The effect of loop curvature on coronal loop kink oscillations

Noname manuscript No. (will be inserted by the editor)

The effect of loop curvature on coronal loop kink oscillations Tom Van Doorsselaere · Erwin Verwichte · Jaume Terradas

Received: date / Accepted: date

Abstract We will review analytical and numerical efforts in modelling the influence of curvature on coronal loop oscillations. We will mainly focus our attention on fast kink mode oscillations. A curved slab model will be presented, where it becomes clear that curvature introduces wave leakage into the system, because of changes in the equilibrium. The importance of leakage will be assessed through the use of a slab and cylindrical model where lateral leakage is allowed. A full analytical model for a semi-toroidal loop will be constructed for a system with no leaking waves but with an inhomogeneous layer that introduces damping due to the process of resonant absorption. The model for a semi-toroidal loop will be extended to also include leakage, and will be studied numerically. The numerical results will be compared to the analytical model. Keywords Sun: corona · Sun: oscillations · Sun: magnetic field

1 Introduction Traditionally, coronal loop oscillations have been modelled using cylindrical loop models. After the identification of fundamental kink modes in coronal loops (Nakariakov et al. 1999), it became clear that the wavelength of these kink oscillations is of the same order of magnitude as the length scale of the structure it supports, and thus also comparable to the radius of curvature of the loop. From this comparison of length scales, it can be expected that coronal loop oscillations are influenced by curvature. In the classical loop model (Edwin & Roberts 1983), only one dimension (the radial direction) is treated as not-ignorable. In the other ignorable directions, longitudinal (z) and T. Van Doorsselaere · E. Verwichte Centre for Fusion, Space and Astrophysics, Physics department, University of Warwick, Coventry, CV4 7AL, UK E-mail: [email protected] J. Terradas Centre for Plasma Astrophysics and Leuven Mathematical Modeling and Computational Science Centre, Katholieke Universiteit Leuven, Celestijnenlaan 200B, B-3001 Leuven, Belgium

2

azimuthal (θ), the problem is subject to a Fourier analysis. It is well known that the introduction of an inhomogeneity in one of the ignorable directions couples the Fourier components in that direction. For example, introduction of longitudinal stratification couples the fundamental kink mode with the third harmonic, giving it a mixed character (see Andries et al. 2009, in this issue). Likewise, since curvature introduces an inhomogeneity in the θ-direction, a specific azimuthal Fourier component will couple to other azimuthal Fourier numbers. For instance, we can predict that a kink mode (m = 1) will take characteristics of the sausage mode (m = 0) and the fluting mode (m = 2). This has been proposed by Roberts (2000), who also postulated that the coupling to the sausage mode would lead to a leaky character for the kink mode. Another important modification will be that the curvature requires the equilibrium conditions to change when compared to the cylindrical model. It will no longer be possible to take a homogeneous magnetic field. This more complicated magnetic field configuration will introduce a wave leakage and introduces a damping of the wave mode. This damping is not of a dissipative nature, but merely an apparent decrease of the wave amplitude as the energy goes away from the system. In the following text, we will first focus on this geometrical effect. We will present analytical models for kink oscillations in curved slabs (Verwichte et al. 2006a,b,c). To further capitalise on this leaky nature of the modes, we will construct a straight slab model, and a straight cylinder with a leaky regime, demonstrating the efficiency of this leakage, even in the absence of the curvature. We will then focus on an analytical model for linear oscillations in a curved cylinder (Van Doorsselaere et al. 2004b), when no leakage is allowed. We will present numerical results for a toroidally shaped loop, in the case both with or without leakage (Terradas et al. 2006b). The analytical and numerical calculations will be used to assess the effect of curvature on the resonant absorption mechanism.

2 Two-dimensional curved slab models 2.1 Analytical curved slab models Studies of waves in two-dimensional curved loop models have been performed numerically (e.g. Smith et al. 1997; Brady & Arber 2005; Selwa et al. 2005), without a density enhanced loop (e.g. Oliver et al. 1993, 1998; Del Zanna et al. 2005; Terradas et al. 2008) or in the large aspect ratio limit (e.g. Cargill et al. 1994). Recently, independently, Verwichte et al. at the University of Warwick and Diaz et al. at the University of St Andrews examined analytically fast magnetoacoustic oscillations in a curved magnetic slab geometry in the zero plasmaβ limit (Verwichte et al. 2006a,b,c; D´ıaz et al. 2006; D´ıaz 2006). We shall review those theories here. Such models are limited to considering vertically polarised wave modes. Also, we will extend them and put them into context with numerical simulations using similar geometries. The main new features a curved geometry introduces are compressibility of vertically polarised fast kink modes, the possibility of lateral wave leakage (Brady & Arber 2005) and specific for a two-dimensional slab model avoided crossings between sausage and kink modes. Lateral leakage appears naturally in curved loop models but curvature is not the necessary condition for lateral leakage to occur. Therefore we shall examine separately the mechanism of lateral leakage in specific straight slab and cylinder models in the next section. The main consequence of a two-dimensional curved loop geometry is that a guiding equilibrium potential magnetic field can no longer be uniform but varies in strength across

3

the slab. For a semi-circular slab, it decreases inversely proportional to the radial coordinate. A coronal loop is modelled in the zero plasma-β limit as a semi-circular magnetic slab of radius R and width 2a with the equilibrium magnetic field as B = B0 (R/r)eφ . The loop density is higher than the external plasma density. Waves are assumed to propagate in the slab plane. In such a model, the fast magnetoacoustic wave is associated with an azimuthal magnetic perturbation (parallel to the equilibrium field) and a radial plasma displacement. The slow magnetoacoustic wave disappears in the zero plasma-β limit. The Alfv´en wave has a velocity component out of the plane and is decoupled from the fast wave. It is clear that this model does not contain all essential physics. Alfv´en waves and the coupling with them have been excluded by explicitly putting the out of plane velocity components to zero, and having no variation of the wave variables in the out of plane direction. A fortiori, this means that the slab model is only valid for vertical oscillations of loops. The displacement and magnetic pressure perturbations are taken to be of the form ξr = ξˆr (r) sin(kφ) exp(−iωt) and P = Pˆ T (r) sin(kφ) exp(−iωt), where k is the non-dimensional azimuthal wave number. The longitudinal wave number k has to be integer in the case of a semi-circular loop. The fast wave displacement is described by a Bessel-like equation   2  !  d 1 d  ω2 k2  ξˆr  2 + = 0, (1) +  2 − 2  r dr r dr VA (r) r where the Alfv´en speed VA (r) is a general function of radial distance. Here, the Alfv´en frequency is defined as ωA (r) = VA k/r. This equation is of course, apart from the photospheric boundaries at φ = 0, π, equivalent to the equation describing fast waves in a z-pinch. The total pressure perturbation, here the Eulerian magnetic pressure perturbation, is given by Pˆ T /P0 = −2r d/dr(ξˆr /r). In order to make analytical progress in solving Eq. (1) the Alfv´en frequency is chosen to be a piece-wise continuous power law profile of index σ that is discontinuous at r = R ± a and that distinguishes three plasma regions, i.e. internal, lower external and upper external: ωA (|r − R| ≤ a) = ωAi (r/R)σ , ωA (|r − R| > a) = ωAe (r/R)σ . The radial displacement and Lagrangian total pressure perturbation, given by δPˆ T /P0 = Pˆ T /P0 − 2ξˆr /r, are continuous across these interfaces. Note that σ may take a different value internally and externally. For the leakage, the value of σ in the external region is the most relevant. In the following text, we have only considered a constant value of σ. As discussed in Verwichte et al. (2006a) the physical type of wave modes depends on the slope of the Alfv´en frequency. For a mode with a frequency such that it is oscillatory inside the loop (ω ≥ ωA (|r − R| ≤ a)) and evanescent in the intermediate external vicinity (ω < ωA (|r −R| > a)), there are the following possibilities w.r.t. the slope of the Alfv´en frequency: 1. dωA /dr < 0, σ < 0: The Alfv´en frequency decreases with r and will be less than the wave frequency at a certain radius in the upper external region. The wave mode is no longer perfectly trapped and will leak upwards through an evanescent barrier. 2. dωA /dr = 0, σ = 0: The Alfv´en frequency is piece-wise constant and the wave frequency will remain below the Alfv´en frequency everywhere in the external regions. The wave remains trapped in the loop. 3. dωA /dr > 0, σ > 0: The Alfv´en frequency increases with r and will be less than the wave frequency at a certain radius in the lower external region. The wave mode is no longer perfectly trapped and will leak downwards through an evanescent barrier. The cases of fast magnetoacoustic waves in the above model with σ = 0 and σ = −2 have been investigated by Verwichte et al. (2006a) and D´ıaz et al. (2006), respectively. Verwichte et al. (2006b) described the solutions for all values of σ different from zero.

4

Fig. 1 Density perturbation δρ/ρ0 of the fundamental trapped kink mode. The velocity field is overplotted. The solid (dashed) lines show the (un)perturbed position of the slab edge. The parameters ρ0e /ρ0i = 0.2 and a/R = 0.2 (figure taken from Verwichte et al. 2006a).

2.2 Fast magnetoacoustic waves in a curved slab with a piece-wise constant Alfv´en frequency profile The case σ = 0 is of particular interest as it represents the curved slab equivalent of the straight slab model. In the same fashion, the toroidal loop model with constant Alfv´en frequency represents the curved cylinder equivalent of the straight cylinder model (Van Doorsselaere et al. 2004b). In all these models fast waves can be trapped in the loop. However, due to the different geometry (e.g. limit in space where the wave kinetic energy has to tend to zero is in the curved slab for r → 0, r → ∞ whilst in the straight slab it is for x → ±∞), the wave solution in the curved slab has aq functional form of a power law instead of an expo±µ ˆ nential, i.e. ξr /r ∼ (r/R) with µ = k 1 − ω2 /ω2 . The dispersion relation for trapped A,{i,e}

modes is (Verwichte et al. 2006a) ) !  (  R + a ! µi R + a µi −inπ |µe | − µi tanh 1 µ ln = − , − e = 0 ⇔ coth 2 i |µe | + µi R−a R−a |µe |

(2)

where n is an integer representing the different solution branches. Solutions arising from the tanh and coth functions correspond to n even and odd respectively. By analogy to the symmetric and asymmetric solutions in the straight slab wave problem, these solutions are the sausage and kink modes, respectively (Edwin & Roberts 1982). In fact, the straight slab dispersion relation is recovered in the thin loop, short wavelength limit, i.e. a/R ≪ 1 and k ≫ 1. This can be seen by comparing Eq. (2) with Eq. (11) from Edwin & Roberts (1982) and considering that for a ≪ R, ln(R + a) − ln(R − a) ≈ 2a/R. Furthermore and similar to the straight slab model, for small a/R all the mode branches, except the fundamental kink mode branch become purely leaky, i.e. the wave is oscillatory everywhere.

5

Note that the sausage and kink modes found here are different from these modes in a cylindrical geometry. They only refer to the symmetry of the mode around the axis, and have no dependence on the third dimension. The curved and straight slab model crucially do differ when considering the density perturbations associated with the wave modes. In the straight slab model, the kink mode is practically incompressible. However, the Alfv´en speed varies with height as rσ+1 . Therefore, even though the Alfv´en frequency is constant, the Alfv´en speed increases linearly with height. For a kink mode, the top of the curved slab will move faster upwards compared with the bottom of the slab. This causes the slab width to increase and hence the density to decrease. This is illustrated in Fig. 1. Mathematically, the Lagrangian density perturbation is given as ! ρ 1 ξr 1 P δρ (ξ.∇) ρ0 = = + −2 , (3) ρ0 ρ0 ρ0 2 P0 r where ρ0 is the equilibrium density. For the kink mode, the total pressure perturbation is approximately axisymmetric w.r.t. the slab axis and averages out when integrating the density perturbation across the loop. The integrated density perturbation for a kink mode is then equal to < δρ/ρ0 >≈ −2 < ξr /r >, which is in anti-phase to the displacement and is independent of σ. For a vertically polarised transverse loop oscillation with a displacement amplitude of 5 Mm in a loop 200 Mm long (< ξr /r > = 8%), we expect relative density and intensity variations of -16% and -32%, respectively. This is observable. In fact, Wang & Solanki (2004) reported intensity variations associated with a vertically polarised transverse loop oscillation with a relative amplitude of -100%. This larger value may be due to the limitation of the slab model or due to observational factors such as time-integration (Verwichte et al. 2006a). D´ıaz (2006) extended the above theory to fast waves in elliptically curved magnetic slabs with a constant Alfv´en frequency. Such a model incorporates magnetic field expansion as the slab is wider at the top compared with at the foot points. This model shares many similarities with the circularly curved slab model. However, it does have a reduced damping rate for the pure leaky modes.

2.3 Laterally leaking fast magnetoacoustic waves in curved slab models If the Alfv´en frequency is not constant (σ , 0), the fast waves can no longer remain trapped in the loop and will leak into the external medium directly or by means of tunnelling through an evanescent barrier. The rate of wave damping due to tunnelling is in the short wavelength limit approximately of the form ℑω ∼ − exp(−Ck∆r), where C is a constant w.r.t. k. This shows that the damping rate is determined by the product of the evanescent barrier width and the azimuthal degree k, so that even if ∆r remains finite, k∆r tends to infinity in the short wavelength limit and the damping rate tends to zero. The general solution of wave equation (1) is in terms of Bessel functions of fractional order (Verwichte et al. 2006b), ξˆr r

!

= A{i,e} J|k/σ| (Ω{i,e} s) + B{i,e} Y|k/σ| (Ω{i,e} s) ,

(4)

where s = (r/R)−σ /|σ| and Ω{i,e} = ωR/VA,{i,e} . The appropriate solution in the lower and upper external regions is in terms of Bessel and Hankel functions of the first kind,

6

respectively. For σ = −2 the Bessel functions have the order k/2 as shown by D´ıaz et al. (2006). The dispersion relation associated with modes with σ < 0 is then n o  W J|k/σ| (Ωi s+ ), H(1) (Ωe s+ ) W J|k/σ| (Ωi s− ), J|k/σ| (Ωe s− ) |k/σ| o , (5) n  = W Y|k/σ| (Ωi s− ), J|k/σ| (Ωe s− ) W Y|k/σ| (Ωi s+ ), H(1) (Ωe s+ ) |k/σ|

df −σ where the Wronskian W{ f (x), g(y)} = f (x) dg dr (y) − dr (x) g(y) and s± = (1 ± a/R) /|σ|. This dispersion relation describes wave modes that are not trapped and leak into the upper external medium. It depends on the longitudinal wave number k, dimensionless frequency Ωi , power index σ, the density contrast ρ0e /ρ0i and the aspect ratio a/R. The damping rate increases as the density profile steepens, as the loop becomes more contrasted or as the loop becomes thinner. For modes with σ > 0, which leak into the lower external medium, an equivalent dispersion relation can be derived (Verwichte et al. 2006b). D´ıaz et al. (2006); Verwichte et al. (2006b) showed that in the curved slab models for long wavelengths, i.e. a/R small, the kink and sausage mode branches interact and interchange identities in avoided crossings in the dispersion diagram. This interaction is due to the presence of the finite, closed lower external regions acting as a second resonator. As a/R varies the number of extrema in the lower and inner regions may change as a mode extremum from the lower region shifts into the inner region. The avoided crossings also manifest themselves in the damping rate, which is oscillatory for small a/R but then monotonically tends to zero as a/R increases. The curved slab model of Verwichte et al. (2006a) can be used for seismological purposes (Verwichte et al. 2006c). From the observations, the loop length and width can be measured, which determines the ratio a/R. Furthermore, the mode harmonic can be determined, which sets k. There are three remaining parameters in the model that cannot directly be measured in the observation: σ, ρ0e /ρ0i and VAi . Wave observables offer three measurements that are function of these parameters, namely the oscillation period P, quality factor τ/P and the ratio of the relative intensity and displacement amplitudes µ. However, the last two observables do not depend on VAi . Therefore, we can examine the two-dimensional parameter space of σ and ρ0e /ρ0i in which the two observables τ/P and µ form curves. Each position along one of these curves represents a different value of VAi . Where the two curves cross, a solution for the unknown parameters is found that is consistent with both the theory and the observations. An example of the parameter space for the observation by Wang & Solanki (2004) is shown in Fig. 2. This systematic seismological method is similar to the approach taken by Arregui et al. (2007) and Goossens et al. (2008), which studied another model involving resonant mode conversion. The curved slab model was also applied to all the existing transverse loop oscillations, even though they are mostly horizontally polarised modes for which this two-dimensional model is strictly speaking not valid. However, the application of the seismological method revealed that the damping rate due to lateral leakage is too strong to account for the observations because it predicts a range for σ that is too narrow near the value of zero (for which the modes are trapped). As we shall see later, the efficiency of lateral leakage is reduced when considering the more realistic cylindrical and toroidal loop geometries.

2.4 Numerical studies of MHD waves in two-dimensional curved slab models Numerical simulations of MHD waves in curved loop geometries in two dimensions have been popular in the last decade to a large extent due to the lower computational expense

7

Fig. 2 Parameter space σ − ρ0e /ρ0i for the observation by Wang & Solanki (2004). The solid curves represent constant quality factor τ/P, calculated using solution of Eq. (5) and matching the observed values of 3.05 (τ=714s) or alternatively 0.85 (τ=200s). The aspect ratio a/R=0.03. Along these curves the value of VAi varies monotonically, as indicated by key values of VAi in units of km s−1 are indicated. The dashed curves represent constant values of µ, the ratio of the relative intensity and displacement amplitudes. Unfortunately, a good measurement of µ is not available. If we assume that the density ratio is larger than, say, 0.05, then σ is larger than -0.6 and the Alfv´en speed is larger than 700 km s1 . (figure taken from Verwichte et al. 2006c)

compared with three-dimensional simulations. Smith et al. (1997) investigated fast magnetoacoustic waves in a density enhanced loop, embedded in a potential coronal arcade with an exponentially increasing Alfv´en speed profile. The authors found evidence of lateral wave leakage and concluded that the damping rate due to leakage is inversely proportional to the wave period. This result is not supported by the analytical studies presented above. Also, it is counter-intuitive because it is expected that the effect of curvature and the leakage to decrease as the wavelength (and also the period) decreases. Brady & Arber (2005) pointed out that their result may be a consequence of the proximity of the numerical boundaries to the modelled loop. Brady & Arber (2005) performed a similar type of simulation with a semicircular loop and an inverse linear Alfv´en speed profile and found that the damping rate due to wave leakage through an evanescent barrier is proportional to the wave period and is of the same order as observed kink oscillation damping times. Taking into account the differences in the Alfv´en speed profiles used, their results are in good agreement with the analytical

8

Fig. 3 Damping rate ℑΩ as a function of normalised period VAi P/R. The analytical results of Verwichte et al. (2006b) are shown as circles, with the lowest degree mode visible indicated with a number. The curves are solid (dashed) where the modes are tunnelling (leaky). Curves A and B correspond to σ=-2.0 and a/R=0.25 & ρ0e /ρ0i =0.1 and a/R=0.44 & ρ0e /ρ0i =0.16, respectively. Curve B is the best fit of the profile used by Brady & Arber (2005). The corresponding Alfv´en speed profiles are shown in the inset. The numerical simulations by Brady & Arber (2005) are shown as diamonds, where the diamond with largest period indicates the mode of degree m=5. The dashed line is a best linear fit. Brady & Arber (2005) used a curved slab model with an Epstein density profile with a/R = 0.25 and ρ0e /ρ0i =0.1. (figure taken from Verwichte et al. 2006b)

work as is shown in Fig. 3. However, because of the foot point driving mechanism to excite the waves, the leakage of the low oscillation harmonics could not be investigated. To further support the concept of lateral wave leakage by tunnelling through an evanescent barrier, the authors performed numerical simulations with a straight slab model that incorporates such a barrier (Brady et al. 2006). Studies of leaky modes, which are oscillatory everywhere, have earlier been performed (e.g. Murawski & Roberts 1993). Numerical simulations by Murawski et al. (2005); Selwa et al. (2005), motivated by the observation of a vertically polarised kink mode oscillation by Wang & Solanki (2004) showed that an initial pressure pulse symmetrically located below the top of a curved loop (with an exponentially decreasing Alfv´en speed profile) excites the fundamental (k = 1) and third harmonic (k = 3) kink modes. The ratio of power in the third harmonic compared with the fundamental mode is proportional to the height of the pulse above the photosphere. Asymmetrically located pulses excite also the second harmonic mode (Selwa et al. 2006). Also, Selwa et al. (2006, 2007a) confirmed that the mechanism of lateral wave leakage is responsible for the damping of the oscillation and that the damping mechanism is too efficient

9

Fig. 4 Left: normalised phase speed, Vph /VAi (top), damping rate, γ/VAi k (middle) and quality factor, τ/P (bottom), as a function of k∆r for the fundamental fast kink harmonic in cylindrical geometry, with k = π/L, m=1, a/L=0.02, for three values of ρ0t /ρ0i = ρ0t /ρ0e : 0.01 (solid), 0.05 (dashed) and 0.1 (dot-dashed). The plasma-β is taken to be zero. The thick and thin curves correspond to the numerical solution of Eq. (17) and the solution using the analytical approximation from Eq. (24), respectively. Right: same as previous but for the fundamental fast kink harmonic in slab geometry. The thick and thin curves correspond to the numerical solution of Eq. (8) and the solution using the analytical approximation from Eq. (13), respectively.

to explain the observations (as also concluded in Verwichte et al. 2006c). The basic numerical model of Selwa et al. (2006) has been extended to include multiple strands, gravitational stratification and a photospheric layer (Gruszecki et al. 2007; Gruszecki & Murawski 2008; Gruszecki et al. 2008). Selwa et al. (2006) showed that also slow magnetoacoustic oscillations are excited in certain cases, when the plasma-β is non-zero. The fundamental slow mode is excited if the initial pulse is localised asymmetrically. However, if the pulse is symmetrically localised (under the loop top), then the second harmonic is generated (Nakariakov et al. 2004; Tsiklauri et al. 2004). Selwa et al. (2007b) investigated the excitation of slow mode oscillations further and concluded that they are more easily set up in curved loops than in onedimensional loop models because the fast wave generated by a pressure pulse will excite not only a slow wave at the nearest foot point but, due to its faster propagation speed, also at other places along the loop before the first slow wave itself has reached them. Gruszecki et al. (2007) studied the lateral leakage of Alfv´en waves in a curved loop arcade where the wave is polarised in the ignorable direction and found that the Alfv´en waves damp due to nonlinear coupling to laterally leaking fast magnetoacoustic waves (Nakariakov et al. 1997).

3 Damping of magnetoacoustic oscillations by lateral leakage To separate the physical process of lateral wave leakage of fast magnetoacoustic oscillations from the effects of curvature, a simplified model is introduced with a straight loop model that

10

includes a finite evanescent barrier (Mikhalyaev & Solov’ev 2005; Brady et al. 2006). We shall discuss here two such models, one in slab geometry and one in cylindrical geometry.

3.1 Slab model with an evanescent barrier The coronal loop is modelled as a magnetic slab in the zero plasma-β limit, of half-width a at x ≤ |a| with an evanescent barrier at a < x ≤ a + ∆x. We consider an evanescent barrier only on one side of the slab to compare with the curved slab where wave leakage occurs either on the top or bottom side of the slab. The equilibrium magnetic field, B0 , is directed along the z-direction. All equilibrium quantities are constant, except for the equilibrium density, which is a piece-wise constant function, i.e.   ρ0t x < −a      x ≤ |a|  ρ0i , (6) ρ0 (x) =    ρ a < x ≤ a + ∆x 0t     ρ0e x > a + ∆x with ρ0t < ρ0i ≤ ρ0e . We take the transverse displacement, ξ x and the total pressure perturbation, PT of the fast kink oscillation to be of the form f = fˆ(x) sin(kz) exp(−iωt), where ω is the angular frequency of a mode and k is the longitudinal component of the wave vector. As before, we have ignored all the out of plane components and variations of the perturbations. This means that several physical effects are excluded from the model, such as Alfv´en waves and the coupling with it. The governing wave equation for this particular model is [d2 /dx2 − κ2 ] Pˆ T = 0, where 2 κ = k2 − ω2 /VA2 . The transverse displacement is related to the total pressure perturbation through ξˆ x = (q/κ)dPˆ T /dx where q = κ/ρ0 (ω2 − VA2 k2 ). The solution for body modes is x < −a : Pˆ T (x) = Ao eκt x , ξˆ x (x) = qt Ao eκt x −a ≤ x ≤ a : h i Pˆ T (x) = Ai ei˜κi x + Bi e−i˜κi x , ξˆ x (x) = iq˜ i Ai ei˜κi x − Bi e−i˜κi x a < x ≤ a + ∆x :   Pˆ T (x) = At eκt x + Bt e−κt x , ξˆ x (x) = qt At eκt x − Bt e−κt x x > a + ∆x : Pˆ T (x) = Ae ei˜κe x , ξˆx (x) = iq˜ e Ae ei˜κe x

(7)

where κ˜ i = (−κi2 )1/2 , κt = (κt2 )1/2 and κ˜ e = (−κe2 )1/2 sgn(ℜ(−κe2 )1/2 ). The form of κ˜ e is chosen to ensure that the wave solution in the external region always represents an outward propagating wave. Continuity of ξˆ x and Pˆ T across the interfaces at x = ±a and x = a + ∆x leads to a dispersion relation D0s (ω)D0k (ω) + D1 (ω) = 0 , (8) with D0s (ω) = q˜ i tan(˜κi a) − qt , D0k (ω) = q˜ i cot(˜κi a) + qt , qt + iq˜ e 2 D1 (ω) = (q + q˜ 2i ) e−2κt ∆x . qt − iq˜ e t

(9) (10) (11)

11

The term D1 (ω) is the modification to the dispersion relation due to the presence of an evanescent barrier of finite width. In the limit of κt ∆x → ∞ (i.e. the barrier is much larger than the wave penetration depth), D1 (ω) tends to zero and the dispersion relation for trapped magnetoacoustic body modes in a slab geometry (Edwin & Roberts 1982) is recovered, where D0s (ω) and D0k (ω) representing the dispersion relations for the trapped sausage and kink modes, respectively. We examine the case of κt ∆x finite but large, such that e−2κt ∆x ≪ 1. The solution of Eq. (8) can then be approximated by ω = ω0 + δω, where ω0 is a kink mode solution of D0k (ω0 ) = 0. The small correction δω is of order O(e−2κt ∆x ) (same order as D1 (ω0 )). Retaining the leading terms in a Taylor expansion of Eq. (8) around ω = ω0 , the solution for δω is D1 (ω0 ) , (12) δω ≈ − 0k D0s (ω0 ) ∂D ∂ω ω=ω 0

which, using Eqs (9) and (11) and D0k (ω0 ) = 0 becomes δω ≈

qt qt + iq˜ e 2 qt − iq˜ e

e−2κt ∆x 1 , ∂D0k ω0 ∂ω2 ω2 =ω2

(13)

0

and the damping rate, defined as γ = −ℑδω, of a kink mode due to an evanescent barrier is γ ≈ −

q2t q˜ e + q˜ 2e

q2t

1

∂D0k ∂ω2 ω2 =ω20

e−2κt ∆x . ω0

(14)

3.2 Cylinder model with an evanescent barrier The coronal loop is modelled in the zero plasma-β limit as a magnetic cylinder of radius a surrounded by an evanescent barrier of thickness ∆r. In order to keep the problem analytically tractable, the evanescent barrier is taken to be azimuthally symmetric, although for a curved loop, the barrier’s thickness will be a function of the azimuthal angle. The equilibrium magnetic field, B, is directed along the positive z-direction. All equilibrium quantities are constant, except for the equilibrium density, which is a piece-wise constant function, i.e.   ρ    0i ρ0t ρ0 (r) =    ρ

0e

r≤a a < r ≤ a + ∆r , r > a + ∆r

(15)

with ρ0t < ρ0i ≤ ρ0e . Similar as before, we take the transverse displacement, ξr and the total pressure perturbation, PT of the fast kink oscillation to be of the form f = fˆ(r) sin(kz) cos(mθ) exp(−iωt), where m is the azimuthal wave number. In this cylindrical configuration, additional physics is possible, when compared to the slab model in the previous sections. The extra dimension (θ) allows for the occurrence of Alfv´en waves, and the coupling thereto. Also, the azimuthal wave number m determines the nature of the fast mode: sausage (m = 0), kink (m = 1) or fluting (m ≥ 2). In the slab case, only two main modes were distinguished: kink (odd) and sausage (even). The governing wave equation for the total pressure perturbation is of the form of a Bessel equation, i.e. r2 d2 Pˆ T /dr2 + rdPˆ T /dr − (κ2 r2 + m2 )Pˆ T = 0. The radial displacement is derived

12

from Pˆ T by ξˆr = (q/κ)dPˆ T /dr. The solution for body modes and for an equilibrium density profile given by Eq. (15) is r≤a : Pˆ T (r) = Ai Jm (˜κi r) , ξˆr (r) = q˜ i Ai J′m (˜κi r) a < r ≤ a + ∆r :   Pˆ T (r) = At Km (κt r) + Bt Im (κt r) , ξˆr (r) = qt At K′m (κt r) + Bt I′m (κt r) r > a + ∆r : ′ κe r) κe r) , ξˆr (r) = q˜ e Ae H(1) Pˆ T (r) = Ae H(1) m (˜ m (˜

(16)

where Jm , Km , Im and H(1) m are the Bessel, first and second modified Bessel and first Hankel functions of order m. The quantities κ˜ i , κt and κ˜ e are the same as before. Continuity of ξˆr and Pˆ T across the interfaces at r = a and r = a + ∆r leads to a dispersion relation D0 (ω) + D1 (ω) = 0 , (17) with D0 (ω) = qt Jm (˜κi a) K′m (κt a) − q˜ i J′m (˜κi a) Km (κt a) ,

(18)

D1 (ω) = − F (ω) E(ω, ∆r) ,

(19)

where F (ω) = qt Jm (˜κi a) I′m (κt a) − q˜ i J′m (˜κi a) Im (κt a) , E(ω, ∆r) =

κe b) K′m (κt b) qt H(1) m (˜ κe b) I′m (κt b) qt H(1) m (˜

− −

′ q˜ e H(1) κe b) Km (κt b) m (˜ (1) ′ q˜ e Hm (˜κe b) Im (κt b)

(20) ,

(21)

and b = a+∆r. Eq. (17) is the same dispersion relation as derived by Mikhalyaev & Solov’ev (2005) in the context of a shell model of a coronal loop. The term D1 (ω) is the modification to the dispersion relation due to the presence of an evanescent barrier of finite width. Note that the contribution from the evanescent barrier is described by modified Bessel functions instead of exponentials as is the case in the slab geometry. The modified Bessel function Km (κt r) decays faster as a function of distance than the exponential, which lead Ruderman (2005) to conclude that lateral leakage is much less effective in the more realistic toroidal geometry compared with the curved slab geometry. How much exactly the leakage effectiveness differs depends on κt , a and ∆r. In the limit of κt ∆r → ∞, Eq. (17) tends to zero and the dispersion relation for trapped magnetoacoustic body modes is recovered (Edwin & Roberts 1983). This can be seen by using the asymptotic expansion of the Bessel functions for large arguments (Abramowitz & Stegun 1964) to approximate Eq. (21) for large values of κt ∆r as ′

E(ω, ∆r) ≈ −π

κe b) κe b) + q˜ e H(1) qt H(1) m (˜ m (˜ ′

qt H(1) κe b) − q˜ e H(1) κe b) m (˜ m (˜

e−2κt a e−2κt ∆r ,

(22)

which shows that E(ω), and hence also D1 (ω), tends to zero as κt ∆r → ∞. We examine the case of κt ∆r and κ˜ e ∆r finite but large, such that e−2κt ∆r ≪ 1 and e−2˜κe ∆r ≪ 1. The solution of Eq. (17) can then be approximated by ω = ω0 + δω, where ω0 is the solution of D0 (ω0 ) = 0

13

and is real. The small correction δω is of order O(e−2κt ∆r ) (same order as D1 (ω0 )). Retaining the leading terms in a Taylor expansion of Eq. (17) around ω = ω0 , the solution for δω is δω ≈ −

D1 (ω0 ) ∂D0 ∂ω ω=ω0

.

(23) ′

κe b) κe b) ≈ iH(1) Substituting Eqs. (19) and (22) into Eq. (23), using the approximation H(1) m (˜ m (˜ for large arguments, and D0 (ω0 ) = 0 to rewrite Eq. (20), we find π qt + iq˜ e e−2κt a J′m (˜κi a) δω ≈ − q˜ i 2 qt − iq˜ e κt a K′m (κt a)

e−2κt ∆r 1 . ∂D0 ω0 ∂ω2 ω2 =ω2

(24)

0

The damping rate γ = −ℑδω is then equal to γ ≈ π

q˜ i qt q˜ e e−2κt a J′m (˜κi a) q2t + q˜ 2e κt a K′m (κt a)

e−2κt ∆r 1 . ∂D0 ω0 ∂ω2 ω2 =ω2

(25)

0

Physically, the damping rate is positive to represent wave damping through the lateral leakage (wave tunnelling) of wave energy through the evanescent barrier. The normalised phase speed, damping rate and quality factor of the fundamental fast kink harmonic is shown in Fig. 4 as a function of k∆r. For comparison, the same is shown for the fundamental fast kink harmonic in the slab geometry. For large values of k∆r the full solution indeed tends to the analytical approximation. As the width of the evanescent barrier diminishes, the wave damping increases. The damping rate in the slab geometry is very efficient, damping the oscillation within one wave period for small values of k∆r. The damping rate in the cylindrical geometry is in this example up to hundred times slower than in the slab geometry, with a quality factor of about ten. From Eq. (25) we see that the damping time is proportional to P−1 e2κt ∆r . The same relation was found by Brady et al. (2006) for wave tunnelling in a straight magnetic slab model. The short wavelength limit (ka ≫ 1), where ω0 = VAi k, is equivalent to κt ∆r ≫ 1. Hence, γ tends to zero in this limit. However, Eq. (25) is not a monotonic function of ka. In the long wavelength limit (ka ≪ 1 ≪ k∆r), ω0 = C K k for m > 0, where C K is the kink speed defined as C K2 = 2 2 (ρ0i VAi + ρ0t VAt )/(ρ0i + ρ0t ). Using the asymptotic expansion of the Bessel functions for small arguments (Abramowitz & Stegun 1964) in Eq. (25), it is possible to show that for m>0 J′m (˜κi a) ∂D0 2m ∼ (ka)−4 , (26) ∼ (ka) and K′m (κt a) ∂ω2 ω2 =ω20

together with κ ∼ ka, ω0 ∼ ka and q ∼ (ka)−1 , we see that γ ∼ (ka)2m+1 (see also the results in Spruit 1982; Goossens & Hollweg 1993). Therefore, the damping rate tends to zero in this limit as well. In between these two limits, the damping rate must have at least one maximum. Fig. 5 shows that the damping rate, calculated by solving Eq. (17) numerically, goes to zero in the limit of ka → 0 and has one maximum. Therefore, qualitatively the behaviour predicted by the analytical approximation extends to the regime where 1 ≪ ka ≪ k∆r. The presence of the maximum means that the fundamental fast kink harmonic is no longer the fastest damped mode. This is in contrast with the case of a slab geometry where the damping rate is a monotonic function of ka, even though the analytical approximation shows a maximum. Note also that in the slab geometry, the longitudinal phase speed for small values of ka exceeds the external Alfv´en speed. In this case, the wave does not need to

14

Fig. 5 Left: normalised phase speed, Vph /VAi (top), damping rate, γ/VAi k (middle) and quality factor, τ/P (bottom), as a function of ka for fast kink modes in cylindrical geometry, with k = π/L, m=1, a/L=0.02, for three values of ρ0t /ρ0i = ρ0t /ρ0e : 0.01 (solid), 0.05 (dashed) and 0.1 (long-dashed). The plasma-β is taken to be zero. The thick and thin curves correspond to the numerical solution of Eq. (17) and the solution using the analytical approximation from Eq. (24), respectively. The vertical dotted line corresponds to the value of ka of the fundamental kink harmonic. Right: same as previous but for fast kink modes in slab geometry. The thick and thin curves correspond to the numerical solution of Eq. (8) and the solution using the analytical approximation from Eq. (13), respectively.

Fig. 6 Quality factor, τ/P , as a function of loop aspect ratio, a/L, for four values of ∆r/a: 0.1 (solid), 1 (dashed), 2 (dot-dashed) and 5 (long-dashed). The plasma-β is taken to be zero and ρ0t /ρ0i = ρ0t /ρ0e = 0.1. The shaded area represents the values of the quality factor the model cannot attain.

15

Fig. 7 The model used for loops modelled as a semi-torus.

tunnel through an evanescent barrier but can radiate out directly. This behaviour also occurs in a curved slab geometry (Verwichte et al. 2006b). Fig. 6 shows the quality factor as a function of the loop aspect ratio for the cylinder model for different values of the barrier thickness. It shows the lower limit of possible quality factors resulting from lateral leakage, which can be seen as the best case scenario in terms of choice of loop parameters and density profiles. Coronal loops have a typical aspect ratio in the range 0.01-0.1. Therefore, this plot reveals that the damping rate of fast magnetoacoustic kink modes due to lateral leakage can be for specific loop parameter values within one magnitude of the observed rates. Note that the minimum quality factor decreases with loop aspect ratio. Hence, especially for thick loops lateral leakage can be a key ingredient in explaining the rapid damping of transverse loop oscillations, even if it is not the main physical damping mechanism. It has to be added that this study was restricted to somewhat artificial models and a more detailed and systematic study using a three-dimensional curved loop should be undertaken that examines the efficiency of lateral wave leakage as a function of Alfv´en frequency profiles, density contrast and loop aspect ratio.

4 Analytical models for curved cylinders We will now concentrate on curved cylindrical models. It is possible to find analytical solutions in a semi-torus, in the case that no wave leakage occurs. We discuss and correct the results of Van Doorsselaere et al. (2004b).

16

4.1 Models & equations Following Van Doorsselaere et al. (2004b), the linearised MHD equations are solved in a toroidal coordinate system for which the transformation formulae are given by: d sinh u , cosh u − cos v x = r cos ϕ, y = r sin ϕ, d sin v z= , cosh u − cos v r=

with u the dimensionless radial coordinate, v the poloidal angle and ϕ the toroidal (longitudinal) angle. The u-coordinate decreases monotonically from infinity at the focus (r=d) to zero as r tends to infinity. Surfaces of constant u form toroids, which have a circular crosssection of radius a=d cosech u and a central axis at distance R=d coth u from the origin. We define a curved coronal loop as the region inside the surface of constant u0 with ϕ between 0 and π. The aspect ratio of the loop can be defined in two ways: as the ratio a/d=cosech u0 or as the ratio a/R=sech u0 . The first definition has been utilised in Van Doorsselaere et al. (2004b). However, we shall use the second definition because R is directly related to the loop length as measured at the loop axis, i.e. L=πR, in contrast to d. Also, the aspect ratio lies conveniently between zero (infinitely narrow loop at r = d) and one (infinitely large and wide loop occupying the whole semi-space y ≥ 0).

4.2 Equilibrium As in Sect. 2, we adopt a force-free equilibrium magnetic field B pointing in the toroidal ϕ-direction (B = B(u, v)eϕ ),  r −1 B(u, v) = B0 , (27) d where B0 is the magnitude of the magnetic field at the focus of the torus. Similarly, as a profile for the plasma density ρ, we take:   −4   for u < u0 , ρ0e dr     −4   r (28) ρ(u, v) =  ρ0i f (u) d for u0 ≤ u ≤ u0 + ul ,        ρ0i r −4 for u > u0 + l. d

Here, ρ0e and ρ0i are the normalisation constants of the density, respectively in the exterior (subscript e) and interior (subscript i) of the coronal loop. The density profile in the two regions are continuously connected in a layer u0 ≤ u ≤ u0 + ul , with a profile f (u). The inclusion of such a smooth variation of the Alfv´en speed causes a resonant coupling between the global mode frequency and the local Alfv´en mode, a mechanism called resonant absorption. This resonant coupling is a very efficient sink of energy, and results in a large damping rate of the global mode. The adopted density profile corresponds to σ = 0 in the slab case, except for the region with the continuous profile. We Fourier-analyse all perturbed quantities in the ϕ-direction and perform a normal b v) exp i(kϕ − ωt). mode analysis in time. We thus put all perturbed quantities Q(u, v, ϕ, t) as Q(u, Consequently, the hats are dropped to simplify the notation. Taking the above profiles (Eqs. 27-28) for the magnetic field and the density, we obtain two regions with a constant squared Alfv´en frequency ω2A = VA2 k2 /r2 .

17

4.3 Linearised equations, analytical solutions and dispersion relation In the two regions with constant Alfv´en frequency, a single equation can be built for the longitudinal component of the perturbed magnetic field bϕ :   !  k2 ω2 (cosh u − cos v)2  bϕ 2 bϕ   . = −  ∇ B B ω2A d2 sinh2 u This equation can be solved analytically and the most general form of the solution is r +∞ bϕ cosh u − cos v X = C i/e,m Fi/e,m (u) exp (imv). B sinh u m=−∞

In this general solution, C i/e,m are unknown amplitudes in the interior and exterior. Fi/e,m (u) is the radial solution in the internal and external region and can be expressed in terms of hypergeometric functions (see Van Doorsselaere et al. 2004b, for more details). In contrast to the solution found in Van Doorsselaere et al. (2004b), the inclusion of an arbitrary phase v0 is not necessary, because it can be absorbed in C i/e,m . The internal and external solution have to be matched across the inhomogeneous layer (u0 ≤ u ≤ u0 + ul ). When ul ≪ u0 this matching condition can be expressed in terms of jump conditions (see e.g. Sakurai et al. 1991; Tirry & Goossens 1996). In Van Doorsselaere et al. (2004b) an error was made when Fourier-analysing the jump conditions, resulting in a wrong dispersion relation. In appendix A, the correct matching condition is calculated, yielding: ! iπω2A 2 1 3 exp (−uA ) (αi − 1)(α2e − 1)C i,m m2 + (coth uA − 1) + |∆A | 2 4 sinh uA X iπω2A 2 Fi,m+δ (u0 ) + (α − 1)(α2e − 1) C i,m+δ |∆A | i Fi,m (u0 ) δ,0

! 3 3 exp (−sgn(δ)uA ) exp (−|δ|uA ) coth uA − |δ − 1| + 2 2 2 sinh uA d log Fe,m d log Fi,m = C i,m (α2i − 1) (u0 ) − (α2e − 1) (u0 ) du du ! 1 − coth u0 2 (αi − α2e ) + 2 X exp (−|δ|u0 ) Fi,m+δ (u0 ) 2 C i,m+δ + (α − α2e ). 2 Fi,m (u0 ) i δ,0

(29)

In this equation, the subscript A expresses that the quantity is evaluated at the resonant position uA , where the oscillation frequency equals the local Alfv´en frequency ω2 = ω2A . We define ∆ = −∂ω2A /∂u, α2i/e = ω2 /ω2Ai/e and sgn(x) returns the sign of the argument x. An analytical dispersion relation is obtained by requiring that the system is actually solvable and thus that the determinant of the system is 0. In Van Doorsselaere et al. (2004b), this dispersion relation was solved analytically in the case that ε = a/d = 1/ sinh (u0 ) ≪ 1. In that article, using a wrong dispersion relation, a first order correction was found for the imaginary part of the frequency. The off-diagonal terms of this equation are all at maximum of the order ε = a/d. Thus, up to the first order in ε, the eigenvalues of the system can be found as zeroes of the diagonal

18

terms. In general, the first order correction ω1 of the frequency ω = ω0 + εω1 can be found, similar as in Sect. 3, as D1 (ω0 ) , ω1 = − ∂D 0 ∂ω (ω0 ) where D(ω) = D0 (ω) + εD1 (ω) is the first order expansion of the diagonal term D(ω). In the corrected matching condition (Eq. 29), since coth uA = 1 − ε2 /2 and exp (−uA ) = ε, the diagonal terms do not have a first order contribution. This means that the curvature has no first order effect on the frequency and damping time of quasi-mode kink oscillations, in contrast to what was erroneously claimed in Van Doorsselaere et al. (2004b). Also, unlike Van Doorsselaere et al. (2004b), the analytical expressions of Goossens & Hollweg (1993) for the frequency and damping time are recovered in the thin tube (a ≪ L) and thin boundary (l/a ≪ 1) limit. For coronal loops observed by Aschwanden et al. (2002), ε is never larger than 24% and generally smaller than 10%. Because the correction to the quasi-mode frequency due to curvature is at most ∼ ε2 , the frequency is changed at maximum 6% and in most cases less than 1%. The same holds for the complex part of the frequency, which is directly related to the damping by resonant absorption. We thus conclude that neither the frequency, nor the damping is changed by curvature.

5 Numerical models for curved cylinders We will now present the results from numerical calculations of eigenmodes in a loop modelled by a semi-torus. First, we seek to confirm the results in the previous section with σ = 0, and then we will expand the model to more general σ, in order to compare with the slab case (Sect. 2).

5.1 Toroidal loop model and equations We will use an equilibrium configuration similar to the one studied in Sect. 4. We consider a magnetic field aligned with a semi-torus, dropping off as 1/r (Eq. 27). We generalise the earlier density profile (Eq. 28) to   −2(2+σ)   for u < u0 , ρ0e dr     −2(2+σ)   r ρ(u, v) =  (30) ρ0i f (u) d for u0 ≤ u ≤ u0 + ul ,     −2(2+σ)   r ρ0i for u > u0 + ul , d

where the parameter σ allows to study different density profiles with height. Such an equilibrium density is the natural toroidal extension of the slab model used in Sect. 2. As before, we have that the Alfv´en frequency, ωA (r) ∼ r−σ , increases (stays constant, decreases) with height when σ < 0 (σ = 0, σ > 0). The parameter ul can be translated to a spatial scale l which is a measure for the thickness of the inhomogeneous layer. As before, the linearised MHD equations are solved. The main difference with the previous section is that we allow the resistivity η to be different from zero and thus there are additional terms in the induction equation. The inclusion of resistivity in the model is simply to avoid the singular behaviour of the equations at the resonances. As in the previous section, we assume that perturbations are of the form exp i(kφ − ωt), where ω = ωR − iωI , and

19

only need to solve the MHD equations in the xz-plane. The photospheric line-tying effect is then included by selecting the appropriate toroidal number, which has been set to k = 1 (fundamental mode). The results presented in this section correspond to a density contrast between the loop and the corona ρ0i /ρ0e = 3. As evidenced in the previous section, it is very difficult to find analytical solutions to the eigenvalue problem in this toroidal configuration. For this reason, it is instructive to numerically solve the two-dimensional eigenvalue problem. For details about the employed numerical eigenvalue solver, we would like to point the reader to the article where these results have been presented for the first time (Terradas et al. 2006b). In the calculations regularity conditions for the velocity at the loop centre and vanishing conditions at the photosphere and at r → ∞ have been imposed.

5.2 Resonantly damped modes: uniform Alfv´en frequency As in the slab case (Sect. 2), in the toroidal model the Alfv´en frequency changes with position. Since the frequency is different for each magnetic field line there is a continuous spectrum of Alfv´en modes. The case σ = 0 is special since the Alfv´en frequency is constant and there is only coupling in the inhomogeneous layer (u0 ≤ u ≤ u0 + ul ). This case is discussed first since it is the most simple situation and has been analytically studied in the previous section. One of the main results for this configuration is that curvature introduces preferential directions of oscillation. In contrast, the polarisation of the oscillation is completely arbitrary in a cylindrical configuration, and solely determined by the initial conditions. This degeneracy is broken in a toroidal configuration. Using the numerical eigenvalue solver, two eigenfunctions and corresponding frequencies are obtained. The magnetic pressure perturbation and velocity fields are displayed in Fig. 8. Clearly, these two modes represent the horizontal oscillation (left panel) and the vertical oscillation (right panel). It is also clear that these modes still correspond to the classical kink mode in a straight cylinder, i.e. a transverse displacement of the tube as a whole. It turns out that these two different polarisations have very similar frequencies. This has been anticipated in Sect. 4 since it has been analytically shown that curvature has no first order effect on the frequency. For example, the full eigenmode calculation of the two horizontally and vertically polarised solutions shown in Fig. 8 have frequencies whose real and imaginary parts differ by only 0.04% and 0.32%, respectively. This small frequency difference is in contrast to the influence of other effects, such as multiple loop strands (Luna et al. 2008; Van Doorsselaere et al. 2008), where the frequency splitting is much larger. When the effect of the loop aspect ratio (a/L) on the frequency splitting is studied, the results of the previous section are recovered. In the limit of a/L ≪ 1, the frequencies of the horizontal and vertical polarisation both tend to the straight cylinder limit (see Fig. 9). This is as expected, because in this case, the nearly perpendicularly propagating fast waves do not “feel” the curvature of the loop. As a/L increases, a maximum frequency splitting of 5% may be obtained (see Fig. 9b). A comparison of the frequency with the thin tube, thin boundary (TTTB) expressions for the frequency in a cylinder (Eqs. (4) and (5) in Goossens et al. 2002) is also instructive. The values of the TTTB expression have been overplotted on Fig. 9 with a full line. It is clear that these analytical expressions are recovered with the numerical results when the aspect ratio a/L → 0. However, for a/L = 0.05 (where the formulae are not strictly valid), ωR and ωI differ from the TTTB values by 5% and 30%, respectively.

20

Fig. 8 Velocity field (arrows) and magnetic pressure perturbation (shaded contours, where light and dark represent positive and negative magnetic pressure perturbations) of the two kink modes for σ = 0, a/L = (20π)−1 and l/a = 0.2. Both solutions are similar to the kink mode in a straight cylindrical loop, although they display well-defined horizontal and vertical orientations of plasma motions. The photosphere is located at z = 0. Large arrows due to resonant absorption are visible in the resonant layer around the loop. The edge of the loop is represented with a continuous circular line (figure taken from Terradas et al. 2006b).

Fig. 9 Real and imaginary part of the frequency as a function of the inverse loop length a/L. The continuous line corresponds to the frequency calculated using the TTTB approximation. Squares and triangles represent the horizontally and vertically polarised modes respectively. In this plot l/a = 0.5 and σ = 0.

We now turn our attention to the variation of the frequency with the thickness of the inhomogeneous layer. We study a loop with a/L = (20π)−1 . The results are displayed in Fig. 10. As can be seen from the vertical scaling of the left panel of that figure, the real part of the frequency only changes very little with the thickness of the inhomogeneous layer. This confirms the earlier numerical results (Van Doorsselaere et al. 2004a; Terradas et al. 2006a) for curved loops. As expected, the variation of the damping rate ωI follows the TTTB approximation reasonably well (right panel of Fig. 10). The maximum difference between the current results and the analytical formula is 12%. This can be seen as the difference between the triangles and squares and the full line. This difference grows with l/a, where the applicability of the analytical formula breaks down.

21

Fig. 10 Real and imaginary part of the frequency as a function of the dimensionless width of the inhomogeneous layer l/a. The continuous line corresponds to the frequency calculated using the TTTB approximation. Squares and triangles represent the horizontally and vertically polarised modes respectively. In this plot σ = 0 and a/L = (20π)−1 .

Fig. 11 Real and imaginary part of the frequency as a function of the parameter σ. The horizontal axis is given in terms of α = −2(2 + σ). Squares and triangles represent the horizontally and vertically polarised modes respectively. In this plot l/a = 0.3 and a/L = (20π)−1 .

5.3 Resonantly damped and leaky modes: non-uniform Alfv´en frequency When we look at the more generalised problem with σ , 0, the physics of the problem is changed drastically, because the Alfv´en frequency changes with the vertical position. Two new ingredients appear. First, an extra resonance is created at the position rA where the frequency equals the global mode frequency ωA (rA ) = ωR . This position is below/above the loop for σ < 0/σ > 0. Second, as in Sect. 2, the eigenmode will show an oscillatory character at a certain distance away from the loop, because its frequency is above the local cutoff frequency ωc . The change from evanescent behaviour to propagating behaviour happens at a position rT where ωc (rT ) = ωR . This position, in combination with the loop boundary, creates a wave tunnelling barrier, where the wave cannot propagate. To illustrate this point let us consider a simple cylindrical straight tube model but with a situation where the Alfv´en speed smoothly increases with the radial coordinate r.pThe Alfv´en frequency is simply given by VA (r)k while the (local) cut-off frequency is VA (r) k2 + (m/r)2 . It is clear that for this situation the external resonant position and the tunnelling position are separated in space. Thus, the important point here is that, for an inhomogeneous external medium we expect the modes to be resonantly damped and leaky at the same time, with the resonant layers (internal and/or external) acting, on one hand, as sinks of energy and, at the same time, the energy being radiated away from the loop due to wave tunnelling.

22

Fig. 12 The real and imaginary part of the frequency as a function of the width of the inhomogeneous layer for the kink mode with horizontal velocity polarisation. The solid line corresponds to the thin loop, thin boundary approximation (Eq. (5) in Goossens et al. 2002), whereas circles, triangles, and squares correspond to σ = −.5, 0, .5. In this plot a/L = (20π)−1 ≃ 0.016 (figure taken from Terradas et al. 2006b).

In order to study the effect of a non-uniform external Alfv´en frequency we have represented in Fig. 11 the dependence of the frequency of the calculated eigenmodes with σ (for a fixed value of l/a). We still see that the differences between the real (Fig. 11a) and imaginary parts of the frequency (Fig. 11b) of the kink-like modes with horizontal and vertical polarisation are rather small (the maximum shifts in ωR and ωI are 0.25% and 6%, respectively), but larger than in the case of σ = 0. This should not be a surprise, because the physical properties of the medium in the different directions are brought forward when σ deviates from 0. This difference becomes especially pronounced for the damping rate ωI when σ > −1 (see Fig. 11b). To understand this difference, we have plotted ωI versus l/a for σ = −.5, 0, .5 in the right panel of Fig. 12 for the horizontal mode only. The behaviour of σ = ±.5 follows the behaviour of σ = 0 relatively well for l/a sufficiently large. However, the different physics comes into play when l/a tends to 0. In that region, the damping rate for σ = 0 tends to the classical damping formula (full line in Fig. 12). In contrast, the damping rates for σ , 0 tend to a constant, non-zero value. This means that the wave tunnelling and the external resonance only have an effect on the eigenmode when the damping in the internal resonant layer is not strong, i.e. when l/a is small.

5.4 Discussion These numerical calculations have confirmed and extended the results from the previous analytical sections. It has been shown that the period and damping time of a curved toroidal loop surrounded by a smooth transition layer are similar to those of a straight cylinder with a density transition to the corona, especially in the limit of thin tube and thin boundary. Although curved loops attenuate slightly faster than the straight ones (note that ωI is always larger that the predictions of the TTTB), in general, curvature does not significantly affect the efficiency of the resonant absorption mechanism. Hence, the oscillatory and damping features of the kink modes in the curved model are determined basically by the loop length, density contrast and width of the inhomogeneous layer. Also, the phenomenon of wave leakage has been confirmed. Similar to the slab case, a wave tunnelling barrier exists through which the wave energy may leak out of the loop system. Nevertheless, the wave leakage in the toroidal configuration is expected to be weaker than in the slab case. As it has been shown in Sect. 3, this is caused by the radial dependence of

23

the wave eigenfunction. In the slab configuration, the eigenfunction is not very confined to the density enhancement, but in the cylindrical configuration a much better confinement, i.e. steeper exponential behaviour, is found. Because of this better confinement, the efficiency of the tunnelling barrier is effectively increased, and allows less energy to leak away. More importantly, the numerical calculations have shown several new effects. It is found that the curvature breaks the degeneracy of the oscillation direction in the straight cylinder. Because of the curvature, two preferential oscillation directions are introduced: vertically and horizontally polarised modes. These modes have a slightly different frequency, but the difference is not large enough to be observationally noticeable. The frequency difference is proportional to the aspect ratio a/L, and will thus be largest for short low-lying loops. It has also been shown that the wave leakage and the resonant absorption in the external layer cannot be viewed as independent mechanisms. They are switched on when an inhomogeneity is present in the external medium. Nevertheless it was shown that their effect is only noticeable when the damping due to the internal resonance is very weak. This is in line with the analytical results by Goossens & Hollweg (1993) and indicates that for practical, i.e. seismological, purposes the oscillation is mainly damped by the resonant conversion of the wave power to Alfv´en modes in the loop boundary layer, because observed coronal loops have smooth density profiles (Aschwanden et al. 2003) with an associated strong damping. Other numerical work, based on the solution of the time-dependent problem, has been done by Miyagoshi et al. (2004), who have performed three-dimensional simulations of horizontally polarised fast magnetoacoustic waves in a potential arcade, triggered by a velocity field localised near the top of a magnetic flux tube. They concluded that the damping of the oscillation is explained as energy transport by fast magnetoacoustic waves propagating away from the excited field lines. They also concluded that an initially imposed density enhancement in the flux tube, which is not in equilibrium, does not substantially enhance the trapping of the excited waves. This result, though, is based only on three choices of initial density contrast. Another full 3D numerical simulation has been performed by McLaughlin & Ofman (2008). They find a very strong leakage of wave energy away from the toroidal loop, and they claim that it is due to curvature. These authors also find that the damping time is proportional to the density contrast, which is in contradiction with the work of Brady & Arber (2005), where the damping time was found to be proportional to the square root of the density contrast. Unfortunately, the eigenmodes of such a complex equilibrium, which would provide a definitive answer, are unknown. The strong damping found in the simulations is slightly surprising since the results shown in this review indicate that the leakage in a 3D model is in general expected to be small. On the other hand, the role of the external resonance, unavoidable in three-dimensional configurations, has not been analysed yet in these numerical studies. 6 Conclusions We have given an overview of the effects of curvature on coronal loop kink oscillations. Three main results have been discussed. – Because of the curved configuration, it is impossible to consider homogeneous magnetic fields. As a result, lateral wave leakage from the coronal loop is introduced, through a wave tunnelling barrier. We have reviewed results in a curved slab which quantified the wave leakage analytically. To separate the effect of curvature and wave leakage, we have built a model (straight slab, straight cylinder with a tunnelling region) to study the

24

effect of leakage without the curvature. From such a model, it has become clear that the leakage is much stronger in the slab case. This is caused by the different radial shape of the eigenfunction: the cylindrical eigenfunction decreases more steeply than the slab eigenfunction, and thus tunnels less wave energy through the barrier. – Next, we focused on an analytical model for a toroidal loop without leakage. We have presented the eigenfunctions of such a system, and have shown that azimuthal wave numbers are coupled, because of the symmetry breaking in the azimuthal direction. We have corrected a previously obtained dispersion relation, and found that the frequency is not modified in the first order of the curvature (ε =minor radius/major radius). The changes are at most ε2 , which means at maximum 6% in observed coronal loops. On the other hand, the eigenfunction itself has first order modifications. – We have reviewed numerical results using a generalised model of a toroidal loop including leakage. In the case of no leakage, the model reproduces the analytical results. Moreover, it finds that preferential oscillation directions are introduced. Two eigenmodes are found, one with a vertical polarisation, the other with a horizontal velocity field. It is found that the frequency difference of these modes is very small, and not observable. Efforts to observationally distinguish between these modes have been done by Wang et al. (2008). When leakage is included in the numerical model, it was shown that the resonant absorption in the loop edge is a much more efficient energy sink than the lateral wave leakage due to tunnelling. It is found that wave leakage is only important for loops with a very sharp radial density profile. In the case that the Alfv´en frequency is constant (σ = 0), the curvature does not have a large effect on the nature of the kink mode. As was shown, the eigenfrequency of the kink mode does not change significantly. On the other hand, the inclusion of curvature introduces preferred oscillation polarisations: horizontal and vertical, which may be distinguished observationally. In all other cases, curvature will have a significant effect. Among others, it introduces lateral wave leakage caused by the change of geometry. This effect is much less pronounced in cylindrical configurations than in slab geometry, but this would have to be verified in a general three dimension geometry. We have also shown that, in the case of a weakly changing radial Alfv´en frequency profile, the damping due to resonant absorption is much more pronounced than the damping due to leakage. However, when the radial Alfv´en frequency profile changes very rapidly near the loop, it may introduce significant damping and even change the kink mode period (because of the modified dispersion relation).

A Correction of the jump conditions To calculate the jump conditions and the correct left hand side of the matching condition (29), we start from Eq. (5) in Van Doorsselaere et al. (2004b) and use the u-component of Eq. (4, with the sign error corrected) to eliminate Vu . Expanding the resulting equation around the resonance, we obtain that [Vu ]u=uA ≡ VuA ,i − VuA ,e = −πdω

ω2A k 2 ∆A

sinh2 uA

1 ∂ bφ ∂ ∂v cosh uA − cos v ∂v B

Multiplying this equation with −

ik2 cosh uA − cos v ωd sinh2 uA

r

sinh uA exp (−im′ v), cosh uA − cos v

!!

.

25 and integrating over v ∈ [0; 2π], the left hand side of the matching condition becomes:  Z 2π ∞ ( iπω2A  2 X 1 1 cos v −m + exp (iδv) dv ∆A 2π 2 cosh u A − cos v 0 δ=−∞ )# Z 2π 1 3 sin2 v − dv , exp (iδv) 2π 0 4 (cosh uA − cos v)2

(31)

when δ = m − m′ and dropping the primes. To calculate the correct expressions for the integrals, some complex analysis is needed. Let’s concentrate on the first integral. Substitute Z = eiv to rewrite the integral to: −

1 1 2 2πi

I

Z δ−1

Γ1

Z2

Z2 + 1 dZ, − 2Z cosh uA + 1

where Γ1 is the unit circle in the complex plane. The integrand has 2 simple poles at Z = exp (±uA ), and, depending on the value of δ, a pole of higher order at Z = 0. We have to separate between cases. First consider δ − 1 ≥ 0. In that case, only a simple pole is found within the integration contour. Calculating the residue at Z = exp (−uA ), the integral becomes: 1 exp (−δuA ) coth uA . 2 When δ − 1 < −1, the integrand is decreasing as Z δ−1 ≤ Z −2 for Z → ∞. Thus, the contour can be deformed to infinity and the integral can be calculated by taking a tiny contour around the pole at Z = exp uA going in the clockwise direction. Now the residue at Z = exp uA has to be calculated and the integral is 1 exp (δuA ) coth uA . 2 When δ = 0, an exception has to be made. The integral now has two poles inside the integration contour. Calculating the residues at Z = 0 and Z = exp (−uA ), the integral can be found to be: 1 (coth uA − 1). 2 These expressions for the integral can be generalized for all δ: 1 exp (−|δ|uA )(coth uA − δk (δ)), 2

(32)

where δk is the Kronecker-delta. Now concentrate on the second integral. Using the same substitution as in the previous integral, the current integral can be rewritten to: 3 1 − 4 2πi

I

Γ1

Z

δ−1



2 Z2 − 1


2 dZ. Z 2 − 2Z cosh uA + 1

As before, we have to seperate between cases. First, we assume that δ − 1 ≥ 0. In this case, the integrand only has a single pole of the order 2 at Z = exp (−uA ). Calculating the residue at that pole, the integral becomes: " # 3 exp (−uA ) − exp (−δuA ) δ − 1 − . 4 sinh uA Whe δ − 1 < −1, the integrand is decreasing as Z δ−1 ≤ Z −2 for Z → ∞. Thus, similar to the first integral, the contour can be deformed to infinity and the integral can be calculated by taking a tiny contour around the pole at Z = exp uA going in the clockwise direction. After calculating the residue at the pole of order 2 Z = exp uA , the integral can be evaluated to be: " # exp uA 3 exp (δuA ) δ − 1 + . 4 sinh uA

26 Again an exception has to be made for δ = 0. Now the integrand has two poles (Z = 0 and Z = exp (−uA )) inside the integration contour. For this case, the integral can be evaluated to be: 3 exp (−uA ) . 4 sinh uA These results for the second integral can be generalized to: " # exp (−sgn(δ)uA ) 3 exp (−|δ|uA ) δk (δ) − |δ − 1| + , 4 sinh uA

(33)

if it is assumed that sgn(0) = 1. Substituting the obtained expression for the integrals (Eqs. 32-33) in Eq. (31), the left hand side of the matching condition (Eq. 29) becomes: iπω2A |∆A |

! 3 exp (−uA ) 1 (coth uA − 1) + 2 4 sinh uA 2 X iπωA 2 Fi,m+δ (u0 ) Ci,m+δ (αi − 1)(α2e − 1) + |∆A | Fi,m (u0 ) δ,0

(α2i − 1)(α2e − 1)Ci,m m2 +

! 3 3 exp (−sgn(δ)uA ) exp (−|δ|uA ) coth uA − |δ − 1| + . 2 2 2 sinh uA

Acknowledgements JT acknowledges support from K.U.Leuven via GOA/2009-009. The research leading to these results has received funding from the European Community’s Seventh Framework Programme (FP7/2007-2013) under grant agreement n˚220555. It was inspired by a workshop held at ISSI.

References Abramowitz, M. & Stegun, I. A. 1964, Handbook of Mathematical Functions (National Bureau of Standards, Washington DC) Andries, J., Roberts, B., Van Doorsselaere, T., et al. ???? Arregui, I., Andries, J., Van Doorsselaere, T., Goossens, M., & Poedts, S. 2007, Astron. Astrophys., 463, 333 Aschwanden, M. J., De Pontieu, B., Schrijver, C. J., & Title, A. M. 2002, Solar Phys., 206, 99 Aschwanden, M. J., Nightingale, R. W., Andries, J., Goossens, M., & Van Doorsselaere, T. 2003, Astrophys. J., 598, 1375 Brady, C. S. & Arber, T. D. 2005, Astron. Astrophys., 438, 733 Brady, C. S., Verwichte, E., & Arber, T. D. 2006, Astron. Astrophys., 449, 389 Cargill, P. J., Chen, J., & Garren, D. A. 1994, Astrophys. J., 423, 854 Del Zanna, L., Schaekens, E., & Velli, M. 2005, Astron. Astrophys., 431, 1095 D´ıaz, A. J. 2006, Astron. Astrophys. D´ıaz, A. J., Zaqarashvili, T., & Roberts, B. 2006, Astron. Astrophys. Edwin, P. M. & Roberts, B. 1982, Solar Phys., 76, 239 Edwin, P. M. & Roberts, B. 1983, Solar Phys., 88, 179 Goossens, M., Andries, J., & Aschwanden, M. J. 2002, Astron. Astrophys., 394, L39 Goossens, M., Arregui, I., Ballester, J. L., & Wang, T. J. 2008, Astron. Astrophys., 484, 851 Goossens, M. & Hollweg, J. V. 1993, Solar Phys., 145, 19 Gruszecki, M. & Murawski, K. 2008, Astron. Astrophys., 487, 717 Gruszecki, M., Murawski, K., & McLaughlin, J. A. 2008, Astron. Astrophys., 489, 413 Gruszecki, M., Murawski, K., Solanki, S. K., & Ofman, L. 2007, Astron. Astrophys., 469, 1117 Luna, M., Terradas, J., Oliver, R., & Ballester, J. L. 2008, Astrophys. J., 676, 717 McLaughlin, J. A. & Ofman, L. 2008, Astrophys. J., 682, 1338 Mikhalyaev, B. B. & Solov’ev, A. A. 2005, Solar Phys. Miyagoshi, T., Yokoyama, T., & Shimojo, M. 2004, Publ. Astron. Soc. Japan, 56, 207 Murawski, K. & Roberts, B. 1993, Solar Phys., 143, 89 Murawski, K., Selwa, M., & Rossmanith, J. A. 2005, Solar Phys., 231, 87 Nakariakov, V. M., Arber, T. D., Ault, C. E., et al. 2004, Mon. Not. R. Astron. Soc., 349, 705

27 Nakariakov, V. M., Ofman, L., DeLuca, E. E., Roberts, B., & Davila, J. M. 1999, Sci., 285, 862 Nakariakov, V. M., Roberts, B., & Murawski, K. 1997, Solar Phys., 175, 93 Oliver, R., Ballester, J. L., Hood, A. W., & Priest, E. R. 1993, Astron. Astrophys., 273, 647 Oliver, R., Murawski, K., & Ballester, J. L. 1998, Astron. Astrophys., 330, 726 Roberts, B. 2000, Solar Phys., 193, 139 Ruderman, M. S. 2005, in ESA Special Publication, Vol. 600, The Dynamic Sun: Challenges for Theory and Observations Sakurai, T., Goossens, M., & Hollweg, J. V. 1991, Solar Phys., 133, 227 Selwa, M., Murawski, K., Solanki, S. K., & Wang, T. J. 2007a, Astron. Astrophys., 462, 1127 Selwa, M., Murawski, K., Solanki, S. K., Wang, T. J., & T´oth, G. 2005, Astron. Astrophys., 440, 385 Selwa, M., Ofman, L., & Murawski, K. 2007b, Astrophys. J. Lett., 668, L83 Selwa, M., Solanki, S. K., Murawski, K., Wang, T. J., & Shumlak, U. 2006, Astron. Astrophys., 454, 653 Smith, J. M., Roberts, B., & Oliver, R. 1997, Astron. Astrophys., 317, 752 Spruit, H. C. 1982, Solar Phys., 75, 3 Terradas, J., Oliver, R., & Ballester, J. L. 2006a, Astrophys. J., 642, 533 Terradas, J., Oliver, R., & Ballester, J. L. 2006b, Astrophys. J. Lett., 650, L91 Terradas, J., Oliver, R., Ballester, J. L., & Keppens, R. 2008, Astrophys. J., 675, 875 Tirry, W. J. & Goossens, M. 1996, Astrophys. J., 471, 501 Tsiklauri, D., Nakariakov, V. M., Arber, T. D., & Aschwanden, M. J. 2004, Astron. Astrophys., 422, 351 Van Doorsselaere, T., Andries, J., Poedts, S., & Goossens, M. 2004a, Astrophys. J., 606, 1223 Van Doorsselaere, T., Debosscher, A., Andries, J., & Poedts, S. 2004b, Astron. Astrophys., 424, 1065 Van Doorsselaere, T., Ruderman, M. S., & Robertson, D. 2008, Astron. Astrophys., 485, 849 Verwichte, E., Foullon, C., & Nakariakov, V. M. 2006a, Astron. Astrophys., 446, 1139 Verwichte, E., Foullon, C., & Nakariakov, V. M. 2006b, Astron. Astrophys., 449, 769 Verwichte, E., Foullon, C., & Nakariakov, V. M. 2006c, Astron. Astrophys., 452, 615 Wang, T. J. & Solanki, S. K. 2004, Astron. Astrophys., 421, L33 Wang, T. J., Solanki, S. K., & Selwa, M. 2008, Astron. Astrophys., 489, 1307