1

Abstract. Interaction of incompressible and compress-ible MHD fluctuations on a one-dimensional plasma inho-mogeneity, transverse to the magnetic field, is consideredin the three-dimensional regime. An initially incompress-ible (Alfvenic) MHD fluctuation generates, by linear cou-pling, compressible (magnetoacoustic) fluctuations. In theinitial stage of the interaction, the efficiency of the cou-pling is enhanced by phase mixing of the incompressiblemode. In the developed stage of phase mixing, the interac-tion saturates and both incompressible and compressiblefluctuations approach a quasi-stationary state. The sat-uration levels of density fluctuations and incompressiblefluctionations tend to become equal as the linear couplingstrength increases.

Key words: Magnetohydrodynamics(MHD)– waves –Sun: activity – Sun: Solar wind

A&A manuscript no.(will be inserted by hand later)

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ASTRONOMYAND

ASTROPHYSICSFebruary 27, 2002

A Three Dimensional Alfvenic Pulse in a TransverselyInhomogeneous Medium

D. Tsiklauri and V.M. Nakariakov

Physics Department, University of Warwick, Coventry, CV4 7AL, U.K.

Received ???? 2001 / Accepted ???? 2001

1. Introduction

Problems of heating of the open corona of the Sun andacceleration of the solar wind are closely related with theinteraction of linearly incompressible Alfven waves withcompressible magnetoacoustic waves. Investigation of cou-pling mechanisms between the modes is of prime impor-tance because, on one hand, the Alfven waves are usualcandidates of energy transport from the lower layers ofthe solar atmosphere to the corona, and, on the other,compressible perturbations are subject to much more ef-ficient dissipation than the incompressible Alfven waves(see, e.g., Narain & Ulmschneider (1996)). Also, the com-pressive waves, as opposed to the Alfven waves, can trans-port energy across the magnetic field, due to the fact thattheir propagation in space is not constrained by the fieldlines.

The original idea of the phase mixing of incompressibleAlfven waves (Hayvaerts & Priest (1983)) was based onthe following argument: when plasma has a densitygradient perpendicular to the magnetic field, local Alfvenspeed is a function of the transverse coordinate. Thus,when an Alfven wave propagates along the field itsperturbations on the adjacent field lines become outof phase. This stretching of Alfven wave front createsprogressively smaller spatial scales across the field. Inturn, because the dissipation is proportional to the wavenumber squared, phase mixing leads to enhanced dissi-pation of the Alfven wave directly. However, in addition,as demonstrated by Malara, Petkaki & Veltri (1996),Nakariakov et al. (1997), Nakariakov et al. (1998),Botha et al. (2000), Tsiklauri, Arber & Nakariakov (2001),Tsiklauri, Nakariakov & Arber (2002), phase mixing oflinearly polarized, plane Alfven waves in the compressibleplasma, in the nonlinear regime leads to the generationof fast magnetoacoustic waves. However, it was estab-lished, in 2.5D geometry, for the harmonic Alfven waveBotha et al. (2000), and for a wide spectrum Alfven pulseby Tsiklauri, Arber & Nakariakov (2001) that compres-sive perturbations, which are initially absent from the

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system, do not grow to a substantial fraction of the initialAlfven wave due the destructive wave interference effect.

Since observations established the exis-tence of large amplitude Alfven waves in thesolar wind, we further extended our previ-ous work Tsiklauri, Arber & Nakariakov (2001)for the strongly non-linear amplitudesTsiklauri, Nakariakov & Arber (2002). We simulatednumerically, in 2.5D geometry, the interaction of astrongly non-linear, linearly polarized, plane Alfvenicpulse with a one-dimensional, perpendicular to the mag-netic field, plasma density inhomogeneity. Among otherfacts we established then that the maximum generatedamplitude of transverse compressive wave, up to about40 % of the initial Alfven wave amplitude, is reached forabout β = 0.5.

There is yet another possibility to generate fast (andslow) magnetoacoustic waves, which becomes possibleeven in the linear regime – that is a linear coupling ofan oblique Alfven wave to fast and slow magnetoacousticwaves. This becomes possible in the presence of non-zerogradients in the wave field in the third direction, perpen-dicular to both the magnetic field and the gradient of theAlfven speed. The efficiency of coupling is proportional toa product of the characteristic scales of the waves in twoperpendicular directions. As the gradient of the trasverseto the magnetic field inhomogeneity is increased by thephase mixing, the efficiency of the linear coupling betweenthe Alfven and compressive modes should also be increasedaccordingly. One of the aims of this work is to investigate,in 3D geometry, whether the linear coupling efficiency isaffected by the destructive wave interference effect as ithas been established in the 2.5D weakly non-linear caseTsiklauri, Arber & Nakariakov (2001).

In this work we consider fully three dimensionalgeometry and study the linear coupling of phase mixedAlfven waves to the longitudinal and transverse compres-sive waves. Similar coupling mechanism was studied byDe Groof & Goossens (2000) in the context of resonantabsorption of MHD waves in coronal loops as a heatingmechanism. However, our treatment is more general as weperform a direct numerical 3D simulation without resort-

Tsiklauri & Nakariakov: A 3D Alfvenic Pulse ... 3

ing to Fourier transform in time and y direction. Besides,they used a harmonic form of the initial perturbationas opposed to our spatially localized, Gaussian, one.Recently, Hood, Brooks & Wright (2002) showed thatthe wide spectrum regime of phase mixing can be quitedifferent from the harmonic one, as in fact, phase mixingof localized Alfven pulses results in a slower, algebraic,damping as opposed to the standard exponential dampingof harmonic Alfven waves, which suggest that localizedAlfvenic perturbations will transport energy higher intothe corona than harmonic ones. Such Alfvenic perturba-tions could be generated e.g. by some transient events suchas solar flares, coronal mass ejections, etc. While mostof the phase-mixing studies concentrated on harmonicAlfven perturbations (e.g. Hayvaerts & Priest (1983),Hood, Ireland & Priest (1997),Hood et al. (1997), Nakariakov et al. (1998),Ruderman, Nakariakov & Roberts (1998),DeMoortel et al. (1999), DeMoortel, Hood & Arber (2000),Grappin, Leorat & Buttighoffer (2000),Botha et al. (2000)) only few considered spa-tially localized ones (Nakariakov et al. (1997),Tsiklauri, Arber & Nakariakov (2001),Tsiklauri, Nakariakov & Arber (2002),Hood, Brooks & Wright (2002)).

An additional motivation to this study is connectedwith the growing interest to the problem of interaction ofMHD waves with 2D and 3D plasma structures and irreg-ularities, as the coronal and wind plasmas are observed tobe structured in all three dimensions. 2D and 3D structur-ing can dramatically affect properties of MHD waves. Inparticular, as it was shown in the incompressible regime bySimilon & Sudan (1989) and confirmed in numerical ex-periments performed by Petkaki, Malara & Veltri (1998)and Malara, Petkaki & Veltri (2000), the structuring dra-matically increases the efficiency of wave dissipation. How-ever, in those studies, the effects of MHD wave couplingwere not taken into account, as only the incompressibleregime has been considered. From this point of view, ourinvestigation of the interaction of a 3D Alfvenic pulse witha 1D plasma inhomogeneity in the compressible regimeprovides the necessary building element of the generaltheory of MHD wave interaction with plasma inhomo-geneities.

We show that when initial Alfvenic perturbation in-teracts with plasma inhomogeneity, it generates fast andslow magnetosonic waves (through linear coupling). TheAlfvenic pulse, initially incompressible, evolves to anMHD pulse which contains both the compressible and in-compressible components. In spite of the continuous ac-tion of phase mixing, the MHD pulse reaches a new quasi-steady saturated state. We found that the saturation leveldepends on the plasma-β parameter, and the saturationlevels of density fluctuations and incompressible fluctiona-tions tend to become equal as the linear coupling strengthincreases.

The paper is organized as follows: in section 2 we for-mulate our model. In section 3 we present the results ofnumerical simulation, while we close in section 4 with thediscussion of main results.

2. The model

In our model we use equations ideal MHD

ρ∂V

∂t+ ρ(V · ∇)V = −∇p− 1

4πB × curlB, (1)

∂B

∂t= curl(V ×B), (2)

∂p

∂t+ V · ∇p + γp∇ · V = 0, (3)

where B is the magnetic field, V is plasma velocity, ρ isplasma mass density, and p is plasma thermal pressure. Inwhat follows we use 5/3 for the value of γ.

We solve equations (1)-(3) in Cartesian coordinates(x, y, z). Note that as we solve a fully 3D problem weretain variation in the y-direction, i.e. (∂/∂y 6= 0). Theequilibrium state is taken to be an inhomogeneous plasmaof density ρ0(x) and a uniform magnetic field B0 in thez-direction. We consider a plasma configuration similar tothe one investigated in Malara, Petkaki & Veltri (1996),Nakariakov et al. (1997), Nakariakov et al. (1998),Botha et al. (2000), Tsiklauri, Arber & Nakariakov (2001),i.e. the plasma has a one-dimensional inhomogeneity inthe equilibrium density ρ0(x) and temperature T0(x).The unperturbed thermal pressure, p0, is taken to beconstant everywhere.

Next, we do usual linearization of the Eqs. (1)-(3) andwrite them in component form as following

ρ0(x)∂Vx

∂t+

∂p

∂x− B0

4π

(∂Bx

∂z− ∂Bz

∂x

)= 0, (4)

ρ0(x)∂Vy

∂t+

∂p

∂y− B0

4π

(∂Bz

∂y− ∂By

∂z

)= 0, (5)

ρ0(x)∂Vz

∂t+

∂p

∂z= 0, (6)

∂Bx

∂t−B0

∂Vx

∂z= 0, (7)

∂By

∂t−B0

∂Vy

∂z= 0, (8)

∂Bz

∂t+ B0

(∂Vx

∂x+

∂Vy

∂y

)= 0, (9)

∂p

∂t+ γp0

(∂Vx

∂x+

∂Vy

∂y+

∂Vz

∂z

)= 0. (10)

It is useful to re-write Eqs. (4)-(10) in a form of threecoupled wave equations as following[∂2

tt −(c2s(x) + c2

A(x))∂2

xx − c2A(x)∂2

zz

]Vx (11)

− [(c2s(x) + c2

A(x))∂2

xy

]Vy −

[c2s(x)∂2

xz

]Vz = 0,

[∂2

tt −(c2s(x) + c2

A(x))∂2

yy − c2A(x)∂2

zz

]Vy (12)

− [(c2s(x) + c2

A(x))∂2

xy

]Vx −

[c2s(x)∂2

yz

]Vz = 0,

[∂2

tt − c2s(x)∂2

zz

]Vz −

[c2s(x)∂2

xz

]Vx (13)

− [c2s(x)∂2

yz

]Vy = 0,

4 Tsiklauri & Nakariakov: A 3D Alfvenic Pulse ...

Fig. 1. Contour plots of Vy through three cross-sections at three time snapshots. Here, β = 0.5, ky = 0.6, kz = 1.0.

where cA(x) = B0/√

4πρ0(x) and cs(x) =√

γp0/ρ0(x)denote local Alfven and sound speeds respectively.

We solve Eqs.(4)-(10) numerically after re-writingthem in a dimensionless form using following normaliza-tion: Bx,y,z = B0Bx,y,z, (x, y, z) = a0(x, y, z), cA(x) =B0/

√4πρ0(x) = c0

A/√

3− 2 tanh(λx), t = (a0/c0A)t,

Vx,y,z = c0AVx,y,z. Note, that cs(x) =

√γβ/2cA(x) ≡√

χcA(x), where β stands for the ratio of thermal andmagnetic pressures β = p0/(B2

0/8π). Here, λ is a free pa-rameter which controls the steepness of the density profilegradient. In our simulations we use λ = 0.5. In what fol-lows we omit bars on top of the physical quantities.

3. Numerical Results

In order to solve Eqs.(4)-(10) numerically we have writ-ten a new numerical code dt4dx10, which uses a highorder finite difference scheme. Namely, it evaluates 10-th order centered spatial derivatives, and advances so-lution in time using 4-th order Runge-Kutta algorithm.Therefore, dt4dx10 is O[(∆t/T )4] accurate in time andO[(max(∆x, ∆y, ∆z)/L)10] accurate in space, with T andL denoting run time and linear size of the simulation do-main. The use of such a high-order numerical approach

was motivated by the very nature of the problem consid-ered, as development of phase mixing leads to the gener-ation of very steep profiles in the wave front.

The simulation cube size is set by the limits −25.0 ≤x ≤ 25.0, −25.0 ≤ y ≤ 25.0 and −25.0 ≤ z ≤ 25.0. Onlywhen ky-parameter was fixed at 0.1 (see below) we had todouble our simulation cube size and double the run-timein order to reach the saturation in the generated densityperturbation amplitude. In this case we doubled the num-ber of grid points each direction in order to preserve thesame simulation resolution. Boundary conditions used inall our simulations are zero-gradient in all three spatialdimensions.

We have performed calculation on various resolutionsin attempt to achieve convergence of the results. Thegraphical results presented here are for the spatial res-olution 1283, which refers to number of grid points in x, yand z directions respectively. We have also performed cal-culation on the spatial resolution 2563 and we found thatthe results converge perfectly. This is understandable dueto the high order of the scheme even for the 1283 resolu-tion time-error, O[(∆t/T )4] ≈ 7×10−8, and spatial-error,O[(max(∆x, ∆y, ∆z)/L)10] ≈ 8×10−22. Since a 2563 res-olution run takes about 8 hours on Compaq ES40 with

Tsiklauri & Nakariakov: A 3D Alfvenic Pulse ... 5

Fig. 2. Snapshots of ε(ρ′/ρ0(x)) at t = 15 through x = 0 cross-section. Here, β = 0.5, ky = 0.6, kz = 1.0.

Fig. 3. Snapshots of ε(ρ′/ρ0(x)) at t = 15 through z = 0 cross-section. Here, β = 0.5, ky = 0.6, kz = 1.0.

Fig. 4. Snapshots of ε(ρ′/ρ0(x)) at t = 15 through y = 0 cross-section. Here, β = 0.5, ky = 0.6, kz = 1.0.

Fig. 6. Normalized kinetic, magnetic, internal and total ener-gies as a function of time. Here, β = 0.5 and kz = 1.0. Solidlines correspond to ky = 0.6, while dashed and dash-dottedlines show cases of ky = 0.4 and ky = 0 respectively.

eight EV6 500-MHz processors, while a 1283 resolutiontakes only 1 h on Compaq ES40 with four EV6 500-MHzprocessors producing the same results, we opted for thelatter, less CPU-consuming, alternative.

We set up the code in such a way that initially longitu-dinal and transverse compressive perturbations as well asdensity perturbation are absent. In the numerical simula-tions the Alfven perturbation is initially a plane (with re-spect to x-coordinate) pulse, which has a Gaussian struc-ture in y and z-coordinatesVy(x, y, z, t = 0) = cA(x) exp

(−(kyy)2 − (kzz)2). (14)

Here, ky and kz are free parameters which control thestrength of gradients in y and z direction of the initialAlfvenic perturbation. As the problem considered is lin-ear, the wave amplitude can be taken to be normalized tounity.

In Fig. 1 we present time evolution of the initialAlfvenic pulse (Vy) through three cross-sections. Namely,left column shows three snapshots of Vy through y = 0cross-section, middle column shows three snapshots of Vy

through x = 0 cross-section, while right column showsthree snapshots of Vy through z = 0 cross-section. Whatwe gather from this graph is as following:

– y = 0 cross-section: the initial Alfvenic pulse, whichis plain with respect to x-coordinate, is split in twoD’Alambert’s solutions, with half of the amplitudes,traveling in two opposite directions along the magneticfield. Because the unperturbed density is inhomoge-neous across the magnetic field (in x-direction) the lo-cal Alfven speed depends upon x. Thus, the Alfvenicpulse is phase-mixed and its initially plain front iscontinuously distorted creating transverse gradients.Note, that the right wing of the pulse is asymmetric tothe left one, this is due to the fact that its amplitude

6 Tsiklauri & Nakariakov: A 3D Alfvenic Pulse ...

Fig. 5. Maximum of the absolute values over the whole 3D simulation box of all physical quantities as a function of time. Here,β = 0.5 and kz = 1.0. Thick solid lines correspond to the resolution 2563, while thin solid lines to that of 1283, both for ky = 0.6.Note that on all seven panels these two curves practically overlap, which serves as a proof of convergence of our simulationresults. Dashed lines represent ky = 0.4, while dash-dotted lines show ky = 0.

drops from 1.0 in the x > 0 domain to 1/√

5 in thex < 0.

– x = 0 cross-section: the initial Alfvenic pulse whichhas an asymmetric as ky 6= kz Gaussian bell shape issplit in two smaller amplitude bells traveling in twoopposite directions along the magnetic field plus someelliptic-shaped wake expanding outwards in all direc-tions. Note, that because the unperturbed density ishomogeneous across the magnetic field in y-directionwe do not see any distortions on the wave front withrespect to the y = 0 line as we saw in the previouscase.

– z = 0 cross-section: the Alfvenic pulse evolution is sim-ilar to the case of y = 0 cross-section. The notable dif-ference is that through z = 0 cross-section pulse lookswider as ky = 0.6 while kz = 1.0.

Perturbation of By is initially absent from the system,but as it is a potential component of the kinetic coun-terpart, Vy, of the wave, it is soon generated and furtherevolves similarly to Vy in a form of two negative and posi-

tive phase-mixed pulses traveling into two opposite direc-tions.

At time t = 0 transverse and longitudinal compressivewaves as well as density perturbation were absent fromthe system, while at time t = 15 as we see from Figs. 2, 3and 4, relative density perturbation grew up to a sizeablefraction of the initial Alfven wave amplitude. Note, thatin all figures pre-factor ε stands to underscore the factthat the linear problem, which we study, does not dependon the initial amplitude. Thus, we use A = 1, while wehave to bear in mind that linear approximation itself isvalid for small amplitudes that us why we use small, arbi-trary pre-factor ε in our notation. A fairly good quantitydescribing the generation of transverse and longitudinalcompressive waves as well as density perturbations is themaximum of absolute value over the whole simulation do-main. We plot this quantity for all physical variables inFig. 5. In this figure, thick solid lines correspond to theresolution 2563, while thin solid lines to that of 1283, bothfor ky = 0.6. It is remarkable that on all panels these twocurves practically do overlap, which serves as a proof of

Tsiklauri & Nakariakov: A 3D Alfvenic Pulse ... 7

Fig. 7. All physical quantities as a function of time in a given point of space. Here, β = 0.5, ky = 0.6, kz = 1.0. Left column:at a point (x=0,y=-8,z=0), mid column: at a point (x=0,y=-8,z=4), right column at a point (x=0,y=-8,z=8)

convergence of our simulation results. The last figure inthe bottom row presents the maximum of absolute valueover the whole simulation domain of divB, and we in-deed observe almost perfect fulfillment of the fundamen-tal law, divB = 0, which comes as a bonus of high-order,centered, finite difference numerical scheme. As expected,

Fig. 5 shows that when ky = 0 Alfvenic perturbations (Vy

and By) are decoupled from the rest of physical quan-tities. This conclusion can be deducted either from an-alyzing Eqs.(11)-(13) or resorting to a classic mechani-cal analogy of coupled pendulums. In effect, Eqs.(11)-(13)also describe three inter-coupled mathematical pendulums

8 Tsiklauri & Nakariakov: A 3D Alfvenic Pulse ...

Fig. 8. All physical quantities as a function of time in a given point of space. Here, β = 0.5, ky = 0.6, kz = 1.0. Left column:at a point (x=16,y=-8,z=0), mid column: at a point (x=16,y=-8,z=4), right column at a point (x=16,y=-8,z=8)

which are located in xOz plane. Thus, as long as we donot perturb these pendulums such that ky 6= 0, they willalways oscillate in the xOz plane. Thus, what we see inFig. 5 when ky = 0 (dash-dotted lines) is that initial ki-netic Alfven perturbation (Vy) is split in half-amplitudeD’Alambert’s solutions (both for Vy and By) and no other

physical quantity is generated. However, when we switchon the coupling, ky 6= 0, all transverse and longitudinalcompressive physical quantities are generated. What isnoteworthy is that we observe formation of a new quasi-steady MHD state, i.e. initial Alfvenic perturbation inter-acts with plasma inhomogeneity, generates fast and slow

Tsiklauri & Nakariakov: A 3D Alfvenic Pulse ... 9

magnetosonic waves (through ky coupling) and asymptot-ically all three modes propagate without change in am-plitude, even though phase mixing effect (kx → ∞) is inaction all the time. Also, it can be gathered from Fig. 5that the saturation levels of density fluctuations and in-compressible fluctionations (Vy and By) tend to becomeequal as the linear coupling strength increases.

In Fig. 6 we investigate the energetics of our numeri-cal simulation. In fact, we observe nearly perfect (±0.2%error) conservation of total energy by dt4dx10 numericalcode. The major conclusion which can be drawn from thisgraph is that when ky = 0 there is no internal (compres-sive) energy generation, while with the increase of ky itssaturation levels increase progressively.

In order to investigate the effect of phase mixing on thecreated quasi-steady MHD state, we produced time seriesof all physical quantities in several points of the simulation3D box, Figs. 7 and 8. Namely, at points (x=0,y=-8,z=0,4, 8), which are located in the phase mixing (spatial in-homogeneity) domain, and (x=16,y=-8,z=0, 4, 8), whichare far away from it. The three panels (from left to theright), in effect, trace the dynamics in the z-direction (i.e.along the regular magnetic field lines). These types of datawould be obtained from, for example, two satellites whichare located in two different regions of solar wind. There aretwo noteworthy features that can be gathered from theseplots: First, we see that in the phase mixing region, Vx,which is associated with the fast magnetosonic wave, at-tains about 40 times larger values than in the region that isfar from the phase mixing region. That is a sensible result,since it is known that phase mixing efficiently generatesoblique fast magnetosonic waves. Second, total magneticfield perturbation |B −B0| is positively correlated withthe pressure (as well as density, which is proportional tothe gas pressure) perturbation, which indeed is what wasexpected from the fast magnetosonic wave.

We have also performed studies of parametric space ofthe problem, by calculating final saturation levels of rela-tive density perturbation as a function of linear couplingstrength parameter, ky and plasma β. We gather fromFig. 9 that the final saturation levels depend weakly onky, while there is somewhat stronger dependence on β.

4. Conclusions

In this paper we describe in detail the effect of phase mix-ing on a spatially localized 3D Alfvenic perturbation.

Our numerical study of the full MHD equations estab-lished the following:

– non-zero ky in the initial Alfvenic perturbation leadsto a linear coupling to the longitudinal and transversecompressive perturbations.

– then the initial Alfvenic perturbation interacts withplasma inhomogeneity, generates fast and slow mag-netosonic waves (through ky coupling) and asymp-totically all three modes propagate without change

Fig. 9. Top panel: Saturation levels of maximum of the abso-lute values over the whole simulation box of the density per-turbation as a function of ky. Bottom panel: the same but asa function β.

in amplitude, even though phase mixing effect(kx → ∞) is in action all the time. Thus, sim-ilarly to Roberts & Wiltberger (1995)’s results weobserve a formation of a new quasi-steady MHDstate. Roberts & Wiltberger (1995) performed numer-ical simulation of 1.5D, compressible MHD in thestrongly non-linear regime. There is substantial dif-ference, however, between our results as in their casesaturation can be attributed to the action of non-linear effects, while we show that even in the lin-ear regime, coupling between the MHD modes, whichis possible only in fully 3D geometry, can resultin the formation saturated quasi-steady MHD state.Roberts & Wiltberger (1995) claim that under a widevariety of conditions B and ρ become anti-correlatedon average, while we show that this is not always thecase (cf. Figs. 7 and 8)

– when ky = 0 there is no internal (compressive) energygeneration, while with the increase of ky its saturationlevels increase progressively.

– the final saturation levels of relative density pertur-bation depend weakly on ky (linear coupling strengthparameter), while there is somewhat stronger depen-dence on β.

– the saturation levels of density fluctuations and incom-pressible fluctionations tend to become equal as thelinear coupling strength increases.

The results obtained provide a necessary basis for fur-ther studies of MHD weak turbulence in the solar wind.In the solar context, the interaction of Alfven and mag-netoacoustic modes in the three- dimensional regime ofphase mixing suggests an efficient mechanism for conver-sion of incompressible perturbations to compressible per-turbations. The latter are subject to more efficient dissipa-tion by collisional or collisionless mechanisms. This makes

10 Tsiklauri & Nakariakov: A 3D Alfvenic Pulse ...

the results obtained relevant to the problem of the heatingof open coronal structures.

Acknowledgements. The authors are grateful to Tony Arberfor valuable advise when dt4dx10 code was written and fora number of valuable discussions. DT acknowledges financialsupport from PPARC. Numerical calculations of this work wereperformed using the PPARC funded Compaq MHD Cluster atSt Andrews and Astro-Sun cluster at Warwick.

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