Resonant Absorption of Axisymmetric Modes in Twisted Magnetic Flux Tubes
Ioannis Giagkiozis, Marcel Goossens, Gary Verth, Viktor Fedun, Tom Van Doorsselaere
aa r X i v : . [ a s t r o - ph . S R ] J un Resonant Absorption of Axisymmetric Modesin Twisted Magnetic Flux Tubes
I. Giagkiozis , M. Goossens , G. Verth , V. Fedun , T. Van Doorsselaere Received ; accepted Solar Plasma Physics Research Centre, School of Mathematics and Statistics, Universityof Sheffield, Hounsfield Road, Hicks Building, Sheffield, S3 7RH, UK Centre for mathematical Plasma Astrophysics, Mathematics Department, KU Leuven,Celestijnenlaan 200B bus 2400, B-3001 Leuven, Belgium Department of Automatic Control and Systems Engineering, University of Sheffield,Mappin Street, Amy Johnson Building, Sheffield, S1 3JD, UK 2 –
ABSTRACT
It has been shown recently that magnetic twist and axisymmetric MHD modesare ubiquitous in the solar atmosphere and therefore, the study of resonant ab-sorption for these modes have become a pressing issue as it can have importantconsequences for heating magnetic flux tubes in the solar atmosphere and the ob-served damping. In this investigation, for the first time, we calculate the dampingrate for axisymmetric MHD waves in weakly twisted magnetic flux tubes. Ouraim is to investigate the impact of resonant damping of these modes for solaratmospheric conditions. This analytical study is based on an idealized config-uration of a straight magnetic flux tube with a weak magnetic twist inside aswell as outside the tube. By implementing the conservation laws derived bySakurai et al. (1991a) and the analytic solutions for weakly twisted flux tubesobtained recently by Giagkiozis et al. (2015), we derive a dispersion relation forresonantly damped axisymmetric modes in the spectrum of the Alfv´en contin-uum. We also obtain an insightful analytical expression for the damping ratein the long wavelength limit. Furthermore, it shown that both the longitudinalmagnetic field and the density, which are allowed to vary continuously in theinhomogeneous layer, have a significant impact on the damping time. Given theconditions in the solar atmosphere, resonantly damped axisymmetric modes arehighly likely to be ubiquitous and play an important role in energy dissipation.We also suggest that given the character of these waves, it is likely that theyhave already been observed in the guise of Alfv´en waves.
Subject headings: axisymmetric modes, mhd, resonant absorption
1. Introduction
Inhomogeneities, such as a density variation across a magnetic flux tube, produce acontinuous spectrum of eigenfrequencies. For instance, consider a straight magnetic fluxtube of radius r e and constant temperature, where the density varies smoothly from itscenter to its boundary, such that cylindrical surfaces have constant density. This meansthat also the sound and Alfv´en speeds within every cylindrical surface are constant. Theseconcentric cylindrical sheaths comprise the flux tube. Due to the difference in characteristicspeeds, every surface will have its own eigenfrequency. This results in an infinite set ofeigenfrequencies, a continuum. One of the consequences of this continuum in driven systemsis resonant absorption, assuming the driving frequency is within the continuum.Given that inhomogeneities are the rule rather than the exception in the solaratmosphere, resonant absorption is bound to occur there. This has long been recognized,from the first suggestion by Ionson (1978) to subsequent studies motivated by advancesin solar observations, see for example the following works (Poedts et al. 1989, 1990;Ruderman & Roberts 2002; Goossens et al. 2002; Andries et al. 2005; Goossens et al.2009; Van Doorsselaere et al. 2009; Verth et al. 2010; Terradas et al. 2010; Antolin et al.2015; Okamoto et al. 2015) to name but a few. In general, resonant absorption inmagnetohydrodynamic (MHD) modes is important for the solar atmosphere. Some ofthe many reasons for this are the following. Resonant damping of Alfv´en waves is anatural and efficient mechanism for energy dissipation of MHD waves in inhomogeneousplasmas (Ionson 1978, 1985; Hollweg & Yang 1988). It can also provide an explanationfor the observed loss of power of acoustic modes in sunspots (Hollweg 1988; Sakurai et al.1991a,b; Goossens & Poedts 1992; Keppens et al. 1994), and, it has been shown that it isof importance in transverse oscillations (kink mode), see for example (Aschwanden et al.1999; Nakariakov et al. 1999; Ruderman & Roberts 2002; Goossens et al. 2002). Resonant 4 –Alfv´en waves can be an energy conduit between photospheric motions at the footpointsof coronal loops (see for example De Groof & Goossens 2000; De Groof et al. 2002;De Groof & Goossens 2002), and, resonant dissipation plays an important role in theobserved damped oscillations in prominences (see Terradas et al. 2008; Arregui et al. 2012).For an in depth review of resonant absorption in the solar atmosphere see Goossens et al.(2011).Since 1999, when the first post-flare standing mode transverse oscillations weredetected using the Transition Region and Coronal Explorer (TRACE) (Aschwanden et al.1999; Nakariakov et al. 1999) there has been a growth in studies of resonant absorptionfor the kink mode. Ruderman & Roberts (2002) produced relations describing theexpected damping for coronal loops using the long wavelength and pressure-less plasma approximations, a result that was previously obtained by Goossens et al. (1992) using theconnection formulae derived by Sakurai et al. (1991a,b) for the driven problem and byTirry & Goossens (1996) for the eigenvalue problem. Later Goossens et al. (2002) andAschwanden et al. (2003) used these results and calculated the expected damping times fora sequence of observed parameters for coronal flux tubes. Goossens et al. (2002) concludedthat for the parameter sample used, resonant absorption can explain the observed dampingtimes well, provided that the density contrast is allowed to vary from loop to loop. Anotherimportant result in this work is that the observed damping does not require modificationof the order of magnitude estimates of the Reynolds number (10 ) as suggested byNakariakov et al. (1999). Aschwanden et al. (2003) also arrived at the conclusion that,on average, the theoretical predictions of the damping rate derived by Goossens et al.(1992) and Ruderman & Roberts (2002), are consistent with observations and suggestedthat damping times of coronal loops can be used to infer their density contrast with the Also referred to as cold plasma approximation. 5 –surrounding plasma. Coronal flux tubes tend to deform in their middle section due tobuoyancy, effectively resulting in cross-sections that are approximately elliptical. Ruderman(2003) studied the damping of the kink mode in flux tubes with an elliptical cross-sectionand found that for moderate ratios of the minor to major semi-axis the difference of thedamping rate for resonant absorption compared with flux tubes with circular cross-sectionis not very large. Another deviation from the ideal straight magnetic flux tube is axialcurvature. Van Doorsselaere et al. (2004) studied the effect of this curvature and also foundthat the longitudinal curvature of flux tubes does not significantly alter the damping timeof kink modes. Progressively the theoretical models for kink oscillations have become moreelaborate, for example, Andries et al. (2005) considered longitudinal density stratification.Also, methods for kink wave excitation have been studied, see for example Terradas (2009).The increased body of observations of kink waves allowed Verwichte et al. (2013) to performa statistical study to constrain the free parameters present in theoretical models of resonantabsorption in kink modes.In contrast to this avalanche of theoretical and observational advances related to thekink mode, resonant absorption for axisymmetric modes has not received much attention.One reason for this is that it was believed that the sausage mode had a long wavelengthcutoff (e.g. Edwin & Roberts 1983) which suggested that observation of the sausage modewould be quite challenging. Furthermore, it was correctly believed that for a straightmagnetic field, axisymmetric modes could not be resonantly damped. However, it isapparent, even in early works in resonant absorption (see for example Sakurai et al. 1991a,b;Goossens et al. 1992), that for weakly twisted magnetic field axisymmetric modes can andare resonantly damped. What was not known until recently, however, was that the longwavelength cutoff for these modes is also removed in the presence of weak magnetic twist(Giagkiozis et al. 2015). Therefore these modes can freely propagate for all wavelengths.And so, at least in principle, these modes should be observable. Additionally, recent works 6 –suggest that magnetic twist and axisymmetric modes are ubiquitous throughout the solaratmosphere. Therefore, the study of these modes has become quite relevant and important.Some examples of magnetic twist in the solar atmosphere are, flux tubes emerging fromthe convection zone (see for example Hood et al. 2009; Luoni et al. 2011), sunspot rotationcan result in twisted magnetic fields (Brown et al. 2003; Yan & Qu 2007; Kazachenko et al.2009), spicules are observed to have twist (De Pontieu et al. 2012; Sekse et al. 2013) as wellas solar tornadoes (Wedemeyer-B¨ohm et al. 2012). Lastly observations of axisymmetricmodes have been recently reported in Morton et al. (2012) and Grant et al. (2015).In this work, we focus on the resonant absorption of axisymmetric MHD modes inweakly twisted magnetic flux tubes. Axisymmetric modes correspond to modes withazimuthal wavenumber m = 0. We accomplish this using the following sequence. Firstwe recall recent results for axisymmetric modes in magnetic flux tubes with weak twist(Giagkiozis et al. 2015). In that work the longitudinal component of the magnetic field, andthe density were discontinuous across the flux tube boundary. This choice was intentionalas it avoids the MHD continua and simplifies the analysis. However, this also left outrelevant physics. Then having as a starting point the setup in Giagkiozis et al. (2015) weintroduce an intermediate layer about the flux tube boundary. Within this layer, we allowthe magnetic field and density to vary smoothly, resulting in an overall continuous profilefor the longitudinal magnetic field and density. This in turn allows for the existence of thetwo MHD continua, the slow and Alfv´en continuum. Next, we assume that the layer thatconnects the internal and external quantities, is thin, namely we assume that ℓ ≪ r e where ℓ is the width of the layer and r e is the flux tube radius. Then we use the conservation laws,and the resulting jump conditions, for the Alfv´en continuum by Sakurai et al. (1991a), andwe derive the resulting complex dispersion relation. We then solve this dispersion relationnumerically. Lastly, to better understand the predicted damping times we apply the longwavelength limit approximation to the resulting complex dispersion relation. These simpler 7 –relations allow us to compare our results with the expected damping for the kink modepredicted using the results by Goossens et al. (1992) and Ruderman & Roberts (2002).We conclude this investigation with a statistical analysis of the resulting approximationsto further understand the necessary conditions for the observation of resonantly dampedaxisymmetric modes. The main contributions of this work can be summarized as follows. • For the first time, we uncover a dispersion relation for axisymmetric modes inmagnetic flux tubes with internal and external twist, including the resonance with theAlfv´en continuum. We produce simplified expressions for the frequency and dampingtime in the long wavelength limit, for which the axisymmetric modes are no longerleaky. • Given that there are four parameters required for the evaluation of the aforementionedrelation, namely density contrast, magnetic field contrast, thickness of theinhomogeneous layer and magnetic twist, we present a statistical framework to inferwhat can be drawn from observations. • We use this statistical framework and show that the predictions of our theoreticalmodel are in agreement with observed damping times that are in agreement withobserved damping times of quasi periodic pulsations (QPPs). QPPs are interpretedas axisymmetric modes (sausage modes) (Kolotkov et al. 2015).The plan of this paper is as follows. In Section 2 we present the model, and includeprior theoretical results required for the derivation of the dispersion relation leading toresonant absorption. In Section 3, using the jump relations in Sakurai et al. (1991a) wederive a dispersion equation. In Section 4 we use the dispersion relation derived in Section 3to obtain an expression for the damping rate in the long wavelength limit and then inSection 5 we elaborate on the significance of the results in this work for the observation 8 –of axisymmetric modes in the solar atmosphere. Lastly, in Section 6 we summarize andconclude this work.
2. Model
In this work we assume an idealized cylindrically symmetric magnetic flux tube instatic equilibrium. We employ cylindrical coordinates r, ϕ and z , with the z coordinatealong the axis of symmetry of the flux tube. The linearized ideal MHD equations are, (1a) ρ ∂ ξ ∂t + ∇ p ′ + 1 µ ( B ′ × ( ∇ × B ) + B × ( ∇ × B ′ )) = 0 , (1b) p ′ + ξ · ∇ p + γp ∇ · ξ = 0 , (1c) B ′ + ∇ × ( B × ξ ) = 0 , where ρ, p and B are the density, plasma kinetic pressure and magnetic field, respectively, atequilibrium, ξ is the Lagrangian displacement, p ′ and B ′ are the Eulerian variations of thepressure and magnetic field, γ is the ratio of specific heats (taken to be 5 / µ is the permeability of free space. In what follows an index, i , indicates quantitiesinside the flux tube ( r < r i ) while variables indexed by, e , refer to the environment outsidethe flux tube ( r > r e ). The inhomogeneous layer has a width equal to ℓ = r e − r i and itis assumed that ℓ ≪ r e . Note that in Giagkiozis et al. (2015), r a , was used to denote thetube radius, this is equivalent to r e in this work. The model configuration is illustratedin Figure 1 when B ϕe ∝ /r . The quantities ρ, p and B are assumed to have only an r -dependence, therefore, the following balance equation must be satisfied when ℓ = 0, (2) ddr (cid:18) p + B ϕ + B z µ (cid:19) = − B ϕ µ r . The equilibrium magnetic field is taken to be B = (0 , B ϕ , B z ), with B ϕi = Sr , B ϕe = r κe S/r κ and B zi , B ze constant. By substituting B ϕi and B ϕe into Eq. (2) and 9 –Fig. 1.— Illustration of the model used in this paper. Straight magnetic cylinder withvariable twist inside ( r < r i ) and outside ( r > r e ) the tube. The region where r i < r < r e is the inhomogeneous layer, where the B z component of the magnetic field and the densityare varying continuously across this layer. The parameters ρ i , p i and T i are respectively thedensity, kinetic pressure and temperature at equilibrium inside the tube, i.e. for r < r i . Thecorresponding quantities outside the tube ( r > r e ) are denoted with a subscript e . Also, r A is the radius at the resonance. The dark blue surface emanating radially outwards inside thetube represents the influence of B ϕ ∝ r . The yellow surface outside the tube correspondsto B ϕ ∝ /r dependence. The dashed red rectangle depicts a magnetic surface which wouldcorrespond to a magnetic field with only a longitudinal ( z ) magnetic field component. Theinhomogeneous layer is bounded between r i and r e and is of width ℓ . Note that the radiusof the tube with the inhomogeneous layer is r e . 10 –defining B ϕA = B ϕ ( r e ) = Sr e , we obtain: (3) p ( r ) = B ϕA µ (cid:18) − r r e (cid:19) + p e for r ≤ r e ,r κe B ϕA (1 − κ )2 µ κ (cid:18) r κ − r κe (cid:19) + p e for r > r e , where, p e , is the pressure at the boundary of the magnetic flux tube and the parameter κ → /r while κ → p ( r ) is continuous, for solar atmospheric conditionsand for weak magnetic twist (sup( B ϕ /B z ) ≪
1) its variation is much smaller than p e and therefore can be assumed to be constant (Giagkiozis et al. 2015). However, in themodel used by Giagkiozis et al. (2015) the equilibrium density and the z component of themagnetic field are discontinuous, therefore the Alfv´en continuum was avoided. Note that inGiagkiozis et al. (2015) the equivalent to Eq. (3) had a typographical error, (1 − κ ) shouldread (1 − κ ).In the present investigation both the density and the magnetic field are continuous,see Figure 2, which introduces the slow and fast continua into our model. Specifically, thedensity is assumed to be a piecewise linear function of the following form, (4) ρ ( r ) = ρ i for r < r i ,ρ i + r − r i ℓ ( ρ e − ρ i ) for r i ≤ r ≤ r e ,ρ e for r > r e , a similar form for the variation in the longitudinal component of the magnetic field isassumed, namely, (5) B z ( r ) = B zi for r < r i ,B zi + r − r i ℓ ( B ze − B zi ) for r i ≤ r ≤ r e ,B ze for r > r e .
11 –Note that the assumption here is that ℓ ≪ r e , so that pressure balance is maintained (seeEq. (2)). Also note that allowing both the density and the magnetic field to vary results ina non-monotonic variation in the Alfv´en frequency across the inhomogeneous layer as seenin Figure 3.The equilibrium quantities depend only on r and therefore the perturbed quantitiescan be Fourier analyzed with respect to the ϕ and z coordinates, namely, (6) ξ , p ′ T ∝ e i ( mϕ + k z z − ωt ) . Here, ω is the angular frequency, m is the azimuthal wavenumber, k z is the longitudinalwavenumber, and p ′ T is the Eulerian total pressure perturbation defined as p ′ + BB ′ /µ . Ourfocus is on axisymmetric modes (sausage waves) and therefore the azimuthal wavenumber istaken to be m = 0. The Lagrangian displacement vector in flux coordinates is ξ = ( ξ r , ξ ⊥ , ξ k )where, ξ ⊥ = B z ξ ϕ − B ϕ ξ z | B | , ξ k = B ϕ ξ ϕ + B z ξ z | B | , (7)assuming B r = 0. Using Eq. (6), Eq. (1) can be transformed to the following two coupledfirst order differential equations, (8a) D d ( rξ r ) dr = C ( rξ r ) − rC p ′ T , (8b) D dp ′ T dr = 1 r C ( rξ r ) − C p ′ T . (8c) ρ ( ω − ω A ) ξ ⊥ = ı | B | C A , (8d) ρ ( ω − ω c ) ξ k = ıf B | B | v s v s + v A C S , (8e) ∇ · ξ = − ω C S ρ ( v s + v A )( ω − ω c ) 12 –Fig. 2.— Density profile as a function of r in the inhomogeneous layer of the magnetic fluxtube. Here, r i and r e are the radius at which the inhomogeneous begins and ends respectively,also, r e is the flux tube radius. Lastly, r A , is the radius at the resonance. 13 –and, (9a) D = ρ ( ω − ω A ) C , (9b) C = 2 B ϕ µ r (cid:16) ω B ϕ − mr f B C (cid:17) , (9c) C = ω − (cid:18) k z + m r (cid:19) C , (9d) C = ρD (cid:20) ω − ω A + 2 B ϕ µ ρ ddr (cid:18) B ϕ r (cid:19)(cid:21) + 4 ω B ϕ µ r − ρC B ϕ ω A µ r , (9e) C = ( v s + v A )( ω − ω c ) , (9f) C A = g B p ′ T − f B B ϕ B z ξ r µ r , C S = p ′ T − B ϕ ξ r µ r where, v s = γ pρ , v A = B µ ρ ,ω c = v s v A + v s ω A , ω A = f B µ ρ ,f B = k · B = mr B ϕ + k z B z , g B = ( k × B ) r = mr B z − k z B ϕ . Here, k = (0 , m/r, k z ) is the wavevector, C A and C S are the coupling functions, v s isthe sound speed, v A is the Alfv´en speed, ω c is the cusp angular frequency and ω A is theAlfv´en angular frequency. Eq. (8) was initially derived by Hain & Lust (1958) and later byGoedbloed (1971); Sakurai et al. (1991a). The first order coupled ODEs in Eq. (8) can bereduced to a single second order ODE for ξ r , (10) ddr (cid:20) DrC ddr ( rξ r ) (cid:21) + (cid:20) D (cid:18) C − C C (cid:19) − r ddr (cid:18) C rC (cid:19)(cid:21) ξ r = 0 .
14 –The assumption of axisymmetry ( m = 0) leads to, f B = k z B z , g B = − k z B ϕ , C A = − k z B ϕ (cid:18) p ′ T + 2 B z µ r ξ r (cid:19) . (11)And therefore, (12a) ρ ( ω − ω A ) ξ ⊥ = − ı k z B ϕ | B | (cid:18) p ′ T + 2 B z µ r ξ r (cid:19) , (12b) ρ ( ω − ω c ) ξ k = ı k z B z | B | v s v s + v A (cid:18) p ′ T − B ϕ µ r ξ r (cid:19) . Note that Eq. (12) suggests that the solutions for the components of the Lagrangiandisplacement vector are coupled. Coupled in the sense that elimination of one component,e.g. by setting it to be identical to zero, has direct implications to the remainingcomponents. To see this, consider a solution for which ξ r = 0, then by Eq. (10), p ′ T mustalso be equal to zero and as a consequence of Eq. (12a) and Eq. (12b) it follows immediatelythat ξ ⊥ and ξ k must also be identically equal to zero. Namely setting ξ r = 0 leads to thetrivial solution. Alternatively, let us assume that ξ ⊥ = 0. In this case, by Eq. (12a) thefollowing relation must hold, (13) p ′ T = − B z µ r ξ r . This in turn implies, (14) ρ ( ω − ω c ) ξ k = − ı k z B z | B | µ r v s v s + v A ξ r , which in general is non-zero. Now, if we assume ξ k = 0 then, (15) p ′ T = 2 B ϕ µ r ξ r which leads to, (16) ρ ( ω − ω A ) ξ ⊥ = − ı k z B ϕ | B | µ r ξ r .
15 –In the case where B ϕ = 0 then ξ ⊥ decouples from ξ r and ξ k . At this point it is instructiveto mention the interpretation of the three components of ξ in flux coordinates byGoossens et al. (2011). Goossens et al. (2011) suggest that ξ ⊥ is the dominant componentfor Alfv´en waves and for low plasma- β the slow and fast magnetoacoustic waves ξ k and ξ r isthe dominant component, respectively. A quick check, by setting B ϕ = 0 in Eq. (7), renders ξ ⊥ equivalent to ξ ϕ . This illuminates the connection of ξ ⊥ with torsional Alfv´en waves.Giagkiozis et al. (2015) solved Eq. (10) for weak internal and external magnetictwist, albeit with the density profile assumed piecewise constant. With the help of theconservation relations for the Alfv´en continuum derived by Sakurai et al. (1991a), thesesolutions, which are for ideal MHD, can be used to produce a dispersion relation for MHDwaves that undergo damping in the continuum. The solutions by Giagkiozis et al. (2015)are as follows, (17a) ξ ri ( s ) = A i s / E / e − s/ M ( a, b ; s ) , (17b) p ′ T i ( s ) = A i k a D i n i − k z e − s/ (cid:20) n i + k z k z sM ( a, b ; s ) − M ( a, b − s ) (cid:21) , and, (18a) ξ re ( r ) = A e K ν ( k re r ) , (18b) p ′ T e = A e (cid:18) µ (1 − ν ) D e − B ϕA n e µ r ( k z − n e ) K ν ( k re r ) − D e k re K ν − ( k re r ) (cid:19) .M ( · ) is the Kummer function and K ( · ) is the modified Bessel function of the second kind(Abramowitz & Stegun 2012). The solutions in Eq. (17a) and Eq. (17b) were initially 16 –derived by Erd´elyi & Fedun (2007). The parameters in Eq. (17) and Eq. (18) are, a = 1 + k ri k z E / , b = 2 , (19) k a = k z (1 − α ) / , α = 4 B ϕA ω Ai µ r e ρ i ( ω − ω Ai ) , (20) s = k a E / r , E = 4 B ϕA n i µ r e D i k z (1 − α ) , (21) k r = k z (cid:18) − n k z (cid:19) , k r = ( k z v s − ω )( k z v A − ω )( v A + v s )( k z v T − ω ) , (22) n = k z ω ( ω s + ω A )( ω − ω c ) , v T = v A v s v A + v s , (23) D i = ρ i ( ω − ω Ai ) , D e = ρ e ( ω − ω Ae ) . (24)and ν is, (25) ν ( κ ; r ) = 1 + 2 r κe B ϕA µ D e r κ (cid:26) r κe B ϕA n e k z r κ + µ ρ e (cid:2) ω Ae ( n e (3 + κ ) − k z (1 − κ )) − ( n e + k z )(1 + κ ) ω (cid:3)(cid:27) . This function in Giagkiozis et al. (2015) is evaluated for κ = 0, resulting in an exactsolution for constant twist outside the flux tube which is also a zero order approximationfor the external solution when magnetic twist is proportional to 1 /r : (26) ν (0; r ) = 1 + 2 B ϕA µ D e (cid:8) B ϕA n e k z + µ ρ e (cid:2) ω Ae (3 n e − k z ) − ω ( n e + k z ) (cid:3)(cid:9) . Using ν = ν (0; r ), i.e. constant external magnetic twist, results in solutions, namely (18a)and (18b), that have approximately 5% root mean squared error when compared with theexact solution corresponding to ν = ν (1; r ), that corresponds to external magnetic twist ∼ /r . For more details see (Giagkiozis et al. 2015). 17 –Imposing continuity for the Lagrangian displacement in the radial direction and totalpressure continuity across the flux tube, (27a) ξ ri | r = r e = ξ re | r = r e , (27b) p ′ T i − B ϕi µ r ξ ri (cid:12)(cid:12)(cid:12)(cid:12) r = r e = p ′ T e − B ϕe µ r ξ re (cid:12)(cid:12)(cid:12)(cid:12) r = r e , the following dispersion relation was derived, r e D e k re K ν − ( k re r e ) K ν ( k re r e ) = ρ i v Aϕi (cid:20) k ri ( n i + k z ) − k re ( n e + k z ) (cid:21) + (1 − ν ) D e k re − D i k ri M ( a, b − s ) M ( a, b ; s ) , (28)where v Aϕi = B ϕA /µ ρ i and ω Aϕi = k z B ϕA /µ ρ i . The long wavelength limit of Eq. (28) is needed for the approximation of the locationof the resonant point used in subsequent sections and is obtained as follows. From Eq.(13.5.5) in Abramowitz & Stegun (2012) we have, (29) lim ǫ → M ( a, b − s ) M ( a, b ; s ) = 1 , furthermore, rewriting ν (0; r ) as, (30) ν (0; r ) = 1 + 2 ρ e B ϕA µ D e (cid:18) ω Ae (cid:18) n e k z − (cid:19) − ω (cid:18) n e k z + 1 (cid:19)(cid:19) k z + 4 B ϕA µ D e n e k z k z , becomes apparent that ν = 1 + O ( ǫ ), where ǫ = r e k z . Therefore using Eq. (9.6.8) and(9.6.9) in Abramowitz & Stegun (2012) we obtain that, (31) lim ǫ → K ( k re r e ) K ( k re r e ) = 0 .
18 –Using Eq. (29) and 31 in Eq. (28) we have, (32)2( ω − ω Ai ) = v Aϕi (cid:20) ( n i + k z ) − k ri k re (cid:0) n e − k z (cid:1)(cid:21) . Expanding the part in square brackets on the right hand side of this equation about ǫ = 0leads to, (33)( n i + k z ) − k ri k re (cid:0) n e − k z (cid:1) = 2 ω ( v Ai + v si ) / k z + O (cid:0) ǫ (cid:1) . Using this approximation in Eq. (32) the positive solution of the dispersion relation Eq. (28)in the long wavelength limit to first order is, (34) ω = 12 " ω Aϕi ( ω Ai + ω si ) / + (cid:18) ω Aϕi ω Ai + ω si + 4 ω Ai (cid:19) / . For notational convenience Eq. (34) is rewritten as follows, ω = ω Ai h, (35) h = 12 " q i (1 + d ) / + (cid:18) q i d (cid:19) / (36)where, q i = B ϕA /B zi and d = v si /v Ai . This ω is used as an approximation to the resonancefrequency, ω , in Section 4. Lastly, we should note that given this value for ω , althoughthe variation of the Alfv´en speed across the inhomogeneity in the flux tube is quadratic (seeFigure 3), since ω Ai < ω < ω Ae , there will only be a single resonance point.
3. Alfv´en Continuum
For an equilibrium with magnetic twist, such as the model used in this work, thetotal pressure perturbation is no longer a conserved quantity and therefore Eq. (27b) andEq. (27a) require modification. Sakurai et al. (1991a) derived new conserved quantities for 19 –Fig. 3.— An example of Alfv´en frequency variation across the resonant layer when B z = B z ( r ) and ρ = ρ ( r ), for χ = ρ e /ρ i = 0 . ζ = B ze /B zi = 0 .
35 and ℓ/r e = 0 .
2. Here r = 1 isthe tube boundary and ω A is the normalised normalized Alfv´en frequency, the normalizationis with respect to the internal Alfv´en frequency, ω Ai . 20 –the Alfv´en and slow continua. Specifically for the Alfv´en continuum the conserved quantityis, (37) C A = g B p ′ T − f B B ϕ B z ξ r µ r . Using this conserved quantity they derived jump conditions for ξ r and p ′ T , namely aprescription on how the radial component of the Lagrangian displacement and the totalpressure perturbation can vary across the inhomogeneous layer connecting the internal withthe external solutions. This prescription then implies, that the following conditions mustbe satisfied, (38) ξ ri ( r ) | r = r i + J ξ r ( r ) K = ξ re ( r ) | r = r e and (39) p ′ T i ( r ) | r = r i + J p ′ T ( r ) K = p ′ T e ( r ) | r = r e , where J ξ r K and J p ′ T K are the jump conditions across the resonant layer in the inhomogeneoussection of the flux tube, in the radial displacement and total pressure perturbation(Sakurai et al. 1991a). They are given by (40) J ξ r K = − ıπ | ∆ A | g B µ ρ v A C A , and (41) J p ′ T K = − ıπ | ∆ A | T B z µ ρ v A r C A , where T = f B B ϕ /µ = k z B z B ϕ /µ and, (42)∆ A = ddr (cid:0) ω − ω A ( r ) (cid:1) .
21 –Taking into account that m = 0 and B ϕ = 0 and Eq. (11) the jump conditions, Eq. (40)and (41), can be written as, (43) J ξ r K = ıπ | ∆ A | k z B ϕ µ ρ v A C A = − ıπ | ∆ A | k z B ϕ µ ρ v A (cid:12)(cid:12)(cid:12)(cid:12) r = r A · (cid:26) p ′ T + 2 B z µ ξ r r (cid:27)(cid:12)(cid:12)(cid:12)(cid:12) r = r i , and, (44) J p ′ T K = − ıπ | ∆ A | k z B ϕ B z µ ρ v A r C A = 2 ıπr | ∆ A | (cid:18) k z B ϕ B z µ ρv A (cid:19) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) r = r A · (cid:26) p ′ T + 2 B z µ ξ r r (cid:27)(cid:12)(cid:12)(cid:12)(cid:12) r = r i . Given that B z = B z ( r ) and ρ = ρ ( r ) in the inhomogeneous layer, see Eq. (4), Eq. (5) andFigure 2, we have, (45)∆ A = ω A ( r ) ∆ AF = ω A ( r ) (cid:20) ρ dρdr − B z dB z dr (cid:21) . Obviously when B z is constant across the inhomogeneous layer, (46)∆ A = ω A ( r ) 1 ρ dρdr . Substituting Eq. (43), 44 and 45 into Eq. (38) and Eq. (39) we obtain the dispersion relationfor axisymmetric MHD waves that undergo resonant absorption in the Alfv´en continuum offrequencies due to the twist in the magnetic field: (47) D AR ( ω, k z ) + ı D AI ( ω, k z ) = 0 , where, (48) D AR = 2 D i k ri M ( a, b − s i ) M ( a, b ; s i ) − ρ i n i ( n i + k z ) v Aϕi k ri + r i D e k re K ν − ( k re r e ) K ν ( k re r e ) − r i r e k re (cid:8) (1 − ν ) D e − ρ e n e v Aϕe (cid:9) ,
22 –and, (49) D AI = πρ | ∆ AF | v Aϕ v A (cid:12)(cid:12)(cid:12)(cid:12) r = r A (cid:20) k ri (cid:18) D i M ( a, b − s i ) M ( a, b ; s i ) − n i ( n i + k z ) ρ i v Aϕi (cid:19) + 2 ρ i v Ai (cid:21)" B z µ r (cid:12)(cid:12)(cid:12)(cid:12) r A + 1 r e k re (cid:8) (1 − ν ) D e − ρ e n e v Aϕe (cid:9) − D e k re K ν − ( k re r e ) K ν ( k re r e ) . In these equations the following definitions were used, v AϕA = B ϕA µ ρ A , v AA ( r A ) = B zA µ ρ A , (50) | ∆ AF ( r A ) | = 1 ℓ (cid:12)(cid:12)(cid:12)(cid:12) ρ e − ρ i ρ A − B ze − B zi B z ( r A ) (cid:12)(cid:12)(cid:12)(cid:12) , (51)where ρ A = ρ ( r A ), v AA = v A ( r A ) and B zA = B z ( r A ). To find the radius at the resonancepoint, namely the radius where v A ( r ) = v (= ω /k z ) , we can express r A as a convexcombination of the radius r i and the width of the inhomogeneous layer ℓ since r A must bewithin the interval ( r i , r e ). Therefore we can write r A = r i + wℓ , where w ∈ (0 , r A to a problem where we have to solvefor w , the convex combination parameter. Given this formulation for r A and Eq. (4) andEq. (5) we can write B zA = B z ( r A ) = B zi + w ( B ze − B zi ) and ρ A = ρ ( r A ) = ρ i + w ( ρ e − ρ i ).Equipped with these definitions, the equation that we need to solve to find w becomes,(52) v AA = ( B zi + w ( B ze − B zi )) µ ( ρ e + w ( ρ e − ρ i )) = v Using the definitions χ = ρ e /ρ i , ζ = B ze /B zi Eq. (52), simplifies to (53) v Ai (1 + w ( ζ − w ( χ −
1) = v . This equation is solved for w in the next section. For the definition of ω see Eq. (35). 23 –
4. Long Wavelength Limit - Alfv´en Continuum E1 Taking the long wavelength limit, ǫ ≪
1, of Eq. (48) and Eq. (49) and using Eq. (29),Eq. (31) then, Eq. (48) and Eq. (49) reduce to, (54) D AR = ω − ω Ai k ri − n i k ri v Aϕi + χ n e k re v Aϕe , (55) D AI = πr e " π ∆ AF v Aϕ v A (cid:12)(cid:12)(cid:12)(cid:12) r = r A − ρ e ρ A n e k re v Aϕe v AA (cid:21)(cid:20) ω − ω Ai k ri − n i k ri v Aϕi + v Ai (cid:21) . These equations can be solved if we allow a complex frequency ω = ω r + ıγ A , andwhen γ A ≪ ω r we can obtain the damping rate, γ A in the Alfv´en continuum frequencies(Goossens et al. 1992) to second order is given by, (56) γ A = −D AI ( ω ) ∂ D AR ∂ω (cid:12)(cid:12)(cid:12)(cid:12) ω = ω ! − . This equation results in an expression that is difficult to interpret, and for this reason, giventhat we seek an expression for the damping rate in the long wavelength limit we expand itin a series about ε = 0 where ε = r e k z . This expansion results in, (57) γ A = ω πZ ℓr e ρ A ρ i B ϕA B zA (cid:18) B ϕA B zA (cid:19) (cid:18) B ϕA B zi (cid:19) + O (cid:0) ε (cid:1) , where (58) Z = (cid:12)(cid:12)(cid:12)(cid:12) ( χ − − ρ A ρ i B zi B zA ( ζ − (cid:12)(cid:12)(cid:12)(cid:12) . E1 NOTE TO EDITOR: Figure 4 and Figure 5 should appear side-by-side in the text. 24 –Fig. 4.— Contour map of the damping time τ d (see Eq. (66)) as a multiple of the period τ , plotted for density contrast in the interval χ ∈ (0 , ζ /h ), and longitudinal magnetic fieldcontrast in the interval ζ ∈ (0 , ℓ/r e = 0 . B ϕA /B zA = 0 .
15. The red line marks ζ /h above which the resonancefrequency is outside of the continuum. The gray region in this plot denotes damping timesof 30 and above. 25 –Fig. 5.— Contour map of the ratio τ d ( χ, ζ ) versus τ d ( χ, χ ∈ (0 , ζ /h ), and longitudinal magnetic field contrast inthe interval ζ ∈ (0 , red line is the same as in Figure 4, whilst values within the grayregion correspond to ratios larger than 4. 26 –Now, in this investigation we assume weak magnetic twist ( q = B ϕA /B zA ≪
1) andtherefore, (59) B ϕA B zA (cid:18) B ϕA B zA (cid:19) (cid:18) B ϕA B zi (cid:19) = B ϕA B zA + O ( q ) , and so Eq. (57) can be simplified to, (60) γ A = ω πZ ℓr e ρ A ρ i B ϕA B zA . Here ω is approximated by (34), i.e. ω ≈ ω Ai h , and the radius at the resonance point, r A ,is obtained analogously to Eq. (53), by solving, (61)(1 + w ( ζ − w ( χ −
1) = h . There are two cases to be considered. First, when ζ = 1, that is B zi = B ze , and assuming χ ∈ (0 , /h ), the solution for w is (62) w = h − h (1 − χ ) , when 1 /h < χ < χ → ζ ∈ (0 ,
1) and χ ∈ (0 , ζ /h ) for which the admissible solution is, (63) w = 2(1 − ζ ) + h (4( ζ − ζ − χ ) + h ( χ − ) / + h ( χ − − ζ ) . When χ > ζ /h , similarly to the first case there is no resonant absorption since theresonance frequency ( ω ) is outside of the continuum. For ζ /h < χ < ζ there existundamped propagating waves, however, when χ > ζ the external Alfv´en speed is smaller 27 –than the internal and no waves propagate. Lastly note, that in this investigation we assumethat B zi ≥ B ze and therefore ρ i = ρ e has no admissible solution for w when B zi = B ze .Now, when B z is assumed to be constant, i.e. B zi = B ze = B z , using Eq. (62), ρ A /ρ i , B zA /B zi and Z simplify to, ρ A ρ i = 1 h , B zA B zi = 1 , Z = | − χ | , (64)resulting in γ A (Eq. (60)) γ A = ω h π | − χ | ℓr e B ϕA B zA = ω Ai h π | − χ | ℓr e B ϕA B zA . (65)To obtain the damping time normalized by the period of the wave, we use a typicalwavelength k z = π/L , where L is the characteristic length of the tube, the associated periodis τ = 2 L/hv Ai (see Eq. (35)) the damping time (1 /γ A ) for modes in the continuum as amultiple of the wave period is, (66) τ d = Z π r e ℓ ρ i ρ A B zA B ϕA τ, a contour map of this equation for ζ ∈ (0 ,
1) and χ ∈ (0 , ζ /h ), can be seen in Figure 4.When B zi = B ze the damping time becomes, (67) τ d = h | − χ | π r e ℓ B zA B ϕA τ. The long wavelength limit approximation of the damping rate γ A in Eq. (57), is accurate to ≈ − at k z r e = 1 when compared with the numerical solution of the dispersion relationin Eq. (47). This accuracy is better than 10 − for k z r e < . N − N X i =1 (cid:18) γ A − ˆ γ A γ A (cid:19) ! / .
28 –In this equation, γ A is the numerically calculated damping rate, ˆ γ A is the theoreticalapproximation in Eq. (57) and N is the number of samples. For this error estimate we used10 samples in the parameter space ( χ, ζ , ℓ/r e , B ϕ /B z ), uniformly distributed .Works investigating resonant absorption in the context of solar atmospheric conditionstend to consider solely a radial non-uniformity in either the magnetic field or density.However, accounting for radial variation in both the magnetic field and density can lead tosignificant variation in the estimated damping times. The ratio of Eq. (66) over Eq. (67) is,(69) τ d ( χ, ζ ) τ d ( χ,
1) = Z | − χ | ρ i ρ A h and in Figure 5 a contour map is shown for ζ ∈ (0 ,
1) and χ ∈ (0 , ζ /h ).It can be seen from Figure 4 that the behavior of the damping rate with respect tochanges in the density contrast is in some regions exactly the opposite to that for the kinkmode (Goossens & Poedts 1992, see for example). Namely, in a roughly triangular regionin Figure 4 the damping rate is proportional to ∼ /χ , in contrast to the kink mode wherethe damping rate is proportional to ∼ χ . Similar behavior has been been shown to existin the leaky regime for sausage modes (Vasheghani Farahani et al. 2014). The factor inEq. (66) that determines this behavior is Zρ i /ρ A . We approximate the local minimum inthe χ direction by evaluating the partial derivative of Z with respect to χ , (70) ∂Z∂χ = − ζ + h ( χ −
1) + 2 h p ζ − ζ − χ ) + h ( χ − , which is subsequently equated to 0. From this we obtain a relation χ = (2 /h ) ζ + b and b isidentified by noting that at ζ = 1, the maximum value for χ is 1 /h , thus the approximation Since the parameter space is not a hypercube, e.g. see Figure 5, we used rejection sam-pling for invalid parameter combinations until the desired number of samples was achieved. 29 –is, (71) χ = 2 h ζ − h . As the remaining terms in Eq. (66) do not vary with χ and ζ (note the ratio B ϕ /B z isheld fixed) this approximation holds for all valid parameters. This approximation allows usto estimate in which regime a specific parameter combination is. Namely, for parametercombinations that are below the line described by Eq. (71), for increasing density contrast( χ ↓ ), damping will be slower ( τ d ↑ ). For parameter combinations that result in pointsabove this line, increasing the density contrast ( χ ↓ ) results in decreasing damping time( τ d ↓ ), namely waves will decay faster. This is illustrated in Figure 4 as a yellow line ((71))and the exact inflection points are marked with a green line.Given the form of Eq. (66), and especially that of Eq. (67), a comparison withprevious results for the kink mode is in order, particularly the expression for the dampingrate obtained by Goossens et al. (1992) and later by Ruderman & Roberts (2002). In(Ruderman & Roberts 2002), and Equation (73) in that work, using the notation in thiswork, reads as follows, (72) τ d = 2 π r e ℓ − χ χ τ. The relative magnitude of the damping time shown in Eq. (72) and Eq. (67) is, (73) τ d,Axisymmetric τ d,Kink = h π (1 − χ ) χ B zA B ϕA . It is evident that there exists a region in the parameter space of ( χ, B ϕ /B z ) for which τ d,Axisymmetric is smaller when compared with τ d,Kink , however, this comparison is givenhere just as a reference and caution should be exercised in its interpretation since thedamping, τ d,Kink in Ruderman & Roberts (2002) was calculated for the kink mode withoutmagnetic twist. It is possible that magnetic twist amplifies dissipation in the kink mode, 30 –and therefore still, dissipation for the kink mode may be larger than that of axisymmetricmodes. We have solved Eq. (47) numerically, using ω obtained in Eq. (34) as an initial pointin the solver. Additionally, by means of investigating whether another solution exists, wesolved the dispersion relation again with a random ω in the range ( v Ai , v Ae ). The solutionsand their associated damping rates can be seen in Figure 6. It is interesting that thereexists another solution in the long wavelength limit that we could not obtain from ouranalysis in Section 4. However, given that for this solution τ d ≪ τ , it is unlikely that thismode will be observed.Now, it has been shown that the singularity about the resonance point at r A islogarithmic for ξ r ( ln ( | r − r A | )) and 1 / ( r − r A ) for ξ ⊥ , so the dynamics will be governed by ξ ⊥ since ξ r /ξ ⊥ → r → r A and therefore ξ ⊥ ≫ ξ re in the neighbourhood of the resonantpoint (Poedts et al. 1989; Sakurai et al. 1991a). Also the ξ r component provides its energyto the resonant layer (Goossens et al. 2011) and therefore the characteristic expansion andcontraction of axisymmetric modes will be reduced. These facts, along with the proximityof the solution corresponding to the long wavelength limit approximation to Section 4 tothe internal Afv´en speed suggest that these waves would appear in observations to haveproperties similar to Alfv´en waves. Given that pure Alfv´en waves require ∇ · ξ to beidentically zero, and, a driving mechanism that is solely torsional, we argue that observed Alfv´en waves are much more likely to be axisymmetric waves, as these do not have thesestrict requirements. In Figure 6 panels (a) through (d) show solutions for different valuesof χ while panels (e) through (h) solutions are shown when q is allowed to vary. Thedamping time for the solution for which we have an analytical approximation (see panels 31 –Fig. 6.— Numerical solutions of the dispersion equation Eq. (47) for χ = { . , . , . , . , . } , q = 0 . ζ = 1 and ℓ/r e = 0 . (a) to (d) ( χ = ρ e /ρ i , ζ = B ze /B zi , q = B ϕA /B zA ) and χ = 0 . q = { . , . , . , . , . } , ζ = 1 and ℓ/r e = 0 . (e) to (h) . The panels (a) , (c) , (e) and (g) depict the normalizedphase velocity and the panels (b) , (d) , (f ) and (h) the corresponding normalized dampingrates. The bottom panel shows a logarithmic plot of the damping time versus magnetic twistfor different values of k z r e . All solutions have been obtained numerically by solving Eq. (47). 32 – (c) ) and (d) ) increases ( τ d ↑ ) for increasing density contrast ( χ ↓ ) while the other solutionexhibits the opposite behaviour (see panels (a) and (b) ) namely τ d ↓ for χ ↓ . However,the damping time for both solutions decreases ( τ d ↓ ) for increasing magnetic twist q ↑ . Thebottom panel of Figure 6 shows a different view of the damping times as a function of themagnetic tiwst ( q ) shown in panels (f) and (h) at k z r e = { . , . , . , . , . } . Fromthis view it can be seen that the solutions in (e) are much more sensitive to variations inthe mangetic twist when compared with the solutions in panel (g). This sensitivity, incombination with the fact that for extremely small twist the sausage cut-off is reintroduced(Giagkiozis et al. 2015), means that this mode will be observable for a very small intervalof magnetic twist. The mode shown in panels (c) and (d) does not present this difficulty,and therefore we expect that observation of this mode is more likely. In both cases thesolution corresponding to the analytic approximation remains very close to the internalAlfv´en speed which is equal to 1 in Figure 6. Since ω from Eq. (35) depends on q , B zA and the internal sound speed, these modes will appear to have a strong Alfv´en characterfor virtually all valid parameter combinations. Lastly, k r can be likened to the wavenumberin the radial direction, and, since in the long wavelength limit k r is proportional to k z , as k z increases, the wavelength in the radial direction decreases and couples with the thininhomogeneous layer more closely and therefore more energy per wavelength is absorbedand thus the damping time is reduced (see Figure 6).
5. Connection to Observations E2 Reports of observations of axisymmetric modes (sausage modes) are increasing E2 NOTE TO EDITOR: Figure 7 and Figure 8 should appear side-by-side in the text. 33 –Fig. 7.— Contour map of the estimated probability (see Eq. (74)) that an axisymmetricmode can be observed to have a normalized damping time ¯ τ d in the range (1 , ζ , q ), i.e. magnetic field contrast and twist respectively. The free parametersare ζ ∈ (0 . ,
1) and q = B ϕA /B zA ∈ (0 , .
3) and the integration parameters are χ ∈ (0 . , ℓ/r e ∈ (0 . , . τ d in the range (1 ,
3) for a point in ( ζ , χ ), i.e. magneticfield and density contrast respectively. The free parameters are ζ ∈ (0 ,
1) and χ ∈ (0 ,
1) andthe integration parameters are q = B ϕA /B zA ∈ (0 , .
3) and ℓ/r e ∈ (0 . , . , ≈ M m . Furthermorethe results by Morton et al. (2012) suggest that axisymmetric modes are ubiquitous andthat they appear to be coexistent with kink modes. This coexistence further supports theargument by Arregui et al. (2015); Arregui & Soler (2015); Arregui (2015) that Bayesiananalysis is an essential tool for the identification of the likely wave modes present inobservations as well as a more systematic way for the appropriate model selection. Theuncertainty in determining the parameters for the kink mode led Verwichte et al. (2013) toperform a statistical analysis as a way to narrow the range of their values. This departurefrom certainty and convergence towards probabilistic inference models for solar observationsis, in our view, long overdue.However, despite this increase in interest in axisymmetric modes, the relation thatapproximates their expected damping rate, see Eq. (66), requires knowledge of fourparameters. Namely, the density and magnetic field contrast, the relative magnetic twistand the ratio of the thickness of the inhomogeneous layer versus the tube radius, i.e.( χ, ζ , q = B ϕA /B zA , ℓ/r e ). In contrast to the large body of observational evidence for thekink mode, observations of sausage waves are relatively scarce. This makes impossiblean analysis similar to Verwichte et al. (2013) for these modes. Therefore, we adopt adifferent approach, a probabilistic approach which is related to the use of Bayesian inferencesuggested by Arregui et al. (2015). 36 –As a first step towards improving this situation we provide a way to estimate theprobability that an observed sausage wave has a damping rate within a specified range,given that, one or more of the four parameters in Eq. (57) are known. The assumptionsrequired for the validity of this estimate are the following: • The four parameters in Eq. (57) are independent, i.e. no parameter is a function ofthe others. • The likelihood of any combination in the parameter space is the same. That is to saythat there exists no preferred combination of parameters.These assumptions are difficult to prove, especially given that there exist no statisticalanalyses of the properties of sausage waves and reliable estimates of all four parameters.Since we do not know if there is, in fact, a set of preferred parameters, these assumptionsare required for an unbiased estimate. Acknowledging these uncertainties, we make afirst attempt in identifying the probability predicted by our model that a wave with thecharacteristics described in this investigation is resonantly damped in the long wavelengthlimit with a damping rate given by Eq. (66) for a given parameter combination.The aforementioned probability can be estimated as follows. First, we identify theparameters for which reasonably good estimates are available. These parameters we referto as free parameters denoted by f . The remaining parameters we refer to as integration parameters and are denoted by i . Subsequently, a domain is defined for the integrationparameters. Then the probability of the damping rate being within the open interval ( a, b ) 37 –is given by, P ( a, b ; f , . . . , f n ) = R C di . . . di − n w ( i , . . . , i − n ) I ¯ τ d >a, ¯ τ d a, ¯ τ d a, ¯ τ d a, ¯ τ d a, ¯ τ d
5) and the observed normalized damping rates were in the interval (1 , q parameter has been selected in an interval that ensures that the magnetic twist is small.As can be seen in both Figure 7 and Figure 8 the probability for the resonantly absorbedaxisymmetric modes for a wide range in parameters is significantly high. Also, as seen inFigure 7 normalised damping times in the interval (1 ,
3) are possible even for extremelysmall magnetic twist ( ≈ . ζ inthe interval ζ ∈ (0 . , − , − ).Since Morton et al. (2012) do not provide an estimate for the width of the inhomogenouslayer we allow it to vary uniformly in (0 . , . . , . times. The estimated PDF for this set of parameters isthe blue curve in Figure 9. The blue vertical line is the expectation value which is equal to E [ τ d /τ ] = 7 .
49. 39 –Fig. 9.— Rescaled probability density functions (PDF) of the normalized damping time usingparameter estimates from Morton et al. (2012) ( blue ) and Van Doorsselaere et al. (2011)( red ). For illustration purposes the scaling in both PDFs is such that their maximum isequal to 1. The support for the blue
PDF is (1 . , .
72) and the expected value for thedamping time is E [ τ d /τ ] = 7 .
49. Similarly the support for the red
PDF is (0 . , . E [ τ d /τ ] = 5 .
58. The intervals used for theparameters ( χ = ρ e /ρ i , ζ = B ze /B zi , q = B ϕA /B zA ) and the associated assumptions aredetailed in Section 5. 40 –In the second example we use parameter estimates from Van Doorsselaere et al. (2011).Assuming a H plasma, ρ = N m p , p = N k B T where N is the number density, m p the protonmass and k B is the Boltzmann constant. With the plasma- β equal to β = 2 µ p/B and theassumption that β e ≪ β i (Van Doorsselaere et al. 2011) we obtain (79) β i β e = ζ χ T i T e ≫ . Assuming a lower limit for β i /β e ≥
100 and a hot flux tube, T i /T e = 10, we can restrict ζ and χ to 1200 ≤ χ ≤ β e β i T i T e , (80) (cid:18) χ T e T i β i β e (cid:19) / ≤ ζ ≤ . (81)The lower limit for χ is considered as a minimum contrast in Van Doorsselaere et al. (2011)to avoid the sausage cut-off. However, in the presence of very weak magnetic twist thiscut-off is removed, and therefore we don’t need to assume extreme values for the densitycontrast. The upper limit for χ and lower limit for ζ are taken so that Eq. (79) is satisfied.The resulting PDF can be seen in Figure 9. It is interesting that the expected value forthe damping time in this case is 5 .
58 which is very close to observed damping ( τ d /τ = 6)of a mode that is believed to be a fast sausage mode (Kolotkov et al. 2015). It is apparentfrom Figure 9 that the PDFs cannot be approximated well using a normal distribution, andtherefore their use for obtaining estimates of the damping time from results like Eq. (66) inthis work, and, similar equations (e.g. Goossens et al. 1992; Ruderman & Roberts 2002) canbe misleading. In contrast, Monte Carlo simulation and non-parametric density estimationcan be quite useful tools for exploring this type of problems. 41 –
6. Discussion and Conclusions
Theoretically, it has been known for some time, that, in the presence of weak magnetictwist, axisymmetric modes will be resonantly damped (see for example Goossens et al.1992). In this work we have calculated, for the first time, a dispersion relation for resonantlydamped axisymmetric modes, in the spectrum of the Alfv´en continuum and derived anapproximation of the damping time in the long wavelength limit. We have shown thatthe damping time can be comparable to that observed for the kink mode in the case thatthere is no magnetic twist. Furthermore, we solved the resulting equation (see Eq. (47)and Eq. (57)) analytically and, i) we confirmed the validity of our approximation, and,ii) we found an additional solution that decays much faster in comparison. The resultingapproximation in the long wavelength limit shows that the damping time is proportional tothe magnetic twist and inversely proportional to the density contrast. It is interesting tonote that Vasheghani Farahani et al. (2014) who investigated the damping of fast sausagemodes in the leaky regime found a similar relation between the damping time and thedensity contrast. However, in that work a very large density contrast is required to allowobservation of the sausage mode. This is not the case for one of the results of this work,which even for modest density contrasts (see Figure 7 and Figure 8) the damping time iswithin one to three periods of the wave.Of the two solutions that we have uncovered, only the one with a phase velocity closethe internal Alfv´en speed has, for some parameter combinations, damping times that wouldallow observation. The other solution is found to be damped on time scales ≈ − − − of the wave period as seen in Figure 6, which means that its observation would be extremelychallenging. On the other hand although the predicted damping times for the solutionwhose phase speed is close to the Alfv´en speed are large enough to allow observation. Also,the fact that its phase speed is so close the internal Alfv´en speed along with the dominance 42 –of the ξ ⊥ component in the wave dynamics means that the character of this wave will bepredominantly Alfv´enic (Goossens et al. 2011). Because of this, we argue that it is possiblethat resonantly damped sausage waves have already been observed, albeit in the guise ofAflv´en waves, see for example Jess et al. (2009).Lastly, we estimated the damping time for the parameters presented by Morton et al.(2012) and Van Doorsselaere et al. (2011) and interestingly the expected damping time isvery close to the observed damping in quasi-periodic pulsations by Kolotkov et al. (2015)that are believed to be fast sausage waves. We find, subject to certain assumptions, thataxisymmetric modes appear to be quite important conduits for energy transfer in thesolar atmosphere. Perhaps even more important than pure Alfv´en waves, given that theexcitation mechanism for sausage modes in weakly twisted magnetic flux tubes appears tobe more readily available in comparison to the required purely torsional drivers for Alfv´enwaves (Giagkiozis et al. 2015).I.G. would like to acknowledge the Faculty of Science of the University of Sheffield forthe SHINE studentship. I.G. also thanks T.V.D. for financial support during his visit to KULeuven. M.G. is grateful to Belspo’s IAP P7/08 CHARM and KU Leuven GOA-2015-014.G.V. and V.F. would like to acknowledge the STFC for funding received (Grant numberST/M000826/1). T.V.D. has received funding from the Odysseus programme of theFWO-Vlaanderen, and would also like to acknowledge the framework of Belspo’s IAP P7/08CHARM and the GOA-2015-014 of the Research Council of the KU Leuven. 43 – REFERENCES