Coronal Flux Tubes Research Articles

Simulations of fully deformed oscillating flux tubes

Context. In recent years, a number of numerical studies have been focusing on the significance of the Kelvin–Helmholtz instability in the dynamics of oscillating coronal loops. This process enhances the transfer of energy into smaller scales, and has been connected with heating of coronal loops, when dissipation mechanisms, such as resistivity, are considered. However, the turbulent layer is expected near the outer regions of the loops. Therefore, the effects of wave heating are expected to be confined to the loop’s external layers, leaving their denser inner parts without a heating mechanism. Aim. In the current work we aim to study the spatial evolution of wave heating effects from a footpoint driven standing kink wave in a coronal loop. Methods. Using the MPI-AMRVAC code, we performed ideal, three dimensional magnetohydrodynamic simulations of footpoint driven transverse oscillations of a cold, straight coronal flux tube, embedded in a hotter environment. We have also constructed forward models for our simulation using the FoMo code. Results. The developed transverse wave induced Kelvin–Helmholtz (TWIKH) rolls expand throughout the tube cross-section, and cover it entirely. This turbulence significantly alters the initial density profile, leading to a fully deformed cross section. As a consequence, the resistive and viscous heating rate both increase over the entire loop cross section. The resistive heating rate takes its maximum values near the footpoints, while the viscous heating rate at the apex. Conclusions. We conclude that even a monoperiodic driver can spread wave heating over the whole loop cross section, potentially providing a heating source in the inner loop region. Despite the loop’s fully deformed structure, forward modelling still shows the structure appearing as a loop.

Guided flows in coronal magnetic flux tubes

Context. There is evidence that coronal plasma flows break down into fragments and become laminar. Aims. We investigate this effect by modelling flows confined along magnetic channels. Methods. We consider a full magnetohydrodynamic (MHD) model of a solar atmosphere box with a dipole magnetic field. We compare the propagation of a cylindrical flow perfectly aligned with the field to that of another flow with a slight misalignment. We assume a flow speed of 200 km s-1 and an ambient magnetic field of 30 G. Results. We find that although the aligned flow maintains its cylindrical symmetry while it travels along the magnetic tube, the misaligned one is rapidly squashed on one side, becoming laminar and eventually fragmented because of the interaction and back-reaction of the magnetic field. This model could explain an observation made by the Atmospheric Imaging Assembly on board the Solar Dynamics Observatory of erupted fragments that fall back onto the solar surface as thin and elongated strands and end up in a hedge-like configuration. Conclusions. The initial alignment of plasma flow plays an important role in determining the possible laminar structure and fragmentation of flows while they travel along magnetic channels.

The Effect of a Twisted Magnetic Field on the Phase Mixing of the Kink Magnetohydrodynamic Waves in Coronal Loops

Abstract There is observational evidence for the existence of a twisted magnetic field in the solar corona. This inspires us to investigate the effect of a twisted magnetic field on the evolution of magnetohydrodynamic (MHD) kink waves in coronal loops. With this aim, we solve the incompressible linearized MHD equations in a magnetically twisted nonuniform coronal flux tube in the limit of long wavelengths. Our results show that a twisted magnetic field can enhance or diminish the rate of phase mixing of the Alfvén continuum modes and the decay rate of the global kink oscillation depending on the twist model and the sign of the longitudinal (k z ) and azimuthal (m) wavenumbers. Also, our results confirm that in the presence of a twisted magnetic field, when the sign of one of the two wavenumbers m and k z is changed, the symmetry with respect to the propagation direction is broken. Even a small amount of twist can have an important impact on the process of energy cascading to small scales.

Heating by transverse waves in simulated coronal loops

Recent numerical studies of oscillating flux tubes have established the significance of resonant absorption in the damping of propagating transverse oscillations in coronal loops. The nonlinear nature of the mechanism has been examined alongside the Kelvin-Helmholtz instability, which is expected to manifest in the resonant layers at the edges of the flux tubes. While these two processes have been hypothesized to heat coronal loops through the dissipation of wave energy into smaller scales, the occurring mixing with the hotter surroundings can potentially hide this effect. We aim to study the effects of wave heating from driven and standing kink waves in a coronal loop. Using the MPI-AMRVAC code, we perform ideal, three dimensional magnetohydrodynamic (MHD) simulations of both (a) footpoint driven and (b) free standing oscillations in a straight coronal flux tube, in the presence of numerical resistivity. We have observed the development of Kelvin-Helmholtz eddies at the loop boundary layer of all three models considered here, as well as an increase of the volume averaged temperature inside the loop. The main heating mechanism in our setups was Ohmic dissipation, as indicated by the higher values for the temperatures and current densities located near the footpoints. The introduction of a temperature gradient between the inner tube and the surrounding plasma, suggests that the mixing of the two regions, in the case of hotter environment, greatly increases the temperature of the tube at the site of the strongest turbulence, beyond the contribution of the aforementioned wave heating mechanism.

Observational Signatures of Transverse Magnetohydrodynamic Waves and Associated Dynamic Instabilities in Coronal Flux Tubes

Abstract Magnetohydrodynamic (MHD) waves permeate the solar atmosphere and constitute potential coronal heating agents. Yet, the waves detected so far may be but a small subset of the true existing wave power. Detection is limited by instrumental constraints but also by wave processes that localize the wave power in undetectable spatial scales. In this study, we conduct 3D MHD simulations and forward modeling of standing transverse MHD waves in coronal loops with uniform and non-uniform temperature variation in the perpendicular cross-section. The observed signatures are largely dominated by the combination of the Kelvin–Helmholtz instability (KHI), resonant absorption, and phase mixing. In the presence of a cross-loop temperature gradient, we find that emission lines sensitive to the loop core catch different signatures compared to those that are more sensitive to the loop boundary and the surrounding corona, leading to an out-of-phase intensity and Doppler velocity modulation produced by KHI mixing. In all of the considered models, common signatures include an intensity and loop width modulation at half the kink period, a fine strand-like structure, a characteristic arrow-shaped structure in the Doppler maps, and overall line broadening in time but particularly at the loop edges. For our model, most of these features can be captured with a spatial resolution of 0.″33 and a spectral resolution of 25 km s−1, although we do obtain severe over-estimation of the line width. Resonant absorption leads to a significant decrease of the observed kinetic energy from Doppler motions over time, which is not recovered by a corresponding increase in the line width from phase mixing and KHI motions. We estimate this hidden wave energy to be a factor of 5–10 of the observed value.

Magnetic shuffling of coronal downdrafts

Channelled fragmented downflows are ubiquitous in magnetized atmospheres, and have been recently addressed from an observation after a solar eruption. We study the possible back-effect of the magnetic field on the propagation of confined flows. We compare two 3D MHD simulations of dense supersonic plasma blobs downfalling along a coronal magnetic flux tube. In one, the blobs move strictly along the field lines; in the other, the initial velocity of the blobs is not perfectly aligned to the magnetic field and the field is weaker. The aligned blobs remain compact while flowing along the tube, with the generated shocks. The misaligned blobs are disrupted and merged by the chaotic shuffling of the field lines, and structured into thinner filaments; Alfven wave fronts are generated together with shocks ahead of the dense moving front. Downflowing plasma fragments can be chaotically and efficiently mixed if their motion is misaligned to field lines, with broad implications, e.g., disk accretion in protostars, coronal eruptions and rain.

Star-disk interaction in classical T Tauri stars revealed using wavelet analysis

The extension of the corona of classical T Tauri stars (CTTS) is under discussion. The standard model of magnetic configuration of CTTS predicts that coronal magnetic flux tubes connect the stellar atmosphere to the inner region of the disk. However, differential rotation may disrupt these long loops. The results from Hydrodynamic modeling of X-ray flares observed in CTTS confirming the star-disk connection hypothesis are still controversial. Some authors suggest the presence of the accretion disk prevent the stellar corona to extent beyond the co-rotation radius, while others simply are not confident with the methods used to derive loop lengths. We use independent procedures to determine the length of flaring loops in stars of the Orion Nebula Cluster previously analyzed using Hydrodynamic models. Our aim is to disentangle between the two scenarios proposed. We present a different approach to determine the length of flaring loops based on the oscillatory nature of the loops after strong flares. We use wavelet tools to reveal oscillations during several flares. The subsequent analysis of such oscillations is settle on the Physics of coronal seismology. Our results likely confirm the large extension of the corona of CTTS and the hypothesis of star-disk magnetic interaction in at least three CTTS of the Orion Nebula Cluster. Analyzing oscillations in flaring events is a powerful tool to determine the physical characteristics of magnetic loops in coronae in stars other than the Sun. The results presented in this work confirm the star-disk magnetic connection in CTTS.

MHD modelling of coronal loops: injection of high-speed chromospheric flows

Observations reveal a correspondence between chromospheric type II spicules and bright upwardly moving fronts in the corona observed in the EUV band. However, theoretical considerations suggest that these flows are unlikely to be the main source of heating in coronal magnetic loops. We investigate the propagation of high-speed chromospheric flows into coronal magnetic flux tubes, and the possible production of emission in the EUV band. We simulate the propagation of a dense $10^4$ K chromospheric jet upwards along a coronal loop, by means of a 2-D cylindrical MHD model, including gravity, radiative losses, thermal conduction and magnetic induction. The jet propagates in a complete atmosphere including the chromosphere and a tenuous cool ($\sim 0.8$ MK) corona, linked through a steep transition region. In our reference model, the jet's initial speed is 70 km/s, its initial density is $10^{11}$ cm$^{-3}$, and the ambient uniform magnetic field is 10 G. We explore also other values of jet speed and density in 1-D, and of magnetic field in 2-D, and the jet propagation in a hotter ($\sim 1.5$ MK) background loop. While the initial speed of the jet does not allow it to reach the loop apex, a hot shock front develops ahead of it and travels to the other extreme of the loop. The shock front compresses the coronal plasma and heats it to about $10^6$ K. As a result, a bright moving front becomes visible in the 171 \AA\ channel of the SDO/AIA mission. This result generally applies to all the other explored cases, except for the propagation in the hotter loop. For a cool, low-density initial coronal loop, the post-shock plasma ahead of upward chromospheric flows might explain at least part of the observed correspondence between type II spicules and EUV emission excess.

ON THE SUPPORT OF SOLAR PROMINENCE MATERIAL BY THE DIPS OF A CORONAL FLUX TUBE

The dense prominence material is believed to be supported against gravity through the magnetic tension of dipped coronal magnetic field. For quiescent prominences, which exhibit many gravity-driven flows, hydrodynamic forces are likely to play an important role in the determination of both the large and small scale magnetic field distributions. In this study, we present the first steps toward creating three-dimensional magneto-hydrostatic prominence model where the prominence is formed in the dips of a coronal flux tube. Here 2.5D equilibria are created by adding mass to an initially force-free magnetic field, then performing a secondary magnetohydrodynamic relaxation. Two inverse polarity magnetic field configurations are studied in detail, a simple o-point configuration with a ratio of the horizontal field (B_x) to the axial field (B_y) of 1:2 and a more complex model that also has an x-point with a ratio of 1:11. The models show that support against gravity is either by total pressure or tension, with only tension support resembling observed quiescent prominences. The o-point of the coronal flux tube was pulled down by the prominence material, leading to compression of the magnetic field at the base of the prominence. Therefore tension support comes from the small curvature of the compressed magnetic field at the bottom and the larger curvature of the stretched magnetic field at the top of the prominence. It was found that this method does not guarantee convergence to a prominence-like equilibrium in the case where an x-point exists below the prominence flux tube. The results imply that a plasma beta of ~ 0.1 is necessary to support prominences through magnetic tension.

Interplanetary Signatures of Unipolar Streamers and the Origin of the Slow Solar Wind

Unipolar streamers (also known as pseudo-streamers) are coronal structures that, at least in coronagraph images, and when viewed at the correct orientation, are often indistinguishable from dipolar (or “standard”) streamers. When interpreted with the aid of a coronal magnetic field model, however, they are shown to consist of a pair of loop arcades. Whereas dipolar streamers separate coronal holes of the opposite polarity and whose cusp is the origin of the heliospheric current sheet, unipolar streamers separate coronal holes of the same polarity and are therefore not associated with a current sheet. In this study, we investigate the interplanetary signatures of unipolar streamers. Using a global MHD model of the solar corona driven by the observed photospheric magnetic field for Carrington rotation 2060, we map the ACE trajectory back to the Sun. The results suggest that ACE fortuitously traversed through a large and well-defined unipolar streamer. We also compare heliospheric model results at 1 AU with ACE in-situ measurements for Carrington rotation 2060. The results strongly suggest that the solar wind associated with unipolar streamers is slow. We also compare predictions using the original Wang–Sheeley (WS) empirically determined inverse relationship between solar wind speed and expansion factor. Because of the very low expansion factors associated with unipolar streamers, the WS model predicts high speeds, in disagreement with the observations. We discuss the implications of these results in terms of theories for the origin of the slow solar wind. Specifically, premises relying on the expansion factor of coronal flux tubes to modulate the properties of the plasma (and speed, in particular) must address the issue that while the coronal expansion factors are significantly different at dipolar and unipolar streamers, the properties of the measured solar wind are, at least qualitatively, very similar.

The generation of magnetic field via convective motions in the photosphere, Alvén waves, and the origin of chromospheric spicules

The basis is laid out for a theory relating various phenomena in the solar atmosphere, including localized concentrations of magnetic field at the bases of coronal magnetic arches, chromospheric spicules, twisted coronal magnetic flux tubes, and flows of energy carried by Alfven waves propagating upward into the corona. The structure of photospheric currents localized in the vicinity of supergranule boundaries and excited by convective motions is studied. These currents exist primarily in a “dynamo layer” of sharply enhanced transverse conductivity, which forms in the weakly ionized thermal photospheric plasma located in the solar gravitational field. The motions of the electrons and ions in this layer have appreciably different characters: the ions are collisionly driven by the flows of neutral atoms, while the electrons drift in the crossed electric and magnetic fields. The electric field supporting the current arises due to the polarization of the electrons and ions. This field also gives rise to Alfven perturbations that propagate upward into the corona, together with their associated longitudinal currents. The character of this “loading” makes the system of fields and currents uniquely defined. Moreover, the momentum flux carried by these Alfven waves should be transferred to the cool chromospheric gas, facilitating the vertical ejection of this gas in the form of spicules, as was first proposed in 1992 by Haerendel.

CAN PLASMA EXPANSION EXPLAIN THE OBSERVED ACCELERATION OF Ne 7+ IONS IN A CORONAL MAGNETIC FUNNEL?

Tu et al. measured outflow of Ne7+ at a height of 20 Mm above the photosphere demonstrating that the solar wind originates in coronal funnels footed in the chromospheric magnetic networks. We suggest that when the bottom of a coronal flux tube is populated by the chromospheric plasma consisting of protons as the major ions, Ne7+ as minor ions, and heated electrons in the magnetic networks, large-scale plasma expansion could accelerate the Ne7+ ions to a velocity ~10 km s–1 at a height ~ 20 Mm as measured. Two scenarios are discussed here, one with the bulk heating of the electrons to a temperature T e > 64 eV and another with a small fraction of the electrons heated to high temperatures T eh > 158 eV, appearing as an energetic tail to the electrons' velocity distribution function. In the former scenario, the expansion produces weak ambipolar electric fields distributed along the entire length of the funnel. In the latter one the electric field is concentrated in a double layer. The electric fields accelerate the ions against the solar gravity. The required electron temperatures in the above scenarios are large enough to ionize neon atoms yielding Ne7+.

OBSERVATIONAL EVIDENCE OF RESONANTLY DAMPED PROPAGATING KINK WAVES IN THE SOLAR CORONA

In this Letter we establish clear evidence for the resonant absorption damping mechanism by analyzing observational data from the novel Coronal Multi-Channel Polarimeter (CoMP). This instrument has established that in the solar corona there are ubiquitous propagating low amplitude ($\approx$1 km s$^{-1}$) Alfv\'{e}nic waves with a wide range of frequencies. Realistically interpreting these waves as the kink mode from magnetohydrodynamic (MHD) wave theory, they should exhibit a frequency dependent damping length due to resonant absorption, governed by the TGV relation showing that transversal plasma inhomogeneity in coronal magnetic flux tubes causes them to act as natural low-pass filters. It is found that observed frequency dependence on damping length (up to about 8 mHz) can be explained by the kink wave interpretation and furthermore, the spatially averaged equilibrium parameter describing the length scale of transverse plasma density inhomogeneity over a system of coronal loops is consistent with the range of values estimated from TRACE observations of standing kink modes.

Exploiting Laboratory and Heliophysics Plasma Synergies

Recent advances in space-based heliospheric observations, laboratory experimentation, and plasma simulation codes are creating an exciting new cross-disciplinary opportunity for understanding fast energy release and transport mechanisms in heliophysics and laboratory plasma dynamics, which had not been previously accessible. This article provides an overview of some new observational, experimental, and computational assets, and discusses current and near-term activities towards exploitation of synergies involving those assets. This overview does not claim to be comprehensive, but instead covers mainly activities closely associated with the authors’ interests and reearch. Heliospheric observations reviewed include the Sun Earth Connection Coronal and Heliospheric Investigation (SECCHI) on the National Aeronautics and Space Administration (NASA) Solar Terrestrial Relations Observatory (STEREO) mission, the first instrument to provide remote sensing imagery observations with spatial continuity extending from the Sun to the Earth, and the Extreme-ultraviolet Imaging Spectrometer (EIS) on the Japanese Hinode spacecraft that is measuring spectroscopically physical parameters of the solar atmosphere towards obtaining plasma temperatures, densities, and mass motions. The Solar Dynamics Observatory (SDO) and the upcoming Solar Orbiter with the Heliospheric Imager (SoloHI) on-board will also be discussed. Laboratory plasma experiments surveyed include the line-tied magnetic reconnection experiments at University of Wisconsin (relevant to coronal heating magnetic flux tube observations and simulations), and a dynamo facility under construction there; the Space Plasma Simulation Chamber at the Naval Research Laboratory that currently produces plasmas scalable to ionospheric and magnetospheric conditions and in the future also will be suited to study the physics of the solar corona; the Versatile Toroidal Facility at the Massachusetts Institute of Technology that provides direct experimental observation of reconnection dynamics; and the Swarthmore Spheromak Experiment, which provides well-diagnosed data on three-dimensional (3D) null-point magnetic reconnection that is also applicable to solar active regions embedded in pre-existing coronal fields. New computer capabilities highlighted include: HYPERION, a fully compressible 3D magnetohydrodynamics (MHD) code with radiation transport and thermal conduction; ORBIT-RF, a 4D Monte-Carlo code for the study of wave interactions with fast ions embedded in background MHD plasmas; the 3D implicit multi-fluid MHD spectral element code, HiFi; and, the 3D Hall MHD code VooDoo. Research synergies for these new tools are primarily in the areas of magnetic reconnection, plasma charged particle acceleration, plasma wave propagation and turbulence in a diverging magnetic field, plasma atomic processes, and magnetic dynamo behavior.

Jet structure of electric currents in coronal magnetic loops

We analyze the properties of the electric-current distribution over the cross sections of fairly dense coronal magnetic flux tubes in which the plasma pressure exceeds the magnetic pressure, so that the equilibrium is maintained by the ambient magnetic field. If the plasma is fully ionized, the distributions of the longitudinal and azimuthal currents over the cross section of the loop have the same spatial scale as the pressure distribution. However, even a small number of neutral atoms in the corona (with a mass fraction of the order of 10−5, taking into account the partial ionization of helium) substantially modifies the current distribution over the tube cross section: in this case, a considerable fraction of the full current flowing along the tube is concentrated in a thin region near the axis with a radius of the order of (10−2–10−3)r0 (where r0 is the characteristic scale of the plasma-pressure distribution over the tube), thus forming a sort of a jet current. This comes about because the pattern of the conductivity anisotropy is substantially modified in the presence of ion-atom collisions in the magnetoactive plasma of the tube, and the Cowling conductivity dominates over the Hall and Pedersen conductivities. The high current density near the axis of the tube can ensure heating of the plasma to coronal temperatures via Joule dissipation.

Open magnetic flux tubes in the corona and the transport of solar energetic particles

We investigate how magnetic fields guide energetic particles through the corona into interplanetary space and eventually to a spacecraft near the Earth. A set of seven simple particle events is studied, where energetic electrons (30–500 keV; Wind spacecraft) or protons (5–55 MeV; SoHO) were released together with low-energy electron beams producing metric-to-kilometric type III emission. Imaging of the coronal (metre-wave) part of this emission with the Nancay Radioheliograph is used to identify the open flux tubes that guide these electrons – and by inference all particles detected at 1 AU. Open coronal field lines are also computed using potential magnetic field extrapolations, constrained by a source surface and by SoHO/MDI measurements in the photosphere (code by Schrijver and DeRosa). We find that in all events the type III radio sources lie in open flux tubes in the potential magnetic field extrapolations. The open flux tubes are rooted in small parts of the parent active region, covering a heliocentric angle of a few degrees in the photosphere. But they expand rapidly above the neighbouring closed magnetic structures and cover several tens of degrees in longitude on the source surface. Some of these open field lines are found to connect the parent active region to the footpoint of the nominal Parker spiral on the source surface, within the uncertainty of about ±10 ◦ inherent to the evaluation of its connection longitude. This is so even when the parent active region is as far as 50 ◦ away. In two cases where the coronal flux tubes point to high heliolatitudes, the detection of Langmuir waves at the Wind spacecraft in the ecliptic plane suggests that the interplanetary field lines curve down to the ecliptic before reaching 1 AU. We conclude that non-radial open flux tubes in the corona can transport particles over several tens of degrees in longitude even in simple impulsive particle events. In all events we studied, potential magnetic field models give an adequate description of these structures.

Coronal loop slow mode oscillations driven by the kink instability

Aims. To establish the dominant wave modes generated by an internal, m = 1 kink instability in a short coronal flux tube. Methods. The 3D MHD numerical simulations are performed using Lare3d to model the kink instability and the subsequent wave generation. The initial conditions are a straight, zero net current flux tube containing a twist higher than the kink instability threshold. Results. It is shown that the kink instability initially sets up a 1st harmonic (1st overtone) that is converted through the rearrangement of the magnetic field into two out-of-phase fundamental slow modes. These slow modes are in the two entwined flux tubes created during the kink instability. Conclusions. The long-lived oscillations established after a kink instability provide a possible way to identify whether sudden, short coronal loop brightenings may have resulted from a confined kink instability. The mode oscillation structure changes from the 1st harmonic to fundamental due to field line relaxation. The subsequent decay in the fundamental mode is comparable to observations and is caused by shock dissipation. This result has important consequences for the damping of the slow mode oscillations observed by SUMER.

On Solar Intermediate Drift Radio Bursts at Decimeter and Meter Wavelength

Fiber – or intermediate drift – bursts are a continuum fine structure in some complex solar radio events. We present the analysis of such bursts in the X17 flare on 28 Oct. 2003. Based on the whistler wave model of fiber bursts we derive the 3D magnetic field structures that carry the radio sources in different stages of the event and obtain insight into the energy release evolution in the main flare phase, the related paths of nonthermal particle propagation in the corona, and the involved magnetic field structures. Additionally, we test the whistler wave model of fiber bursts for the meter and the decimeter wave range. Radio spectral data (Astrophysikalisches Institut Potsdam, Astronomical Observatory Ondřejov) show a continuum with fibers for ≈ 6 min during the main flare phase. Radio imaging data (Nancay Radio Heliograph) yield source centroid positions of the fibers at three frequencies in the spectrometer band. We compare the radio positions with the potential coronal magnetic field extrapolated from SOHO/MDI data. Given the detected source site configuration and evolution, and the change of the fiber burst frequency range with time, we can also extract those coronal flux tubes where the high-frequency fiber bursts are situated even without decimeter imaging data. To this aim we use a kinetic simulation of whistler wave growth in sample flux tubes modeled by selected potential field lines and a barometric density model. The whistler wave model of fiber bursts accurately explains the observations on 28 Oct. 2003. A laterally extended system of low coronal loops is found to guide the whistler waves. It connects several neighboring active regions including the flaring AR 10486. For varying source sites the fiber bursts are emitted at the fundamental mode of the plasma frequency over the whole range (1200 – 300 MHz). The present event can be understood without assuming two different generation mechanisms for meter and decimeter wave fiber bursts. It gives new insight into particle acceleration and propagation in the low flare and post-CME corona.

Effect of coronal structure on loop oscillations: exponential profiles

Aims. The role of longitudinal structuring of the surrounding corona on the modes of oscillation of a coronal magnetic flux tube was studied in Donnelly et al. (2006) for a piecewise uniform profile. Here we investigate whether a more realistic continuous exponential profile changes the conclusions drawn from that paper. Methods. A partial differential equation is derived for the total pressure perturbation of the fast modes, which is then decomposed by separation of variables. The longitudinal part is solved numerically, obtaining a dispersion relation. These results are supported by an analytical investigation in terms of Bessel functions of purely imaginary order. Results. Structure in the interior of the loop shifts the frequencies of the modes (and may trap higher harmonics), an effect which can be understood by taking an averaged profile with a suitable weight. Structure in the environment modifies only slightly the frequencies, but displaces the cutoff frequency. The shift due to the structure in the fundamental period is small, but the ratio between the periods of the fundamental mode and its harmonics can be used to probe the structure. Conclusions. The results support our previous study in a more realistic, continuously varying profile and provide limits to the conclusions drawn in coronal seismology if an unstructured loop is used. Also, the ratio between the period of the fundamental kink (even) mode and its first (odd) harmonic is proven as an extra seismological tool for coronal loops.

A tenuous X-ray corona enveloping AE Aquarii

In this paper we propose that the observed unpulsed X-ray emission in AE Aquarii is the result of a very tenuous hot corona associated with the secondary star, which is pumped magnetohydrodynamically by the propeller action of the fast rotating white dwarf. It is shown that the closed coronal field of the secondary star envelops a substantial portion of the binary system, including the fast rotating magnetized white dwarf. This implies that the propeller outflow of material in AE Aquarii is initiated inside an enveloping magnetic cavity. The outflow crossing the secondary dead-zone field constitutes a β gen = (8πρv 2 /B 2 ) � 1 plasma, acting as a magnetohydrodynamic generator resulting in the induction of field-aligned currents in these closed magnetospheric circuits where βcir = (8πnkT/B 2 ) � 1. The Ohmic heating of the coronal circuit can readily account for a Tx � 10 7 K plasma in the coronal flux tubes connecting the generator and the stellar surface. Further, the bremsstrahlung losses of the thermal electrons in the coronal circuit can readily drive the observed unpulsed X-ray luminosity of L x ∼ 10 31 erg s −1 , which correlates with the luminosity and relatively large source implied by recent