The Oscillations of Coronal Loops Including the Shell

We investigate the MHD waves in a double magnetic flux tube embedded in a uniform external magnetic field. The tube consists of a dense hot cylindrical cord surrounded by a co-axial shell. The plasma and the magnetic field are taken to be uniform inside the cord and also inside the shell. Two slow and two fast magnetosonic modes can exist in the thin double tube. The first slow mode is trapped by the cord, the other is trapped by the shell. The oscillations of the second mode have opposite phases inside the cord and shell. The speeds of the slow modes propagating along the tube are close to the tube speeds inside the cord and the shell. The behavior of the fast modes depends on the magnitude of Alfvén speed inside the shell. If it is less than the Alfvén speed inside the cord and in the environment, then the fast mode is trapped by the shell and the other may be trapped under the certain conditions. In the opposite case when the Alfvén speed in the shell is greater than those inside the cord and in the environment, then the fast mode is radiated by the tube and the other may also be radiated under certain conditions. The oscillation of the cord and the shell with opposite phases is the distinctive feature of the process. The proposed model allows to explain the basic phenomena connected to the coronal oscillations: i) the damping of oscillations stipulated in the double tube model by the radiative loss, ii) the presence of two different modes of perturbations propagating along the loop with close speeds, iii) the opposite phases of oscillations of modulated radio emission, coming from the near coronal sources having sharply different densities.     

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.     

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.     

SIMULATION STUDIES OF ACCELERATION OF HEAVY IONS AND THEIR ELEMENTAL COMPOSITIONS

By using a one-dimension (one spatial coordinate and three-velocity components), electromagnetic particle simulation code with full ion and electron dynamics, we have studied the acceleration of heavy ions by a nonlinear magnetosonic wave in a multi-ion-species plasma. First, we describe the mechanism of heavy ion acceleration by magnetosonic waves. We then investigate this by particle simulations. The simulation plasma contains four ion species: H, He, O, and Fe. The number density of He is taken to be 10% of that of H, and those of O and Fe are much lower. Simulations confirm that, as in a single-ion-species plasma, some of the hydrogen can be accelerated by the longitudinal electric field formed in the wave. Furthermore, they show that magnetosonic waves can accelerate all the particles of all the heavy species (He, O, and Fe) by a different mechanism, i.e., by the transverse electric field. The maximum speeds of the heavy species are about the same, of the order of the wave propagation speed. These are in good agreement with theoretical prediction. These results indicate that, if high-energy ions are produced in the solar corona through these mechanisms, the elemental compositions of these heavy ions can be similar to that of the background plasma, i.e., the corona.     

Nonlinear Evolution of a 3D Inertial Alfvén Wave and Its Implication in Particle Acceleration

A simulation based on a pseudo-spectral method has been performed in order to study particle acceleration. A model for the acceleration of charged particles by field localization is developed for the low-\(\upbeta\) plasma. For this purpose, a fractional diffusion approach has been employed. The nonlinear interaction between a 3D inertial Alfvén wave and a slow magnetosonic wave has been examined, and the dynamical equations of these two waves in the presence of ponderomotive nonlinearity have been solved numerically. The nonlinear evolution of the inertial Alfvén wave in the presence of slow magnetosonic wave undergoes a filamentation instability and results in field intensity localization. The results obtained show the localization and power spectrum of inertial Alfvén wave due to nonlinear coupling. The scaling obtained after the first break point of the magnetic power spectrum has been used to calculate the formation of the thermal tail of energetic particles in the solar corona.     

On the Excitation of Leaky Modes in Cylindrical Loops

The role of leaky waves in the coronal loop oscillations observed by TRACE is not yet clearly understood. In this work, the excitation of fast waves in solar coronal loops modelled as dense plasma cylindrical tubes in a uniform straight magnetic field is investigated. We study the trapped and especially leaky modes (whose energy escapes from the tube) that result from an initial disturbance by solving the time-dependent problem numerically. We find that the stationary state of the tube motion is given by the trapped normal modes. By contrast, the transient behaviour between the initial and the stationary phase is dominated by wave leakage. The so-called trig leaky modes are clearly identified since the transient behaviour shows periods and damping times that are in agreement with the values calculated from the normal-mode analysis. Consequently, these radiating modes have physical significance. However, we have not found any evidence for the excitation of other types of modes, such as the principal leaky kink mode. J. Andries is postdoctoral Fellow of the National Fund for Scientific Research – Flanders (Belgium) (F.W.O.-Vlaanderen).     

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.     

Nonlinear magnetosonic waves propagating perpendicular to a magnetic neutral sheet

Nonlinear magnetosonic waves propagating in a magnetic neutral sheet are investigated within the framework of a fluid model. It is shown that the behavior of the magnetosonic waves is governed by a ‘modified Burgers equation’ with an additional termc(η)? due to the relevant slowly varying background plasma parameter (density or magnetic field), $$\frac{{\partial \phi }}{{\partial \eta }}$$ where ?(ξ, η) is the amplitude of the wave, \(\xi = \int {k_x } {\text{d}}x + k_y y - \omega t\) , and η=εx is the coordinate stretched by a smallness parameter ε. When we consider fast magnetosonic waves propagating toward the neutral region across the magnetic field, they grow and undergo rapid steepening after passing through the neutral region; i.e., shock formation is promoted by the background inhomogeneity. By the numerical computation of the above equation, the time evolution is examined for two initial disturbances, the pulse type (gaussian) and the wave train type (sinusoidal wave). The relevance of the interactions between the magnetosonic shock waves and the neutral sheet plasma to a triggering mechanism of sympathetic flares is also suggested.     

Alfvén-magnetosonic waves interaction in the solar corona

The nonlinear propagation of the Alfvén and magnetosonic waves in the solar corona is investigated in terms of model equations. Due to viscous effects taken into account the propagation of the fast wave itself is governed by Burgers type equations possessing both expansion and compression shock solutions. Numerical simulations show that both parallely and perpendicularly propagating fast waves can steepen into shocks if their amplitudes are in excess of some sizeable fraction of the Alfvén velocity. However, if the magnetic field changes linearly in the perpendicular direction, then formation of perpendicular shocks can be hindered. The Alfvén waves exhibit a tendency to drive both the slow and fast magnetosonic waves whose propagation is described by linearized Boussinesq type equations with ponderomotive terms due to the Alfvén wave. The limits of the slow and fast waves are investigated.     

Guided waves in magnetospheric tubes of enhanced density

Properties of a guided MHD-wave propagating in a magnetic field tube with the plasma density differing from the ambient density are studied. Like the Alvén wave this wave propagates along the magnetic field and is connected with the field-aligned currents flowing at the periphery of the oscillating tube. The guided wave is accompanied by the magnetic field compression, nevertheless the wave moves without attenuation. The guided wave velocity is between the Alvén velocities inside and outside the oscillating tube. In a tube of elliptical cross-section the propagation velocity depends on the polarization of the wave.     

Formation of fast magnetosonic shock waves during a two-current-loop collision in solar flares

We present a numerical simulation of the fast magnetosonic shock wave formation during a two-current-loop collision by using a magnetohydrodynamical model. It is shown that the rarefaction waves are generated in the initial stage when the two current loops start to collide. After the rarefaction waves propagate away from the excited region, the fast magnetosonic waves with density enhancement can be produced for the simulation when the current strength of the loop is weak. As the current becomes strong enough, the magnetosonic shock waves can be generated in the direction perpendicular to that of the two-loop collision.     

Alfvén Wave Phase Mixing as a Source of Fast Magnetosonic Waves

The nonlinear excitation of fast magnetosonic waves by phase mixing Alfvén waves in a cold plasma with a smooth inhomogeneity of density across a uniform magnetic field is considered. If initially fast waves are absent from the system, then nonlinearity leads to their excitation by transversal gradients in the Alfvén wave. The efficiency of the nonlinear Alfvén–fast magnetosonic wave coupling is strongly increased by the inhomogeneity of the medium. The fast waves, permanently generated by Alfvén wave phase mixing, are refracted from the region with transversal gradients of the Alfvén speed. This nonlinear process suggests a mechanism of indirect plasma heating by phase mixing through the excitation of obliquely propagating fast waves.     

Simulations of jet formation from a magnetized accretion disk

Axisymmetric magnetohydrodynamic (MHD) simulations have been made of the formation of jets from a Keplerian disk threaded by a magnetic field. The disk is treated as a boundary condition, where matter with high specific entropy is ejected with a Keplerian azimuthal speed and a poloidal speed less than the slow magnetosonic velocity, and where boundary conditions on the magnetic fields correspond to a highly conducting disk. Initially, the space above the disk, the corona, is filled with high specific entropy plasma in the thermal equilibrium in the gravitational field of the central object. The initial magnetic field is poloidal and is represented by the superposition of the fields of monopoles located below the plane of the disk.The rotation of the disk twists the initial poloidal magnetic field lines, and this twist propagates into the corona pushing matter into jet-like outflow in a cylindrical region. After the first switch-on wave, which originates during the first rotation period of the inner radius of the disk, the matter outflowing from the disk starts to flow and accelerate in thez-direction owing to both the magnetic and pressure gradient forces. The flow accelerates through the slow magnetosonic and Alfvén surfaces and at larger distances through the fast magnetosonic surface. The flow velocity of the jet is approximately parallel to thez-axis, with the collimation mainly a result of the pinching force of the toroidal magnetic field. The energy flux of the flow increases with increasing magnetic field strength on the disk. Some of the cases studied have been run for long times, 60 rotation periods of the inner radius of the disk, and show indications of approaching a stationary state.     

Generation of Type U Bursts in Solar Radio Emission

If plasma waves propagate in the direction of the plasma density decrease, their spectrum shifts to large wave numbers (to small phase velocities). This means that the spectrum of plasma waves excited by an electron beam concentrates near the distribution function (“plateau”) border, which shifts in the region of low velocities in the process of quasilinear relaxation. As the spectrum of excited plasma waves shifts in the region of large wave numbers, their frequency grows in accordance with the dispersion equation, which describes these waves. When the growth of the plasma wave frequency exceeds the decrease of the frequency owing to the regular inhomogeneity in the corona, the branch with positive frequency drift appears on the dynamic spectrum of the radio emission. Our computations allow us to estimate the density and energy of electron beams generating type U bursts.     

Understanding the Physical Nature of Coronal “EIT Waves”

For almost 20 years the physical nature of globally propagating waves in the solar corona (commonly called “EIT waves”) has been controversial and subject to debate. Additional theories have been proposed over the years to explain observations that did not agree with the originally proposed fast-mode wave interpretation. However, the incompatibility of observations made using the Extreme-ultraviolet Imaging Telescope (EIT) onboard the Solar and Heliospheric Observatory with the fast-mode wave interpretation was challenged by differing viewpoints from the twin Solar Terrestrial Relations Observatory spacecraft and data with higher spatial and temporal resolution from the Solar Dynamics Observatory. In this article, we reexamine the theories proposed to explain EIT waves to identify measurable properties and behaviours that can be compared to current and future observations. Most of us conclude that the so-called EIT waves are best described as fast-mode large-amplitude waves or shocks that are initially driven by the impulsive expansion of an erupting coronal mass ejection in the low corona.     

Numerical Simulations of Impulsively Generated Magnetosonic Waves in a Coronal Loop

We consider impulsively excited magnetosonic waves in a highly magnetized coronal loop that is approximated by a straight plasma slab of enhanced mass density. Numerical results reveal that wavelet spectra of time signatures of these waves possess characteristic shapes that depend on the position of the initial pulse: in the case of a pulse launched inside the slab, these spectra are of a tadpole shape, while for a pulse excited in the ambient medium these spectra display more complex structures with branches of long and short-period waves. These short period oscillations correspond to waves that are trapped inside the slab, and the long-period oscillations are associated with waves that propagate through the ambient medium and reach the detection point. These findings are compatible with recent theoretical studies and observations by the solar eclipse coronal imaging system (SECIS).     

Peculiarities of entropy and magnetosonic waves in optically thin cosmic plasma

The stability problem for small magnetohydrodynamic (MHD) perturbations in an optically thin, perfectly conducting uniform plasma with a cosmic abundance of elements is solved in the linear approximation. The electron heat conduction along the magnetic field and the proton heat conduction across the field are taken into account. We have shown for the first time that the entropy waves can grow exponentially, while the magnetosonic waves are damped in a wide range of physical conditions closest to the conditions in stellar coronae with the proper allowance for radiative losses. Slow magnetosonic waves are damped particularly rapidly. For the solar corona, the calculated damping decrement of slow magnetosonic waves agrees well with the averaged one in 11 quasi-periodic events observed from the TRACE satellite in extreme ultraviolet radiation. Other possible astrophysical applications of the results obtained are briefly discussed.     

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.     

Enhanced emission of Alfvén waves from sunspots during proton flares

Thomas (1978) has shown that, if Alfvén waves exist in a sunspot umbra, they are normally reflected so strongly by the temperature minimum as to be essentially undetectable in the upper solar atmosphere. However, it is known that in many proton flares, chromospheric emission overlies the umbra of a sunspot, indicating that the transition region (TR) between chromosphere and corona in the umbral flux tube has moved down to lower altitudes. As a result of this lowering, umbral Alfvén waves have readier access to the corona: the coronal leakage depends exponentially on the altitude of the TR. We find that the Alfvén wave flux which leaks out of the umbra into the corona can exceed 10⁷ ergs cm^-2 s^-1. A flux of this magnitude is expected to dissipate rapidly in the corona, thereby contributing to a positive feedback loop which ensures prolonged (1 hr) leakage of the umbral Alfvén waves into the corona. We propose that these Alfvén waves may contribute significantly to prolonged energization of proton flares in which umbral coverage occurs.     

Damping of slow magnetoacoustic waves in an inhomogeneous coronal plasma

We study the propagation and dissipation of slow magnetoacoustic waves in an inhomogeneous viscous coronal loop plasma permeated by uniform magnetic field. Only viscosity and thermal conductivity are taken into account as dissipative processes in the coronal loop. The damping length of slow-mode waves exhibit varying behaviour depending upon the physical parameters of the loop in an active region AR8270 observed by TRACE. The wave energy flux associated with slow magnetoacoustic waves turns out to be of the order of 10⁶ erg cm^?2 s^?1 which is high enough to replace the energy lost through optically thin coronal emission and the thermal conduction below to the transition region. It is also found that only those slow-mode waves which have periods more than 240s provide the required heating rate to balance the energy losses in the solar corona. Our calculated wave periods for slow-mode waves nearly match with the oscillation periods of loop observed by TRACE.