Resonant absorption: transformation of compressive motions into vortical motions
AAstronomy & Astrophysics manuscript no. 38394final c (cid:13)
ESO 2020September 18, 2020
Resonant absorption:Transformation of compressive motions into vortical motions
M. Goossens , I. Arregui , , R. Soler , , and T. Van Doorsselaere Centre for mathematical Plasma Astrophysics, KU Leuven, Celestijnenlaan 200B bus 2400, B-3001 Leuven, Belgium Instituto de Astrofísica de Canarias, Vía Láctea s / n, E-38205 La Laguna, Tenerife, Spain Departamento de Astrofísica Universidad de La Laguna, E-38206 La Laguna, Tenerife, Spain Departament de Física, Universitat de les Illes Balears, E-07122 Palma de Mallorca, Spain Institut d (cid:48)
Aplicacions Computacionals de Codi Comunitari (IAC ), Universitat de les Illes Balears, E-07122 Palma de Mallorca,Spaine-mail: [email protected] September 18, 2020
ABSTRACT
This paper investigates the changes in spatial properties when magnetohydrodynamic (MHD) waves undergo resonant damping in theAlfvén continuum. The analysis is carried out for a 1D cylindrical pressure-less plasma with a straight magnetic field. The e ff ect ofthe damping on the spatial wave variables is determined by using complex frequencies that arise as a result of the resonant damping.Compression and vorticity are used to characterise the spatial evolution of the MHD wave. The most striking result is the huge spatialvariation in the vorticity component parallel to the magnetic field. Parallel vorticity vanishes in the uniform part of the equilibrium.However, when the MHD wave moves into the non-uniform part, parallel vorticity explodes to values that are orders of magnitudehigher than those attained by the transverse components in planes normal to the straight magnetic field. In the non-uniform part ofthe equilibrium plasma, the MHD wave is controlled by parallel vorticity and resembles an Alfvén wave, with the unfamiliar propertythat it has pressure variations even in the linear regime. Key words.
Magnetohydrodynamics (MHD) – Waves – Sun: corona – Sun: magnetic fields
1. Introduction
In a recent paper, Goossens et al. (2019) have studied the prop-erties of magnetohydrodynamic (MHD) waves in non-uniformplasmas. They pointed out that in non-uniform plasmas, MHDwaves behave di ff erently from their counterparts in uniform plas-mas of infinite extent. In the latter case, the MHD waves canbe separated into Alfvén waves and magneto-acoustic waves.The Alfvén waves propagate vorticity and are incompressible.In addition, they have no parallel displacement component. Themagneto-acoustic waves are compressible and in general have aparallel component of displacement, but do not propagate par-allel vorticity. Compression, parallel vorticity, and parallel dis-placement are the characteristic quantities. In a uniform plasmaof infinite extent, compression and parallel displacement on onehand and parallel vorticity on the other hand are mutually exclu-sive. In a pressure-less plasma, the parallel displacement is zerobecause the Lorentz force has no component along the magneticfield. Hence, in a pressure-less plasma, the waves have only twocharacteristic quantities: compression and parallel vorticity. Thedistinction between the waves is then based on compression orparallel vorticity.The situation is di ff erent in a non-uniform plasma, as waspointed out by Goossens et al. (2019). The clear division be-tween Alfvén waves and magneto-acoustic waves is no longerpresent. The MHD waves have mixed properties. The generalrule is that MHD waves propagate both parallel vorticity, as inclassic Alfvén waves, and compression, as in classic magneto-acoustic waves. The present paper focusses on the properties of MHD waves that undergo resonant absorption. Here we concen-trate on resonant absorption in the Alfvén continuum. In order tokeep the mathematical analysis as simple as possible while stillretaining the essential physics, we consider a straight magneticfield, and in addition, we assume that the plasma is pressure-less.This assumption removes the slow magneto-acoustic part of thespectrum and resonant absorption in the slow continuum.Goossens et al. (2019) also studied what happens with com-pression and vorticity for frequencies in the slow and Alfvéncontinuum. The analysis was restricted to the driven problem ofstationary waves with real frequencies. The authors determinedthe dominant singularities in the ideal MHD solutions and thedominant dynamics for stationary waves. For another study inwhich resonant absorption in both Alfvén and slow continua areconsidered, see Soler et al. (2009). Studies on resonant absorp-tion have mainly focussed on the components of the displace-ment, the amount of absorbed energy, and the damping rate.Analytical solutions for the components of the Lagrangian dis-placement in the dissipative layer have for example been de-rived by Sakurai et al. (1991) and Goossens et al. (1995) forresonant MHD waves in ideal and dissipative stationary MHD.Ruderman et al. (1995) studied non-stationary incompressibleresonant MHD waves in non-ideal MHD for a planar equilib-rium. Tirry & Goossens (1996) studied non-stationary resonantMHD waves for cylindrical plasmas in visco-resistive MHD.Soler et al. (2013) studied non-stationary MHD waves for cylin-drical plasmas both in resistive MHD and ideal MHD. Theirmathematical scheme for non-stationary ideal MHD followedthe scheme devised by Hollweg (1990b) for planar plasmas. Lit- Article number, page 1 of 9 a r X i v : . [ a s t r o - ph . S R ] S e p & A proofs: manuscript no. 38394final tle or no attention has been given to the change in the spatialbehaviour of fundamental quantities such as compression andvorticity. Goossens et al. (2012) were the first to point out thatthe fundamental radial mode of kink waves propagates parallelvorticity in the non-uniform part of the loop required for reso-nant absorption to operate. In the present investigation, we fo-cus on the eigenvalue problem and try to understand what hap-pens when the wave is actually damped in non-stationary MHD.We take the frequency to be complex and relate the spatial be-haviour to the damping properties of the MHD wave. Soler et al.(2013) concentrated on the components of Lagrangian displace-ment and the Eulerian perturbation of total pressure. Goossenset al. (2012) took the existence of a parallel vorticity componentas the base for the physical interpretation of the fundamental ra-dial mode of kink waves in terms of surface Alfvén waves. Ouranalytic investigation first presents the components of vorticityin the case of non-stationary ideal MHD close to the resonantposition. Then, the semi-analytic approach of Soler et al. (2013)in non-stationary MHD is used to verify the predictions basedon approximate analytic theory. The aim is to obtain a simpleunderstanding of how the fundamental quantities compressionand vorticity are a ff ected by the non-stationary behaviour of theresonantly damped wave.
2. Resonant absorption
This section collects results on resonant absorption that are usedin our discussion of the spatial solutions of the MHD wavesthat undergo resonant damping. Resonant absorption has a longhistory in fusion plasma physics, space plasma physics, so-lar physics, and astrophysics. A characterisation was given byParker (1991), who noted that resonant absorption in the Alfvéncontinuum is to be expected when a wave with a phase velocity ω/ k spans a region in which the variation of the Alfvén velocity v A across the region provides the resonance condition ω/ k = v A .Non-uniformity is key to the process. The Alfvén velocity v A andAlfvén frequency ω A depend on position. The resonance occursat the position r A , where the frequency of the wave ω is equalto the local Alfvén frequency ω = ω A ( r A ). The local Alfvén fre-quency ω A is assumed to be an analytic function.Since 2002 (Ruderman & Roberts 2002; Goossens et al.2002a), resonant absorption of kink waves is a popular and plau-sible mechanism for explaining the rapid damping of standingand propagating MHD waves in coronal loops (see e.g. Montes-Solís & Arregui 2017, for a discussion). The simple model in-vokes MHD waves superimposed on a cylindrical plasma col-umn in static equilibrium. Cylindrical coordinates ( r , ϕ, z ) areused. The inhomogeneity necessary for resonant absorption tooperate is usually provided by the equilibrium density ρ ( r ) thatvaries from ρ i to ρ e in the interval [ R − l / , R + l / ρ is constant in the internal and the external parts of theloop with values ρ i and ρ e , ρ = (cid:40) ρ i for 0 ≤ r ≤ R − l / ,ρ e for R + l / ≤ r < + ∞ . In the non-uniform transitional layer of thickness l , the equi-librium density varies continuously from its internal value ρ i toits external value ρ e .The equilibrium magnetic field is assumed to be axial andconstant B = B z , with z the unit vector in the z - direc-tion. The temporal dependence and the spatial dependence onthe ignorable ( ϕ, z )- coordinates are given by exp( − i ω t ) and exp( i ( m ϕ + k z z )), with m and k z the azimuthal and axial wavenumbers, respectively. The local Alfvén frequency ω A and theAlfvén velocity v A are defined as ω = ( k · B ) µρ = k v = k (cid:107) v and v = B µρ . (1)The fundamental radial mode of kink ( m =
1) waves has itsfrequency in the Alfvén continuum and is always resonantlydamped. Here we consider standing waves, which means thatthe axial wave number k z is real and the frequency ω is complex.Hence ω = ω R + i γ, exp( − i ω t ) = exp( − i ω R t ) exp( γ t ) = exp( − i ω R t ) exp( − t /τ D ) . (2)The Period = π/ω R , γ is the decrement, and τ D = / | γ | the damping time. The variation in density is confined to a layerof thickness l and has a steepness α . The steepness α is related tothe factor F introduced by Arregui et al. (2007, 2008a) and hasbeen used many times before, for example, by Goossens et al.(2008), Arregui et al. (2008b), and Soler et al. (2009) as F = α π . (3)When we aim to arrive at analytical expressions for the periodand damping time, we can use the thin tube and thin boundary(TTTB) approximation so that k z R (cid:28) l / R (cid:28)
1. Analyticalexpressions for the damping time τ D were derived by Hollweg(1990a), Goossens et al. (1992), Ruderman & Roberts (2002),Van Doorsselaere et al. (2004), Soler et al. (2013), and Arregui& Goossens (2019). We use the expression given by Arregui &Goossens (2019), τ D Period = | m | π α l / R ρ i + ρ e ρ i − ρ e . (4)In this equation, the quantities l / R and α arise from the adop-tion of variation in the equilibrium density in a non-uniformlayer of thickness l and with steepness α so that the derivativeof the density at the resonant position r A is given by d ρ dr ] r = Re( r A ) = − α ρ i − ρ e l . (5)We first recall that ω is complex, with an imaginary part γ, and strictly speaking, the position r A is in the complex plane.The thin boundary (TB) approximation assumes that the abso-lute value of γ is low in comparison with ω R and that Im( r A ) isneglected. In what follows, Im( r A ) is neglected unless otherwisestated.The steepness α is 1 for a linear variation of density and π/ m =
1, but any non-axisymmetric wave ( m (cid:44)
0) with its frequency in the Alfvén con-tinuum is resonantly damped. Soler (2017) studied the resonantdamping of fluting modes ( | m | ≥
2) and showed that these modescan be heavily damped ( τ D / Period (cid:28) l / R (cid:28) Article number, page 2 of 9. Goossens et al.: Transformation of compressive motions into vortical motions
3. MHD waves with mixed properties for a straightfield
We recall from Goossens et al. (2019) that the mixed propertiesarise because the MHD equations are coupled. The coupling ofthe MHD equations is controlled by the coupling functions C A and C S , which were first introduced by Sakurai et al. (1991). Fora straight field B = B ( r ) z , they take the simple form C A = g B P (cid:48) = mr B P (cid:48) , C S = P (cid:48) , (6)with P (cid:48) the Eulerian perturbation of total pressure. This meansthat the equations are coupled because of P (cid:48) . This was alreadyknown by Hasegawa & Uberoi (1982). They noted thatThe basic characteristic of the ideal Alfvén wave is thatthe total pressure in the fluid remains constant during thepassage of the wave as a consequence of the incompress-ibility condition. For inhomogeneous medium, however,the total pressure, in general, couples with the dynamicsof the motion, and the assumption of neglect of pressureperturbations becomes invalid.We now concentrate on compression and parallel vorticity.We also take non-axisymmetric waves m (cid:44) m = C A = , and there is no coupling of equations, no mixed proper-ties, and no resonant damping. Because we study non-stationaryMHD waves, the frequency ω that appears in the following equa-tions is a complex quantity, as defined in Eq. (2). For a pressure-less plasma, the velocity of sound v S = ξ z = ξ (cid:107) =
0. We take v = ξ and for compression ∇ · ξ ξ r = ρ ( ω − ω ) dP (cid:48) dr ,ξ ⊥ = ξ ϕ = im / r ρ ( ω − ω ) P (cid:48) ,ξ (cid:107) = ξ z = , ∇ · ξ = − P (cid:48) ρ v = − P (cid:48) B /µ . (7)Similarly, we take v = ∇ × ξ ( ∇ × ξ ) r = k z mr ρ ( ω − ω ) P (cid:48) . ( ∇ × ξ ) ⊥ = ( ∇ × ξ ) ϕ = ik z ρ ( ω − ω ) dP (cid:48) dr , ( ∇ × ξ ) (cid:107) = ( ∇ × ξ ) z = − i mr (cid:110) ρ ( ω − ω ) (cid:111) ddr (cid:110) ρ ( ω − ω ) (cid:111) P (cid:48) . (8)Equations (7) and (8) clearly show that P (cid:48) plays the role ofthe coupling function. The transverse components of vorticity ( ∇× ξ ) r and ( ∇× ξ ) ϕ are always non-zero. The parallel component( ∇ × ξ ) (cid:107) = ( ∇ × ξ ) z is non-zero when ddr (cid:110) ρ ( ω − ω ) (cid:111) (cid:44) . (9)All wave variables are non-zero in a non-uniform plasma.Except that here ξ (cid:107) = v = ff erently in di ff erent parts ofthe plasma because of the inhomogeneity of the plasma, as al-ready emphasised, for example, by Goossens (2008), Goossenset al. (2002a), Goossens et al. (2006), Goossens et al. (2011),Goossens et al. (2012), and Goossens et al. (2019).We note that for a straight constant field B = B z , thespatial variation of ω is solely due to the spatial variation of theequilibrium density ρ , hence d ρ dr (cid:44) ∇ × ξ ) (cid:107) = ( ∇ × ξ ) z = − i mr ω (cid:110) ρ ( ω − ω ) (cid:111) d ρ dr P (cid:48) . (11)In order to make clear, as was done by Goossens et al. (2019)in their equations (62, 63, and 64), that parallel vorticity andcompression go together in a non-uniform plasma, the previousEq. (11) can be rewritten as( ∇ × ξ ) (cid:107) = ( ∇ × ξ ) z = i mr ω (cid:110) ρ ( ω − ω ) (cid:111) d ρ dr B µ ∇ · ξ . (12)We note that ρ v A = B µ = constant , ρ ω A = k z B µ = constant .
4. Analysis close to the ideal resonant point
This section investigates the behaviour of the wave variables fora standing damped resonant wave close to the ideal resonantpoint r A . For a resonantly damped standing wave, Eq. (2) reads ω = ω R + i γ, γ = ω I , ω R ≈ ω A ( r A ) , | γ | ω R = π Period τ D . (13)An immediate consequence of the fact that the frequency, ω , is complex in non-stationary ideal MHD is that the resonantpoint, r A , is also a complex quantity. The higher the dampingrate, γ , the larger the imaginary part of r A . The following analyticinvestigation assumes weak damping, so that the imaginary partof r A can be ignored. We therefore treat r A as a real quantity asin stationary MHD. However, we are aware that this is just anapproximation. In Section 5 we show that we need to consider r A to be complex in order to understand the behaviour of theperturbations when wave damping is not weak. Article number, page 3 of 9 & A proofs: manuscript no. 38394final
It is instructive to summarise the main results on the be-haviour of resonant MHD waves in ideal MHD. The fundamen-tal conservation law for resonant Alfvén waves for a straight fieldwas obtained by Sakurai et al. (1991), P (cid:48) = constant , [ P (cid:48) ] = . (14)This conservation law was later confirmed in dissipative station-ary MHD by Goossens et al. (1995) for cylindrical plasmas, indissipative non-stationary MHD for incompressible plasmas byRuderman et al. (1995) for planar geometry, and by Tirry &Goossens (1996) for compressible visco-resistive MHD in cylin-drical geometry. Soler et al. (2013) showed that in non-stationaryideal MHD, P (cid:48) displays a logarithmic jump at the resonant point.The jump is proportional to the imaginary part of r A , so that theconservation law of P (cid:48) = constant is approximately valid if thedamping is only weak (see Section 5).In view of the observed periods and damping times, it is anaccurate approximation to use | γ | ω R (cid:28) . (15)In the first instance we concentrate on the parallel componentof vorticity. The analysis in Goossens et al. (2019) has shownthat in ideal MHD, the dominant dynamics resides in the per-pendicular component of the displacement and the parallel com-ponent of vorticity in ideal MHD. We use the following approx-imate results by retaining the dominant terms, that is, the lowestorder terms in | γ | /ω R : ω d ρ dr = ω (1 + i γω R ) d ρ dr ≈ ω d ρ dr ,ω = ( ω R + i γ ) ≈ ω + i γω R ,ω − ω ≈ i γω R , (cid:110) ρ ( ω − ω ) (cid:111) ≈ − ω ( ρ ( r A )) (cid:32) | γ | ω R (cid:33) , − i mr ω (cid:110) ρ ( ω − ω ) (cid:111) d ρ dr ≈ i mr A ρ ( r A ) ω × ρ ( r A ) d ρ dr π (cid:26) τ D Period (cid:27) . (16)With the use of Eq. (16), we obtain for the parallel compo-nent of vorticity( ∇ × ξ ) z ≈ i mr A π k ρ ( r A ) d ρ dr B /µ (cid:26) τ D Period (cid:27) P (cid:48) . (17)The aim is to relate vorticity to compression. In particular,we are interested in parallel vorticity and compression. The rea-son is that for a uniform plasma of infinite extent, these twoquantities are mutually exclusive and characterise Alfvén wavesand fast magneto-acoustic waves, respectively. Goossens et al.(2019) have already pointed out that all wave variables are non-zero in a non-uniform plasma so that there are no pure magneto-acoustic waves and no pure Alfvén waves. For a pressure-less plasma with a straight magnetic field, theexpression for ∇ · ξ is simply ∇ · ξ = − P (cid:48) / ( B /µ ). Hence at r = r A | ( ∇ × ξ ) z || ∇ · ξ | ] r A ≈ | m | r A π k ρ ( r A ) | d ρ dr | (cid:26) τ D Period (cid:27) . (18)So far, we have not made any assumption about the variationin the equilibrium density in the non-uniform layer or about thethickness of the non-uniform layer. We try to obtain an estimatefor 1 ρ ( r A ) | d ρ dr | . We assume that the variation in density is confined to a non-uniform layer of thickness l and has a steepness α, r A ≈ R , ρ ( r A ) ≈ ρ i + ρ e . | d ρ dr | ≈ α ρ i − ρ e l / R R , ρ ( r A ) | d ρ dr |≈ ρ i − ρ e ρ i + ρ e α l / R R . Hence | ( ∇ × ξ ) z || ∇ · ξ | ] r A ≈| m | π ( k z R ) ρ i − ρ e ρ i + ρ e α l / R (cid:26) τ D Period (cid:27) . (19)This expression gives the dependence of | ( ∇ × ξ ) z | on k z R , l / R , and density contrast for a variation in density with steep-ness α . Other prescriptions for the variation of density can beused. We note that there is a hidden dependence on k z R , l / R ,and density contrast because τ D / Period also depends on thesequantities. Nevertheless, this simple formula is relatively goodfor the interpretation of numerical results obtained, for example,by Goossens et al. (2012). We improve this if we cheat slightly.In the TT and TB approximation, analytic expressions exist for τ D / Period. With the use of expression (4) for τ D / Period, we canrewrite Eq. (19) as | ( ∇ × ξ ) z || ∇ · ξ | ] r A ≈ | m | π k z R ) ρ i + ρ e ρ i − ρ e α ( l / R ) . (20)We now concentrate on the transverse components ( ∇ × ξ ) r and ( ∇ × ξ ) ϕ . Expressions for these two components are given inthe first two equations of expression (8).These two quantities are non-zero everywhere. They do notrequire non-uniformity to be non-zero. We now try to determinetheir values at and close to the position r = r A where ω = ω A ( r A ).We recall that at this position, ρ ( ω − ω ) ≈ i γω R ρ ( r A ) ω , so that for the radial component( ∇ × ξ ) r ≈ − i mk z r A π τ D Period 1 B /µ P (cid:48) ≈ i mk z r A π τ D Period ∇ · ξ . (21) Article number, page 4 of 9. Goossens et al.: Transformation of compressive motions into vortical motions
Hence | ( ∇ × ξ ) r || ∇ · ξ | ] r A ≈ | m | π k z R τ D Period , (22)where we have taken r A ≈ R .With the approximate expression (4) for τ D / Period for a vari-ation in density of steepness α, this can be rewritten as | ( ∇ × ξ ) r || ∇ · ξ | ] r A ≈ k z R π ρ i + ρ e ρ i − ρ e α ( l / R ) . (23)The azimuthal component of vorticity can be written as( ∇ × ξ ) ϕ ≈ π k z τ D Period dP (cid:48) dr B /µ (24)and | ( ∇ × ξ ) ϕ || ∇ · ξ | ] r A ≈ π k z τ D Period 1 P (cid:48) | dP (cid:48) dr | . (25)For a straight field the Eulerian perturbation of total pres-sure is a conserved quantity (see e.g. Sakurai et al. 1991; Tirry& Goossens 1996). In non-stationary ideal MHD, it is approxi-mately conserved when the damping is weak (Soler et al. 2013)and does not undergo strong spatial variations. This is also veri-fied by numerical computations. This leads us to use dP (cid:48) dr ≈ P (cid:48) / R as an estimate for the derivative. In Fig. 1 the compression,which is proportional to P (cid:48) , is plotted as a function of the ra-dial position for the fundamental radial mode. This figure showsthat a linear function is a very good representation for P (cid:48) . Withthis approximation we obtain | ( ∇ × ξ ) ϕ || ∇ · ξ | ] r A ≈ π k z R τ D Period . (26)With the use of expression (4) for τ D / Period, we can rewriteEq. (26) as | ( ∇ × ξ ) ϕ || ∇ · ξ | ] r A ≈ | m | k z R π ρ i + ρ e ρ i − ρ e α ( l / R ) . (27)We recall that these expressions only apply in the non-uniform part of the loop, and strictly speaking, they apply onlyclose to the position r A where ω = ω A ( r A ). These expressionsgive us a good description for understanding what happens withthe spatial solutions when a wave undergoes resonant damping.
5. Spatial variation for resonant absorption for acoronal loop
Here we apply our results from the previous two sections to un-derstand and predict the spatial behaviour of MHD waves thatundergo resonant absorption in the equilibrium coronal loop de-fined in Section 3. The aim here is to obtain a simple understand-ing of how the fundamental quantities compression and vorticityare influenced by the non-stationary behaviour of the resonantlydamped wave.Compression and transverse vorticity do not require non-uniformity. They are non-zero everywhere in the plasma col-umn. However, parallel vorticity is zero in a uniform plasma.
Fig. 1.
Absolute value of the compression as a function of the radialposition in a flux tube with l / R = . k z R = .
1, and ρ i /ρ e =
3. Thenormalisation max {| ∇ · ξ |} = It requires a spatial variation of the equilibrium magnetic fieldand / or the equilibrium density. For a constant axial magneticfield, the spatial variation in density causes the non-zero parallelvorticity. Hence we arrive at the following situation for the res-onant damping of transverse waves in the Alfvén continuum inthe equilibrium model of Section 3: In the uniform internal partof the loop, 0 ≤ r ≤ R − l / , and in the uniform external part ofthe loop, R + l / ≤ r < ∞ , we expect compression and transversevorticity. We do not expect parallel vorticity. In the non-uniformpart of the loop, we expect compression and both transverse andparallel vorticity. In the vicinity of the resonant point r = r A , theparallel vorticity is expected to dominate the transverse vorticity, | ( ∇ × ξ ) z |(cid:29)| ( ∇ × ξ ) r |≈| ( ∇ × ξ ) ϕ | , (28)so that the MHD wave is almost a pure Alfvén wave.In order to verify these predictions based on approximate an-alytic theory, now we consider the semi-analytic approach ofSoler et al. (2013) in non-stationary ideal MHD. The methodwas inspired by an early work by Hollweg (1990b) in a sim-plified Cartesian configuration. We give a brief summary of thetechnique. We express the perturbations in the non-uniform partof the loop as a Frobenius series that includes a singular termaccounting for the e ff ect of the Alfvén resonance. This seriesexpansion allows us to connect through the non-uniform layerthe perturbations in the internal medium to those in the externalmedium. Thus, we obtain a dispersion relation for all trappedmodes with any value of the azimuthal wavenumber, m . How-ever, we focus on m =
1. The dispersion relation is valid forarbitrary values of l / R . When the nonuniform layer is thin, thatis, l / R (cid:28)
1, the obtained dispersion relation consistently pro-vides the same results as those of the TTTB approximation. Thenumerical part of the method consists of numerically solvingthe transcendental dispersion relation to obtain the complex fre-quency, ω , for a fixed k z . When the frequency is known, we cancompute the spatial behaviour of the perturbations. We refer toSoler et al. (2013) for a more detailed explanation of the method.Soler et al. (2013) studied the components of Lagrangian dis-placement ξ r , ξ ϕ, and the Eulerian perturbation of total pressure, P (cid:48) . Compression and the components of vorticity have now beencomputed and are presented in Figs. 1-6. A similar exercise was Article number, page 5 of 9 & A proofs: manuscript no. 38394final
Fig. 2.
Absolute value of the radial (left), azimuthal (centre), and parallel (right) vorticity components as functions of the radial position in thesame flux tube as in Fig. 1. made for the parallel vorticity in non-stationary resistive MHDby Goossens et al. (2012) (see their figure 4). The normalisationmax {| ∇ · ξ |} = r / R ∈ [0 ,
3] for a loop with a relativelythin non-uniform layer with l / R = .
5. We also used k z R = . ρ i /ρ e =
3. This case can be directly compared with the ap-proximate TTTB results obtained in the previous sections. Fig-ure 1 shows that nothing particular happens for compression.The behaviour does not substantially di ff er from that found in thestepwise constant case. Figure 2 shows that the transverse com-ponents of vorticity are non-zero in the whole domain; the par-allel component of vorticity is only non-zero in the non-uniformpart of the loop. The parallel component of vorticity is far largerthan the transverse components. In Fig. 2 the maximum valueof the parallel component is at least four orders of magnitudehigher than the maximum values of the transverse components.In turn, the amplitudes of the two transverse components are ofthe same order of magnitude. All these results agree with theanalytic predictions in the TTTB approximation.Next we consider l / R = . k z R and ρ i /ρ e . In Fig. 3we considered three values of k z R = ρ i /ρ e =
2, while in Fig. 4 we considered three values of ρ i /ρ e =
2, 5,and 10 and k z R = r ∈ [ R − l / , R + l /
2] alone. The resultsare plotted in both linear and logarithmic scales for better visu-alisation, and we recall that max {| ∇ · ξ |} = ff ected when either k z R or ρ i /ρ e are modified. This is so because the spatial behaviour ofvorticity is largely determined by the spatial variation of density,which is the same in all cases. However, the relative amplitudesof the three components of vorticity depend upon the considered k z R and ρ i /ρ e . The larger k z R , the lower the vorticity. The ampli-tude of the field-aligned component of vorticity decreases with k z R as ( k z R ) − , approximately, as Eq. (19) predicts, while bothradial and azimuthal components behave as ( k z R ) − accordingto Eq. (22) and (26). Equivalently, the higher the density con-trast, ρ i /ρ e , the lower the vorticity. The dependence on densitycontrast is also consistent with the analytic TTTB formulas, al-though there is no such clear dependence as in the case of k z R .We have confirmed so far that the analytic predictions in theTTTB approximation are consistent with the full solution pro-vided by the method of Soler et al. (2013) when the non-uniform transition is thin. Now, we can be more ambitious and fully ex-ploit the method by computing results beyond the range of ap-plicability of the approximations. Figures 5 and 6 display com-pression and the components of vorticity, respectively, for a loopwith a thick non-uniform layer of l / R = .
5. The other parame-ters are the same as in Figs. 1 and 2.When we compare the plots of compression for a thin (Fig. 1)and a thick (Fig. 5) non-uniform transition, a distinct feature isobvious. In the case of a thick transition, compression displaysa small jump at the resonance position. In order to understandthe behaviour of compression, we recall that compression is pro-portional to P (cid:48) and we resort to the Frobenius series of P (cid:48) . Ingeneral, the full solution provided by the method of Frobenius isdi ffi cult to handle (see Soler et al. 2013). To illustrate the presentdiscussion, it su ffi ces to consider the first non-zero terms in theFrobenius series of P (cid:48) in the nonuniform layer, P (cid:48) ≈ S + A + S m r ln ( r − r A ) ( r − r A ) , (29)where A and S are constants. The neglected terms in Eq. (29)are of order ( r − r A ) and higher. We note that in non-stationaryideal MHD, r A as defined by the resonant condition ω = ω A ( r A )is a complex quantity because ω is a complex quantity. As aconsequence of this, there is not an actual resonance but a quasi-resonance at the radial position where r = Re( r A ). At the quasi-resonance position, r − r A = − i Im( r A ) (cid:44)
0. The imaginary partof r A owes its existence to the non-zero damping rate, γ . Equa-tion (29) has a term proportional to ( r − r A ) ln ( r − r A ). Thisterm does not vanish at r = Re( r A ) as it would do at r = r A if r A were real. Instead, the logarithm jumps when r − r A crossesthe imaginary axis. In the case of a thin non-uniform layer, γ is small, so that Im( r A ) is small and can be ignored. Then, P (cid:48) ≈ constant as in stationary ideal MHD and as the thin bound-ary approximation assumes (see Hollweg & Yang 1988). Con-versely, when the nonuniform layer is thick the wave is stronglydamped, non-stationarity is important, γ is not small, and Im( r A )is not negligible. Hence, the jump in P (cid:48) . This explains the pres-ence of the jump in compression seen in Fig. 5. Of course, if adissipative process is taken into account, the jumps obtained innon-stationary ideal MHD are replaced by smooth variations innon-stationary dissipative MHD (see e.g., Mok & Einaudi 1985;Ruderman et al. 1995; Wright & Allan 1996; Vanlommel et al.2002; Soler et al. 2012, 2013).On the other hand, by comparing Figs. 2 and 6, we see thatthe components of vorticity have a smaller amplitude in the case Article number, page 6 of 9. Goossens et al.: Transformation of compressive motions into vortical motions
Fig. 3.
Absolute value of the radial (left), azimuthal (centre), and parallel (right) vorticity components in the non-uniform part of a flux tube with l / R = . ρ i /ρ e =
2. The top panels are in linear scale, and the bottom panels are in logarithmic scale. The di ff erent line styles denote k z R = . k z R = . k z R = . Fig. 4.
Same as Fig. 3, but with k z R = . ff erent values of the density contrast: ρ i /ρ e = ρ i /ρ e = ρ i /ρ e =
10 (dashed blue line). of a thick transition. Remarkably, the amplitude of the field-aligned component is more than an order of magnitude smaller.The analytic TTTB approximations, although not strictly validnow, are helpful to understand this result. According to Eq. (20), the parallel component should behave as ( l / R ) − , while both ra-dial and azimuthal components should behave as ( l / R ) − accord-ing to Eqs. (23) and (27). The predicted behaviour in the TTTBapproximation explains the decrease in amplitude of the vorticity Article number, page 7 of 9 & A proofs: manuscript no. 38394final
Fig. 5.
Same as Fig. 1, but with l / R = . components when l / R increases and why the parallel componentdecreases more than the perpendicular components. Finally, wealso note that in Fig. 6 all the components of vorticity jump atthe resonance position, whereas the jumps were not apparent inFig. 2.
6. Conclusions
We used linear non-stationary ideal MHD to investigate the spa-tial behaviour of compression and vorticity for MHD waves thatundergo resonant absorption in the Alfvén continuum. In linearMHD there is no interaction between waves, and the behaviourthat we discussed is associated with a single MHD wave thatlives in the whole space. Pure Alfvén waves and pure magneto-acoustic waves exist in a uniform plasma of infinite extent. Ina non-uniform plasma, the MHD waves combine the propertiesof the classic Alfvén waves and of the magneto-acoustic wavesin a uniform plasma of infinite extent. In a non-uniform plasma,MHD waves propagate both compression and parallel and trans-verse vorticity. The properties of the MHD wave depend on theproperties of the background plasma. As an MHD wave prop-agates through a non-uniform plasma, its properties thereforechange. When an MHD wave moves from a uniform into a non-uniform plasma, it is transformed from a fast magneto-acousticwave into a mixed fast - Alfvén wave.Resonant absorption is a clear example of the phenomenonthat the properties of an MHD wave change when it travelsthrough an inhomogeneous plasma. In the case of resonantAlfvén waves, the MHD wave eventually arrives at a positionwhere it behaves as an almost pure Alfvén wave, but with theunfamiliar property that it has pressure variations. The totalpressure perturbation and compression are non-zero everywhere.The pressure variations are essential for resonant absorption be-cause the amount of absorbed energy and the damping rate aredirectly related to the pressure variations (see e.g. Thompson& Wright 1993; Tirry & Goossens 1995; Andries & Goossens2001; Goossens et al. 2002b; Goossens 2008; Goossens et al.2011; Arregui et al. 2011).Classic Alfvén waves are not the only waves to propagatevorticity from the photosphere to outer space. MHD waves thatundergo resonant absorption in a non-uniform plasma can alsoplay this role.
Acknowledgements.
M.G. was supported by the C1 grant TRACEspace of In-ternal Funds KU Leuven (number C14 / / / AEI / FEDER, UE) and from the Ministerio de Economía, In-dustria y Competitividad and the Conselleria d (cid:48)
Innovacio, Recerca i Turisme delGovern Balear (Pla de ciència, tecnologia, innovació i emprenedoria 2013-2017)for the Ramón y Cajal grant RYC-2014-14970. T.V.D. was supported by theEuropean Research Council (ERC) under the European Union’s Horizon 2020research and innovation programme (grant agreement No 724326) and the C1grant TRACEspace of Internal Funds KU Leuven (number C14 / / References
Andries, J. & Goossens, M. 2001, A&A, 375, 1100Arregui, I., Andries, J., Van Doorsselaere, T., Goossens, M., & Poedts, S. 2007,Astron. Astrophys., 463, 333Arregui, I., Ballester, J., & Goossens, M. 2008a, Astrophys. J. Lett., 676, L77Arregui, I. & Goossens, M. 2019, A&A, 622, A44Arregui, I., Soler, R., Ballester, J. L., & Wright, A. N. 2011, A&A, 533, A60Arregui, I., Terradas, J., Oliver, R., & Ballester, J. 2008b, ApJ, 682, L141Goossens, M. 2008, in IAU Symposium, Vol. 247, IAU Symposium, ed. R. Erdé-lyi & C. A. Mendoza-Briceño, 228–242Goossens, M., Andries, J., & Arregui, I. 2006, Royal Society of London Philo-sophical Transactions Series A, 364, 433Goossens, M., Andries, J., & Aschwanden, M. J. 2002a, A&A, 394, L39Goossens, M., Andries, J., Soler, R., et al. 2012, ApJ, 753, 111Goossens, M., Arregui, I., Ballester, J. L., & Wang, T. J. 2008, A&A, 484, 851Goossens, M., de Groof, A., & Andries, J. 2002b, in ESA Special Publication,Vol. 505, SOLMAG 2002. Proceedings of the Magnetic Coupling of the SolarAtmosphere Euroconference, ed. H. Sawaya-Lacoste, 137–144Goossens, M., Erdélyi, R., & Ruderman, M. S. 2011, Space Sci. Rev., 158, 289Goossens, M., Hollweg, J. V., & Sakurai, T. 1992, Sol. Phys., 138, 233Goossens, M., Ruderman, M. S., & Hollweg, J. V. 1995, Sol. Phys., 157, 75Goossens, M. L., Arregui, I., & Van Doorsselaere, T. 2019, Frontiers in Astron-omy and Space Sciences, 6, 20Hasegawa, A. & Uberoi, C. 1982, The Alfvén wave, Tech. rep., Technical infor-mation Center, U.S. Department of Energy, Springfield, VA, USAHollweg, J. V. 1990a, Computer Physics Reports, 12, 205Hollweg, J. V. 1990b, J. Geophys. Res., 95, 2319Hollweg, J. V. & Yang, G. 1988, J. Geophys. Res., 93, 5423Mok, Y. & Einaudi, G. 1985, Journal of Plasma Physics, 33, 199Montes-Solís, M. & Arregui, I. 2017, ApJ, 846, 89Parker, E. N. 1991, ApJ, 376, 355Pascoe, D. J., Hood, A. W., & Van Doorsselaere, T. 2019, Frontiers in Astronomyand Space Sciences, 6, 22Ruderman, M. S. & Roberts, B. 2002, ApJ, 577, 475Ruderman, M. S., Tirry, W., & Goossens, M. 1995, Journal of Plasma Physics,54, 129Sakurai, T., Goossens, M., & Hollweg, J. V. 1991, Sol. Phys., 133, 227Soler, R. 2017, ApJ, 850, 114Soler, R., Andries, J., & Goossens, M. 2012, A&A, 537, A84Soler, R., Goossens, M., Terradas, J., & Oliver, R. 2013, ApJ, 777, 158Soler, R., Goossens, M., Terradas, J., & Oliver, R. 2014, The Astrophysical Jour-nal, 781, 111Soler, R., Oliver, R., Ballester, J. L., & Goossens, M. 2009, ApJ, 695, L166Thompson, M. J. & Wright, A. N. 1993, J. Geophys. Res., 98, 15541Tirry, W. J. & Goossens, M. 1995, J. Geophys. Res., 100, 23687Tirry, W. J. & Goossens, M. 1996, ApJ, 471, 501Van Doorsselaere, T., Andries, J., Poedts, S., & Goossens, M. 2004, ApJ, 606,1223Vanlommel, P., Debosscher, A., Andries, J., & Goossens, M. 2002, Sol. Phys.,205, 1Wright, A. N. & Allan, W. 1996, J. Geophys. Res., 101, 17399
Article number, page 8 of 9. Goossens et al.: Transformation of compressive motions into vortical motions
Fig. 6.
Same as Fig. 2, but with l / R = ..