Nonlinear Instability of kink oscillations due to shear motions

First results from a high-resolution three-dimensional nonlinear numerical study of the kink oscillation are presented. We show in detail the development of a shear instability in an untwisted line-tied magnetic flux tube. The instability produces significant deformations of the tube boundary. An extended transition layer may naturally evolve as a result of the shear instability at a sharp transition between the flux tube and the external medium. We also discuss the possible effects of the instability on the process of resonant absorption when an inhomogeneous layer is included in the model. One of the implications of these results is that the azimuthal component of the magnetic field of a stable flux tube in the solar corona, needed to prevent the shear instability, is probably constrained to be in a very specific range

Kelvin-Helmholtz instability in coronal magnetic flux tubes due to azimuthal shear flows

Transverse oscillations of coronal loops are often observed and have been theoretically interpreted as kink magnetohydrodynamic (MHD) modes. Numerical simulations by Terradas et al. (2008, ApJ 687, L115) suggest that shear flows generated at the loop boundary during kink oscillations could give rise to a Kelvin-Helmholtz instability (KHI). Here, we investigate the linear stage of the KHI in a cylindrical magnetic flux tube in the presence of azimuthal shear motions. We consider the basic, linearized MHD equations in the beta = 0 approximation, and apply them to a straight and homogeneous cylindrical flux tube model embedded in a coronal environment. Azimuthal shear flows with a sharp jump of the velocity at the cylinder boundary are included in the model. We obtain an analytical expression for the dispersion relation of the unstable MHD modes supported by the configuration, and compute analytical approximations of the critical velocity shear and the KHI growth rate in the thin tube limit. A parametric study of the KHI growth rates is performed by numerically solving the full dispersion relation. We find that fluting-like modes can develop a KHI in time-scales comparable to the period of kink oscillations of the flux tube. The KHI growth rates increase with the value of the azimuthal wavenumber and decrease with the longitudinal wavenumber. However, the presence of a small azimuthal component of the magnetic field can suppress the KHI. Azimuthal motions related to kink oscillations of untwisted coronal loops may trigger a KHI, but this phenomenon has not been observed to date. We propose that the azimuthal component of the magnetic field is responsible for suppressing the KHI in a stable coronal loop. The required twist is small enough to prevent the development of the pinch instability.Comment: Submitted in Ap

Transverse kink oscillations in the presence of twist

Magnetic twist is thought to play an important role in coronal loops. The effects of magnetic twist on stable magnetohydrodynamic (MHD) waves is poorly understood because they are seldom studied for relevant cases. The goal of this work is to study the fingerprints of magnetic twist on stable transverse kink oscillations. We numerically calculated the eigenmodes of propagating and standing MHD waves for a model of a loop with magnetic twist. The azimuthal component of the magnetic field was assumed to be small in comparison to the longitudinal component. We did not consider resonantly damped modes or kink instabilities in our analysis. For a nonconstant twist the frequencies of the MHD wave modes are split, which has important consequences for standing waves. This is different from the degenerated situation for equilibrium models with constant twist, which are characterised by an azimuthal component of the magnetic field that linearly increases with the radial coordinate. In the presence of twist standing kink solutions are characterised by a change in polarisation of the transverse displacement along the tube. For weak twist, and in the thin tube approximation, the frequency of standing modes is unaltered and the tube oscillates at the kink speed of the corresponding straight tube. The change in polarisation is linearly proportional to the degree of twist. This has implications with regard to observations of kink modes, since the detection of this variation in polarisation can be used as an indirect method to estimate the twist in oscillating loops

The effect of longitudinal flow on resonantly damped kink oscillations

The most promising mechanism acting towards damping the kink oscillations of coronal loops is resonant absorption. In this context most of previous studies neglected the effect of the obvious equilibrium flow along magnetic field lines. The flows are in general sub-Alfv\'enic and hence comparatively slow. Here we investigate the effect of an equilibrium flow on the resonant absorption of linear kink MHD waves in a cylindrical magnetic flux tube with the aim of determining the changes in the frequency of the forward and backward propagating waves and in the modification of the damping times due to the flow. A loop model with both the density and the longitudinal flow changing in the radial direction is considered. We use the thin tube thin boundary (TTTB) approximation in order to calculate the damping rates. The full resistive eigenvalue problem is also solved without assuming the TTTB approximation. Using the small ratio of flow and Alfv\'en speeds we derive simple analytical expressions to the damping rate. The analytical expressions are in good agreement with the resistive eigenmode calculations. Under typical coronal conditions the effect of the flow on the damped kink oscillations is small when the characteristic scale of the density layer is similar or smaller than the characteristic width of the velocity layer. However, in the opposite situation the damping rates can be significantly altered, specially for the backward propagating wave which is undamped while the forward wave is overdamped

The statistical significance of the N-S asymmetry of solar activity revisited

The main aim of this study is to point out the difficulties found when trying to assess the statistical significance of the North-South asymmetry (hereafter SSNSA) of the most usually considered time series of solar activity. First of all, we distinguish between solar activity time series composed by integer or non-integer and dimensionless data, or composed by non-integer and dimensional data. For each of these cases, we discuss the most suitable statistical tests which can be applied and highlight the difficulties to obtain valid information about the statistical significance of solar activity time series. Our results suggest that, apart from the need to apply the suitable statistical tests, other effects such as the data binning, the considered units and the need, in some tests, to consider groups of data, affect substantially the determination of the statistical significance of the asymmetry. Our main conclusion is that the assessment of the statistical significance of the N-S asymmetry of solar activity is a difficult matter and that an absolute answer cannot be given, since many different effects influence the results given by the statistical tests. In summary, the quantitative results about the statistical significance of the N-S asymmetry of solar activity provided by different authors, as well as the studies about its behaviour, must be considered with care because they depend from the chosen values of different parameters or from the considered units.Comment: Astronomy and Astrophysics Latex, 9 pages, 4 figure

Transverse oscillations of two coronal loops

We study transverse fast magnetohydrodynamic waves in a system of two coronal loops modeled as smoothed, dense plasma cylinders in a uniform magnetic field. The collective oscillatory properties of the system due to the interaction between the individual loops are investigated from two points of view. Firstly, the frequency and spatial structure of the normal modes are studied. The system supports four trapped normal modes in which the loops move rigidly in the transverse direction. The direction of the motions is either parallel or perpendicular to the plane containing the axes of the loops. Two of these modes correspond to oscillations of the loops in phase, while in the other two they move in antiphase. Thus, these solutions are the generalization of the kink mode of a single cylinder to the double cylinder case. Secondly, we analyze the time-dependent problem of the excitation of the pair of tubes. We find that depending on the shape and location of the initial disturbance, different normal modes can be excited. The frequencies of normal modes are accurately recovered from the numerical simulations. In some cases, because of the simultaneous excitation of several eigenmodes, the system shows beating and the phase lag between the loops is $\pi/2$.Comment: Accepted for publication in The Astrophysical Journa

Transverse oscillations of systems of coronal loops

We study the collective kinklike normal modes of a system of several cylindrical loops using the T-matrix theory. Loops that have similar kink frequencies oscillate collectively with a frequency which is slightly different from that of the individual kink mode. On the other hand, if the kink frequency of a loop is different from that of the others, it oscillates individually with its own frequency. Since the individual kink frequency depends on the loop density but not on its radius for typical 1 MK coronal loops, a coupling between kink oscillations of neighboring loops take place when they have similar densities. The relevance of these results in the interpretation of the oscillations studied by \citet{schrijver2000} and \citet{verwichte2004}, in which transverse collective loop oscillations seem to be detected, is discussed. In the first case, two loops oscillating in antiphase are observed; interpreting this motion as a collective kink mode suggests that their densities are roughly equal. In the second case, there are almost three groups of tubes that oscillate with similar periods and therefore their dynamics can be collective, which again seems to indicate that the loops of each group share a similar density. All the other loops seem to oscillate individually and their densities can be different from the rest