# Linear MHD Wave Propagation in Time-Dependent Flux Tube

## Abstract

In this article, we evaluate the time-dependent wave properties and the damping rate of propagating fast magneto-hydrodynamic (MHD) waves when energy leakage into a magnetised atmosphere is considered. By considering a cold plasma, initial investigations into the evolution of MHD wave damping through this energy leakage will take place. The time-dependent governing equations have been derived previously in Williamson and Erdélyi (2014a, *Solar Phys.***289**, 899 – 909) and are now solved when the assumption of evanescent wave propagation in the outside of the waveguide is relaxed. The dispersion relation for leaky waves applicable to a straight magnetic field is determined in both an arbitrary tube and a thin-tube approximation. By analytically solving the dispersion relation in the thin-tube approximation, the explicit expressions for the temporal evolution of the dynamic frequency and wavenumber are determined. The damping rate is, then, obtained from the dispersion relation and is shown to decrease as the density ratio increases. By comparing the decrease in damping rate to the increase in damping for a stationary system, as shown, we aim to point out that energy leakage may not be as efficient a damping mechanism as previously thought.

### Keywords

Magnetohydrodynamics Waves Waves, modes## 1 Introduction

The understanding of propagating magneto-hydrodynamic (MHD) waves, whilst just one facet of the investigation into the nature of the multitude of observed activity within the solar atmosphere, is an important tool for the description of energy transfer within solar environments and other astrophysical plasmas. The study of MHD waves has, for the most part, been restricted to the study of trapped waves, where the wave propagation is considered to be confined to the interior of the magnetic structure under consideration; for a full review of this theoretical work see *e.g.* Roberts (2000). Such an analysis is performed in this manner for good reason, the magnetic structures found in the chromosphere or the corona are unlikely to leak significant quantities of energy through the process as investigated and presented in this article. The wave propagation within such non-leaky structures is well described by the series of works by Roberts (1981) through to Edwin and Roberts (1983). Many other studies have expanded upon this framework, and reviews of MHD wave propagation in the context of solar magneto-seismology (SMS) are given by Ruderman and Erdélyi (2009), Wang (2011) and Mathioudakis, Jess, and Erdélyi (2013) for transverse, longitudinal, and Alfvén waves, respectively.

The introductory work, by *e.g.* Defouw (1976) or Roberts (1981), on MHD wave propagation laid the foundations for the analysis of propagating waves and is widely used in their current format. Most of these early works restricted themselves to models with purely real wave frequency, \(\omega\), representing a conservation of energy within the magnetic structure. Complex frequencies, to account for damping of these MHD waves, were considered by Wilson (1979) for a magnetic field-free environment, whilst Spruit (1982) and Cally (1986) made some of the first rigorous investigations into damping as a result of energy losses into the surrounding magnetic atmosphere. Later works have expanded upon the results found therein; perhaps the most relevant to this article were the efforts made of combining the damping through energy leakage and resonant absorption as investigated by Goossens and Hollweg (1993), Stenuit *et al.* (1999), and Goossens *et al.* (2009). For a review of the resonant absorption as discussed in these works, see *e.g.* Goossens, Erdélyi, and Ruderman (2011).

Time-dependent background plasmas are a growing area of study within the solar physics community. A number of recent studies (*e.g.*, Morton and Erdélyi, 2009, 2010; Morton, Hood, and Erdélyi, 2010; Erdélyi, Al-Ghafri and Morton, 2011) have made a series of investigations into dynamic plasmas and the standing wave modes found therein, whilst the first two works in this series, Williamson and Erdélyi (2014a, 2014b; Paper I and Paper II hereafter, respectively) discussed time-dependent propagating waves in coronal environments, showing that both the fast kink and the slow sausage wave modes are amplified exponentially in time for a temporally decreasing background density. However, both works assumed evanescent wave propagation. This assumption is now relaxed in the first part of this work, allowing for energy leakage into the magnetised atmosphere and allowing for a more generalised description of MHD wave propagation in astrophysical plasmas.

The work performed earlier in this series was carried out under the assumption of a coronal environment, *i.e.* the over-dense loop and the low plasma-beta assumptions (Papers I and II, respectively). Whilst such work is useful in describing the plentiful wave propagation in the solar atmosphere, it is less applicable in describing wave activity in the photosphere or other regions with a density ratio close to, or greater than, one. We here aim to generalise the study of MHD waves in time-dependent plasmas, allowing for the concept of energy leakage to be taken into account. Energy leakage is of primary concern in under-dense structures, such as coronal holes or sunspots, but will be present in most of the flux tubes found in solar structures.

## 2 Governing Equations

*i.e.*\(p_{\mathrm{i}}=p_{\mathrm{e}}\approx0\). In both regions, the unperturbed magnetic field (\(B_{0z}\)) is constant and vertical. Neglecting the plasma pressure directly implies that the magnetic fields are identical inside and outside the waveguide. In this work, however, we considered an exterior plasma density that is greater than or equal in magnitude to the initial interior density, \(\rho_{\mathrm{i}0}\) allowing for the propagation of leaky MHD waves. We considered such solutions as are appropriate in the analysis of

*e.g.*coronal holes and sunspots. The main emphasis of this analysis is on the time-dependent background density, where the density is assumed to have the form

*e.g.*Hain and Lüst, 1958). At this point we can, once again, solve the two governing equations for the perturbed total pressure and the radial displacement. The differential equation for the perturbed total pressure can therefore be written as

## 3 Dispersion Relation, Frequency, and Wavenumber

*i.e.*the radial displacement and the total pressure perturbation, across the discontinuity in the magnetic field, we can now express the dispersion relation for leaky waves in a steady temporally evolving, time-dependent flux tube as

*i.e.*\(V_{0z}\ll V_{\mathrm{A}}\)), we can reduce the real part of the dispersion relation from

*i.e.*\(\theta (0,t)=-\omega t\), Equation (8) becomes

*e.g.*Corless

*et al.*(1996).

*i.e.*past the point of rapid decrease, can be compared to the point at which the interior of the flux tube is largely evacuated of density and, hence, the change in frequency past that point is negligible. A similar property can be derived for the dynamic wavenumber in that there is the initial rapid decrease, before tending to a settled limit after the point where the tube has been effectively evacuated of plasma, see Figure 2.

*i.e.*flute) wave modes, \(m\geq\)2, have the same manner of evolution as their counterparts in the over-dense loop and, hence, we can conclude that all the wave modes, \(m\geq3\) will be damped without the need for the energy losses into the magnetic atmosphere. The amplitude of the \(m=\)2 wave mode is plotted in Figure 4 for comparison with the \(m=\)1 mode.

## 4 Damping Coefficient

*i.e.*that the damping rate is low. An algebraic approach would see us solve Equation (8) for the damping coefficient \(\gamma\), ignoring terms of order \(\gamma^{2}\). However, for this work we used the following differential approach, as detailed in

*e.g.*Krall and Trivelpiece (1973). In what follows, only the \(m=1\) wave mode is considered; this is a result of the increasingly insignificant levels of damping known to exist in the higher-order wave modes,

*i.e.*the damping is of the order of \((M_{\mathrm{e}}R)^{m}\). The quantity \(\gamma\) can be written as

With the explicit forms of the frequency and wavenumber determined, we now make analytical progress with determining the evolution of the damping coefficient, \(\gamma\). Substituting explicit forms of the dynamic frequency and wavenumber into Equation (15) gives an expression from which it is difficult to clearly see the temporal changes as a result of the decreasing density in time. Hence, we plot the evolution of the damping coefficient with respect to time and height above the driving point.

As a result of the moderate-activity approximation, the analysis with regard to the height above the driving point is potentially inaccurate, therefore, different ratios of the initial Alfvén speed and the background flow can change the initial evolution of the quantities discussed above. However, all the cases tend to the same limit and, therefore, we can consider them to be consistent.

*e.g.*Equation (66) in Goossens

*et al.*(2009). By rewriting Equation (16) in terms of the density ratio, \(\chi\), it is possible to write

## 5 Conclusions

We here relaxed the assumption of evanescent wave propagation in the magnetised atmosphere for temporally evolving MHD waveguides. The algebraic form of both the interior and the exterior wave propagation, as well as the damping coefficient were determined and their evolution in time and height explored. The moderate-activity approximation was applied to find the real part of the frequency and the wavenumber. Using the expressions for the dynamic frequency and wavenumber, the full evolution of the propagating wave modes can be simply shown. The various wave modes all follow the same manner of propagation as found in Paper I for the over-dense loop approximation. The \(m=\)1 kink wave is amplified in an approximately exponential manner. This amplification continues without any limiting factor, as was suggested in the case of the over-dense loop, beyond the possibility of wave propagation in an empty flux tube. The \(m=2\) flute wave mode is amplified to a constant level after a few characteristic density change periods. The higher order, \(\geq3\), wave modes are damped after small number of characteristic time periods of density change, with the damping rate increased for higher values of the azimuthal wavenumber, \(m\).

The main focus of this work was on the damping coefficient \(\gamma\) and its evolution in time for time-dependant waveguides. The damping coefficient \(\gamma\) was shown to decrease exponentially in time as the flux tube evacuates. The damping as a result of leaky waves can therefore be considered to be negligible after a small number of characteristic density change periods. By varying several characteristic parameters, it was possible to obtain results that matched the stationary case in their evolution, including an initial increase in the damping rate before the exponential decrease. Analytical attempts to further determine a critical value for this turning point have yet to yield results and as such require further investigation.

Comparison of these results to those found in a stationary system showed a distinct difference between the damping in the case of a stationary plasma and a dynamic plasma. This difference can be mainly attributed to the constant vertical wavenumber in the stationary model, whereas in this dynamic model the temporal evolution of the wavenumber results in a decrease of the damping coefficient.

Further investigations to expand these findings are required, *e.g.*, inclusion of a finite plasma pressure and relaxing of the thin tube approximation, in order to make this an even more useful tool for investigating the magneto-seismology of photospheric structures.

## Notes

### Acknowledgements

RE is grateful to NSF, Hungary (OTKA, Ref. No. K83133) for support received.

### References

- Cally, P.S.: 1986, Leaky and non-leaky oscillations in magnetic flux tubes.
*Solar Phys.***103**, 277. ADSCrossRefGoogle Scholar - Corless, R.M., Gonnet, G.H., Hare, D.E.G., Jeffrey, D.J., Knuth, D.E.: 1996, On the Lambert W function.
*Adv. Comput. Math.***5**, 329. MathSciNetCrossRefMATHGoogle Scholar - Defouw, R.J.: 1976, Wave propagation along a magnetic tube.
*Astrophys. J.***209**, 266. ADSCrossRefGoogle Scholar - Edwin, P.M., Roberts, B.: 1983, Wave propagation in a magnetic cylinder.
*Solar Phys.***88**, 179. ADSCrossRefGoogle Scholar - Erdélyi, R., Al-Ghafri, K.S., Morton, R.J.: 2011, Damping of longitudinal magneto-acoustic oscillations in slowly varying coronal plasma.
*Solar Phys.***272**, 73. ADSCrossRefGoogle Scholar - Goossens, M., Hollweg, J.V.: 1993, Resonant behavior of MHD waves on magnetic flux tubes. IV – Total resonant absorption and MHD radiating eigenmodes.
*Solar Phys.***145**, 19. ADSCrossRefGoogle Scholar - Goossens, M., Erdélyi, R., Ruderman, M.: 2011, Resonant MHD waves in the solar atmosphere.
*Space Sci. Rev.***158**, 289. ADSCrossRefGoogle Scholar - Goossens, M., Terradas, J., Andries, J., Arregui, I., Ballester, J.L.: 2009, On the nature of kink MHD waves in magnetic flux tubes.
*Astron. Astrophys.***503**, 213. ADSCrossRefMATHGoogle Scholar - Hain, K., Lüst, R.: 1958, Zur Stabilität zylindersymmetrischer Plasmakonfigurationen mit Volumenströmen.
*Z. Naturforsch. A***13**, 936. ADSCrossRefMATHGoogle Scholar - Krall, N.A., Trivelpiece, A.W.: 1973,
*Principles of Plasma Physics*,*International Student Edition*, McGraw-Hill, New York. Section 8.6.2. Google Scholar - Mathioudakis, M., Jess, D.B., Erdélyi, R.: 2013, Alfvén waves in the solar atmosphere. From theory to observations.
*Space Sci. Rev.***175**, 1. ADSCrossRefGoogle Scholar - Morton, R.J., Erdélyi, R.: 2009, Transverse oscillations of a cooling coronal loop.
*Astrophys. J.***707**, 750. ADSCrossRefGoogle Scholar - Morton, R.J., Erdélyi, R.: 2010, Application of the theory of damping of kink oscillations by radiative cooling of coronal loop plasma.
*Astron. Astrophys.***519**, A43. CrossRefGoogle Scholar - Morton, R.J., Hood, A.W., Erdélyi, R.: 2010, Propagating magneto-hydrodynamic waves in a cooling homogeneous coronal plasma.
*Astron. Astrophys.***512**, A23. ADSCrossRefGoogle Scholar - Roberts, B.: 1981, Wave propagation in a magnetically structured atmosphere I.
*Solar Phys.***69**, 27. ADSCrossRefGoogle Scholar - Roberts, B.: 2000, Waves and oscillations in the corona (invited review).
*Solar Phys.***193**, 139. ADSCrossRefGoogle Scholar - Ruderman, M., Erdélyi, R.: 2009, Transverse oscillations of coronal loops.
*Space Sci. Rev.***149**, 199. ADSCrossRefGoogle Scholar - Spruit, H.C.: 1982, Propagation speeds and acoustic damping of waves in magnetic flux tubes.
*Solar Phys.***75**, 3. ADSCrossRefGoogle Scholar - Stenuit, H., Tirry, W.J., Keppens, R., Goossens, M.: 1999, Leaky and resonantly damped flux tube modes reconsidered.
*Astron. Astrophys.***342**, 863. ADSGoogle Scholar - Wang, T.: 2011, Standing slow-mode waves in hot coronal loops: observations, modeling, and coronal seismology.
*Space Sci. Rev.***158**, 397. ADSCrossRefGoogle Scholar - Williamson, A., Erdélyi, R.: 2014a, Linear MHD wave propagation in time-dependent flux tube. I. Zero plasma-\(\beta\).
*Solar Phys.***289**, 899. ADSCrossRefGoogle Scholar - Williamson, A., Erdélyi, R.: 2014b, Linear MHD wave propagation in time-dependent flux tube. II. Finite plasma beta.
*Solar Phys.***289**, 1193. ADSCrossRefGoogle Scholar - Wilson, P.R.: 1979, Hydromagnetic wave modes in magnetic flux tubes.
*Astron. Astrophys.***71**, 9. ADSMATHGoogle Scholar

## Copyright information

**Open Access** This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.