Resonant and phase-mixed magnetohydrodynamic waves in the solar atmosphereMarcel Goossens and Anik De Groof Citation: Physics of Plasmas (1994-present) 8, 2371 (2001); doi: 10.1063/1.1343090 View online: http://dx.doi.org/10.1063/1.1343090 View Table of Contents: http://scitation.aip.org/content/aip/journal/pop/8/5?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Heating of the Solar Corona by Alfvén Waves: SelfInduced Opacity AIP Conf. Proc. 1356, 123 (2011); 10.1063/1.3598101 An application of the turbulent magnetohydrodynamic residual-energy equation model to the solar wind Phys. Plasmas 14, 112904 (2007); 10.1063/1.2792337 Absorption of fast magnetosonic waves in the solar atmosphere in the limit of weak nonlinearity AIP Conf. Proc. 537, 144 (2000); 10.1063/1.1324934 Coronal heating by quasi-2D MHD turbulence driven by non-WKB wave reflection AIP Conf. Proc. 471, 361 (1999); 10.1063/1.58773 Observations of the parametric decay instability of nonlinear magnetohydrodynamic waves Phys. Plasmas 4, 846 (1997); 10.1063/1.872183

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Resonant and phase-mixed magnetohydrodynamic wavesin the solar atmosphere *

Marcel Goossens† and Anik De GroofCentre for Plasma-Astrophysics, K. U. Leuven, Celestijnenlaan 200B, 3001 Leuven, Belgium

~Received 23 October 2000; accepted 28 November 2000!

The magnetic field in the solar atmosphere is not uniformly distributed but organized in typicalconfigurations: e.g., intense flux tubes in the photosphere, magnetic loops in the corona, plumes inthe solar wind. Each of these magnetic configurations can support magnetohydrodynamic~MHD!waves and observations show that this is indeed the case. The intrinsic inhomogeneity of themagnetic configurations enables local~slow and! Alfven waves to exist on individual magneticsurfaces. These local Alfve´n waves provide a means for dissipating wave energy which is far moreefficient in a weakly dissipative plasma than classical resistive or viscous MHD wave damping ina uniform plasma. This property has inspired a lot of work on the dissipation of driven Alfve´n wavesand wave heating in the solar atmosphere by resonant absorption and phase mixing. This reviewconcentrates on the interaction between fast magnetosonic waves, local Alfve´n waves andquasimodes and discusses recent results on the time evolution of phase mixing of resonant wavesdriven by footpoint motions. ©2001 American Institute of Physics.@DOI: 10.1063/1.1343090#

I. INTRODUCTION

Magnetohydrodynamic~MHD! waves transport energyand, when part of this energy is dissipated, they can heatplasmas. This property, combined with the fact they are ob-served in the solar atmosphere, makes MHD waves and, inparticular, Alfven waves a natural mechanism for heating~part of! the solar corona~see, e.g., Refs. 1–4!. MHD wavesare also important in their own right. They reflect the stabledynamic behavior of the plasma objects in which they resideand can be used as probes for investigating the structure andcomposition of the plasma objects in which they are ob-served~see, e.g., Ref. 5!.

This review concentrates on resonant and phase-mixedAlfven waves. The physical basis of resonant and phase-mixed Alfven waves can be understood in the context oflinear ideal MHD. In linear ideal MHD, each individualmagnetic surface can oscillate at its own local Alfve´n fre-quency without interaction with neighboring magnetic sur-faces. Nonuniformity creates a range of local Alfve´n fre-quencies known as the Alfve´n continuum. In ideal MHD,these local Alfve´n waves are confined to the resonant mag-netic surfaces on which their dispersion relations are satisfiedlocally. Dissipative effects produce coupling to the neighbor-ing surfaces, but the local Alfve´n waves still have steep gra-dients across the magnetic surfaces. Because of these steepgradients, excitation of local Alfve´n waves provides a meansfor dissipating wave energy which is far more efficient inweakly dissipative plasmas in the solar atmosphere than clas-sical resistive or viscous MHD wave damping in a uniformplasma. This excitation can be direct by driving at the photo-spheric footpoints of the magnetic field lines~see, e.g., Ref.6! or indirect by lateral driving~see, e.g., Refs. 7–13!. Indi-rect driving involves a carrier wave~fast magnetosonic

wave! which transports energy across the magnetic surfacesand couples to local Alfve´n oscillations at the resonant posi-tion xA wherevd5vA(xA). This review concentrates on theinteraction between fast magnetosonic waves, local Alfve´nwaves, and quasimodes and discusses recent results on thetime evolution of phase mixing of resonant waves driven byfootpoint motions.

II. MHD WAVES IN LINEAR VISCORESISTIVE MHD

The study of driven Alfve´n waves involves forced oscil-lations in dissipative MHD. This means that the time depen-dent nonlinear equations of dissipative MHD have to be in-tegrated in the presence of a time varying force term. Moststudies have used linear theory of wave motions superim-posed on a static~or stationary! background. The simplestequilibrium state that still contains the basic physics is aone-dimensional planar slab of pressureless plasma with aconstant vertical magnetic field and with a densityr varyingin one direction. Hence, in a system of Cartesian coordinatesx,y,z: B5Bz1z , p50 and r(x). The assumption of aplasma without pressure removes the slow waves, the vari-able density introduces a variable Alfve´n velocity vA

2(x)5@B2/mr(x)#. In our simple equilibrium state the magneticsurfaces are the planesx5constant and the photosphere is atthe positionsz50 and z5L. y is an ignorable coordinatewhich enables us to Fourier analyze with respect toy and putthe wave quantities proportional to exp(ikyy) with ky 5 theazimuthal wave number.

The basic time-dependent equations for linear waves inviscoresistive MHD are

S ]2

]t22vA

2 ]2

]z22vA

2 ]2

]x22~h1n!

]3

]x2]tD jx5 ikyvA

2 ]jy

]x,

S ]2

]t22vA

2 ]2

]z21ky

2vA22~h1n!

]3

]x2]tD jy5 ikyvA

2 ]jx

]x,

~1!*Paper JI2 2, Bull. Am. Phys. Soc.45, 186 ~2000!.†Invited speaker.

PHYSICS OF PLASMAS VOLUME 8, NUMBER 5 MAY 2001

23711070-664X/2001/8(5)/2371/6/$18.00 © 2001 American Institute of Physics

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j is the Lagrangian displacement.h is the coefficient ofmagnetic diffusivity andn is the isotropic viscosity.jy is thecomponent in the magnetic surfaces and perpendicular to themagnetic field lines. It is the component that characterizesthe Alfven waves.jx is the component normal to the mag-netic surfaces and characterizes the fast waves. The azi-muthal wave numberky plays an important role as it controlsthe linear coupling between Alfve´n waves and fast magneto-sonic waves. Forky50 Eqs.~1! for jx andjy are decoupledso that the Alfve´n waves and fast waves do not interact inlinear MHD. However, for kyÞ0 these equations arecoupled and, consequently, the Alfve´n waves and fast wavesdo interact.

For the variation of the wave quantities along the loopwe have to distinguish two cases. When viewed as an eigen-value problem and also as a driven problem with lateral driv-ing, the classic approach is to treat the loop as infinitely long.In that casez is an ignorable coordinate and the wave quan-tities can be put proportional to exp(ikzz) with kz the axialwave number. However, in the case of footpoint driving inclosed loops there are boundary conditions in the axial direc-tion and the axial dependence is now given by a Fourierseries. Hence,

exp~ ikzz!⇒( Xn sinnp

Lz⇒kz5

np

L, nPN.

When the interest is in the eigenmodes of the system or inthe stationary state of a single frequency harmonically drivensystem, the time can be factored out as exp(2ivt) with vunknown in the eigenvalue problem and prescribed and realin the stationary state of driven problem. When the aim is tostudy the temporal evolution of the system under given ini-tial and boundary conditions, the only option is to solve thetime dependent equations.

III. RESONANT ALFVE N WAVES IN LINEAR IDEALMHD

Dissipative MHD is only necessary to treat the behaviorof the waves when they developed very short length scalesand dissipative effects can no longer be ignored. This hap-pens in a dissipative layer around the ideal resonance posi-tion. Since the viscous and magnetic Reynolds numbers arevery large in the solar atmosphere, this dissipative layer isnecessarily very thin. Part of the basic physics of resonantAlfven waves can be understood in linear ideal MHD. Let uslook at the eigenvalue problem or the stationary state ofsingle frequency driven MHD waves. The governing equa-tions are two first-order ordinary differential equations forjx

and the Eulerian perturbation of total pressure,P8 and analgebraic equation forjy :

rvA2~v22vA

2 !djx

dx5@ky

2vA22~v22vA

2 !#P8,

dP8

dx5r~v22vA

2 !jx , ~2!

r~v22vA2 !jy5 ikyP8,

vA is the local Alfven frequency. Its square is defined as

vA2~x!5kz

2vA2~x!. ~3!

The local Alfven frequency is a function of position anddefines a continuum of resonant Alfve´n waves

AC5@kz minvA~x!,kz maxvA~x!#.

The variation ofvA with position plays a key role for theresonant Alfve´n waves. When there are no boundary condi-tions in the axial direction, there is one Alfve´n continuumrelated to the specifiedkz . For a line-tied loop the situationis more involved. Because of the Fourier sum inz, we havean infinite set of axial wave numberskz5np/L, an infiniteset of equations and an infinite number of Alfve´n continua.The equations are singular and the waves show resonant be-havior at the Alfven resonance pointx5xA where the localdispersion relation for Alfve´n waves is satisfiedv2

5vA2(xA). For a frequencyv in the Alfven continuum the

solutions forjx andjy are singular. They are characterized,respectively, by a logarithmic singularity and a jump injx ,and by a 1/s singularity andd(s) contribution injy ~see, forexample, Ref. 14! wheres is the distance to the ideal singu-larity s5x2xA . The solutions have very steep~infinite! gra-dients in thex direction, i.e., normal to the magnetic surfacesand the dominant dynamics is in they components, i.e., inthe magnetic surfaces and perpendicular to the magnetic fieldlines, as expected for Alfve´n waves.

The spectrum of eigenmodes consists of Alfve´n and fasteigenmodes. The azimuthal wave number plays the role ofcoupling factor between Alfve´n eigenmodes and fast eigen-modes. Forky50, the eigenmodes are decoupled into Alfve´ncontinuum eigenmodes withjx50,P850,jyÞ0 and discretefast eigenmodes withjxÞ0,P8Þ0,jy50. The ~real! eigen-values of the fast eigenmodes can lie in the Alfve´n con-tinuum, but there is no coupling forky50. However, forkyÞ0, the fast eigenmodes with an eigenfrequency in theAlfven continuum couple to a local Alfve´n continuum eigen-mode and produce the famous quasimodes. These quasi-modes are the natural wave modes of the system.15,16 Theycombine the properties of global and local oscillation modes.They are damped for a static equilibrium state and play aprominent role in MHD wave heating scenarios~see, e.g.,Refs. 12, 13, 17, and 18!.

IV. RESONANT ALFVE N WAVES IN VISCORESISTIVEMHD

The singular solutions found in ideal MHD are trans-formed into well-behaved regular solutions when dissipationis included. Dissipation is important in a dissipative layeraround the ideal resonant position with a thickness measuredby the quantitydA ~see, e.g., Refs. 12, 19, and 20!: dA

5(v(h1n)/uDu)1/3}(R)21/3. D5d/dx@v22vA2(x)# mea-

sures the variation of the local Alfve´n frequency;R is thetotal Reynolds number. In the solar atmosphereR@1 and

2372 Phys. Plasmas, Vol. 8, No. 5, May 2001 M. Goossens and A. De Groof

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this results in very small length scales required for efficientdamping. Let us look at the stationary state of driven oscil-lations. Close to the ideal resonance point~s! the solutions forthe resonant contributions tojx and jy can be expressed interms of the universal functionsG(t) andF(t) ~see Ref. 21!

G~t!5E0

`e2u3/3

u$exp@ iut sign~D!#21%du,

F~t!5E0

`

exp@ iut sign~D!2u3/3#du,

wheret5s/dA is a scaled variable which is of order unity inthe dissipative layer.G(t) andF(t) contain the remnants ofthe ideal singular behavior, but are finite everywhere. Theycan be used to compute the jumps of the physical variablesover the dissipative layer and the amount of absorbed waveenergy.22 The dominant dynamics is in the perpendicularcomponent like in ideal MHD.

Let us now look at the eigenmodes. ForkyÞ0, the dis-crete fast eigenmodes with an eigenfrequency in the Alfve´ncontinuum couple to a local Alfve´n eigenmode and becomedamped eigenmodes with a complex eigenfrequencyv5vR1 iv I ,v I,0. Damped quasimodes were identified asnormal modes of the dissipative MHD operator for fusionplasmas by Ref. 23 and by Ref. 24 for solar plasmas. Tirryand Goossens24 obtained the solutions forjx andjy of thesequasimodes close to the ideal resonance point~s! in terms ofthe modifiedF(t) andG(t) functions. Compared withG(t)andF(t), the integrands ofF(t), andG(t) contain an extrafactor exp(2uL) with L52v IvA /dAuDu.

V. FOOTPOINT DRIVING IN CLOSED LOOPS

Phase mixing of Alfve´n waves in open and closed loopsby footpoint motions was first studied as a possible source ofcoronal heating in a classic paper~Ref. 6!. Since then, phasemixing has been studied in more detail~see, e.g., Ref. 25 andreferences therein!. So far all work on phase mixing ofAlfven waves in open loops is forky50 ~or m50 in thecylindrical case! and involves direct excitation of Alfve´nwaves without any interaction with fast waves. Here, we fo-cus on Alfven waves driven by footpoint motions in closedloops. Strauss and Lawson26 were the first to set up a numeri-cal simulation on the excitation of Alfve´n waves in closedloops. They restricted their analysis to incompressible mo-tions and considered both the stationary state and the initialvalue solutions.

A. Recapitulation of basic results

Results from the previous sections help us to understandbasic properties of Alfve´n waves driven by footpoint mo-tions in closed loops. In our view the following results areessential:

There is an infinite number of Alfve´n continua becauseof the Fourier sum for describing the axial dependence:

;nPN: 'ACn5S pn

LminvA~x!,

pn

LmaxvA~x! D .

The componentsjx and jy characterize the fast wavesand the Alfven waves, respectively.

For ky50 the fast waves and the Alfve´n waves are fullyseparated.

For kyÞ0 the fast waves and the Alfve´n waves arecoupled and a fast wave with a frequency in the Alfve´n con-tinuum excites a local resonant Alfve´n wave. Also forky

Þ0 discrete fast eigenmodes with a frequency in the Alfve´ncontinuum are transformed in quasimodes and these quasi-modes are the natural oscillations modes of the system.

Driving normal to the magnetic surfaces excites fastwaves. Forky50 that is the end of the story. There are notany resonant waves excited in the system. ForkyÞ0 the fastwaves excite resonant Alfve´n waves and also the quasimode.This can be summarized as follows~Channel I!:

jx :Fast waves⇒resonant Alfve´n wavesvd5vA~xA!,

⇒quasimodevQ , ~4!

wherevd is the frequency present in the driving spectrumandvQ is the frequency of one of the quasimodes.

Driving in the magnetic surfaces perpendicular to themagnetic field lines excites Alfve´n waves. Forky50 theseare torsional Alfve´n waves which are resonant on the mag-netic surface where their local dispersion is satisfied. Thereare no quasimodes excited. ForkyÞ0, the Alfven waves, inaddition to being resonant on the magnetic surface wheretheir local dispersion relation is satisfied, also excite fastwaves. These fast waves excite resonant Alfve´n waves andalso the quasimode. This can be summarized as follows~Channel II!:

jy :Alfven waves⇒Resonant Alfve´n wavesvd5vA~xA!

⇒Fast waves

⇒H resonant Alfve´n waves

quasimodevQ .~5!

B. Stationary state in viscoresistive MHD

The stationary state in viscoresistive MHD of drivenresonant Alfve´n waves for single frequency driving is de-scribed by Eq.~1! with the time derivative]/]t replaced by2 ivd . For a closed loop, use of a Fourier series inz resultsin an infinite system of~uncoupled! equations. The footpointmotions are included as inhomogeneous boundary condi-tions. References 27–29 studied single frequency driven tor-sional Alfven waves. Ruderman30 showed that the steadystate of directly driven axisymmetric Alfve´n waves in coro-nal loops found in Ref. 28 is the asymptotic state of a non-stationary solution. In addition, Refs. 31 and 32 determinedthe stationary state solutions excited by footpoint motionsnormal to the magnetic surfaces having a nonzero divergenceand a nonzero vorticity. Reference 33 considered asymmetricdriving (kyÞ0) in the magnetic surfaces perpendicular to themagnetic field lines. The results in Ref. 33 are particularlyinteresting as they reveal a complicated dependence of ab-sorption on the combinationky ,vd with a prominent role forthe quasimodes. In addition to enhanced absorption forquasimode frequencies, they found an antiresonance line in

2373Phys. Plasmas, Vol. 8, No. 5, May 2001 Resonant and phase-mixed magnetohydrodynamic waves . . .

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the (ky ,vd) plane of reduced absorption. They related theseresults to the interaction of the direct Poynting flux associ-ated with the directly driven Alfve´n waves and the indirectPoynting flux due to the Alfve´n waves driven by fast waves.

C. Temporal evolution in ideal MHD

In principle, the stationary state of driven MHD wavesdescribed by Eqs.~1! is attained fort→`. The importantquestion is how long it takes for the waves to create suffi-ciently short length scales. The time required for phase mix-ing to attain these short length scales should be shorter thanthe lifetime of the loops that are to be heated. The evolutiontowards short length scales can be studied in ideal MHD upto the point where the length scales become too short fordissipation to be further ignored. The equations that governthe temporal evolution of MHD waves in ideal MHD areobtained by dropping the dissipative terms in Eqs.~1!. Thetemporal evolution in ideal MHD forky50 was studied inRefs. 34 and 35. We focus here on asymmetric driving withkyÞ0.

1. Monoperiodic driving normal to the magneticsurfaces

Here channel I applies. The fast wave with frequencyvd

excites a resonant Alfve´n wave at the positionxA wherevd

5vA(xA) and also the quasimode at the positionxQ wherevA(xQ)5vQ . Figure 1 shows the temporal evolution ofjy

for vdÞvQ ~left-hand side inset! and for vA(xQ)5vQ

~right-hand side inset!. The left-hand side inset shows thatinitially the quasimode is more prominently present than theresonant Alfve´n wave. The Alfve´n wave that is excited is nota natural oscillation mode of the system. This implies that apersistent monoperiodic driving at the frequencyvd is re-quired for building up this resonant Alfve´n wave with thecreation of increasingly smaller length scales. The quasi-mode is initially clearly present but is phase-mixed away astime progresses. The right-hand side inset shows that there is

a resonant coupling between the excited quasimode and thelocal Alfven continuum mode. The quasimode is excitedright from the start with a much larger amplitude than in theprevious case. Moreover the evolution towards increasinglylarger amplitudes and smaller length scales is clearly muchfaster than in the previous case. Since the excited quasimodeis a natural oscillation mode which combines the propertiesof a global fast eigenmode and a local Alfve´n wave, there isalso an enhanced excitation ofjx . The solution forjy oscil-lates between a smoothed delta function and a smoothed 1/sfunction while the solution forjx oscillates between asmoothed logarithm and a smoothed jump.36 The lengthscale of the resonance defined as the full width at half maxi-mum, decreases inversely proportional to timet for bothcases, although the difference in reduction is clear.36,37

According to a recent study on the transition to steadystate in the approximation of a thin magnetic cavity,38 twocharacteristic transitional times are relevant. The first one isinversely proportional to the decrement of the global mode;the second time, which often is the longest of the two, isproportional toR1/3.

2. Random driving normal to the magnetic surfaces

Channel I applies again. De Groofet al.39 have driven aloop by a series of 30 pulses with randomly distributedwidths and randomly distributed time intervals in betweenthe pulses. In contrast to monoperiodic driving, the drivercontains a whole frequency interval~broad band driver! sothat there is no need for fine tuning to hit a quasimode fre-quency. As a matter of fact, all the quasimodes lying in thefrequency interval covered by the driver can be excited. Asshown in Fig. 2~left-hand side inset! where the Alfven en-ergy is plotted againstx, indeed resonances are built up at themagnetic surfaces corresponding to these quasimodes. Themodes corresponding to the first eigenfunctions and smallvalues ofkz are the most dominant. As expected, resonant

FIG. 1. jy component as function ofx at z5L/2 forvdÞvQ ~left-hand side inset! andvd5vQ ~right-handside inset!.

2374 Phys. Plasmas, Vol. 8, No. 5, May 2001 M. Goossens and A. De Groof

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peaks are growing in time at the magnetic surfaces wherevA(xA)5vQ ~vertical lines in the figure!. De Groofet al.39

also calculated the time scale in which the necessary smalllength scales are formed. They concluded that even for theslowest decrease, the time needed to generate a length scaleof about 100 m is about 3 h.

In contrast to previous studies where only one dominantresonance peak was formed, this model shows resonantpeaks that are packed rather closely together. Moreover, avariation of the pulse widths of the driving pulses or theconsideration of a longer loop would result in even morequasimodes coupling to Alfve´n modes and consequently theloop can be heated even more globally. In Fig. 2~right-handside inset!, a corresponding result is shown for a mono-periodic driver. Only one peak results from a fast-Alfve´nenergy transfer. The second peak atx50.6 is forced by thedriver.

Apart from the random driver, a second important dif-ference with previous studies is that all the possible values ofkz are taken into account. The classic assumption of a singlekz excludes all but the first harmonic of the fast modes in thedriving spectrum.40

3. Monoperiodic driving in the magnetic surfacesperpendicular to the magnetic field lines

Here the second channel applies with directly drivenAlfven waves and Alfve´n waves driven by fast waves. Thesetwo types of Alfven waves have different amplitudes and areusually out of phase, with the phase difference depending onthe combination ofky ,vd . Whenvd5vQ the two types ofAlfven waves are in phase providing the best conditions forthe creation of large amplitudes and small length scales. Itturns out that there are combinations ofky ,vd for which thedirectly driven Alfven waves and the Alfve´n waves drivenby fast waves are exactly in antiphase. These combinationsare the worst possible for creating resonant Alfve´n wavesand correspond in the stationary state to the antiresonanceline.33,41

The evolution of the amplitude ofjy in time depends onthe frequencyvd and position. As expected, the evolutiontowards larger amplitudes and smaller length scales is fasteras the difference betweenvd andvQ decreases. On a reso-nant surface the increase of the amplitude is monotone; onnon-resonant surfaces the time variation ofjy is character-ized by beats.41

VI. CONCLUSIONS

The theoretical results collected in this paper give con-fidence that the excitation of resonant Alfve´n waves mayindeed play a key role in heating closed~and also open!loops in the solar corona. Small length scales required forefficient dissipation are attained on sufficiently short timespans. The inherent coupling of the fast waves and Alfve´nwaves for nonaxisymmetric motions (kyÞ0,mÞ0) results intwo channels for exciting resonant Alfve´n waves. The quasi-modes play a fundamental role for both channels as theyprovide ~by far! the best conditions for excitation.

For a monoperiodic driver, the efficiency depends on thefrequency in a complicated way, with optimal efficiency fora quasimode frequency and strongly reduced efficiency atfrequencies where the directly driven and indirectly drivenAlfven waves are in antiphase. In addition to the need forfine tuning the frequency of the driver to the quasimode fre-quency, monoperiodic driving results in a single resonantlayer. A random driver covering a broad band spectrum rem-edies these two weak points in addition to being a betterapproximation of physical reality. The random drivingsource is effectively filtered by the loop producing a distur-bance that is essentially a superposition of all quasimodeswith frequencies in the broad band spectrum. Therefore it isimportant to take into account all possible values ofkz toprevent from excluding all but the first harmonic of the fastmodes in the driving spectrum. In this way, random drivingresults in multiple resonant peaks at the different magneticsurfaces wherevA(xA)5vQ . Consequently the loop isheated more globally. Moreover, if larger lengths are consid-

FIG. 2. Left-hand side inset: Alfve´n energy as function ofx ~integrated overz) for t510– 50 Alfven crossing times after driving with a randomly varyingpulse train. Right-hand side inset: comparison to monoperiodic driver@qualitatively comparable to (t540) left-hand side inset#.

2375Phys. Plasmas, Vol. 8, No. 5, May 2001 Resonant and phase-mixed magnetohydrodynamic waves . . .

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ered, more valuable quasimodes are present and the loop isheated even more globally. Thus we can conclude that ran-dom footpoint driving can act as an efficient dissipationmechanism to heat the loop globally!

Of course, this is not the end of the story. Within thepresent model, random driving in the magnetic surfacesneeds to be explored. Furthermore, there is a need to gobeyond the present model. The backreaction of the chromo-sphere, effects of flow, and effects of kinetic theory need tobe considered.

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