A. J. Díaz

Analytical determination of coronal parameters using the period ratio P1/2P2

Context. In transverse coronal loop oscillations, two periodicities have been measured simultaneously and are interpreted as the fundamental kink mode (with period P1) and the first harmonic (with period P2). Deviations of the period ratio P1/2P2 from unity provide information about the extent of longitudinal structuring within the loop. Aims. Here we develop an analytical approximation that describes the shift in P1/2P2 in terms of the ratio L/Λc of the length 2L of a coronal loop and the density scale height Λc. Methods. We study the MHD wave equations in a low β plasma using the thin tube approximation. Disturbances are described by a differential equation which may be solved for various equilibrium density profiles, obtaining dispersion relations in terms of Bessel functions. These dispersion relations may be used to obtain analytical approximations to the periods P1 and P2 .W e also present a variational approach to determining the period ratio and show how the WKB method may be used. Results. Analytical approximations to the period ratio P1/2P2 are used to shed light on the magnitude of longitudinal structuring in a loop, leading to a determination of the density scale height. We apply our formula to the observations in Verwichte et al. (2004) and Van Doorsselaere et al. (2007), obtaining the coronal density scale height. Conclusions. Our simple formula and approximate approaches highlight a useful analytical tool for coronal seismology. We demonstrate that P1/2P2 is linked to the density scale height, with no need for estimates of other external parameters. Given the accuracy of current observations, our formula provides a convenient means of determining density scale heights.

Instability of twisted magnetic tubes with axial mass flows

Context. Recent observations of various kinds of jets in the solar atmosphere motivate studying the influence of mass flow on the stability of solar magnetic structures. Aims. We study the influence of axial mass flows on the stability of twisted magnetic flux tubes. Methods. We use the incompressible magnetohydrodynamic equations to get the dispersion relation governing the behaviour of normal modes in uniformly twisted magnetic tubes with sub-Alfvenic flows. The dispersion relation is then solved analytically and numerically to find stability criteria for twisted tubes with flow. Results. Two main important results are found. First, the axial mass flow reduces the threshold of kink instability in twisted magnetic tubes. Second, the twist of magnetic tubes leads to the Kelvin-Helmholtz instability of sub-Alfvenic flows for the harmonics with a large enough azimuthal wave number -m. Conclusions. The observed mass flow may trigger the kink instability in magnetic configurations that are near their stability threshold, leading to solar flares and coronal mass ejections. The effect is more significant for photospheric magnetic tubes than for coronal ones. Sub-Alfvenic flows undergo the Kelvin-Helmholtz instability in slightly twisted magnetic tubes if the azimuthal wavenumber is big enough.

Fast magnetohydrodynamic oscillations in a multifibril Cartesian prominence model

Observations of quiescent filaments show very fine structures which suggests that they can be composed of small-scale threads or fibrils. Two-dimensional, high-resolution observations point out that individual fibrils or groups of fibrils may oscillate independently with their own periods. In this paper, we study the fast magnetohydrodynamic modes of oscillation of multifibril Cartesian systems to represent the oscillations of the fibril structure of a real prominence. In the case of a system made of equal fibrils, our results show that the only non-leaky mode is the symmetric one, which means that all the fibrils oscillate in spatial phase with the same frequency. On the other hand, in a system made of non-equal fibrils, i.e. with different Alfven speeds, the results show that the amplitudes of oscillation are higher in the denser fibrils, that the frequency of oscillation of the only non-leaky mode is slightly smaller than that of the dominant fibril considered alone, and that all the fibrils also oscillate in phase.

Effect of coronal structure on loop oscillations

Aims. We investigate the influence of longitudinal structuring of the surrounding corona on the modes of oscillation of a coronal magnetic flux tube. Methods. A partial differential equation is derived for the total pressure perturbation of the fast modes and it is solved analytically in terms of Bessel functions, obtaining a dispersion relation. Results. The introduction of coronal structuring changes the cutoff frequency, enhancing coronal leakage, so even the fundamental kink mode may become leaky. Structure also modifies the loops oscillatory frequencies and may result in higher harmonics being trapped. Conclusions. Depending on the structuring, two competing effects take place: environmental structuring enhances leakage, while loop structuring helps confine the modes. This has important consequences for coronal seismology, leading to the absence of trapped modes for certain parameters and shifts in frequencies.

Fast MHD oscillations in line-tied homogeneous coronal loops

Loop oscillations have been abundantly reported in recent years. Earlier analytical studies of loop oscillations con- sider freely propagating waves, allowing for line-tying by a quantization of the wavenumber. Here we consider the rich spectrum of fast MHD modes (both standing and leaky) in coronal loops, allowing for line-tying and performing some comparisons with observational data. We point out that in a straight and homogeneous cylindrical flux tube there should be observational signa- tures of the excitation of higher order harmonics. Our results indicate that these modes become leaky with the addition of the chromospheric structure at the base of the loop. Leakage can be quite efficient in damping the oscillations for many of these high frequency (compared to fundamental) modes.

Fast magnetohydrodynamic oscillations in an elliptical coronal arcade

Aims. A model of a elliptically shaped coronal arcade with piecewise constant density is discussed to explore the effects of curvature on radially polarised fast modes. It is important to test whether the main results in the straight and cylindrical geometries can be extrapolated to these more complex equilibria. Methods. An equilibrium model for a force-free, line-tied elliptical arcade is introduced and a partial differential equation is derived for the velocity perturbation of the fast modes, which is solved analytically. The properties of the modes are studied in terms of the dispersion relation, which depends on the eccentricity, the arcade width, and the density contrast. Results. Modes mainly contained in the cavity below the arcade are also present, and have avoided crossings with the modes of the arcade. Even the fundamental mode becomes leaky due to curvature. Approximated relations are deduced for the frequency of the modes and the spatial structure is discussed, focusing on the different families through which a rich mode spectrum can be classified. Conclusions. The different types of modes of the spectrum are described and its relevance to observations is discussed. The periods obtained in Cartesian geometry provide a reasonable approximation, but this geometry lacks some other key ingredients: the damping rates are different and some types of modes present in the elliptical geometry are not sustained in the straight slab.

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.

Slow MHD oscillations in density structured coronal loops

Aims. Signals of stationary slow modes have been detected in observational data and modelled through numerical computations, comparing these results with the modes of a homogeneous tube. Here we explore the effect of structure along the magnetic field on the modes of oscillation of a coronal loop. Methods. We present a limit in which the slow mode is decoupled from the other magnetohydrodynamic modes, describing its behaviour in terms of a relatively simple partial differential equation. This equation is solved analytically and numerically for various longitudinal profiles. Results. For low density contrast between footpoints and apex, the modes of the structured tube are similar to the modes of the homogeneous tube, evolving regularly from them, with small modifications in frequency and spatial structure. As the density contrast is increased, the extrema are displaced towards the dense layers and the frequencies of the higher harmonics are strongly modified. Finally, as the ratio is increased further, two types of modes appear: modes approximately line-tied in the dense layer and modes with high amplitude in them (with avoided crossings between them in the dispersion diagrams). Conclusions. Different regimes can be identified, depending on the density contrast between the loop footpoints and its apex. This allows us to compare apparently different numerical results and understand their various features. Our analytical results are in accordance with current numerical simulations.

Effect of longitudinal density structure on a straight magnetic field modelling coronal arcade oscillations

Aims. Motivated by recent observations of oscillations in coronal arcades, we investigate analytically the influence of longitudinal structuring on the modes of oscillation of a straight coronal loop arcade. As a first step towards more complicated models, we use a simple structure to obtain analytical solutions. Methods. A partial differential equation is derived for the total pressure perturbation of the fast modes in a zero beta plasma and it is solved analytically. We first recover the results for a homogeneous structure, and then study an equilibrium with an exponentially structured density profile, solving it in terms of Bessel functions of non-integer order and exponential argument, thus obtaining a dispersion relation. The properties of this dispersion relation are discussed and some limits studied, leading to analytical approximations to the eigenfrequencies. Results. The introduction of longitudinal structuring results in a modification to the oscillatory frequencies of the modes of oscillation in such structures when compared with the uniform case. Regarding the oscillatory periods Pn, n = 1, 2 , . .. , the period ratios P1/2P2 and P1/3P3 are both shifted from unity. Other properties described in structured coronal loops are also found in an arcade: the occurrence of avoided crossings in the dispersion diagram and the displacement of the extrema towards the footpoints in the spatial structure of the eigenmodes. Conclusions. We show analytically for simple arcade modes that the shift in the fundamental period proves to be small, but the ratio P1/2P2 depends strongly on the density structure. Moreover, transversal propagation also shifts the ratio P1/2P2 from unity, so it can be used in the coronal seismology of arcades in which transversal propagation is present. We use the currently available observational data to illustrate this application.

Twisted magnetic tubes with field aligned flow - I. Linear twist and uniform longitudinal field

Aims. We study the equilibrium and stability of twisted magnetic flux tubes with mass flows along the field lines. Then, we focus on the stability and oscillatory modes of magnetic tubes with uniform twist B0 = B0(r/peϕ + ez) in a zero-β plasma, surrounded by a uniform, purely longitudinal field. Methods. First we investigate the possible equilibriums, and then consider the linearised MHD equations and obtain a system of two first-order differential equations. These are solved numerically, while analytical approximations involving confluent hypergeometric functions are found in the thin tube limit. Finally, new appropriate boundary conditions are deduced and the outer solution considered (with the apparition of cut-off frequencies). We use this to derive a dispersion relation, from which the frequencies of the normal modes can be obtained. Results. Regarding the equilibrium, the only value of the flow that satisfies the equations for this magnetic field configuration is a super-Alfvenic one. Then, we consider the normal modes of this configuration. The thin-tube approximation proves accurate for typical values, and it is used to prove that the equilibrium is unstable, unless the pitch is large. The stability criteria for twisted tubes are significantly lowered. Conclusions. The twisted tube is subject to the kink instability unless the pitch is very high, since the Lundquist criterion is significantly lowered. This is caused by the requirement of having a magnetic Mach number greater than 1, so the magnetic pressure balances the magnetic tension and fluid inertia. This type of instability might be observed in some solar atmospheric structures, like surges.