Axisymmetric Modes in Magnetic Flux Tubes with Internal and External Magnetic Twist
Ioannis Giagkiozis, Viktor Fedun, Robart Erdeliy, Gary Verth
aa r X i v : . [ a s t r o - ph . S R ] J un Axisymmetric Modes in Magnetic Flux Tubes with Internal and ExternalMagnetic Twist
I. Giagkiozis , V. Fedun , , R. Erdélyi , and G. Verth Solar Plasma Physics Research Centre, School of Mathematics and Statistics, University ofSheffield, Hounsfield Road, Hicks Building, Sheffield, S3 7RH, UK Department of Automatic Control and Systems Engineering, University of Sheffield, MappinStreet, Amy Johnson Building, Sheffield, S1 3JD,UK Debrecen Heliophysical Observatory (DHO), Research Centre for Astronomy and Earth Sciences,Hungarian Academy of Sciences, Debrecen, P.O.Box 30, H-4010, Hungary
Abstract.
Observations suggest that twisted magnetic flux tubes are ubiquitous in the Sun’s atmosphere. Themain aim of this work is to advance the study of axisymmetric modes of magnetic flux tubes bymodeling both twisted internal and external magnetic field, when the magnetic twist is weak. In thiswork, we solve the derived wave equations numerically assuming twist outside the tube is inverselyproportional to the distance from its boundary. We also study the case of constant magnetic twistoutside the tube and solve these equations analytically. We show that the solution for a constanttwist outside the tube is a good approximation to the case where the magnetic twist is proportionalto /r , namely the error is in all cases less than . .The solution is in excellent agreement withsolutions to simpler models of twisted magnetic flux tubes, i.e. without external magnetic twist.It is shown that axisymmetric Alfvén waves are naturally coupled with magnetic twist as theazimuthal component of the velocity perturbation is nonzero. We compared our theoretical resultswith observations and comment on what the Doppler signature of these modes is expected to be.Lastly, we argue that the character of axisymmetric waves in twisted magnetic flux tubes can leadto false positives in identifying observations with axisymmetric Alfvén waves.
1. Introduction
There is ample evidence of twisted magnetic fields inthe solar atmosphere and below. For instance, it hasbeen suggested that magnetic flux tubes are twistedwhilst rising through the convection zone (see for exampleMurray & Hood 2008; Hood et al. 2009; Luoni et al. 2011).Brown et al. (2003); Yan & Qu (2007); Kazachenko et al.(2009) have shown that sunspots exhibit a relatively uni-form rotation which in turn twists the magnetic fieldlines emerging from the umbra. Several studies arguethat the chromosphere is also permeated by structuresthat appear to exhibit torsional motion (De Pontieu et al.2012; Sekse et al. 2013). These structures, known as typeII spicules, were initially identified by De Pontieu et al.(2007). De Pontieu et al. (2012) show that spicules exhibita dynamical behavior that has three characteristic compo-nents, i) flows aligned to the magnetic field, ii) torsional mo-tion and iii) what the authors describe as swaying motion.Also, recent evidence shows that twist and Alfvén wavespresent an important mechanism of energy transport fromthe photosphere to the corona (Wedemeyer-Böhm et al.2012). The increasing body of observational evidence ofmagnetic twist in the solar atmosphere, in combination withubiquitous observations of sausage waves (Morton et al.2012), reinforce the importance of refining our theoreticalunderstanding of waves in twisted magnetic and especiallyaxisymmetric modes as these could be easily perceived astorsional Alfvén waves.Early studies of twisted magnetic flux tubes focused onstability analyses. For example, Shafranov (1957) investi- gated the stability of magnetic flux tubes with azimuthalcomponent of magnetic field proportional to r inside thecylinder and no magnetic twist outside. Kruskal et al.(1958) derived approximate solutions for magnetic fluxtubes with no internal twist embedded in an environmentwith B ϕ ∝ /r . Bennett et al. (1999) obtained solutionsfor the sausage mode for stable uniformly twisted mag-netic flux tubes with no external twist and Erdélyi & Fedun(2006) extended the analysis for the uncompressible case ofconstant twist outside the flux tube. The authors also ex-amined the impact of twist on the oscillation periods incomparison to earlier studies (e.g. Edwin & Roberts 1983)considering magnetic flux tubes with no twist. In a sub-sequent work Erdélyi & Fedun (2007) extended their re-sults in Erdélyi & Fedun (2006) to the compressible casefor the sausage mode with no twist outside the tube.Karami & Bahari (2010) investigated modes in incompress-ible flux tubes. The twist was considered to be ∝ r for all r , which is unphysical for r → ∞ , while the density pro-file considered was piecewise constant with a linear func-tion connecting the internal and external densities. Theauthors revealed that the wave frequencies for the kinkand fluting modes are directly proportional to the mag-netic twist. Also, the bandwidth of the fundamental kinkbody mode increases proportionally to the magnetic twist.Terradas & Goossens (2012) investigated twisted flux tubeswith magnetic twist localized within a toroidal region of theflux tube and zero everywhere else. Terradas & Goossens(2012) argue that for small twist the main effect of stand-ing oscillations is the change in polarization of the velocity Article number, page 1 of 11 pJ Preprint:
Axisymmetric Modes with Magnetic Twist, Giagkiozis et al. perturbation in the plane perpendicular to the longitudinaldimension ( z -coordinate).In this work we study axisymmetric modes, namelyeigenmodes corresponding to k ϕ = 0 , where k ϕ is theazimuthal wavenumber in cylindrical geometry . The az-imuthal magnetic field inside the tube is ∝ r , while theazimuthal field outside is constant. If there is a currentalong the tube, according to the Biot-Savart law this cur-rent will give rise to a twist proportional to r inside theflux tube and a twist inversely proportional to r outside.For this reason we start our analysis by assuming a mag-netic twist outside the tube proportional to /r and weinsert a perturbation parameter that can be used to revertto the case with constant twist. Subsequently, we presentan exact solution for the case with constant twist outsidethe tube and solve numerically for the case with magnetictwist proportional to /r . Then, we compare the numericalsolution for /r with the exact solution for constant twist.Based on the obtained estimated standard error the solu-tion corresponding to weak constant twist appears to bea good approximation to the solution with weak magnetictwist that is proportional to /r . In the case where there isa pre-existing twist inside the cylinder, assuming this twistis uniform, this will give rise to a current which in turn willcreate the external twist that is again inversely proportionalto the distance from the cylinder. The latter case may oc-cur, for example, due to vortical foot-point motions on thephotosphere (Ruderman et al. 1997). Recent observationalevidence (e.g. Morton et al. 2013) put the assumptions ofRuderman et al. (1997) on a good basis, however, the vorti-cal motions in Morton et al. (2013) are not divergence freewhich means that the same mechanism can be responsiblefor the axisymmetric modes studied in this work.The rest of this paper is organized as follows. In Sec-tion 2 we describe the model geometry and MHD equationsemployed. In Section 3 we derive the dispersion relationfor k ϕ = 0 , and, in Section 3.3 we explore limiting casesconnecting the results in this work with previous models.Furthermore, in Section 4 we study a number of physicallyrelevant cases and elaborate on the results. In, Section 5we reflect on the applicability and potential limitations ofthe presented model and in Section 6 we summarize andconclude this work.
2. Model Geometry and Basic Equations
The single-fluid linearized ideal MHD equations in the forceformalism are (Kadomtsev 1966), (1a) ρ ∂ δ ξ ∂t + ∇ δp + 1 µ ( δ B × ( ∇ × B ) + B × ( ∇ × δ B )) = 0 , (1b) δp + δ ξ · ∇ p + γp ∇ · δ ξ = 0 , (1c) δ B + ∇ × ( B × δ ξ ) = 0 , where, ρ, p and B are the density, kinetic pressure and mag-netic field, respectively, at equilibrium, δ ξ is the Lagrangiandisplacement vector, δp and δ B are the pressure and mag-netic field perturbation, respectively, γ is the ratio of spe-cific heats (taken to be / in this work), and µ is thepermeability of free space. In this study we employ cylindri-cal coordinates ( r, ϕ, z ) and therefore δ ξ = ( δξ r , δξ ϕ , δξ z ) , δ B = ( δB r , δB ϕ , δB z ) . In what follows an index, i , indicates k ϕ is often denoted as m in a number of other works. Fig. 1.
Model illustration. Straight magnetic cylinder with vari-able twist inside ( r < r a ) and outside ( r > r a ), where r a is thetube radius, ρ i , p i and T i are the density, pressure and tempera-ture at equilibrium inside the tube. The corresponding quantitiesoutside the tube are denoted with a subscript e . B ϕ is continu-ous across the tube boundary. The dark blue surface inside themagnetic cylinder represents the influence of B ϕ ∝ r . The yel-low surface outside the cylinder illustrates the B ϕ ∝ /r depen-dence. The dashed red rectangle represents a magnetic surfacewith only a longitudinal ( z ) magnetic field component. quantities inside the flux tube ( r < r a ) while variables in-dexed by, e , refer to the environment outside the flux tube( r > r a ). The model geometry is illustrated in Figure 1when B ϕe ∝ /r . For static equilibrium, (2a) ∇ · B = 0 , (2b) J = 1 µ ∇ × B , (2c) ∇ p = J × B . We assume that, ρ, p and B have only an r -dependence. Weconsider a magnetic field of the following form, (3) B = (0 , B ϕ ( r ) , B z ( r )) . Notice that in cylindrical coordinates, Equation (2a) isidentically satisfied. Then, Equation (2b) expands to (4) µ J = (cid:18) , − dB z dr , r d ( rB ϕ ) dr (cid:19) , and based on Equation (4), Equation (2c) becomes (5) ∇ p = (cid:18) − B z µ ∂B z ∂r − B ϕ µ r d ( rB ϕ ) dr , , (cid:19) . Therefore, the pressure in the ϕ and z directions is constantand the magnetic field and the plasma pressure must satisfythe following pressure balance equation in the r direction,(6) ddr p + B ϕ + B z µ ! + B ϕ µ r = 0 . Article number, page 2 of 11 or a magnetic flux tube of radius, r a , according to theBiot-Savart law (for κ = 1 in Equation (7)) a reasonableassumption for the form of the magnetic field is, (7) B = (cid:26) (0 , S i r, B zi ) for r ≤ r a , (cid:0) , r κa S e /r κ , B ze (cid:1) for r > r a , where, B zi , B ze , S i and S e are constants and κ is a pertur-bation parameter. The perturbation parameter has beeninserted in Equation (7) in such a way so as to preserve di-mensional consistency. The constant S i can be determinedby application of the Biot-Savart law and is therefore takento be, (8) S i = µ I πr κa , where I is the current. By substituting Equation (7) intoEquation (6) we obtain: p ( r )= S i µ ( r a − r ) + p a for r ≤ r a ,r κ ) a S e (1 − κ )2 µ κ (cid:18) r κ − r κa (cid:19) + p a for r > r a , (9)where, p a , is the pressure at the boundary of the magneticflux tube. The constant, S e , is equal to S i , however wechoose to maintain the notational distinction so that wecan separate the internal and external environments to theflux tube which helps us validate our results with previouswork (e.g. Erdélyi & Fedun 2007). The solution of the system of equations shown in Equa-tion (1), in cylindrical coordinates can be found by Fourierdecomposition of the perturbed components, namely theperturbed quantities are taken to be, (10) δ ξ , δp T ∝ e i ( k ϕ ϕ + k z z − ωt ) , where, ω is the angular frequency, k ϕ is the azimuthalwavenumber for which only integer values are allowed and, k z , is the longitudinal wavenumber in the z direction. TheEulerian total pressure perturbation is, δp T = δp + B δ B /µ ,which is obtained by linearization of the total pressure: p T = ( p + δp ) + ( B + δ B ) / (2 µ ) and p is the equilibriumkinetic pressure. Note that for the sausage mode, consid-ered in this work, the azimuthal wavenumber is k ϕ = 0 .Combining Equation (1) with Equation (10) we obtain theequation initially derived by Hain & Lust (1958) and laterby Goedbloed (1971); Sakurai et al. (1991) to name but afew. This equation can be reformulated as two coupled firstorder differential equations, (11a) D d ( rδξ r ) dr = C ( rδξ r ) − rC δp T , (11b) D dδp T dr = 1 r C ( rδξ r ) − C δp T . and the multiplicative factors are defined as: (12a) D = ρ ( ω − ω A ) C , (12b) C = 2 B ϕ µ r (cid:18) ω B ϕ − k ϕ r f B C (cid:19) , (12c) C = ω − k z + k ϕ r ! C , (12d) 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 , (12e) C = ( v s + v A )( ω − ω c ) , where, ω c = v s v A + v s ω A , ω A = f B µ ρ , f B = k ϕ r B ϕ + k z B z . Here, v s = p γp/ρ is the sound speed, v A = | B | / √ µ ρ is the Alfvén speed, ω c is the cusp angular frequency and ω A is the Alfvén angular frequency. The coupled first orderODEs in Equation (11) can be combined in a single secondorder ODE for, δp T or δξ r . In this work we choose to usethe latter approach, namely: (13) ddr (cid:20) DrC ddr ( rδξ r ) (cid:21) + (cid:20) D (cid:18) C − C C (cid:19) − r ddr (cid:18) C rC (cid:19)(cid:21) δξ r = 0 . Using flux coordinates and assuming k ϕ = 0 , it can beshown that (Sakurai et al. 1991), (14a) ρ ( ω − ω A ) δξ ⊥ = − i k z B ϕ | B | (cid:18) δp T + 2 B z µ r δξ r (cid:19) , (14b) ρ ( ω − ω c ) δξ k = i k z B z | B | v s v s + v A δp T − B ϕ µ r δξ r ! . Here δξ k and δξ ⊥ are the Lagrangian displacement compo-nents parallel and perpendicular to the magnetic field linesrespectively. The dominant component of the Lagrangiandisplacement vector ( δξ r , δξ ⊥ , δξ k ) determines the charac-ter of the mode. For the Alfvén mode the δξ ⊥ componentis dominant, while for the slow and fast magnetoacousticmodes δξ k and δξ r is dominant respectively (Goossens et al.2011). Equation (14) suggests that in the presence of mag-netic twist the slow and fast magnetoacoustic modes arecoupled to the Alfvén mode even when k ϕ = 0 , namely theslow and fast modes do not exist without the Alfvén modeand vice versa. This is because for the Alfvén mode to bedecoupled from the slow and fast magnetoacoustic modes itis required that for δξ ⊥ = 0 , δξ r = 0 and δξ k = 0 . However,it follows trivially from Equation (11) that, if δξ r = 0 thenalso δp T = 0 and therefore from Equation (14) we have that δξ ⊥ = 0 . From this, it follows that the Alfvén mode cannotexist without the components corresponding to the slowand fast magnetoacoustic modes, hence the Alfvén modeis coupled with the slow and fast magnetoacoustic modes.Furthermore, from Equation (14) we can also see that for asolution, i.e. ( ω, k z ) pair, as ω approaches ω A the δξ ⊥ com-ponent is amplified that leads to the azimuthal componentof the displacement to be accentuated. 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3. Dispersion Equation
In this section we follow a standard procedure in deriving adispersion equation, namely we solve Equation (13) insideand outside the flux tube and match the two solutions us-ing the boundary conditions. The boundary conditions thatmust be satisfied are: (15a) δξ ri | r = r a = δξ re | r = r a , (15b) δp T i − B ϕi µ r δξ ri (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) r = r a = δp T e − B ϕe µ r δξ re (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) r = r a , where, Equation (15a) and Equation (15b) are continuityconditions for the Lagrangian displacement and total pres-sure across the tube boundary respectively. The parameters in Equation (12) for the case inside the fluxtube for the sausage mode become, (16a) D i = ρ i ( ω − ω Ai ) , (16b) C = 2 S i rµ n i , (16c) C = n i − k z , (16d) C = ρ i (cid:20) ( ω − ω Ai ) + 4 S i µ ρ i (cid:18) S i r µ ρ i n i − ω Ai (cid:19)(cid:21) , (16e) n i = ω ( v si + v Ai )( ω − ω ci ) , where, ω ci = v si v Ai + v si ω Ai , ω Ai = k z B zi µ ρ i . The B ϕ component is assumed small to avoid the kink in-stability, see for example (Gerrard et al. 2002; Török et al.2004). This implies, B ϕ ≪ B z and since B ϕ is a functionof r inside (and outside) the tube we require sup( B ϕ ) ≪ B z ⇒ Sr a ≪ B z . This condition is satisfied in the solar at-mosphere, so we can use the approximation: v Ai = ( B ϕi + B zi ) / ( µ ρ i ) ∼ B zi / ( µ ρ i ) . Notice that according to Equa-tion (9) the pressure depends on r , however, in this work weassume that the sound speed is constant. This is becausethe term that depends on r in Equation (9) is assumedto be small when compared with p a in solar atmosphericconditions. To see this, consider that sup( Sr a ) = 0 . B z and B z ∼ (10 − − − ) T , T ∼ (10 − ) K and thenumber density n ∼ (10 − ) m − . This means that p a ∼ (10 − − ) N · m − and the term that depends onthe radius is of the order ( Sr a ) /µ ∼ (10 − − ) N · m − and therefore the constant term p a is (10 − ) timeslarger when compared with the term that has an r depen-dence. Hence, to a good approximation, the pressure can beassumed to be constant. Note, that the density is discontin-uous across the tube boundary and therefore we avoid theAlfvén and slow continua that lead to resonant absorption. In the following expressions the left number corresponds totypical values on the photosphere while the right number corre-sponds to typical values of the quantity in the corona. Here we use p = nk B T . Substitution of the parameters in Equation (16) intoEquation (13), leads to the following second order differen-tial equation (see for example Erdélyi & Fedun 2007), (17) R d δξ r dR + R dδξ r dR − (cid:18) k ri k z R + ER (cid:19) δξ r = 0 , where, R = k α r,k ri = ( k z v si − ω )( k z v Ai − ω )( v Ai + v si )( k z v T i − ω ) ,E = 4 S i n i µ D i k z (1 − α ) ,α = 4 S i ω Ai µ ρ i ( ω − ω Ai ) ,v T i = v Ai v si v Ai + v si . Here k α = k z (1 − α ) / is the effective longitudinalwavenumber and v T i is the internal tube speed. Equa-tion (17) was derived and solved before by Erdélyi & Fedun(2007). The solution is expressed in terms of Kummer func-tions (Abramowitz & Stegun 2012) as follows, δξ r ( s ) = A i s / E / e − s/ M ( a, b ; s ) + A i s / E / e − s/ U ( a, b ; s ) , (18)and the parameters, a, b and the variable s are defined as a = 1 + k ri k z E / ,b = 2 ,s = R E / = k α E / r ,A i and A i are constants. Furthermore, the total pressureperturbation, δp T is: δp T ( 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) + A i k a D i n i − k z e − s/ (cid:20) n i + k z k z sU ( a, b ; s ) − − a ) U ( a, b − s ) (cid:21) . Now, considering that solutions at the axis of the flux tube,namely at r = 0 , must be finite, we take A i = 0 and so(19a) δξ ri ( s ) = A i s / E / e − s/ M ( a, b ; s ) , (19b) δp 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) . Note that the corresponding equation to Equation (19b)had a typographical error in Erdélyi & Fedun (2007) (seeEquation (13) in that work).
Article number, page 4 of 11 .2. Solution Outside the Flux Tube
The multiplicative factors in Equation (12) outside of theflux tube for the sausage mode, k ϕ = 0 , become (20a) D e = ρ e ( ω − ω Ae ) , (20b) C = 2 r κ ) a S e µ r κ n e , (20c) C = n e − k z , (20d) C = ρ e ( ω − ω Ae ) + 4 r κ ) a S e µ r κ ) " r κ ) a S e µ r κ n e − ρ e ω Ae − ρ e (1 + κ )2 ( ω − ω Ae ) , (20e) n e = ω ( v se + v Ae )( ω − ω ce ) , where, ω ce = v se v Ae + v se ω Ae , ω Ae = k z B ze µ ρ e . Equation (13) with Equation (20) for κ = 1 corresponds to B ϕ ∼ /r , however, the resulting ODE is difficult to solve.By setting κ = 0 we obtain the case for constant twistoutside the tube, which is also a zeroth-order approxima-tion to the problem with κ = 1 (Bender & Orszag 1999).Note, that it is unconventional to use only the zeroth-orderterm in perturbative methods, and therefore, to establishthe validity of the approximation we estimate the error bysolving for κ = 1 numerically. The estimated error is quotedin the caption of the dispersion diagrams in this work andthe process which we followed to obtain this is described inAppendix B. Substituting the parameters given in Equa-tion (20) into Equation (13) we have (21) r d δξ r dr + r dδξ r dr − (cid:0) k re r + ν ( κ ; r ) (cid:1) δξ r = 0 , where, k re and ν : (22) k re = − ( n e − k z ) , (23) ν ( κ ; r ) = 1 + 2 r κ ) a S e µ D e r κ ( r κ ) a S e 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)) . Notice that ν (0; r ) is independent of r . Therefore, for κ → , Equation (21) is transformed to either the Besselequation for k re < or the modified Bessel equation for k re > . It should be noted that n e = k z , namely k re = 0 ,is prohibited since during the derivation of Equation (21)it was assumed that n e = k z to simplify the resulting equa-tion. Therefore, the solution to Equation (21) for κ → ,and, assuming no energy propagation away from or towardsthe cylinder ( k re > ), is (24) δξ r ( r ) = A e K ν ( k re r ) + A e I ν ( k re r ) . On physical grounds we require the solution to be evanes-cent, i.e. δξ r ( r ) → as r → , and therefore we must have A e = 0 , namely (25) δξ re ( r ) = A e K ν ( k re r ) , and, from Equation (11a), the total pressure perturbation δp T e is (26) δp T e = A e (cid:18) µ (1 − ν ) D e − r a S e n e µ r ( k z − n e ) K ν ( k re r ) − D e k re K ν − ( k re r ) (cid:19) . Note that, for the case k re > and S e → , namely zerotwist outside the cylinder, ν → , thus retrieving the solu-tion for δξ r by Edwin & Roberts (1983). The limiting casesfor Equation (19a) and Equation (19b) have been verified toconverge to the solutions with no twist inside the magneticcylinder in Erdélyi & Fedun (2007) in Section 3.3. Application of the boundary conditions (see Equation (15a)and Equation (15b)) in combination to the solutions for δξ r and δp T inside the magnetic flux tube, Equation (19a) andEquation (19b) as well as the solutions in the environmentof the flux tube, Equation (25) and Equation (26) respec-tively, leads to the following general dispersion equation, forthe compressible case in presence of internal and externalmagnetic twist, r a D e k re K ν − ( k re r a ) K ν ( k re r a ) = r a µ (cid:20) S i k ri ( n i + k z ) − S e 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 ) . (27)In order to validate Equation (27) we consider a numberof limiting cases. First, the case where there is no exter-nal magnetic twist. In this case S e = 0 and from Equa-tion (23) it follows trivially that, ν ( κ ; r ) = 1 . Therefore,Equation (27) for no external twist becomes: r a D e k re K ( k re r a ) K ( k re r a ) = S i r a µ k ri ( n i + k z ) − D i k ri M ( a, b − s ) M ( a, b ; s ) . (28)This equation is in agreement with the dispersion equa-tion obtained by Erdélyi & Fedun (2007) and all the lim-iting cases therein also apply to Equation (27). However,there is one limiting case missing from the analysis inErdélyi & Fedun (2007), namely for no twist inside andoutside the tube with k ri < , which in combinationto k re > corresponds to body wave modes. We com-plete this analysis here. Starting with 13.3.1 and 13.3.2 inAbramowitz & Stegun (2012), in the limit as S i → and k ri > we have (29a) lim S i → ( M ( a, b − s )) = I ( k ri r ) , (29b) lim S i → ( M ( a, b ; s )) = 2 k ri r I ( k ri r ) , while for k ri < : Article number, page 5 of 11 pJ Preprint:
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Characteristic Speeds Ordering Type β i β e v SE > v SI > v AE > v AI Warm dense tube > > v SE > v AI > v AE > v SI Cool evacuated tube ≪ ≫ v SE > v SI > v AI > v AE Weak cool tube > ≫ v AE > v SI > v AI > v SE Intense warm tube ≪ ≪ v AE > v AI > v SE > v SI Intense cool tube ≪ ≪ Table 1.
Equilibrium cases considered in this work. The nor-malized characteristic speeds are defined in Appendix A. (30a) lim S i → ( M ( a, b − s )) = J ( k ri r ) , (30b) lim S i → ( M ( a, b ; s )) = 2 k ri r J ( k ri r ) . Therefore, Equation (28) in conjunction with the facts that J ′ ( s ) = − J ( s ) , I ′ ( s ) = I ( s ) and K ′ ( s ) = − K ( s ) (9.1.28and 9.6.27 in Abramowitz & Stegun 2012) is, in the limitas S i → equal to k ri D e K ( k re r a ) K ′ ( k re r a ) = k re D i I ( k ri r a ) I ′ ( k ri r a ) , for k ri > , (31) | k ri | D e K ( k re r a ) K ′ ( k re r a ) = k re D i J ( | k ri | r a ) J ′ ( | k ri | r a ) , for k ri < , (32)which are in agreement with Edwin & Roberts (1983) anddescribe wave mode for the case with no magnetic twist.
4. Dispersion Equation Solutions
To explore the behavior of the sausage mode Equation (27)was normalized and solved numerically for different solaratmospheric conditions, see Table 1. Normalized quantitiesare denoted with capitalized indices (see Appendix A). Thesolutions of the dispersion relation depend only on the rela-tive ordering of the magnitudes of the characteristic veloc-ities ( v AE , v AI , v SE , v SI ). The sign of k rI and k rE dependson this ordering and this in turn defines the three bandtypes in the dispersion plot, i) bands that contain surfacemodes (for k rI > and k rE > ), ii) bands that containbody modes (for k rI < and k rE > ), and, iii) forbid-den bands corresponding to k rE < . We make additionalcomments on the selection of the characteristic speeds inAppendix C. The non-dimensional dispersion equation isgiven in Appendix A while the solutions for the perturbedquantities in terms of δξ r and δp T are given in Appendix D. β regime Based on the results by Vernazza et al. (1981) and themodel for the plasma- β in the solar atmosphere intro-duced by Gary (2001) we anticipate that the results inthis section are pertinent to conditions typically observedin the upper photosphere, lower chromosphere and mid-chromosphere. The solutions of the dispersion relation inEquation (27), in terms of the non-dimensional phase speed, v F = v ph /v Ai = ω/k z v Ai , and the non-dimensional longitu-dinal wave-vector, K = k z r a , for a warm dense tube (see Ta-ble 1) are shown in Figure 2. For this case the ordering of thecharacteristic speeds is as follows: v SE > v SI > v AE > v AI .In this figure, and in the following, we over-plot two cases,i) B ϕi /B zi = 0 . and, ii) B ϕi /B zi = 0 . , which corre-spond to (practically) no twist and small twist, respec-tively. The reason for using a small, but non-zero twist forthe case corresponding to the dispersion relation with zero ✵(cid:0)✁ ✶ ✶(cid:0)✁ ✷ ✷(cid:0)✁ ✸ ✸(cid:0)✁ ✹ ✹(cid:0)✁ ✁ ✁(cid:0)✁ ✻✵(cid:0)✂✶✶(cid:0)✷✶(cid:0)✹✶(cid:0)✻✶(cid:0)✂✷✷(cid:0)✷✷(cid:0)✹ Fig. 2.
Solutions of the dispersion Equation (27) for a warmdense tube embedded in a dense environment ( β i > , β e > )with speed ordering v SE > v SI > v AE > v AI . The color cod-ing is as follows: blue indicates body waves, red corresponds tosurface waves, orange corresponds to either the internal or ex-ternal Alfvén speeds, note this convention is used consistentlyin this work. The circle with a point corresponds to the case B ϕi /B zi = 0 . while the spiral corresponds to B ϕi /B zi = 0 . .The mean root mean squared error (RMSE) for this speed or-dering is . . twist which we have shown to be equivalent to the resultof Edwin & Roberts (1983), is that the limits of the Kum-mer functions in Equation (29) and Equation (30) requirean increasing number of terms as a i → and their calcu-lation becomes inefficient by direct summation. However, B ϕi /B zi = 0 . is a good approximation to the case withzero azimuthal magnetic field component. Note that in thefollowing we take the twist, namely B ϕ ( r ) , to be continuousacross the flux tube and thus set, S i = S e . The behaviorof the fast sausage body waves (FSBW) is very similar forboth the case with and without twist, and in extension itis very similar to the case with only internal twist studiedby Erdélyi & Fedun (2007). It is worth noting that wheninternal and external twist is present, the different radialharmonics of the FSBW modes have two solutions, onedispersive and one approximately non-dispersive see Fig-ure 2. It is, however, unclear if the non-dispersive solutionremains valid until the next radial harmonic. Nevertheless,it is clear that, in the neighborhood of the v SE singularity we obtain two solutions with comparable phase speeds ( v F )which opens the possibility for beat phenomena and thuswidens the possibility of detection of these waves since thebeat frequency will be smaller than both waves that pro-duce it. This behavior is not present when we consider twistonly inside the flux tube. Otherwise, the overall behaviorof the solutions is virtually identical to Erdélyi & Fedun(2007).In Figure 3 we present the solutions for the secondcase in Table 1. This scenario can occur when the in-ternal plasma- β is very low, β i ≪ , while the externalplasma- β is high, β e ≫ . At this point we would liketo stress the fact that for a specific set of characteris-tic speeds the internal and external plasma- β values areuniquely defined (see Appendix C). We focus here onlyin the region of solutions in ( v AE , v SE ) since the infinitenumber of the slow sausage body waves (SSBW), presentin the ( v T I , v SI ) interval are minimally affected by thetwist and thus are almost identical when compared with Article number, page 6 of 11 ig. 3.
Solutions of the dispersion Equation (27) for a cool evac-uated tube embedded in a dense environment ( β i ≪ , β e ≫ )with speed ordering v SE > v AI > v AE > v SI . The top figurecorresponds to no external twist for r > r a , namely S e = 0 ,while in the lower plot there is twist outside as well as in-side the flux tube. Note that in both figures the solutionsfor B ϕi /B zi = { . , . , . , . , . , . } have been over-plotted to conserve space and illustrate the effect of increasingthe magnetic twist. The axes inside the figures match the pro-gression of twist for the specific regions, for instance, in thetop plot the axis with the arrow to the left indicates that thefirst FSBW from the right corresponds to magnetic twist of . the second to . etc. Note that the vertical axis, for B ϕi /B zi = 0 . has no non-dispersive FSSW (horizontal solu-tions marked in red) which is indicated by the empty parenthesisnear the value . . The mean RMSE is . . the corresponding case with no twist. We plot solutionsfor B ϕi /B zi = { . , . , . , . , . , . } . The upperplot in Figure 3 represents the solutions only for internaltwist. A feature for this set of solutions is that the FSBWwhich is transformed to the fast sausage surface (FSSW) for K ∼ . , for increasing twist the transition becomes discon-tinuous and an interval, in K , is created where there existno solutions. For example, for B ϕi /B zi = 0 . , this intervalextends for K = (2 . , . where no fast body waves exist.This interval becomes larger with increasing twist. How-ever, this not the case in the presence of external twist,see lower figure in Figure 3. The FSBW and FSSW ap-pear to behave similarly, however, in all cases except for B ϕi /B zi = 0 . there is a surface wave solution that isnearly non-dispersive for a wide range of K . Another featureis the s -like set of surface wave solutions that are clearly vis-ible for B ϕi /B zi = 0 . and B ϕi /B zi = 0 . . Note, that this s -like set is also present for the other cases however the cuspis encountered for larger values of K . This structure is quite Fig. 4.
Plots of density perturbations superimposed on thebackground equilibrium density plots for the fast body and sur-face modes shown in Equation (3). The gray lines represent ve-locity perturbation streamlines on the xy -plane. Notice that forvisualization purposes the streamlines contain no information onthe magnitude of the perturbation, only direction information isconveyed. In the density plots red and blue correspond to highand low density respectively. The slices are snapshots at t = 0 atdifferent positions for the wavelength λ of the oscillation. Notethat this does not imply that the wavelength of the two oscil-lations is the same, rather, it is a fraction of the correspondingwavelength. ( Left ) Fast body mode for with magnetic twist for K = 0 . and v F = 1 . . ( Right ) Fast surface mode withmagnetic twist for K = 3 . and v F = 0 . . Notice that inboth cases the azimuthal component of the velocity perturbationat / λ, / λ and / λ is is non-zero. interesting since in some interval of K there exist simul-taneous solutions while outside of this interval exists onlyone. This means that within that interval, for a broadbandexcitation, the power of the driver will be distributed tomore than one solution thus reducing the individual powerspectrum signatures of the individual waves. In essence thiswill result in a interval of solutions that are much more dif-ficult to detect. Another interesting point in respect to this s -like set of solutions is that it seems that a point may ex-ist, for a certain value of B ϕi /B zi and a single K that therewould be a continuum as the s -shape becomes vertical (seeFigure 3). However, the existence or physical significance ofthis point is speculative since it does not appear to exist forsmall twist, namely the regime for which our approximationis valid. In Figure 4 we illustrate a FSBW (left panel) and aSSBW (right panel). In contrast to the kink mode in mag-netic flux tubes with weak twist that exhibit a polarization(Terradas & Goossens 2012), the sausage mode appears tobe the a superposition of an Alfvénic wave and a sausagewave leading by π/ .The last plasma regime with high plasma- β consideredin this work has the following characteristic speed order-ing: v SE > v SI > v AI > v AE . In Figure 5 we plot thesolutions to Equation (27) for this case. The most notablefeature, which seems to be consistent for alternative param-eter sets corresponding to photospheric conditions, is thatthe magnetic twist appears to have only a small effect on Article number, page 7 of 11 pJ Preprint:
Axisymmetric Modes with Magnetic Twist, Giagkiozis et al.
Fig. 5.
Solutions of the dispersion Equation (27) for a weak cooltube embedded in a dense environment ( β i > , β e ≫ ) withspeed ordering v SE > v SI > v AI > v AE . The mean RMSE is . . Fig. 6.
Solutions of the dispersion Equation (27) for an intensewarm tube embedded in a rarefied environment ( β i ≪ , β e ≪ )with the following speed ordering, v AE > v SI > v AI > v SE . Themean RMSE is . . the solutions to the dispersion equation. For example, wehave also used: v AI > v SE > v SI > v AE and there too (plotnot shown as it is identical to the case with no twist) thedeviation of the normalized phase speed was on the orderof . or less for magnetic twist up to B ϕi /B zi = 0 . . β regime In consultation with the results presented byVernazza et al. (1981) and Gary (2001), we expectthe results presented in this section to be most relevant toconditions that are typical of the upper chromosphere thetransition region and corona. The remaining two cases thatwe consider in this work are for low plasma- β conditions(see Table 1).In Figure 6 we consider an intense warm flux tube forwhich the characteristic speeds ordering is the following: v AE > v SI > v AI > v SE . This case was also considered byErdélyi & Fedun (2007) under the assumption that there isonly internal magnetic twist and zero twist in the environ-ment surrounding the flux tube. In that work the influenceof twist was under a percent, however, when the externaltwist is also considered interesting behavior emerges. In thiscase, when there is zero twist, the first SSBW changes char- Fig. 7.
Solutions of the dispersion equation Equation (27) foran intense cool tube embedded in a rarefied environment ( β i ≪ , β e ≪ ) with speed ordering v AE > v AI > v SE > v SI . Themean RMSE is . . acter to a slow sausage surface wave (SSSW) crossing v AI at approximately K = 0 . . When a small twist is intro-duced the first radial harmonic of the SSBW modes nowbecomes bounded by v AI and a SSSW mode appears. Also,a non-dispersive solution with a character similar to a sur-face wave emerges that closely follows v AI . We have namedthis solution as surface-Alfvén wave (SAW) in Figure 6 andwe have expanded the plot to make it visible since it is ex-tremely close to the internal Alfvén speed. Interestingly thehigher radial harmonics of the SSBW appear to be mini-mally affected when the magnetic twist is increased. Also,the correction to the phase velocity for the FSBW withmagnetic twist appears to be small compared with the caseof no magnetic twist. For the first radial harmonic this cor-rection is of the order of . while the correction is lessthan . for higher radial harmonics. However, this doesnot mean that the FSBW for the case with magnetic twistis identical to the case without twist. This is because theazimuthal component of the velocity perturbation in theformer case is non-zero altering the character of these wavessignificantly as compared with its counterpart in the casewithout magnetic twist.Lastly an intense cool tube is considered, i.e. v AE >v AI > v SE > v SI , which corresponds to conditions in thesolar corona. The solutions to the dispersion equation (i.e.Equation (27)) are shown in Figure 7. In this case, mag-netic twist has more pronounced effect on the FSBWs, whilethe SSBW are virtually unaffected. In the long wavelengthlimit, K ≪ , the FSBWs become non-dispersive while forshort wavelength limit, K ≫ , the solutions are identicalto the case of a straight magnetic flux tube with verticalmagnetic field only. It is important to note that, althoughthe effect of magnetic twist appears to be subtle in this case,it has an significant difference compared with the case withno magnetic twist, e.g. Edwin & Roberts (1983), as wellas the case considering only internal magnetic twist, e.g.Erdélyi & Fedun (2007). In both of these cases the sausage Article number, page 8 of 11 ode becomes leaky as
K → . This however, is not thecase when both internal and external twist are considered.Instead, as the magnetic twist increases so does the cutoff ofthe trapped fast sausage waves toward longer wavelengths.For example, for the particular characteristic speeds order-ing considered in Figure 7, the first FSBW ceases to havea cutoff wavelength when B ϕi /B zi > . , approximately.Therefore, the FSBW for a twisted magnetic cylinder abovea certain threshold remains trapped for all wavelengths. Aconsequence of this is that FSBWs remain in the Alfvéncontinuum and therefore may be resonantly damped, see forexample Sakurai et al. (1991). This means that the sausagemode cannot be ruled out as a source of energy to thecorona.
5. Discussion
Although the model we present in this work for a magneticflux tube with internal and external twist is relatively ad-vanced in comparison to recent theoretical models, it con-tains a number of simplifications and therefore we wouldbe remiss not to discuss the potential caveats when used tointerpret observations. Observations suggest that the cross-section of magnetic flux tubes is not circular. Althoughthere are no theoretical studies of magnetic flux tubes withcompletely irregular cross-section, some steps towards thisdirection have been taken by considering flux tubes withelliptical cross-section, see for example Ruderman (2003)and Erdélyi & Morton (2009). The results for the sausagemode presented in Erdélyi & Morton (2009) show that incomparison with the model of Edwin & Roberts (1983) (cir-cular cross-section) the ellipticity of the cross-section tendsto increase the phase speed of the sausage mode for photo-spheric conditions by approximately in the short wave-length limit, and, is negligible in the long wavelength limit.Conversely, in coronal conditions for increasing ellipticitythe phase speed increase is more pronounced for a widerange of wavelengths and is shown to be as much as higher of the predicted phase speed by the model with cir-cular cross-section. This effect is quite important since, forsufficiently large ellipticity, it could counteract the effectthat magnetic twist has on the cutoff frequency for the fastsausage body modes seen in Figure 7. Namely, as can beseen in Figure 7, the fast sausage mode remains trapped inthe long wavelength limit, however, should the phase speedbe increased, then a cutoff frequency for the fast sausagebody modes may be reinstated.Furthermore, although we have studied propagatingwaves in this work, the study of standing modes for k ϕ = 0 is trivially extended. Namely, if the magnetic flux tube isline-tied on both footpoints the longitudinal wavevector willbe quantized according to k z = nπ/L , where n is an integerand L is the length of the magnetic flux tube. If the fluxtube is assumed to be line tied on one end and open on theother, then no quantization takes place and there can bepropagating and standing waves for all k z . Here it shouldbe mentioned that the effect of the magnetic flux tube cur-vature is of the order of ( r a /L ) and therefore has a smalleffect on the eigenfrequencies of magnetic flux tubes in thesolar atmosphere (van Doorsselaere et al. 2004, 2009).Other effects that can alter the eigenfrequencies pre-dicted using the model in this work are, density stratifica-tion, flux tube expansion and resonance phenomena due toneighboring magnetic flux tubes, see Ruderman & Erdélyi (2009) for a more in depth discussion. Of course, more com-plicated magnetic field topologies can have other unforeseeneffects. This can be seen in magneto-convection simulations,e.g. Wedemeyer-Böhm et al. (2012), Shelyag et al. (2013),Trampedach et al. (2014) as well as in simulations withpredefined background magnetic fields, see Bogdan et al.(2003), Vigeesh et al. (2012), Fedun et al. (2011). However,the interpretation of the results from such simulations is amajor challenge which is only increased by considering thatthe initial conditions, which are mostly unknown, play avery important role in their subsequent evolution.
6. Conclusions
In the presence of weak twist the sausage mode has mixedproperties since it is unavoidably coupled to the axisym-metric Alfvén wave. This apparent from the solutions, seefor example Appendix D where the azimuthal velocity per-turbation component is nonzero and is also supported bythe results presented in Section 2.1. The implications of thison the character of surface and body waves are seen clearlyin Figure 4, where the relative magnitude of the radial andazimuthal components of the velocity perturbation alter-nate periodically and waves tend to exhibit Alfvénic char-acter the closer their phase velocity is to one of the Alfvénspeeds. The reason for this behaviour has been explainedin Section 2.1.Observations of Alfvén waves rely on the apparent ab-sence of intensity (i.e. density) perturbations in conjunc-tion with torsional motion observed by alternating Dopplershifts, see for example (Jess et al. 2009). The results of thiswork suggest that there exists at least one more alternativeinterpretation for waves with the aforementioned charac-teristics. Namely, the observed waves by Jess et al. (2009)could potentially be surface sausage waves (see right panelof Figure 4), since the localized character of the densityperturbation also implies localized intensity perturbationsthat can be well below the instrument resolution. Further-more, given the presence of torsional motion (see right panelof Figure 4) the sausage mode will have a Doppler signa-ture similar to that of an Alfvén wave. The Doppler signa-ture in combination with the fact that surface waves canhave a phase velocity very close to the Alfvén speed (seeSAW in Figure 6) suggests that Jess et al. (2009) poten-tially observed a sausage mode in the presence of magnetictwist. This line of reasoning is further supported by the ev-idence in Wedemeyer-Böhm et al. (2012) and Morton et al.(2013), where the authors show that vortical motions areubiquitous in the photosphere. However, the excitation ofthe decoupled Alfvén wave requires vortical motion thatis divergence free, see for example Ruderman et al. (1997),while the vortical motions observed in Morton et al. (2013)are not free of divergence. In our view these vortical mo-tions could be a natural mechanism for the excitation ofthe axisymmetric modes studied in this work.In this work we considered the effect of internal and ex-ternal magnetic twist on a straight flux tube for the sausagemode. It was shown that magnetic twist naturally couplesaxisymmetric Alfvén waves with sausage waves. Some ofthe main results of this coupling are: – Sausage waves can exhibit Doppler signatures similar tothese expected to be observed for Alfvén waves.
Article number, page 9 of 11 pJ Preprint:
Axisymmetric Modes with Magnetic Twist, Giagkiozis et al. – The phase difference between the radial and torsionalvelocity perturbations are π/ , which means that botheffects can be simultaneously observed. – Excitation of these modes can be accomplished with alarger variety of drivers compared to the pure sausageand axisymmetric Alfvén waves. Therefore, we speculatethat these waves should be more likely to be observedcompared with their decoupled counterparts. – For coronal conditions the fast sausage body waves re-main trapped for all wavelengths when the magnetictwist strength surpasses a certain threshold. This ap-pears to be characteristic of magnetic twist and couldpotentially be used to identify the strength of the mag-netic twist.These findings suggest that axisymmetric modes with mag-netic twist can be easily mistaken for pure Alfvén waves.
Appendix A: Dimensionless Dispersion Equation
For completeness we give here the dimensionless version ofthe dispersion equation Equation (27). The following equa-tion is now a function of v F and K , instead of ω and k z .One of the benefits of solving Equation (A.1) instead ofEquation (27) directly is that the former is, usually, nu-merically more stable. Another benefit is that the studyof different plasma conditions is made simpler since it isstraightforward to alter the ordering of the characteristicspeeds ( v si , v Ai etc). ρ e ρ i K ( v F − v AE ) k rE K ν − ( k rE K ) K ν ( k rE K )= (cid:20) v A Φ I k rI (1 + N I ) − ρ e ρ i v A Φ E k rE (1 + N E ) (cid:21) + ρ e ρ i (1 − ν )( v F − v AE ) M E − v F − k rI M ( a, b − s ) M ( a, b ; s ) , (A.1)where, a = 1 + k rI (cid:2) K ( v F − − v A Φ I (cid:3) v A Φ I N I ( v F − , b = 2 , s = 2 v A Φ I N I ( v F − ,v ph = ωk z , v F = v ph v Ai , v SI = v si v Ai , v SE = v se v Ai ,v AI = 1 , v AE = v Ae v Ai , n i = k z N I , n e = k z N E ,k ri = k z k rI , k re = k z k rE v T I = v T i v Ai , v T E = v T e v Ai , K = k z r a , v A Φ I = v Aϕi v Ai , v A Φ E = v Aϕe v Ai , and N I = v F v F + v SI ( v F − , N E = v F v F v AE + v SE ( v F − ,k rI = ( v SI − v F )(1 − v F )(1 + v SI )( v T I − v F ) , k rE = ( v SE − v F )( v AE − v F )( v AE + v SE )( v T E − v F ) ,v T i = v Ai v si v Ai + v si , v T I = v SI v SI , v T e = v Ae v se v Ae + v se ,v T E = v AE v SE v AE + v SE , ν = 1 + 2 v A Φ E ( v F − v AE ) (cid:2) v A Φ I N E + ( v AE (3 N E − − v F ( N E + 1)) (cid:3) . Also the plasma- β inside and outside the flux-tubecan be calculated using: β i = (2 /γ ) v SI and β e =(2 ρ e B z i /γρ i B z e ) v SE respectively. Appendix B: Estimation of the Root Mean SquareError
We argue that the exact solution for constant twist outsidethe magnetic flux tube is a good approximation to the so-lution corresponding to the case where the twist is ∝ /r .However, as we state in the text, we only obtain the zeroth-order term in the perturbation series which corresponds toconstant twist. To justify this statement we estimate theroot mean squared error (RMSE) also referred to as stan-dard error, defined as follows, RM SE = lim L →∞ L − Z Lr a (cid:16) ˆ δξ re ( r ) − δξ re ( r ) (cid:17) dr ! . In this context, ˆ δξ re ( r ) is the solution to the case with con-stant magnetic twist, i.e. κ = 0 in Equation (21), while δξ re ( r ) is a numerical solution to Equation (21) with κ = 1 ,namely magnetic twist proportional to /r . The RMSE isexpected to vary for different parameters, i.e. K , v F and B ϕi /B zi , and for this reason we discretize K and v F us-ing a × grid and also use the following value for B ϕi /B zi = 0 . , since for all values of B ϕi /B zi < . theRMSE is consistently smaller. Subsequently we average theresulting root mean square errors which we then quote inthe corresponding figure caption. Note, that ˆ δξ re ( r ) and δξ re ( r ) are normalized, therefore a value for the meanRMSE of, e.g. . , means that the standard error is onaverage, when comparing ˆ δξ re ( r ) with δξ re ( r ) . Appendix C: Characteristic Speeds OrderingConsiderations
The ordering of characteristic speeds depends on vari-ables: B zi , B ze , T i , T e , n i , n e , where n i and n e are thenumber densities inside and outside the flux tube respec-tively. Starting from and assuming the magnetic twist issmall, v Ai = B zi √ µ ρ i , v Ai = B zi √ µ ρ i , v si = r γ p i ρ i , v se = r γ p e ρ e ,β i = 2 µ p i B zi , β e = 2 µ p e B ze , ρ = nm p , p = nk B T. Taking logs of the speeds and using the following defini-tions: v ⋆Ai = ln( v Ai ) + 12 ln( µ m p ) , v ⋆Ae = ln( v Ae ) + 12 ln( µ m p ) ,v ⋆si = ln( v si ) −
12 ln (cid:18) γ k B m p (cid:19) , v ⋆se = ln( v se ) −
12 ln (cid:18) γ k B m p (cid:19) ,b ⋆i = 12 (ln( β i ) − ln(2 k B µ )) , b ⋆e = 12 (ln( β e ) − ln(2 k B µ )) , Article number, page 10 of 11 nd B ⋆zi = ln( B zi ) , B ⋆ze = ln( B ze ) , T ⋆i = (1 /
2) ln( T i ) ,T ⋆e = (1 /
2) ln( T e ) , n ⋆i = (1 /
2) ln( n i ) , n ⋆e = (1 /
2) ln( n e ) , the speeds and plasma- β parameters can be written as fol-lows, − −
10 0 1 0 0 00 0 0 1 0 0 − − B ⋆zi B ⋆ze T ⋆i T ⋆e n ⋆i n ⋆e = v ⋆Ai v ⋆Ae v ⋆si v ⋆se b ⋆i b ⋆e . Now notice that the above matrix is rank which meansthat the dimension of the null-space is , with basis vectors: y = (1 , , , , , , and, y = (0 , , , , , . This meansin practice that for a given set of parameters resulting in aspecific speed ordering β i and β e are uniquely defined butthere is a dimensional subspace involving B ⋆zi , B ⋆ze , n ⋆i , n ⋆e , that is, all linear combinations of y and y . Also no-tice that the sound speeds, v ⋆si and v ⋆se , depend only on theinternal and external temperature, T ⋆i and T ⋆e respectively.Additionally the null-space of the matrix (see the basis vec-tors y and y ) suggests that the densities, n ⋆i and n ⋆e , aresecondary variables to the magnetic field strength, B ⋆zi and B ⋆ze . Appendix D: Perturbed quantities in terms of δξ r and δp T Given, δξ r and δp T in Equation (19a),Equation (19b)or Equation (25),Equation (26) the remaining per-turbed quantities for the sausage mode ( k ϕ = 0 ) are(Erdélyi & Fedun 2010), δξ ri ( s ) = A i s / E / e − s/ M ( a, b ; s ) ,δp 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) ,δξ re ( r ) = A e K ν ( k re r ) ,δp T e = A e (cid:18) µ (1 − ν ) D e − r a S e n e µ r ( k z − n e ) K ν ( k re r ) − D e k re K ν − ( k re r ) (cid:19) ,δξ ϕ = ik z ρ ( ω − ω A ) B z B ϕ µ ρ " n ω B ϕ µ r δξ r − δp T ! − ρr δξ r ,δξ z = ik z ρω ( ω − ω A ) " ( ω − n v A ) δp T + 2 B ϕ µ r v A n δξ r ,δB r = ik z B z δξ r ,δB ϕ = k z B ϕ D (cid:18) δp T + 2 B z µ r δξ r (cid:19) − ddr ( B ϕ δξ r ) ,δB z = − r ddr ( rB z δξ r ) . Appendix E: Acknowledgments
I.G. would like to acknowledge the Faculty of Science of theUniversity of Sheffield for the SHINE studentship and M.Ruderman, T. Van Doorsselaere and M. Goossens for theuseful discussions on this paper. V.F., G.V. and R.E. wouldlike to acknowledge STFC for financial support. R.E. is alsothankful to the NSF, Hungary (OTKA, Ref. No. K83133).G.V. would like to acknowledge the Leverhulme Trust (UK)for the support he has received.
References