## Abstract

In the last decade we have been overwhelmed by an avalanche of discoveries of magnetohydrodynamic (MHD) waves by the Solar and Heliospheric Observatory and Transition Region and Coronal Explorer observatories. Both standing and propagating versions of fast magnetoacoustic and slow magnetoacoustic MHD waves have been detected. Information on the damping times and damping distances of these waves is less detailed and less accurate than that on periods and amplitudes. Nevertheless, observations show the damping times and damping lengths are often short. Also, different types of MHD waves in different types of magnetic structures likely require different damping mechanisms. The phenomenon of fast damping is well documented for the standing fast magnetosonic kink waves in coronal loops. This paper concentrates on standing fast magnetosonic waves. It reports on results on periods and damping times due to resonant absorption in one-dimensional and two-dimensional models of coronal loops. Special attention is given to multiple modes.

## 1. Introduction

Transverse coronal loop oscillations, triggered by explosive events such as flares or filament eruptions, were first observed by the extreme ultraviolet telescope on board of the Transition Region and Coronal Explorer (TRACE) satellite (Aschwanden *et al*. 1999; Nakariakov *et al*. 1999). Since then, several similar events have been reported and studied (Aschwanden *et al*. 2002; Schrijver *et al*. 2002). These oscillations have been interpreted in terms of magnetohydrodynamics (MHD) fast kink-modes of a cylindrical coronal flux tube by Nakariakov *et al*. (1999) and Nakariakov & Ofman (2001). This interpretation in terms of discrete modes of oscillation opens a way to coronal seismology, i.e. the determination of unknown physical properties of the corona by comparison of observed and predicted properties of waves and oscillations, as first suggested by Uchida (1970) and Roberts *et al*. (1984).

For most observed transverse coronal loop oscillations the amplitude has been found to be strongly damped, with an exponential decay time of a few periods. This observational fact might have some important physical consequences, both for coronal seismology as for wave based coronal heating theories, once we have identified the correct damping mechanism(s). Nevertheless, the cause of the rapid damping is still a matter of considerable debate. Currently, several mechanisms are under study, among them phase mixing of Alfvén waves (Heyvaerts & Priest 1983; Nakariakov *et al*. 1999; Ofman & Aschwanden 2002); and resonant absorption of waves (Hollweg & Yang 1988; Goossens *et al*. 2002; Ruderman & Roberts 2002; Aschwanden *et al*. 2003). Observational tests with TRACE data, by Ofman & Aschwanden (2002), suggested that the scaling law of the damping time as a function of other physical parameters, such as the loop length and the oscillation period, seem to favour phase mixing. However, as pointed out by Goossens *et al*. (2002), damping of quasi-mode kink oscillations by resonant absorption gives a consistent explanation of the rapid decay of the observed coronal loop oscillations in the sense that by assuming reasonable values of the unknown parameters (such as density contrast and inhomogeneity length scale) appropriate damping times can be obtained. An additional attraction of resonant absorption, as a damping mechanism, is that quasi-mode damping is fully consistent with the classical estimates of very large coronal Reynolds numbers of the order of 10^{14} (e.g. Priest 1982; Goedbloed & Poedts 2004).

This paper concentrates on kink eigenmode oscillations in one-dimensional and two-dimensional equilibrium models of coronal loops. Periods and damping times due to resonant absorption are presented. It is seen that the periods and damping times obtained from fully non-uniform models can differ substantially from those obtained for thin boundaries. An observational test of resonant absorption, for a sample of 11 oscillating loops, with results from one-dimensional equilibrium models, leads to values that are commensurable with observations, within the observational limitations. The generalization of the thin boundary theory to longitudinally stratified flux tubes and the very recent numerical results for resonantly damped modes in two-dimensional equilibria in models with thick non-uniform transitional layers are presented. Implications for coronal seismology are discussed. Furthermore, we comment on the importance of recent observations of multi mode coronal loop oscillations. Multi mode oscillations were detected for the first time by Verwichte *et al*. (2004). They found that two loops were oscillating in both the fundamental and the first overtone mode. The loops under study by Verwichte *et al*. (2004) stretch out up to a distance of about one density scale height. It should thus not come as a surprise that they are influenced by longitudinal density stratification. Using the method outlined by Andries *et al*. (2005*a*) we show that the stratification of the density in the longitudinal direction has a different effect on the fundamental as on the first overtone. The main result is that in a stratified atmosphere the oscillation period of the fundamental mode should be less than double that of the first overtone mode. The observed values of the period ratio can thus be used as a seismological tool to determine the longitudinal density stratification in coronal loops.

## 2. Equilibrium and linear MHD waves

In our first attempt we use a straight cylindrical one-dimensional flux tube as a model for the coronal loops. In a system of cylindrical coordinates (*r*, *φ*, *z*) with the *z*-axis coinciding with the axis of the cylinder (loop), the equilibrium quantities, magnetic field * B*=(0,

*B*

_{φ}(

*r*),

*B*

_{z}(

*r*)), pressure

*p*(

*r*) and density

*ρ*(

*r*) are functions of the radial distance only. They satisfy the radial force balance equation(2.1)This is one equation for four scalar functions, which does not involve density. Consequently, the density profile can be chosen arbitrarily. Since the plasma pressure is much smaller than the magnetic pressure in the corona, it is a good approximation to neglect plasma pressure. This classic

*β*=0 approximation removes the slow waves from the analysis. When the magnetic field is straight,

*=*

**B***B*(

*r*)

**1**

_{z}, equation (2.1) also implies that the magnetic field is uniform. The coronal loop is then a density enhancement in an almost homogeneous field. Later on, we shall move towards two-dimensional equilibrium models, where the density depends both on

*r*and

*z*.

Since in one-dimensional equilibrium models the equilibrium quantities depend on *r* only, the perturbed quantities can be Fourier-analysed with respect to the ignorable coordinates, *φ*- and *z*-, and put proportional to . Here, *m* (an integer) and *k*_{z} are the azimuthal and axial wave numbers. The observed coronal loop oscillations show no nodes in the *z*-direction so that *k*_{z}=*π*/*L*, where *L* is the length of the loop. For *m*=1, the waves are called *kink modes*. Since the axis of the loop is displaced, the oscillations have to be kink mode oscillations with *m*=1.

The relevant equations for the linear motions of a pressureless plasma superimposed on a static one-dimensional cylindrical equilibrium model with a straight magnetic field are(2.2)(2.3)(2.4)(2.5)*ξ* is the Lagrangian displacement and *P*′ is the Eulerian perturbation of total pressure, is the Alfvén speed and is the Alfvén frequency. For a one-dimensional cylindrical equilibrium model with a straight magnetic field, *ξ*_{φ} is the relevant component for Alfvén waves and *ξ*_{r} is the relevant component for the fast waves.

Consider now the equations as normal mode equations and note that the differential equations have a regular singular point at the position, where , or, consequently, at the resonant position *r*_{A}, where *ω*=*ω*_{A}(*r*_{A}). This singularity and the fact that *ω*_{A}(*r*) is a function of position, give rise to a continuous range in the spectrum which is associated with resonant Alfvén waves with singular spatial solutions in ideal MHD. This is the so-called Alfvén continuum. The Alfvén continuum waves imply that in ideal MHD each magnetic surface can oscillate at its own Alfvén continuum frequency. In dissipative MHD, the singular solutions are replaced with large but finite solutions (see Goossens *et al*. 1995; Tirry & Goossens 1996).

For *m*=0, the eigenmodes are decoupled into torsional Alfvén continuum eigenmodes with *ξ*_{r}=0, *P*′=0, *ξ*_{φ}≠0 and discrete fast eigenmodes with *ξ*_{r}≠0, *P*′≠0, *ξ*_{φ}=0. There is no interaction between the Alfvén waves and magnetosonic waves. However, for *m*=1 (as a matter of fact ) pure magnetosonic waves do not exist, since waves with *ξ*_{r}≠0, *P*′≠0 necessarily have *ξ*_{φ}≠0. Consequently, fast discrete eigenmodes with an eigenfrequency in the Alfvén continuum couple to a local Alfvén continuum eigenmode and produce quasi-modes.

When the density is uniform in the internal and external region and changes discontinuously at the loop radius *r*=*R*, the dispersion relation for these modes can be written down analytically. In the ‘long-wave’ approximation (), the frequency can be calculated explicitly as(2.6)The indices ‘i’ and ‘e’ refer to internal and external, respectively. In the ‘thin tube’ (or long wave) approximation, the frequencies of the eigenoscillations are independent of the azimuthal wave number *m*. An exception is the sausage mode (*m*=0), in which no long-wavelength limit can be obtained. For small tubes, the sausage mode becomes leaky.

The important point to note from equation (2.6) is that the eigenfrequency of the fundamental kink mode is in between the external and internal Alfvén frequency. Hence, when the discontinuous transition from *ρ*_{i} to *ρ*_{e} is replaced with a continuous variation in a non-uniform transitional layer of length *l*, the fundamental kink mode has its eigenfrequency in the Alfvén continuum and couples to an Alfvén continuum mode. The obvious conclusion is that the classic kink mode is always a resonantly damped quasi-mode. This result is independent of the long-wave assumption that was used to obtain equation (2.6). All fast body kink eigenmodes have frequencies in between the external and internal Alfvén frequency (Edwin & Roberts 1983). Consequently, all these kink modes are resonantly damped quasi-modes when a non-uniform transitional layer connects the internal and external plasmas.

## 3. Damping by resonant absorption

To the best of our knowledge, there are only a limited number of papers that deal with the calculation of the damping rates and decay times of quasi-modes in cylindrical flux tubes, and their dependence on the length scale of the inhomogeneity and the radius of the loop. The earliest paper is by Hollweg & Yang (1988). A more detailed discussion is given by Ruderman & Roberts (2002). Goossens *et al*. (1992) derived an approximate analytical expression for the damping rate of quasi-modes in cylindrical flux tubes with ‘thin’ transitional layers, by using connection formulae. It is important to point out that analytical solutions to equations (2.2)–(2.5) do not exist in general for non-uniform equilibrium models. The dispersion relation cannot be written down in closed analytical form, except possibly for special choices of the equilibrium profiles. Analytic progress is still possible when the non-uniform transitional layer [*R*−*l*/2,*R*+*l*/2] of thickness *l* is considered to be ‘thin’. A thin non-uniform layer means that . Because of the singularity in equations (2.2)–(2.5), large variations occur in the non-uniform transitional layer. In fact, for real frequencies in the Alfvén continuum [*ω*_{A,i},*ω*_{A,e}], the amplitude becomes unbounded, where *ω*=*ω*_{A}(*r*). The variations in the resonant layer can be expressed as jump conditions over the resonant layer (e.g. Goossens *et al*. 1995; Tirry & Goossens 1996)(3.1)(3.2)where is the slope of the density profile normalized to the width of the layer and to the total density difference. It is a parameter that depends only on the shape of the density profile. The dispersion relation can be written as(3.3)We then search for solutions in the neighbourhood of the solution for the discontinuous model *ω*=*ω*_{0}+δ*ω* and develop *D*(*ω*) in a Taylor series to find the imaginary frequency shift(3.4)This relation shows that under the thin tube thin boundary (TTTB) approximation the damping rate is linearly proportional to the length-scale of the inhomogeneity *l*/*R*. It was first obtained by Hollweg & Yang (1988) for nearly perpendicular propagation of a surface wave on a Cartesian interface. Hollweg & Yang (1988) applied the result to coronal loops, as the limit of nearly perpendicular propagation in Cartesian geometry coincides with the thin tube limit in cylindrical geometry. They concluded, more than a decade before the oscillations were observed, that oscillations of coronal loops would be damped effectively with an e-folding time of two periods. An analytic expression for the damping rate can be obtained when the ‘thin tube’ and ‘thin boundary’ approximations are combined. In terms of observable quantities this expression becomes(3.5)Ruderman & Roberts (2002) related equation (3.5) to the observed damping of coronal loop oscillations and used it to estimate the inhomogeneity length-scale from the observed period and damping time for one case.

Considering *ρ*_{e}/*ρ*_{i}=0.1 as an estimate for the density ratio, Goossens *et al*. (2002) calculated the inhomogeneity scale-length for a set of 11 loop oscillations and concluded that quasi-mode damping gives a consistent explanation of the fast decay of the observed coronal loop oscillations, since appropriate damping times can be obtained with reasonable values of the unknown parameters. However, caution is called for as the values found for *l*/*R* are not entirely consistent with the assumption of a thin boundary layer used to obtain equation (3.5). Hence, there is an obvious need to relax the assumption of a thin boundary and to calculate eigenmodes of fully non-uniform loops. The results of such computations are described in §4.

## 4. Eigenmode calculations in fully non-uniform one-dimensional equilibrium models

This section presents results on periods and damping times of resonantly damped kink mode oscillations in fully non-uniform one-dimensional cylindrical equilibrium models. The calculation of quasi-modes for one-dimensional highly non-uniform models is more involved than the procedure described in §3. This is not primarily due to the thickness of the inhomogeneous layer itself. A numerical integration routine could be used to integrate the set of differential equations away from the resonance, while the jump conditions or the analytical dissipative solutions could be used to cross the resonant layer. The real problem resides in the fact that the damping increases as the non-uniform layer becomes thicker. The jump relations, used in the previous section, are derived for weak damping and cannot be generalized easily to strong damping. In the case of strong damping, the dissipative MHD equations need to be solved numerically. For that purpose, Van Doorsselaere *et al*. (2004) used the large-scale eigenvalue solver for the dissipative Alfvén spectrum (LEDA) code. The code is a one-dimensional finite element code with Galerkin method to discretize the linearized dissipative MHD equations. Then, the correspondence between the ideal quasi-mode and the dissipative eigenmode is used, i.e. the fact that for sufficiently small resistivity, *η*, the damping becomes independent of resistivity, as shown by Poedts & Kerner (1991). The meaning of *R*, in equation (3.5), is not straightforward when thick non-uniform layers are used. As shown by Van Doorsselaere *et al*. (2004), linear dependence of the damping rate on *l*/*R* does not imply linear dependence on *l*/*a* nor on *l*/*b* (with *a*=*R*+*l*/2 the outer radius and *b*=*R*−*l*/2 the inner radius). This illustrates that the linear dependence is only valid in the thin boundary regime, i.e. , where the three formulas are equivalent.

Van Doorsselaere *et al*. (2004) computed the period and damping rate of coronal loop oscillations as a function of several loop parameters such as the longitudinal wavenumber, *k*_{z}, the density contrast, , and the length of the inhomogeneous layer, *l*/*R*. The ranges of variation for these parameters where chosen so as to cover the physical properties of all the coronal loops observed by Aschwanden *et al*. (2002).

In order to compare the damping rate of coronal loop oscillations with thick non-uniform transitional layers with the values given by equation (3.5), Van Doorsselaere *et al*. (2004) expressed their results for the normalized damping rate, −*ω*_{i}/*ω*_{r} in terms of the quantity *q*_{TTTB} defined by means of(4.1)hence, *q*_{TTTB} is the damping rate normalized to the value given by the TTTB approximation.

In figure 1*a*, we show the normalized damping rate as a function of the inhomogeneity length-scale, normalized to the TTTB damping rate. Hence, the TTTB results are a horizontal line. For small *l*/*R*, the approximate damping rate given by equation (3.5) and the approximate TB results do not differ very much from the correct LEDA results. The LEDA results start to deviate significantly (over 5%) when *l*/*R* is about 0.5. For intermediate *l*/*R* (around 1), there is a maximal difference of almost 25%. For even larger values of *l*/*R*, the difference diminishes and eventually disappears again. Finally, for extremely large values of *l*/*R* (fully inhomogeneous model), the TTTB and TB approximations overestimate the damping rate calculated by LEDA.

On the other hand, figure 1*b* shows the normalized damping rate as a function of the density contrast. This figure shows that the damping rate is very sensitive to the density contrast between the loop and the ambient plasma. In vacuum, the loop oscillations would be damped within a half oscillation period. The higher the ambient plasma density, the less severe damping by resonant absorption, so that undamped oscillations can only be supported if the density contrast is very small.

## 5. Resonantly damped kink modes in longitudinally stratified loops

This section moves on to two-dimensional equilibrium models. The equilibrium models now have a variation of the density in both the radial and axial directions (Andries *et al*. 2005*a*). An important result of that investigation is that, in the limit of the thin boundary approximation, the frequency is largely unaffected by longitudinal stratification of the density itself, but only depends on the longitudinally weighted mean density.

The equilibrium model used by Andries *et al*. (2005*a*) is similar to that of the previous sections, but the density is allowed to vary in both the radial and axial directions. By expressing the solutions in the two homogeneous regions as a sum of eigenmodes of the local Alfvén operator, the internal and external solutions can be readily obtained and represented as a sum of separable terms. The *z*-dependence of the perturbed quantities can be expressed as a sine series, and, in practice, all computations are done on the sine series coefficient vectors. The stratified density profile is expressed as a sine series plus a constant term taking into account the density at the footpoints(5.1)Here, *α*_{n} are the stratification parameters for profiles with different *n* integer nodes.

Andries *et al*. (2005*a*) have derived a linear expression for the frequency shift due to the presence of density stratification (*k*=1 for the fundamental and higher *k* for the harmonics):(5.2)In order to obtain this expression it was assumed that the footpoint density is the same in the unstratified and in the stratified loop. In general, one can impose that the footpoint density is not the same but rather a weighted mean density:(5.3)where *f* is a different function of *α* depending on the weighting function (*f*=1 for constant footpoint density, for constant top density, and for constant mean density). For the linear prediction one finds in general(5.4)This leads to the conclusion that, at least linearly, the frequency is independent of the stratification, provided that the weighted mean density is kept constant using a weight function that satisfies . The definition of *S*_{n,k,k} is exactly the mean of weighted with , so that it is clear that is the desired weight function. The frequency of the *k*th eigenmode thus only depends on the weighted mean density, weighted with . From the linear analysis this is clear for all *k* linearly in *α*, but by solving the dispersion relation numerically Andries *et al*. (2005*a*) have shown, at least for the fundamental *k*=1, that it is also true for moderately nonlinear values of (see figure 2*a*,*b*).

Another important result obtained by Andries *et al*. (2005*a*) refers to the spatial distribution of the perturbations in longitudinally stratified coronal loops. The fast quasi-modes for large values of the stratification parameter show clear signatures of higher-order sine contributions. However, these signatures are only visible in the pressure perturbation and not in the radial or azimuthal component of the displacement (see figure 2*c*,*d*). As the total pressure perturbation is related to compression, this result suggests that the observations should show intensity oscillations somewhere halfway along the loop legs. However, as the oscillations are linear these intensity oscillations are very small, and maybe not detectable. The clear detections of loop oscillations (including their periods and damping times) are not based on the intensity oscillations due to compression, but simply on the displacement of the more intense loop structure. However, as no visible signatures of higher-order components are present in the displacement, this may explain why higher-order sine components are not yet detected in the observations, although they may be present.

## 6. Coronal seismology

Let us point out that both the period of the oscillation and its damping time provide important sources of information about the corona. With the present results, a frequency inversion can be performed when an oscillating coronal loop has been observed. From the observed quantities (loop length and density contrast), an approximation for the Alfvén speed can be calculated. If the density is known, an estimate of the magnetic field can be obtained, using only the period (Nakariakov & Ofman 2001).

### (a) Observational test of resonant absorption

From the analysis presented in the previous sections it is clear that, in order to use the damping of coronal loop oscillations, once resonant absorption has been assumed, the important parameters when comparing observations with theoretical models are the inhomogeneity length-scale, *l*/*R*, and the density contrast, . These two parameters are, for the moment, difficult to determine from observations. The strong dependence of the damping rate on these parameters can be clearly seen in equation (3.5). By using this equation for the damping rate (which is directly related to the damping per period, *P*/*τ*_{d}), curves of constant damping per period, such as those shown in figure 3, can be constructed for the different observationally obtained values. From these curves, observational values for the damping rate together with the inhomogeneity scale-length may allow one to determine the density contrast (or vice versa). The problem is that, in principle, there are an infinite number of combinations of density contrast and inhomogeneity length-scale that give the same damping rate.

The first full-scale observational test of resonant absorption of quasi-mode kink oscillations was performed by Aschwanden *et al*. (2003), by comparing the oscillatory properties of a set of 11 loop oscillations observed by TRACE with the numerical eigenvalue computations for one-dimensional non-uniform equilibrium models, obtained by Van Doorsselaere *et al*. (2004). The density contrast for these 11 loops was calculated from the observed damping time and compared with the observed density contrast. The observed density contrast as well as the inhomogeneity length-scale were only determined indirectly through the emission measure and are thus subject to considerable modelling. In view of these observational restrictions and of theoretical assumptions, the calculated density contrast can be regarded as commensurable with the observed ones. It is reassuring that the physical mechanism causing the damping, namely, resonant damping due to the spatial variation of the Alfvén speed, is captured with this simple one-dimensional equilibrium model. In this sense, the study carried out by Aschwanden *et al*. (2003) provides new support for the interpretation of the damping mechanism of coronal loop oscillations in terms of the resonant absorption process.

### (b) Determination of the coronal density stratification

Recently, multiple harmonic transverse oscillations have been detected in coronal loops by Verwichte *et al*. (2004). These authors found the simultaneous presence of both the fundamental and the first overtone mode in two coronal loops belonging to a post-flare arcade. According to theory of MHD waves for uniform loops, the ratio of the period of the fundamental to the period of the first overtone is exactly 2. The ratios found by Verwichte *et al*. (2004) are 1.81 and 1.64. Verwichte *et al*. (2004) seem to be worried about the two numbers differing from 2 and from an analysis of the errors they conclude that the observed values are not in conflict with the expected value of 2. In our view, there should not be any concern about the values differing from 2. Rather they are a source of new important information. The difference between the observed ratio and 2 has been used by Andries *et al*. (2005*b*) as a seismological tool to obtain information about the density scale height in coronal loops.

Let us consider the linear expression for the frequency shift obtained by Andries *et al*. (2005*a*) and given by equation (5.2). As , the stratification can be seen to act differently on the frequency of the fundamental mode and that of the first overtone. The ratio of the two periods is found to be(6.1)The ratio can thus be seen to be 2 in an unstratified loop, but it differs from it in a stratified loop. Andries *et al*. (2005*b*) used an exponentially stratified atmosphere, i.e. , where *h* is the height in the solar atmosphere and *H* is the density scale height. This stratification function is projected on a semi-circular loop of length *L* with a height of consequently *L*/*π*. Thus, the relative height of the loop as compared to the density scale height *L*/*πH* is used as a measure of the stratification.

In figure 4, the density scale height (normalized with respect to the loop height) is shown as a function of the difference from two of the observed period ratio for the two cases observed by Verwichte *et al*. (2004). Unfortunately, the independent variable (2−*P*_{1}/*P*_{2}) suffers from extremely large error bars and their observed values are 0.19±0.25 and 0.36±0.23 s. In the first case (figure 4*a*), with a loop height of 70 Mm, the measurement allows for negative values within the error bars, indicating the possibility of larger density at the loop tops. For this case, we obtain an estimate of the density scale height of around 65 Mm and, due to the upper bound of the error bar, most probably not below 27 Mm. This measurement does not exclude the possibility of heavier plasma at the loop tops, but that possibility is certainly limited as the correspondingly negative density scale height is most probably lower than −190 Mm. The error bar on the second measurement is much smaller. With a loop height of 73 Mm this leads to a much more reasonably confined estimate of around 36 Mm, and most likely within 20 and 99 Mm.

## 7. Summary and conclusions

This paper gives an overview of our current understanding on the theoretical modelling of transverse coronal loop oscillations and their damping by means of resonant absorption. Resonant absorption of quasi-mode kink oscillations is due to the spatial variation of the Alfvén speed in the equilibrium configuration.

The eigenmode oscillations and their damping are rather well understood for one-dimensional equilibrium models. A recent exciting development is the theory of eigenmodes of two-dimensional equilibrium models. These models consider longitudinally stratified loops, in which the density varies not only in the radial direction, but also in the direction along the equilibrium magnetic field. Andries *et al*. (2005*a*) have shown that the observable quantity damping per period remains unchanged when stratification does not depend on the radial direction. More importantly, the period and damping time do not depend on the details of the longitudinal stratification, but on the mean density. This study has been generalized to an eigenmode computation of resonantly damped kink modes in longitudinally stratified loops with thick non-uniform boundary layers (Arregui *et al*. 2005).

From the point of view of coronal seismology, the results obtained by Van Doorsselaere *et al*. (2004) have been used to perform the first observational test of resonant absorption. The results of this test (Aschwanden *et al*. 2003) show that the density contrasts obtained from observations are commensurable with those obtained by numerical simulations. The theory of two-dimensional eigenmodes combined with the first observation of multiple modes (Verwichte *et al*. 2004) opens new fascinating opportunities for coronal seismology, as shown by Andries *et al*. (2005*b*) by determining the density scale height in coronal loops by using the ratio between the period of the fundamental mode to the period of the first overtone.

## Discussion

T. J. Bogdan (*Advanced Study Program, National Center for Atmospheric Research, Boulder, CO, USA*). Without the sexy movies the elegance of the presentation cannot be denied. However, I draw your attention to the potential field extrapolations shown in Professor Carlsson's talk. Looking at these I am hard-pressed to see magnetic slabs or even cylinders, nor to attach any sensible meaning to quantities like *m*, *k*_{x}, *k*_{y} and *k*_{z}. Without rigid or reflecting boundaries individual discrete frequencies are not obtained. How then should one expect these results to apply to more complex (i.e. without symmetries) solar atmosphere?

M. Goossens. Of course, the straight cylindrical flux tube model is only a first approximation of reality. It is clear that nature is capable of generating more complicated structures. Nevertheless, it is amazing to see how good the theoretical and numerical results for oscillations of this simple model are at explaining the observations.

E. Verwichte (*University of Warwick, UK*). (i) You mentioned our paper (Verwichte *et al*. 2004) and the detection of two harmonics within two loops, and say that the authors seem to be ‘worried’ to have a period of ratio 2. I don't see how my emotional state can be inferred from the paper. (ii) You show that from the period ratio, information about the quantity ∼*L*/*H* can be obtained, with *L* and *H* the loop length and *H* the density scale height. Although I wholeheartedly encourage such work, I wonder about the accuracy of the result. Beside the errors in periods, the quantity *L* is difficult to obtain accurately as well, with errors of the order of 10–20%. Would such a method be accurate enough in comparison with other diagnostic tools, e.g. the naive method of calculating *H* from assuming that the temperature of the instrument bandpass matches the loop temperature (we see the loops).

M. Goossens. (i) Dr E. Verwichte is right that we are unable to know anything about his emotional state. However, in his paper with V. M. Nakariakov, L. Ofman & E. E. Deluca (Verwichte *et al*. 2004, *Sol. Phys*. **223**, 77) on observations of multimode coronal loop oscillations, the authors went to some length to explain using statistical arguments that their observed value of the ratio of periods is not in disagreement with 2, the value for coronal loops that are uniform in the longitudinal direction. We believe there should not be any concern about values differing from 2 and that, rather, they are a source of important new information for coronal seismology. (ii) As can be seen in the analysis presented in Andries *et al*. (2005*b*) (*Astrophys. J.* **624**, L57), despite the considerable error bars on the observations and the poor condition of the problem, the accuracy of the method progressively improves the more as the ratio of periods differs from 2. In this respect, a reasonably confined estimate for the density scale height can be obtained, for the second case they analysed. This means that the accuracy is better for loops with a larger difference between the density at the footpoints and the density at the top. We believe that the observation of different harmonics in the same coronal loop and the analysis presented in Andries *et al*. (2005*b*) provides an important new and additional tool for the determination of unknown physical parameters in coronal loops.

## Footnotes

↵† Postdocdoral Fellow of the National Fund for Scientific Research—Flanders (Belgium) (F.W.O.-Vlaanderen).

One contribution of 20 to a Discussion Meeting Issue ‘MHD waves and oscillations in the solar plasma’.

- © 2005 The Royal Society