Normal modes of magnetic flux tubes and dissipation

Abstract. Loop oscillations have been abundantly reported in recent years. Earlier analytical studies of loop oscillations consider 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 signatures 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.

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“…This is a particular form of other well-known dispersion relations (Edwin & Roberts 1983;Cally 1986;Karami et al 2002).…”

Section: Analytical Solutionmentioning

confidence: 84%

Fast MHD oscillations in line-tied homogeneous coronal loops

Díaz

Oliver

Ballester

et al. 2004

A&A

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“…The coefficients ε, β, γ and ɛ are determined by the boundary conditions. From both Karami et al (2002) and Erdélyi & Carter (2006), the necessary boundary conditions are that: at the boundaries r = a and r = R , both the total Lagrangian pressure and δ v r should be continuous. These conditions yield to the dispersion relations for surface, m 2 0 > 0, and hybrid, m 2 0 < 0, modes which are same as the results obtained by Erdélyi & Carter (2006) in equations (28a) and (28b), respectively.…”

Section: Equations Of Motionmentioning

confidence: 99%

“…They found that the existence of inhomogeneities in the form of structuring of the magnetic field enables loops to act as wave guides for a variety of different modes. Karami, Nasiri & Sobouti (2002) used the model of Edwin & Roberts (1983), and solved numerically the dispersion relation for each mode in its full generality. They obtained that in the presence of weak viscous and ohmic dissipations, the damping rate is inversely proportional to the Reynolds and Lundquist numbers, R and S , respectively.…”

Section: Introductionmentioning

confidence: 99%

The effects of twisted magnetic field on coronal loop oscillations and dissipation

Karami

Barin

2009

Monthly Notices of the Royal Astronomical Society

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The standing magnetohydrodynamic (MHD) modes in a cylindrical incompressible magnetic flux tube modelled as a straight core surrounded by a magnetically twisted annulus, both embedded in a straight ambient external field, are considered. The dispersion relation for the MHD waves is derived and solved numerically to obtain the frequencies of both the kink (m = 1) and fluting (m = 2, 3) waves. Damping rates due to both viscous and resistive dissipations in the presence of the twisted magnetic field are derived and solved numerically for both the kink and fluting waves.

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“…See, e.g., Karami et al (2002) and Safari et al (2006) for details. Let us also emphasize that the present form of Eq.…”

Section: Equations Of Motionsmentioning

confidence: 99%

Fast kink modes of longitudinally stratified coronal loops

Safari¹,

Nasiri

Sobouti³

2007

A&A

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Aims. We investigate the standing kink modes of a cylindrical model of coronal loops. The density is stratified along the loop axis and changes discontinuously at the surface of the cylinder. The periods and mode profiles are studies with their deviation from those of the unstratified loops. The aim is to extract information on the density scale heights prevailing in the solar corona. Methods. The problem is reduced to solving a single second-order partial differential equation for δB z (r, z), the longitudinal component of the Eulerian perturbations of the magnetic field. This equation, in turn, is separated into two second-order ordinary, differential equations in r and z that are, however, connected through a dispersion relation between the frequencies and the longitudinal wave numbers. In the thin tube approximation, the eigensolutions are obtained by a perturbation technique, where the perturbation parameter is the density stratification parameter. Otherwise the problem is solved numerically. Results. 1) On functional dependencies of the dispersion relation the radial wave number is independent of the longitudinal stratification. 2) We verify the earlier computational finding that the first overtone frequencies increase with increasing stratification and the observational finding (from analysis of TRACE data) that the ratio of the first to the fundamental overtone frequency decreases with increasing stratification. The method we use to arrive at these conclusions, however, is more analytical than computational, and yet our numerical results agree with the earlier results.3) The mode profiles depart from the sinusoidal mode profiles of the unstratified loops. This departure and its dependence on the scale height is obtained, and might serve to determine scale heights once high resolution data become available.

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Normal modes of magnetic flux tubes and dissipation

Cited by 8 publications

References 16 publications

Fast MHD oscillations in line-tied homogeneous coronal loops

Fast MHD oscillations in line-tied homogeneous coronal loops

The effects of twisted magnetic field on coronal loop oscillations and dissipation

Fast kink modes of longitudinally stratified coronal loops

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