, thermal conduction, and optically thin radiation. The classic ideal Alfvén continuum is not altered by these non-ideal effects, but the discrete Alfvén modes, which exist under certain conditions above or below the Alfvén continuum in ideal MHD, are shown to be influenced by non-adiabatic effects.The existence of discrete, non-adiabatic Alfvén waves, and their damping and overstability are examined for 1D cylindrical equilibrium states with twisted magnetic fields. First, analytic results are obtained for modes of high radial order by means of a WKB-analysis. The subspectrum of discrete Alfvén modes is computed with a numerical code, with particular emphasis on the modes of low radial order. The results show that discrete Alfvén waves are of potential importance for solar applications and also that the information obtained with the WKB-analysis is of limited use in this context.Research Assistant of the Belgian National for Scientific Research.     

Thermal stability of a corona heated by fast mode waves

It has been proposed that dissipation of hydromagnetic waves is an important heat source for the solar corona. We consider damping by collisionless processes and by electron thermal conduction and ion viscosity, and calculate the wave energy density such that heating balances the energy radiated by the plasma. We then analyze the thermal stability of the wave heated medium. The fastest growing instabilities are condensations perpendicular to the fieldlines. The instability may be important for producing coronal fine structure, and in loops and streamers.The National Center for Atmospheric Research is sponsored by the National Science Foundation.     

The coupling resonance of torsional alfven wave and fast wave in coronal loops

The method of Orthogonal Function Series Expansion (OFSE) is generalized and applied to the study of the evolution of the coupling of nondissipative torsional Alfven wave and fast wave in coronal loops. Using this method, the intrinsic angular frequency of the overall wave mode can be described mathematically and that of the Alfven waves along the magnetic lines in the coronal loop during the coupling of the Alfven and fast waves can be analyzed both theoretically and numerically. Also with this method, the relation between the coupling driven term and the Alfven wave resonance may be analyzed. Results of computation reveal the place of appearance of coupling resonance as well as the characteristics of the amplitudes of the Alfven and fast waves. As found by the calculations, if the footpoint driven angular frequency is not equal to the intrinsic angular frequency of the overall wave mode of the coronal loop and when a δ section appears at the place of coupled resonance, the radial gradient of the fast wave's amplitude is quite large. Sometimes it approximates to a discontinuity, and this is extremely favorable for the dissipation of the fast wave. If the footpoint driven angular frequency is equal to the intrinsic angular frequency of the overall wave mode and when a δ section occurs in the Alfven wave amplitude, abundant small-scale structures appear in the radial direction. Then the location of resonance approximately becomes a discontinuity, very favorable to the dissipation of the Alfven wave.     

Inclination of large coronal loops observed by TRACE

A TRACE field of view well inside the solar disk shows very well defined large loops, likely to be very inclined to the solar surface. On the other hand there is little evidence of large loops perpendicular to the solar surface. We show that this does not mean that most large loops have such large inclination but that perpendicular loops are much less visible to TRACE, because of density stratification. We quantitatively evaluate this effect by modeling in detail loops with different inclinations.     

Models of dynamic coronal loops

This paper reviews the basic ideas underlying one-dimensional fluid dynamic models of coronal loops and presents some of their most recent applications. These models are an important theoretical support to explore the new scenario provided by the data of Yohkoh, SOHO, and TRACE, and are useful to interpret observations, when supplemented by appropriate spectral synthesis codes. Possible developments are also discussed.     

The structure of coronal loops

With the advent of space telescopes, coronal magnetic loops, both within and outside active regions, are being observed with renewed interest. This paper is an attempt to outline some general physical considerations pertinent to such loops, as a prelude to more sophisticated modelling. For example, a loop that is stretched (or possibly twisted) too much may be subject to a thermal instability that cools its core to a new equilibrium below 10⁵ K. Also a simple consequence of hydrostatic balance along an equilibrium loop is that, under some circumstances, the density inside a cool loop can be comparable with that outside, despite the much smaller scale height. Finally, when the equilibrium loop density is less than the ambient density, several small scale magnetohydrodynamic instabilities are sometimes efficient enough to generate a circulation that tends to equalize the densities.     

The effect of Alfvén waves on MHD fast shocks

The Rankine-Hugoniot equations for hydromagnetic shocks are extended to include the energy and momentum flux due to Alfvén waves incident on the shock. The shock relations are solved numerically for a wide variety of parameters typical of interplanetary conditions. The presence of the waves can cause appreciable changes in the structure of fast shocks of low Mach number.     

Plasma flow around coronal loops

Current dissipation models of coronal loop heating are studied. Turbulent current dissipation is shown to lead to a time dependent process because of an enormous mass motion induced in the current layer. A stationary heating process involves only ohmic heating, which requires a large current layer. To insure MHD stability, the loop must be composed of many elements with the oppositely directed currents. A stationary current dissipation process induces the plasma motion across the magnetic field into the loop and down the loop with the speeds v 10⁴ cm s^–1 and v 10⁴ cm s^–1, respectively. The pressure of the loop is also estimated to be proportional to the current density: p/J=6.3 × 10^-8dyn/statamp.