astron. astrophys. 354, 334–348 (2000) astronomy …aa.springer.de/papers/0354001/2300334.pdf ·...

Astron. Astrophys. 354, 334–348 (2000) ASTRONOMYAND

ASTROPHYSICS

Phase mixing of Alfven waves in a stratifiedand radially diverging, open atmosphere

I. De Moortel, A.W. Hood, and T.D. Arber

School of Mathematical and Computational Sciences, University of St Andrews, North Haugh, St Andrews, Fife KY16 9SS, Scotland

Received 23 September 1999 / Accepted 8 December 1999

Abstract. Phase mixing was proposed by Heyvaerts and Priest(1983) as a mechanism for heating the plasma in open magneticfield regions of coronal holes. Here the basic model is modifiedto include a gravitationally stratified density and a divergingbackground magnetic field. We present WKB solutions and usea numerical code to describe the effect of dissipation, stratifi-cation and divergence on phase mixing of Alfven waves. It isshown that the wavelengths of an Alfven wave is shortened asit propagates outwards which enhances the generation of gradi-ents. Therefore, the convection of wave energy into heating theplasma occurs at lower heights than in a uniform model. Thecombined effect of a stratified density and a radially divergingbackground magnetic field on phase mixing of Alfven wavesdepends strongly on the particular geometry of the configura-tion. Depending on the value of the pressure scale height, phasemixing can either be more or less efficient than in the uniformcase.

Key words: Magnetohydrodynamics (MHD) – waves – Sun:corona

1. Introduction

The mechanism by which the solar corona is heated is still oneof the major unsolved problems in solar physics. Reviews of thecoronal heating problem have been presented by Narain & Ulm-schneider (1990, 1996), Browning (1991) and Zirker (1993). Inthe open field regions of coronal holes, wave heating mecha-nisms remain the most attractive possibility but like all proposedheating theories, magnetic wave heating depends on the creationof sufficiently small lengthscales in order for dissipation to playan efficient role. Since it was first realised that Alfven waves arenot easily damped, various effects of the propagation of MHDwaves have been investigated. An important property of MHDwaves in an inhomogeneous plasma is that individual surfacescan oscillate with their own Alfven frequency. This implies thata global wave motion can be in resonance with local oscilla-tions on a specific magnetic surface. The resonance conditionis that the frequency of the global motion is equal to the localAlfv en frequency of the magnetic surface. In this way, energy

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is transferred from the large scale motion to the small scale os-cillations, i.e. to a lengthscale where dissipation can become ef-fective. This process of resonant absorption was first suggestedby Ionson (1978) as a mechanism for heating coronal loops.Since this original work, a lot of studies, both numerically andanalytically have been done on resonant absorption (e.g. Goed-bloed & Halberstadt 1994; Halberstadt & Goedbloed 1995a, b;Tirry et al. 1997; Berghmans & Tirry 1997; Tirry & Berghmans1997; Poedts & Boynton 1996).

Heyvaerts and Priest (1983) proposed a simple but promis-ing idea for the behaviour of Alfven waves when the localAlfv en speed varies across the magnetic field lines. They sug-gested damping of Alfven waves due to phase mixing could bea possible source of coronal heating. Basically, phase mixingand resonant absorption are two aspects of the same physicalphenomenon, namely that Alfven waves can exist on individ-ual flux surfaces. Examples of the close interplay between therelated phenomena of phase mixing and resonant absorptioncan be found in e.g. Ruderman et al. (1997a, 1997b). How-ever, in this paper we will not consider resonant absorption andconcentrate on damping of Alfven waves due to phase mixing. The propagation and damping of shear Alfven waves in aninhomogeneous medium has been studied in more detail (Ire-land 1996; Cally 1991; Browning & Priest 1984; Nocera etal. 1984) by relaxing the Heyvaerts and Priest limits of weakdamping and strong phase mixing. Recently, Hood et al. (1997a,1997b) have found analytical, self-similar solutions describingphase mixing of Alfven waves in both open (coronal holes)and closed (coronal loops) magnetic configurations. Possibleobservational evidence of coronal heating by phase mixing isdiscussed by Ireland (1996). Numerical simulations of phasemixing in coronal holes have been performed by Poedts et al.(1997) who found that in coronal holes, the phase mixing ofAlfv en waves is speeded up by the flaring out of the magneticfield lines. Ofman & Davila (1995) found that in an inhomo-geneous coronal hole with an enhanced dissipation parameter(S = 103 − 104), the Alfven waves dissipate within severalsolar radii and can provide significant energy for the heatingand acceleration of the high-speed solar wind. Ruderman etal. (1998) considered phase mixing of Alfven waves in planartwo-dimensional open magnetic configurations, using a WKBmethod. However, the validity of the WKB technique requires

I. De Moortel et al.: Phase mixing of Alfven waves in a stratified and open atmosphere 335

a particular relationship between the magnetic Reynolds num-ber, the wavelength of the basic Alfven wave and the coronalpressure scale height. Nakariakov et al. (1997) considered thenon linear generation of fast magnetosonic waves by Alfvenwave phase mixing and showed that transversal gradients in theAlfv en wave, produced by phase mixing, lead to the generationof propagating fast waves which are subject to strong damping.This phenomenon may be considered as indirect heating of thecoronal plasma by phase mixing. The propagation of magneto-hydrodynamic waves in a cold plasma with an inhomogeneousste ady flow directed along a straight magnetic field was stud-ied by Nakariakov et al. (1998). They found that in regionswith transversal gradients in the steady flow, phase mixing ofAlfv en waves takes place similarly to classical phase mixing ina static medium with an inhomo geneity in the Alfven speed.Non linear effects on the dissipation of Alfven waves have beendiscussed by Boynton & Torkelsson (1996), and Torkelsson &Boynton (1998) for spherical Alfven waves, who found that nonlinear Alfven waves can steepen to form current sheets whichenhance the dissipation rate of the Alfven waves by several or-ders of magnitude. This result was confirmed by Nakariakov etal. (1999a) who describe the dynamics of non linear, spherical,linearly polarised small amplitude Alfven waves in the strat-ified and dissipative plasma of coronal holes by the sphericalscalar Cohen-Kulsrud-Burgers equation. From the analysis ofthis equation it was found that linearly polarised Alfven wavesof weak amplitude and relatively long periods are subject to nonlinear steepening and efficient non linear dissipation. Narain &Sharma (1998) found that in the non linear regime the viscousdamping of Alfven waves becomes a viable mechanism of solarcoronal plasma heating when strong spreading of the magneticfield is taken into account. From recent TRACE observations,Nakariakov et al. (1999b) estimate the coronal dissipation co-efficient to be eight or nine orders of magnitude larger than thetheoretically predicted value. This larger dissipation coefficientcould solve some of the existing difficulties with wave heatingand reconnection theories.

In this paper we aim to study the effect of both vertical andhorizontal density stratifications on the phase mixing of Alfvenwaves in an open and (radially) diverging magnetic atmosphere.We restrict ourselves to a study of travelling waves, generatedby photospheric motions that cause disturbances to propagateoutwards from the Sun without total reflection.

In Sect. 2 we describe the basic equilibrium and equations.For simplicity we consider the scale height to be infinity andin Sect. 3 we discuss the effect of the radially diverging back-ground magnetic field on phase mixing of Alfven waves in theabsence of dissipation. In Sect. 4 we add dissipation to our basicmodel. In Sect. 5 we look at the combined effect of the verticalstratification of the density and the divergence of the backgroundmagnetic field, while Sect. 6 contains the discussion and con-clusion.

2. Equilibrium and linearised MHD equations

We consider an inhomogeneous density with an associated in-homogeneous Alfven speed. If we assume that the backgroundAlfv en speed (only) has variations in the horizontal direction,then Alfven waves on neighbouring field lines, driven with thesame frequency, will have different wavelengths. This will causethem to become out of phase as they propagate up in height and,therefore, large transverse gradients will build up. In this way,short lengthscales are created which means dissipation eventu-ally becomes important and allows the energy in the wave todissipate and heat the plasma. To study the effect of a radiallydiverging background magnetic field on phase mixing of Alfvenwaves, spherical coordinates will be the most convenient choiceand we therefore set up the basic equilibrium in spherical coor-dinates.

Assuming a low-β-plasma and an isothermal atmosphere,i.e. T0 uniform, the equilibrium is expanded in powers ofβ.Following Del Zanna et al. (1997), the leading order solution

is a radially diverging field,B0 = B0r20

r2 r, whereB0 is thesurface field strength andr0 is the solar radius. Note that wecannot assume a uniform field because of flux conservation,which shows thatr2Br has to be constant. At orderβ the radialmagnetohydrost atic force balance equation reduces to

∂p0

∂r= −p0r0

H

1r2,

with r = r0r and, therefore,

p0 = p0(θ) exp(

−r0H

(1 − 1

r

)),

and

ρ0 = ρ0(θ) exp(

−r0H

(1 − 1

r

)),

whereH = RTµg is the pressure scale height. The force

balance in theθ-direction determines the finiteβ correction tothe magnetic field (Del Zanna et al. 1997).

We now analyse Alfven waves by considering perturba-tions vφ and bφ in the velocity and the magnetic field. As-suming a time dependence of the formexp(iΩt) for both theperturbed magnetic fieldB1 = b(θ, r)eiΩtφ and the velocityv= v(θ, r)eiΩtφ, the linearised MHD equations become:

iΩρ0v =1µB0

r20r3

∂

∂r(rb) + ρ0ν∇2v ,

and

iΩb = B0r20r

∂

∂r

(vr

)+ η∇2b ,

where the magnetic diffusivityη and the dynamic viscosityρ0ν only depend on the temperature (Priest 1982) and in anisothermal atmosphere are assumed constant. The Alfven speedis given by

v2A(r, θ) = v2

A(θ)1r4

exp(r0H

(1 − 1

r

)), (1)

336 I. De Moortel et al.: Phase mixing of Alfven waves in a stratified and open atmosphere

wherev2A(θ) = B2

0/µρ0(θ). These equations can be combinedto give either one single equation for the perturbed magneticfield b,

b+v2

A(θ)Ω2

r40r

∂

∂r

(e

r0H (1− 1

r ) 1r4

∂

∂r(rb)

)+ i

η

Ω∇2b = 0 , (2)

where we neglect the dynamic viscosityρ0ν, or to give an equa-tion for the perturbed velocityv,

Ω2

v2A(θ)

e− r0H (1− 1

r )v +r40r3

∂2

∂r2

(vr

)+ i

ρ0νΩµB2

0∇2v = 0 , (3)

where we neglect the resistivityη. Including both dissipationterms increases the order of the equations and obscures the phys-ical effects of each term (see De Moortel et al. 1999).

As it will be convenient to use dimensionless variables forthe numerical solutions to Eqs. (2) and (3), we setr = r0r,θ = θ0θ and

v2A(θ) = v2

A0(1 + δ cos

(mπθ

))= v2

A0v2A(θ) , (4)

whereδ regulates the magnitude of the equilibrium density vari-ations in theθ-direction for given radial disctancer, andm isthe number of density inhomogeneities inside the coronal hole.This could be, for example, the number of coronal plumes. Theequation for the perturbed magnetic field then becomes

b +(

λ0

2πr0

)2v2

A

Ω2

1r

∂

∂r

(e

1H (1− 1

r ) 1r4

∂

∂r(rb)

)

+ iΛ2

η

Ω

(θ20r2

∂

∂r

(r2∂b

∂r

)

+1

r2 sin(θ0θ)∂

∂θ

(sin(θ0θ)

∂b

∂θ

)

− θ20b

r2 sin2(θ0θ)

)= 0, (5)

where

λ0 = 2πvA0

Ω0, (6)

is the basic wavelength,H = H/r0 is the pressure scale heightmeasured in units of the solar radius andΛ2

η = ηΩ0r2

0θ20

. Similarlywe can rewrite the equation for the perturbed velocity as

Ω2

v2A

e− 1H (1− 1

r )v +(

λ0

2πr0

)2 1r3

∂2

∂r2

(vr

)

+ iΛ2νΩ

(θ20r2

∂

∂r

(r2∂v

∂r

)

+1

r2 sin(θ0θ)∂

∂θ

(sin(θ0θ)

∂v

∂θ

)

− θ20v

r2 sin2(θ0θ)

)= 0, (7)

whereΛ2ν = ρ0νΩ0µ

B20

(λ0

2πr0θ0

)2.

From now on we will drop the barred variables and work interms of dimensionless variables. We consider either the ohmicheating, and solve Eq. (5) withΛ2

η = η/Ω0r20θ

20, or the viscous

heating, and solve Eq. (7) withΛ2ν = ρ0νΩ0µ

B20

(λ0

2πr0θ0

)2.

The boundary conditions are chosen as

v = 0, θ = 0 andθ = 1, (8)

v = sinπθ, r = 1, (9)

on the photospheric base and an outward propagating wave onthe upper boundary. To obtain this upper boundary conditionwe assume that the density remains constant and the dissipationnegligible outside the computational box. The solution corre-sponding to an outward propagating wave is

v ∼ exp(

−ik(θ)e− 1H (1− 1

rmax) r

3 − 13

),

and matchingv and ∂v∂r onto the solution inside the computa-

tional box gives

∂v

∂r= −ik(θ)e− 1

H (1− 1r )r2v, r = rmax, (10)

where

k(θ) = (1 + δ cos(mπθ))−1/2 . (11)

When dissipation is included and the height of the numericalbox, i.e.rmax, is taken sufficiently large, the waves are dampedand the actual choice of the upper boundary condition is unim-portant.

The spherical geometry allows the effect of flux tube area di-vergence to be studied. We can retrieve the non-diverging Carte-sian case in the following manner. If we assume

r = 1 +λ0

2πr0z and x = rθ , (12)

we see that at low heights, i.e.r ≈ 1, and for small initialwavelengthsλ0, Eqs. (5) and (7) transform to

b+v2

A

Ω2

∂2b

∂z2 + iΛ2

η

Ω

(x2

0

z20

∂2b

∂z2 +∂2b

∂x2

)= 0 , (13)

and

Ω2

v2A

v +∂2v

∂z2 + iΛ2νΩ(x2

0

z20

∂2v

∂z2 +∂2v

∂x2

)= 0 , (14)

with Λ2η = η/Ω0x

20 andΛ2

ν = ρ0νµΩ0/B20

(z0x0

)2wherex0 =

r0θ0 andz0 = λ0/2π with λ0 as in Eq. (6). As we expect thatthe dominant damping terms are the second order derivativesin Eqs. (5) and (7), we dropped the other damping terms. So, atlow heights and for smallλ0, we recover the standard Heyvaertsand Priest case (Heyvaerts & Priest 1983).

We note here that we are looking at torsional Alfven waves,i.e. vr = 0, vθ = 0, vφ /= 0 and ∂

∂φ = 0, in open field re-gions. Examples of studies of torsional Alfven waves in closed


field regions such as coronal loops and arcardes, can be foundin Ruderman et al. (1997a, 1997b). Furthermore, we remarkthat the choice of analysing torsional Alfven waves excludes allpossible coupling with slow and fast magneto-acoustic waves.Coupling of the different types of MHD waves and more com-plicated classes of motions are considered by e.g. Nakariakov etal. (1997, 1998), Berghmans & Tirry 1997; Tirry & Berghmans1997; Ruderman 1999.

3. Diverging magnetic field and no gravitationalstratification: no dissipation

To understand each physical effect clearly, we first examine theeffect of a diverging field on the phase mixing of Alfven waves.Hence, we eliminate gravitational effects by setting the pressurescale height to infinity, i.e.1/H = 0. In the rest of this paper, weassume that the plasma is structured in theθ-direction. Whenreferring to stratification, we mean radial stratification due togravity while a diverging atmosphere refers to the area changedue to spherical geometry.

If we neglect dissipation, i.e.Λ2 = 0, the solution for themagnetic field and velocity perturbations are given by

b = sin(πθ)(r

r0

)3/2

J5/6

(2π r0

λ0k(θ) r3

3

)− iY5/6

(2π r0

λ0k(θ) r3

3

)J5/6

(2π3

r0λ0k(θ)

)− iY5/6

(2π3

r0λ0k(θ)

) , (15)

and

v = sin(πθ)(r

r0

)3/2

J1/6

(2π r0

λ0k(θ) r3

3

)− iY1/6

(2π r0

λ0k(θ) r3

3

)J1/6

(2π3

r0λ0k(θ)

)− iY1/6

(2π3

r0λ0k(θ)

) , (16)

wherek(θ) = ΩvA(θ) andJ is a Bessel function of order either

56 or 1

6 .The ideal MHD solutions in spherical coordinates may be

approximated by a simple WKB solution (see Appendix) of theform

b = sin(πθ) exp(

−2πir0λ0k(θ)ψ

)and

v = sin(πθ) exp(

−2πir0λ0k(θ)ψ

), (17)

where3ψ = r3 − 1. Although the exact analytical solutions forthe perturbed magnetic field and velocity differ, the approximateWKB solutions are both the same since the leading asymptoticterms of the Bessel functions agree on using the large argu-ment approximationJν(φ) ∼√2/πφ cos(φ− 1

2νπ− 14π) and

Yν(φ) ∼√2/πφ sin(φ− 12νπ− 1

4π) (Abramowitz & Stegun).The solutions (15) and (16) show that area divergence decreases

the wavelength in agreement with the numerical solutions shownin Figs. 1 and 2. Figs. 2(a) and 2(b) confirm that the solutionsfor the perturbed magnetic field and the perturbed velocity arethe same as predicted by Eq. (17).

WhenΛ2 = 0, Eq. (14) becomes the standard wave equationfor phase mixing in a Cartesian, non-dissipative system, i.e.

v +v2

A

Ω2

∂2v

∂z2 = 0 . (18)

Therefore, the solutions for the perturbed velocity and magneticfield in Cartesian coordinates are given by

v = sin(πx) exp(−ik(x)z)and

b = sin(πx) exp(−ik(x)z) , (19)

wherek(x) = ΩvA(x) .

From Fig. 2(c) we see that low down, i.e. nearr ≈ 1, andfor small initial wavelengthsλ0 (see Eq. (6)), the spherical andthe Cartesian case indeed agree extremely well. The results inFig. 1 show clearly that, unlike gravitational stratification whichlengthens the wavelengths, area divergence shortens the wave-lengths while the wavelengths remain constant in the Cartesiancase. Indeed, in this case the Alfven speed and the wavelengthλ behave like1

r2 . The amplitude of both the perturbed magneticfield and the perturbed velocity are constant in height as weexpected. Wright & Garman (1998) and Torkelsson & Boynton(1998) showed that in the large wavenumber limit the ampli-tudes of the Alfven waves behave asb ∼ ρ

1/40 andv ∼ ρ

−1/40

and asρ0 is constant with height in this case, the result fol-lows. This suggests that phase mixing will be more efficientin a diverging medium than in a non diverging medium as theshort length scales, necessary for efficient dissipation, will becreated much faster. Therefore, heat could now be deposited atlower heights. Similar results were obtained by Ruderman et al.(1998). These results also show that, unlike the results due tostratification, the effect of the divergence of the background fieldis the same whether resistive or viscous heating is considered.

To obtain an estimate of where this heat would be depositedif dissipation were included, we now consider the current den-sity, j2. In spherical coordinates, only including the dominantterms,j2 is given by

j2 =(

1r

∂b

∂θ

)2

+(∂b

∂r

)2

=1

r20θ20

(1r

∂b

∂θ

)2

+1r20

(∂b

∂r

)2

, (20)

while in Cartesian coordinates,j2 is given by

j2 =(∂b

∂x

)2

+(∂b

∂z

)2

=1x2

0

(∂b

∂x

)2

+1z20

(∂b

∂z

)2

, (21)

wherex0 = r0θ0 andz0 = λ0/2π. The numerical results ob-tained for both the magnetic field and the velocity indicate that


Fig. 1. (left) A contour plot of the perturbed magnetic field in a diverging atmosphere, withδ = 0. (right) The behaviour of the wavelength withheight (atθ = 0.5) with δ = 0. The solid line is the solution for a diverging atmosphere, the dashed line corresponds to the Cartesian case.

the behaviour of the current density and the vorticity will besimilar. Therefore we concentrate on the current density alone.

From Fig. 3(a) we see that even when there is no phase mix-ing, i.e. δ = 0, the current density builds up very rapidly ina diverging magnetic field. In the non-diverging atmosphere,i.e. the Cartesian case, this current density remains constant asgradients in the vertical or horizontal direction do not build up.As there is no phase mixing, the growth inj2 in the divergingatmosphere is entirely due to the flaring out of the backgroundfield lines, i.e. the radial derivatives are building up. When weinclude phase mixing, i.e.δ > 0, we see from Fig. 3(b) that thebuild up of the current density,j2, increases whenδ increases.However, when the magnetic field is diverging, the effect ofincreasingδ is larger than in the Cartesian case. Indeed, fromcomparing Eqs. (17) and (19), we see that a change inδ causesa bigger change in the sphericalθ-derivatives than in the Carte-sianx-derivatives. Atθ = 0.5 both the sphericalr-derivativesand the Cartesianz-derivatives stay the same when the phasemixing parameterδ changes.

Fig. 4(a) shows the change in the current densityj2 whenwe change the initial wavelengthλ0, through changes to eitherthe frequencyΩ0 or the background Alfven velocityvA0. Sinceλ02π = vA0

Ω0(Eq. (6)), doublingλ0 has the same effect as doubling

vA0 or halvingΩ0 and so on. Therefore we concentrate on theeffect of varying just the one parameterλ0. Changingλ0 alsocausesz0 to change asz0 = λ0

2π . We notice thatj2 starts off witha higher initial value when the initial wavelengthλ0 is smaller,which is clear from Eqs. (17) and (19). We still see that thecurrent densityj2 builds up faster in the spherical case and eventhat the difference between the spherical and the Cartesian caseis more pronounced as the initial wavelengthλ0 gets smaller.From Eqs. (17) and (19) we see that all derivatives, apart fromthe Cartesianz-derivative, contain a1

λ0factor. From Eq. (21),

we see that, in the Cartesian case, the dimesionless variablesintroduce a factor1

λ20

in front of the square of thez-derivative.

These means that in both the spherical and Cartesian case,j2 ∼1λ2

0which implies that the difference between the spherical and

Cartesian results gets 4 times bigger when we halve the initial

wavelengthλ0. From Fig. 4(b) we see that the current densitybuilds up higher and that the difference between the sphericaland the Cartesian case gets larger as we decrease the value ofthe parameterθ0. However, from Eqs. (17) and (19) it is clearthat all the derivatives remain unchanged when we varyθ0. Theparameterθ0 only appears in expression (20) and (21) for thecurrent densityj2. We see that the sphericalθ-derivatives andCartesianx-derivatives are multiplied with a factor1

θ20

or thatthese derivatives will become 4 times larger when we halveθ0.Again this implies that the difference between the spherical andCartesianj2 will get larger asθ0 gets smaller.

From studying the non-dissipative case, we expect phasemixing to be enhanced when we increase the phase mixing pa-rameterδ or decrease either the intial wavelengthλ0 or theparameterθ0. We also expect more heat to be deposited into theplasma when we consider a diverging magnetic field comparedto the Heyvaerts and Priest model in Cartesian coordinates. Wenow want to examine if these effects remain the same when weinclude dissipation in the system.

4. Diverging magnetic field and no gravitationalstratification: non-zero dissipation

In this section dissipation is included, i.e.Λ2 /= 0, but gravita-tional stratification is neglected and numerical results to Eqs. (5)and (7) are presented. We use the same numerical code as DeMoortel et al. (1999). We note here that, altough all the resultspresented in this paper are numerical, the WKB solutions ob-tained in the Appendix give very good agreement in all casesconsidered and are useful in understanding the behaviour of thecomputational results.

Only considering resistivity, the WKB solution (see Ap-pendix) for the perturbed magnetic field in spherical coordi-nates, including the dominant second order damping terms isgiven by

b = sin(πθ) exp(

−2πi(r0λ0

)k(θ)ψ

)


Fig. 2a–c. A cross-section of the perturbeda magnetic field andbvelocity for a radially diverging background magnetic field atθ = 0.5with δ = 0.5, λ0 = 0.1 and θ0 = 0.1. c A cross-section of theperturbed magnetic field atθ = 0.5 with δ = 0.5, λ0 = 0.005 andθ0 = 0.1.The dotted lines represent the corresponding solutions inCartesian coordinates atx = 0.5.

exp

(−1

2

(2πr0λ0

)3

k3 Λ2η

Ω

[θ20r7 − 1

7+

19k′2

k2(r7

7− r4

2+ r − 9

14

)]), (22)

Fig. 3a and b.A cross-section of the current densityj2 for a divergingbackground magnetic field withλ0 = 0.1 andθ0 = 0.1 at θ = 0.5for a δ = 0.0 b different values ofδ (solid line: δ = 0.5, dashedline: δ = 0.1) The thin lines represent the corresponding solutions inCartesian coordinates atx = 0.5.

with 3ψ = r3 − 1, Λ2η = η

Ω0r20θ2

0andk(θ) as defined in

Eq. (11). Considering viscosity, the solution for the perturbedvelocity becomes

v = sin(πθ) exp(

−2πi(r0λ0

)k(θ)ψ

)

exp

(−1

2

(2πr0λ0

)3

kΛ2νΩ[θ20r7 − 1

7+

19k′2

k2(r7

7− r4

2+ r − 9

14

)]), (23)

with Λ2ν = ρ0νΩ0µ

B20

(λ0

2πr0θ0

)2.

Again only including the dominant damping terms, corre-sponding non-diverging WKB solutions in Cartesian coordi-nates are given by

b = sin(πx) exp(−ikz)

exp

(−1

2Λ2

η

Ωk3x

20

z20z − 1

6Λ2

η

Ωk′2kz3

), (24)


Fig. 4a and b.A cross-section of the current densityj2 for a divergingbackground magnetic field withδ = 0.5 atθ = 0.5 for a θ0 = 0.1 anddifferent values ofλ0 (solid line:λ0 = 0.2, dot-dashed line:λ0 = 0.1,dashed line:λ0 = 0.05) and b λ0 = 0.1 and different values ofθ0 (solid line: θ0 = 0.2, dot-dashed line:θ0 = 0.1, dashed line:θ0 = 0.05). The thin lines represent the corresponding solutions inCartesian coordinates atx = 0.5.

with Λ2η = η

Ω0x20

and

v = sin(πx) exp(−ikz)exp

(−1

2Λ2

νΩkx2

0

z20z − 1

6Λ2

νΩk′2

kz3), (25)


B20

(z0x0

)2.

From Fig. 5, we see that including dissipation does notchange the effect of an initially diverging magnetic field on theperturbed magnetic field. When the background field is radiallydiverging, wavelengths get shorter as the waves travel outwardfrom the solar surface. This enhances the overall damping of thewave amplitudes and therefore we expect heat to be depositedinto the plasma at lower heights in a diverging atmosphere. Fromcomparing Figs. 5(a) and 5(b) we see again that the perturbedmagnetic field and the perturbed velocity behave similarly whenthe resistive damping coefficientΛ2

η and the viscous dampingcoefficientΛ2

ν have the same value. However, we have to re-mark here that this is only the case atθ = 0.5 (or x = 0.5 in

Fig. 5a–c. A cross-section of the perturbeda magnetic field andbvelocity for a radially diverging background magnetic field atθ = 0.5with Λ2 = 10−4, δ = 0.5 andλ0 = 0.1. c A cross-section of theperturbed magnetic field atθ = 0.5 with Λ2 = 10−4, δ = 0.5 andλ0 = 0.01. The dotted lines represent the corresponding solutions inCartesian coordinates atx = 0.5.

the Cartesian case). At other values ofθ there is a slight differ-ence between the damping rate of the perturbed magnetic fieldand the perturbed velocity as Eqs. (22) and (23) show that forthe magnetic field the damping term is proportional tok3/Ωwhile for the velocity the damping term is proportional tokΩ.


Fig. 6a and b.A contour plot of the current densityj2 with Λ2 = 10−4,δ = 0.5,λ0 = 0.1andθ0 = 0.1 for aa diverging background magneticfield andb a non-diverging background magnetic field.

Fig. 5(c) confirms that at low heights, i.e.r ≈ 1, and for smallinitial wavelengthsλ0 there is a very good agreement betweenthe spherical and the Cartesian case.

We now again look at the current densityj2 and the vorticityω2 to find out how the heat is deposited into the plasma throughohmic or viscous dissipation. From what we know about theno dissipation case and from the behaviour of the perturbedmagnetic field, we expect strong current densities to build upwhen we consider a background field with radially divergingfield lines.

Fig. 6 (a) and (b) are contour plots of the current densityj2 with a diverging and non-diverging background magneticfield. This figure shows clearly that the current density is con-centrated at lower heights in the radially diverg ing geometry.However, although the maximum of the current density occursat lower height in the spherical case, its value is less than thecorresponding Cartesian case. Due to the combination of strongphase mixing and the shortening of the length scales caused bythe divergence of the background magnetic field, the perturbedmagnetic field is damped more quickly.

Fig. 7 (a) shows the behaviour of the current densityj2 fordifferent values of the phase mixing parameterδ in both thespherical and Cartesian geometry. We see that when phase mix-

Fig. 7a. A cross-section of the current densityj2 with Λ2 = 10−4

at θ = 0.5 for different values of delta (solid line:δ = 0.5, dashedline: δ = 0.2). The thin lines represent the corresponding solutions inCartesian coordinates atx = 0.5.

ing is weak, the results are the same as in the non-dissipativecase. The current density builds up to a higher maximum, at alower height when the background magnetic field is radially di-verging. However, when phase mixing is stronger, i.e.δ = 0.5,we see again that the maximum of the current density is situ-ated at a lower height in the spherical case but reaches a highermaximum in the Cartesian case. The vorticityω2 behaves in asimilar manner to the current density so that the deposition ofheat into the plasma will occur at similar heights, whether weconsider ohmic or viscous dissipation. Therefore we concen-trate on the current densityj2 to see the effect of changing theplasma parameters.

Fig. 8(a) shows the variation in the current densityj2 whenwe change the initial wavelengthλ0. Changingλ0 causes thedamping coefficientΛ2 to change asΛ2 ∼ λ0. The resistiv-ity η is kept the same for all cases. From Fig. 8(a) we seein both the spherical and the Cartesian case that the currentdensity j2 builds up higher and quicker. Indeed, by startingoff with a smaller initial wavelength, the small length scalesneeded for dissipation to be effective are created faster, i.e.the current density will build up at lower heights. However,although initially j2 builds up faster and the maxima occurat lower heights in the spherical case, the maxima are smallerthan in the corresponding Cartesian case. The current densityj2 is dominated by the transverse derivatives. In the sphericalcase, we see that1r

∂b∂θ ∼ 1

rr3−1

3 e−r7/7 while in the Cartesian

case,∂b∂x ∼ ze−z3/6 (see Eqs. (22) and (24)). So although the

sphericalθ-derivatives initially build up asr2, the damping terme−r7/7 is significantly stronger thane−z3/6 explaining whyj2

initially builds up faster and why the maxima are situated atlower height and are less high in the spherical case. Fig. 8(b)shows the change in the current densityj2 when we changethe parameterθ0. Again this causes other parameters to changeas well. In this case the damping coefficientΛ2 andx0 willchange asΛ2 ∼ θ−2

0 andx0 = r0θ0. The effect of changing


Fig. 8a and b.A cross-section atθ = 0.5 of the current densityj2 fora different values ofλ0 (solid line:λ0 = 0.2, dot-dashed line:λ0 =0.1, dashed line:λ0 = 0.05) andb different values ofθ0 (solid line:θ0 = 0.2, dot-dashed line:θ0 = 0.1, dashed line:θ0 = 0.05). The thinlines represent the corresponding solutions in Cartesian coordinates atx = 0.5.

θ0 is largely the same as in the zero dissipation case. The cur-rent density builds up stronger at lower heights and again themaxima are higher in the Cartesian case.

Fig. 9 describes the height (in solar radii) at which the max-imum of the current density would occur for different initialwavelengthsλ0. This figure confirms that the maxima are sit-uated lower down in the spherical case but also shows that thedifference gets smaller as the initial wavelengthλ0 gets smaller.If we look back at expression (12), we see that for smallλ0, thespherical and the Cartesian geometry give the same results. Asthe maximum ofj2 is situated lower down asλ0 gets smallerwe can conclude that higher initial frequencies or a lower back-ground Alfven speed will cause the deposition of heat into theplasma to occur at lower heights.

The total ohmic heating in the spherical (η∫ rmax

1 j2r2dr)and the Cartesian (η

∫ zmax

0 j2dz) case, for certain values ofθandx respectively, agree very well. This suggests that simi-lar amounts of heat will be deposited through ohmic dissipa-tion into the plasma in both cases and that the total amount ofheat deposited does not depend on the geometry of the back-ground magnetic field but only on the Poynting flux of magnetic

Fig. 9a.The height in solar radii at which the maximum of the currentdensity is situated for different initial wavelengthsλ0. The dotted linerepresents the corresponding solution in Cartesian coordinates.

energy through the photospheric base in response to footpointmotions. As expected, we also notice that the total ohmic heat-ing η

∫ rmax

1 j2r2dr does not depend onλ0 or θ0 (or z0 andx0in the Cartesian case).

Overall, we can conclude that a diverging background mag-netic field enhances phase mixing of Alfven waves in the sensethat wavelengths get shorter as the waves propagate up makingthe process of phase mixing more efficient as the small length-scales needed for dissipation will build up lower down com-pared to a non-diverging atmosphere. A similar conclusion canbe found in the Ruderman et al. (1998) solution for a uniformdensity in the vertical direction and an exponentially divergingmagnetic field. We cannot make a direct comparison as we con-sidered a truly open magnetic field, meaning that at no pointdo the magnetic field lines connect back to the solar surface.However, we do reach the same conclusion that wave dampingdue to phase mixing will be faster in a diverging atmosphere.

5. Gravitational stratification in a diverging atmosphere

Gravitational stratification was shown by De Moortel et al.(1999) to inhibit phase mixing but the results of Sect. 4 of adiverging magnetic field indicate an enhancement of energy dis-sipation. In this section we investigate the effect of both grav-itational stratification of the density and a radially divergingbackground magnetic field. Therefore we solve Eqs. (5) and (7)with a finite scale heightH. Again considering either resistivityor viscosity and including the dominant second order deriva-tives in the damping term, the WKB solutions (see Appendix)for the perturbed magnetic field and velocity are given by

b = sin(πθ) exp(

−i2πr0λ0

k(θ)R)

exp

[−14H

(1 − 1

r

)− 1

2Λ2

η

Ω

(2πr0λ0

)3

k3


∫ r

1

(θ20r

6e− 32H (1− 1

r ) +k′2

k2 R2e− 1

2H (1− 1r ))dr

], (26)

with R =∫ r

1 r2e− 1

2H (1− 1r )dr andΛ2

η = ηΩ0r2

0θ20

and,

v = sin(πθ) exp(

−i2πr0λ0

k(θ)R)

exp

[1

4H

(1 − 1

r

)− 1

2Λ2

νΩ(

2πr0λ0

)3

k

∫ r

1

(θ20r

6e− 12H (1− 1

r ) +k′2

k2 R2e

12H (1− 1

r ))dr

], (27)


B20

(λ0

2πr0θ0

)2. In the limitH → ∞, these

solutions agree with solutions (22) and (23) for a radially diverg-ing background magnetic field without stratification. By settingr = 1 + λ0

2πr0z andx = rθ, we recover the Cartesian solutions

b = sin(πx)e− z4Hc exp(−ikZ)

exp

(−1

6Λ2

η

Ωk3[x2

0

z202Hc(1 − e−3z/2Hc) +

k′2

k2 Z3])

, (28)

with Z = 2Hc

(1 − e

z2Hc

)andΛ2

η = ηΩ0x2

0and

v = sin(πx)e−z4Hc exp(−ikZ)

exp(

−12Λ2

νΩk[x2

0

z20Z − k′2

k2 (2Hc)3(z

Hc− 2 sinh

(z

2Hc

))]), (29)


B20

(z0x0

)2. Compared with the results in

De Moortel et al. (1999), these solutions have an extra damp-ing term. The extra term arises from the need to include bothsecond order derivatives in the damping term when consideringspherical geometry rather than just the transverse derivatives.However, the contribution from this extra term is almost negli-gible in the Cartesian limit.

5.1. No dissipation

From Fig. 10 we see that the gravitational stratification of thedensity has a very strong influence on the behaviour of both theperturbed magnetic field and velocity. Rather than staying con-stant, as in the radially diverging atmosphere, the amplitude ofthe magnetic field decreases with height due to the stratification.The amplitude of the velocity on the other hand, increases withheight. Indeed, as mentioned earlier,b ∼ ρ

1/40 andv ∼ ρ

−1/40

(Wright & Garman 1998) and as the density decreases withheight in a stratified atmosphere, the amplitude ofb will de-crease with height andv will increase.

The Alfven speedvA and the wavelengthλ now behave as1r2 e

12H (1− 1

r ) in the diverging case and asez

2Hc in the Cartesiancase. This means that the wavelength will increase everywherein the Cartesian case, as we see from Fig. 11(b) but that will

Fig. 10a and b.A cross-section of the perturbed magnetic field andvelocity for a radially diverging background magnetic field atθ = 0.5with Λ2 = 0.0, δ = 0.5 andλ0 = 0.1 for H = 0.2. The dotted linesrepresent the corresponding solution forH = ∞.

only increase everywhere forr < rmax = 1/4H in the di-verging case. This is clearly seen in Fig. 11(a). The wavelengthonly increases for all values ofr for H = 0.1. WhenH > 1

4 ,the wavelengths will decrease everywhere, which in Fig. 11(a)happens forH = ∞ andH = 0.5. For 1

4rmax< H < 1

4 ,the wavelengths will increase till they reach the turning pointr = 1

4H and then decrease. Indeed, forH = 0.2, λ initiallyincreases tillr = 1.0/(4H) = 1.25 and then decreases.

From Fig. 12, it is clear that gravitational stratification in-troduces dramatic changes. When we look at the results for theohmic heatingj2 we retrieve the effects found when studyingeither a purely stratified or a diverging atmosphere. Despite thefact that the amplitude ofb decreases due to the density stratifi-cation,j2 still builds up for most values of the scale heightH,due to the shorter wavelengths caused by the area divergenceof the background magnetic field. It is only when stratificationis very strong thatj2 decreases. We also see thatj2 builds upstronger in the diverging atmosphere that in the Cartesian case,a result already noted in the case without gravitational stratifi-cation of the density. When the value of the initial wavelength


Fig. 12.A cross-section of the (left) current density and (right) vorticity for a (top) radially diverging and (bottom) uniform background magneticfield atθ = 0.5 with λ0 = 0.1 for different values of the scale-height (solid line:H = ∞, dot-dashed line:H = 0.5, dashed line:H = 0.2,dotted line:H = 0.1).

λ0 is decreased, we find that the current density reaches highervalues, as found in the purely diverging atmosphere.

The results are quite different when considering the viscousheatingω2. We see that the vorticity builds up higher than thecorresponding current density and that the effect of changing thescale height is reduced. This different behaviour is due to theincrease of the velocity amplitude, a result which we also foundin the purely stratified atmosphere. The effect of the changingthe value of the initial wavelength is the same for the currentdensity and the vorticity. If we analyse the results for differentgeometries, we see that, unlike the current density, the vorticitybuilds up higher in the Cartesian case. We also notice that whilethe vorticity decreases as the scale height is increased in thediverging atmosphere, the opposite happens in the Cartesiancase. We see that, as expected, the (Cartesian) vorticity initiallybuilds up less high as the stratification increases. This is due tothe lengthening of the wavelengths caused by the stratificationand is in agreement with previous results. However, we see thatvery quickly the vorticity reaches higher values for strongerstratification due to the extremely rapid increase of the velocityamplitude caused by the radially decreasing density. The effectof changing the initial wavelengthλ0 is nevertheless maintained.Decreasing the initial wavelength causes the vorticity to start of

with a higher initial value and to reach higher values as thewaves propagate up.

5.2. Gravitationally stratified, diverging atmosphere,non-zero dissipation

Figs. 13 and 14 show that including dissipation gives familiarresults for the behaviour of the perturbed magnetic field. Wesee that, in both the spherical and the Cartesian case, the mag-netic field initially decays faster when we include gravitationalstratification. But, overall the damping rate is reduced in a strat-ified atmosphere. For weak gravitational stratification the radialdivergence of the background magnetic field still causes thewaves to dissipate faster in the spherical case compared to theCartesian case. We notice an initial increase in the amplitudeof the velocity in the stratified plasma which is the remnant ofthe amplitude increase of perturbed velocity noted in the zerodissipation case. We also see that the differences between thevelocity results for the atmosphere with and without gravity,are considerably smaller than the magnetic field results. Whenconsidering viscous dissipation we see that the perturbed ve-locity decays faster than the perturbed magnetic field dampedby ohmic dissipation. However, in general, the wave amplitudes


Fig. 11a and b. aBehaviour of the wavelengthλ for a radially divergingbackground magnetic field atθ = 0.5 with λ0 = 0.1 for differentvalues of the scale-height (solid line:H = ∞, dot-dashed line:H =0.5, dashed line:H = 0.2, dotted line:H = 0.1). b Behaviour of thewavelengthλ in the corresponding Cartesian case.

decay faster in an atmosphere without gravitational stratifica-tion. For both the perturbed magnetic field and the perturbedvelocity we mainly recover the results we found when studyingthe effect of (only) gravitational stratification on phase mixingof Alfv en waves. The effect of a radially diverging backgroundmagnetic field on phase mixing does not seem to be strongenough to compensate for the stratification of the density whenthe dimensionless pressure scale heightH is smaller than1/4.

The cross section (Fig. 15) of the current densityj2 onlyconfirms the dominant effect of the stratified density. In boththe spherical and the Cartesian case we see that the current den-sity is spread out over a wider area when the pressure scaleheight is smaller. The maximum ofj2 is less high and situatedhigher up. However, we do see that the divergence of the back-ground magnetic field still has some effect. When comparingcorresponding different geometries we see that in the spheri-cal case the maximum of the current density is situated at alower height but is also smaller in magnitude, a result noticedand explained when studying the effect of divergence on phasemixing of Alfv en waves. We also recover the effect of changingthe initial wavelengthλ0. Whenλ0 is decreased,j2 obtaines a

Fig. 13a and b.A cross-section of the perturbed magnetic field andvelocity for a radially diverging background magnetic field atθ = 0.5with Λ2 = 10−4, δ = 0.5 andλ0 = 0.1 for H = 0.2. The dottedlines represent the corresponding solution forH = ∞.

higher maximum at a lower height.The effect of stratificationon the vorticity is a lot smaller than the effect on the currentdensity. The vorticity is only spread out very slightly due tothe lengthening of the wavelengths in the stratified atmosphere.This different behaviour is due to the initial increase in the am-plitude of the perturbed velocity and the fact that thedynamicviscosityρ0ν is constant, rather than the kinematic viscosityν.

Overall, we can make two conclusions about the combinedeffect of a gravitational density stratification and a radiallydiverging background magnetic field on the phase mixing ofAlfv en waves. The stratification generates longer wavelengths,therefore phase mixing is less efficient and heat is deposited intothe plasma at higher heights compared to a purely diverging at-mosphere without gravitational stratification. At the same timethe divergence results in shorter wavelengths which enhancesphase mixing and heat is deposited at lower heights comparedto a non-diverging atmosphere. So, comparing the gravity re-sults with the Heyvaerts and Priest solution, phase mixing canbe more or less efficient depending on the value of the scaleheightH. A similar conclusion can be found in Ruderman et al.(1998) but a direct comparison cannot be made. They assumedan exponentially diverging magnetic field and an exponentially


Fig. 15. A cross-section of the (left) current density and (right) for a (top) radially diverging and (bottom) uniform background magnetic fieldatθ = 0.5 with λ0 = 0.1 for different values of the scale-height (solid line:H = ∞, dot-dashed line:H = 0.5, dashed line:H = 0.2, dottedline: H = 0.1).

decreasing density in such a manner that the resulting Alfvenvelocity was depending on the horizontal coordinate only. Inthis study we have a different Alfven speed and a truly openatmosphere in the sense that no magnetic field lines connectback to the solar surface. However, the general conclusions arebroadly in agreement with Ruderman et al. (1998).

6. Discussion and conclusions

Let us put the results obtained in this paper into typical solarcorona conditions. A coronal hole plasma is strongly inhomo-geneous due to the presence of, for example, plumes, so phasemixing will occur when waves generated by photospheric mo-tions travel outwards from the Sun. For example, if we assumethat the plasma density inside a coronal plume is a factor of4 higher than the surrounding plasma, we findδ ≈ 0.5. If wefurther assume a driver with a 1-minute period and assume theAlfv en speed in a coronal hole to be 1000 km/s, we find thatλ0 ≈ 0.1 solar radii. The pressure scale heightH = RT

µg is

found to be 0.2 solar radii forT = 2 × 106K. With these val-ues, we expect the maximum of the ohmic heating to occur at1.4 solar radii and the maximum of the viscous heating at 1.35solar radii (Fig. 12). When we consider a driver with a 5-minuteperiod, the initial wavelength will be longer, which will causeto maximum of the ohmic and viscous dissipation to be situated

at larger heights. So we find that for T=2 × 106 K, a 1-minuteoscillation will deposit most heat within a few solar radii andcould therefore be a candidate for heating the coronal holes,while e.g. a 5-minute oscillation might be a way to deposit heatinto the solar wind.

From studying the effect of a diverging background mag-netic field on phase mixing of Alfven waves in an open fieldregion we conclude that wavelengths shorten when the wavespropagate outwards. Therefore, phase mixing is more efficientand sufficiently small lengthscales for dissipation to be im-portant will now build up faster. The waves will be dissipatedquicker when compared to the standard non-diverging case. Dueto the combination of the shortening of the wavelenghts and theenhanced efficiency of phase mixing, the maxima of the cur-rent densityj2 occur at lower heights but at the same time arelower than in the Cartesian case. We found that, unlike in anon-diverging atmosphere, the current density and the vorticitybehave in a similar manner and that ohmic and viscous heatingwill have a similar importance in the heating process.

From our study of the combined effect of a gravitationallystratified density and a radially diverging background magneticfield on phase mixing of Alfven waves we found that the effi-ciency of phase mixing depends strongly on the particular ge-ometry of the configuration. Finally, depending on the value of


Fig. 14a and b.A cross-section of the (Cartesian) perturbed magneticfield and velocity atx = 0.5 with Λ2 = 10−4, δ = 0.5 andλ0 = 0.1for H = 0.2. The dotted lines represent the corresponding solution forH = ∞.

the scale height the wave amplitudes can damp either slower orfaster than in the uniform non-diverging model.

Appendix A: calculation of the WKB solutions

The phase mixing equation for the perturbed velocity in a grav-itationally stratified atmosphere with a radially diverging back-ground magnetic field is given by

k2(θ)e− 1H (1− 1

r )v +(

λ0

2πr0

)2 1r3

∂2

∂r2

(vr

)

+ iΛ2νΩ(θ20∂2v

∂r2+

1r2∂2v

∂θ2

)= 0, (A.1)

wherek2(θ) = Ω2

v2A(θ) andΛ2

ν = ρ0νΩ0µB2

0

(λ0

2πr0θ0

)2.

Now setχ = vr so that

k2(θ)e− 1H (1− 1

r )χ+(

λ0

2πr0

)2 1r4∂2χ

∂r2

+ iΛ2νΩ(θ20r

∂2rχ

∂r2+

1r2∂2χ

∂θ2

)= 0 . (A.2)

Settingχ = g(r)X(R, θ) with R = f(r) and only takinginto account second order derivatives in the damping term weget

∂2X

∂R2

+

(k2(

2πr0λ0

)2r4e− 1

H (1− 1r )

f ′2 +g′′

gf ′2

)X

+(f ′′

f ′2 +2g′

gf ′

)∂X

∂R

+iΛ2νΩ(

2πr0λ0

)2(θ20r

4 ∂2X

∂R2 + r21f ′2

∂2X

∂θ2

)= 0

(A.3)

To obtain a second order partial differential equation withconstant coefficients, we eliminate the first order derivative bysettings its coefficient equal to zero:

f ′′

f ′2 +2g′

gf ′ = 0 ⇒ g = (f ′)−1/2 .

To leading order, the coefficient ofX is constant inR if:

f ′2 = r4e− 1H (1− 1

r ) ⇒ R = f(r) =∫ r

1r2e− 1

2H (1− 1r )dr ,

where we have chosen the positive square root to match theoutgoing wave condition. We then find

g = r−1e1

4H (1− 1r ) .

Neglecting the second term in the coefficient ofX which isof smaller order, Eq. (A.1) becomes:

∂2X

∂R2 +(

2πr0λ0

)2

k2X (A.4)

+iΛ2νΩ(

2πr0λ0

)2(θ20r

4 ∂2X

∂R2

1r2e

1H (1− 1

r ) ∂2X

∂θ2

)= 0 .

To obtain a solution to the non-zero dissipation case, we set

X = exp(

−ik(θ)2πr0λ0

R0

)F (R1) , (A.5)

whereR0 = R andR1 = εR0. From Eq. (A.5) we then get[(−ik 2πr0

λ0

)2F + 2ε

(−ik 2πr0

λ0

)∂F∂R1

+ ε2 ∂2F∂R2

1

]

+k2(

2πr0λ0

)2F + iΛ2

νΩ(

2πr0λ0

)2θ20r

4(−ik 2πr0

λ0

)2F

+ e1H (1− 1

r )r2

(−ik′ 2πr0

λ0R0

)2F

= 0 ,

(A.6)

and with

∂F

∂R1=∂F

∂r

∂r

∂R1= ε−1 ∂F

∂r

∂r

∂R0= ε−1 (f ′(r))−1 ∂F

∂r,

we find,

lnF = −12Λ2

νΩ(

2πr0λ0

)3

k

∫ r

1θ20r

6e− 12H (1− 1

r ) + e1

2H (1− 1r ) k

′2

k2 R2dr .


The full solution for the perturbed velocity then becomes:

v = sin(πθ) exp(

−i2πr0λ0

k(θ)R)

exp[

14H

(1 − 1

r

)− 1

2Λ2

νΩ(

2πr0λ0

)3

k∫ r

1

(θ20r

6e− 12H (1− 1

r) +k′2

k2 R2e

12H (1− 1

r))dr

], (A.7)

with R =∫ r

1 r2e− 1

2H (1− 1r )dr

andΛ2ν = ρ0νΩ0µ

B20

(λ0

2πr0θ0

)2.

The solution for the perturbed magnetic field can be obtainedin a similar manner.

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