Absorption of magnetosonic waves in presence of resonant ...

1 Astron. Astrophys. 326, 1241{1251 (1997)ASTRONOMYANDASTROPHYSICSA bsorption of magnetosonic waves in presenceof resonant slow waves in the solar Cade z?, A. Cs k??, elyi???, and M. GoossensCenter for Plasma-Astrophysics, Leuven, Celestijnenlaan 200 B, B-3001 Heverlee, BelgiumReceived 10 March 1997 / Accepted 12 May resonant Absorption of slow and fast magne-tosonic waves in a nonuniform magnetic plasma is studied fora simple planar equilibrium model. Propagating slow and fastmagnetosonic waves are launched upwards in a lower uniformlayer. They are partially absorbed by coupling to local resonantwaves in an overlying nonuniform plasma layer at the magneticsurface where the frequency of the incoming wave equals thelocal Alfv en continuum frequency or the local slow slow magnetosonic waves can only be coupled to res-onant slow continuum waves .}

2 For the fast magnetosonic wavesthere are three possibilities as they can be coupled to reso-nant Alfv en continuum waves alone, resonant Alfv en contin-uum waves combined with resonant slow continuum waves ,and resonant slow continuum waves alone. The present paperfocuses on the Absorption of magnetosonic waves by coupling toresonant slow continuum waves either alone or in combinationwith resonant Alfv en continuum results show that the resonant Absorption of slow andfast magnetosonic waves at the slow resonance position stronglydepends on the characteristics of the equilibrium model and ofthe driving wave. The Absorption can produce ef cient localheating of plasma under conditions as in the solar words:Magnetohydrodynamics (MHD) { methods: nu-merical { Sun: corona { Sun: oscillations1.}}}

3 IntroductionThe solar atmosphere contains a wide variety of magnetic struc-tures which can support MHD waves . waves that are gener-ated either by turbulent motions in the photosphere and chro-mosphere, or by global solar oscillations or by local releasesSend offprint requests to: A. Cs k?On leave from Institute of Physics, Box 57, YU-11000 Beograd, leave from Department of Astronomy, E otv os University Bu-dapest, Ludovika t er 2 H-1083 Budapest, Observatory, College Hill, Armagh, BT61 9DG, energy in reconnection events interact with these waves transport energy away from the regionwhere they are generated into the ambient plasma, while dis-sipation causes a deposition of part of the wave energy in theplasma. The classic viscous or resistive damping of for exam-ple Alfv en waves in a uniform plasma is known to be a veryinef cient way to transform wave energy irreversibly into heatin weakly dissipative plasmas with large values of the viscousand magnetic Reynolds numbers as in the solar , a more ef cient mechanism for dissipating waveenergy can occur in nonuniform magnetic plasmas, where res-onant slow and resonant Alfv en waves can exist.

4 In ideal MHDthese resonant waves are con ned to an individual magneticsurface without any interaction with neighbouring magneticsurfaces. Since each magnetic surface has its own local slowfrequency and its own local Alfv en frequency, a nonuniformmagnetic plasma can have two continuous ranges of frequen-cies corresponding to resonant slow waves and resonant Alfv effects cause coupling of the resonant magneticsurface to neighbouring magnetic surfaces. For large values ofthe viscous and the magnetic Reynolds numbers as in the so-lar atmosphere this coupling is weak and the local resonantslow oscillations and the local resonant Alfv en oscillations arecharacterized by steep gradients accross the magnetic of these local slow oscillations or local Alfv en oscil-lations provides a means for dissipating wave energy which isfar more ef cient in weakly dissipative plasmas than classicalresistive or viscous MHD wave damping in a uniform mechanism of resonant wave damping was rst put for-ward as a possible mechanism for heating the solar corona byIonson (1978).

5 The excitation of the local resonant oscillations can be di-rect by driving motions in the magnetic surfaces or indirect bydriving motions normal to the magnetic surfaces. The indirectexcitation relies on global motions that transfer energy acrossthe magnetic surfaces up to the resonant magnetic surface wherethe frequency of the global motion equals the local Alfv en fre-quency or local slow frequency. The excitation of the Cade z et al.: Absorption of magnetosonic waves in presenceresonant waves is indirect since we need magnetosonic wavesthat propagate across the magnetic surfaces to excite studies on heating by indirect excitation have con-sidered incoming fast magnetosonic waves that couple to lo-cal resonant Alfv en waves in a nonuniform plasma (Poedts,Goossens and Kerner (1989), Poedts, Goossens and Kerner(1990), Okreti c and Cade z (1991)).

6 Fast magnetosonic wavespropagate almost isotropically as long as their high frequenciesare above the cutoff frequency for fast waves . Slow magne-tosonic waves are not considered an important means for trans-ferring energy into the corona and for heating the plasma there,primarily because of the limited range of frequencies of the slowcontinuum waves . However, they can play a role when slow andfast magnetosonic waves that are generated in the photosphereinteract with chromospheric magnetic structures and when slowand fast magnetosonic waves that are generated locally in thecorona interact with coronal magnetic this paper the focus is on the Absorption of slow and fastincoming magnetosonic waves by coupling to local resonantslow magnetosonic waves in a nonuniform plasma.

7 This processis studied for an equilibrium con guration which makes themathematical analysis as simple as possible but still containsthe basic physical equilibrium model is speci ed in Sect. 2. In Sect. 3 wepresent the set of ideal MHD equations that govern the linearmotions in a planar 1D equilibrium model. In this Sect. we alsodiscuss the singularities in the differential equations of linearideal MHD and their role for resonant wave damping. The solu-tions of the differential equation of linear ideal MHD in the twouniform plasma layers are also given. In Sect. 4 we explain howthe analysis of resonant waves in dissipative layers (Sakurai,Goossens and Hollweg (1991), Goossens and Hollweg (1993),Goossens, Ruderman and Hollweg (1995) in what follows SGH,GH and GRH respectively) can be applied to obtain the solu-tions in the vicinity of the mathematical singularity of the idealMHD equations.

8 The de nition of the Absorption coef cient andthe method for its computation are given in Sect. 5. In Sect. 6we present our results and The equilibrium con gurationWe consider a static 1D planar equilibrium model composedof two semi-in nite uniform plasma layers which embrace anonuniform plasma layer. We use a system of Cartesian coordi-nates with thez-axis directed downward. The horizontal planesz= 0 andz=Lbound the nonuniform plasma layer from aboveand from below respectively. The plasma is uniform in the re-gionsz<0 (region 1) andz>L(region 2). In the nonuniformplasma layer (0 z L) the equilibrium quantities depend magnetic eld is assumed to be constant in the wholespace and is oriented along thex axis:B0=(B0;0;0). Gravityis neglected in the present characteristic velocity pro les for the considered equilib-rium state with = andn= the magnetic pressurepm B20=2 0is constant, thethermal pressurep0is also constant and so is the plasma param-eter =p0=pm:p0=1 0(z)v2s(z)=const:(1)In addition we can freely specify either the temperatureT0orthe density course,B0=const:implies that 0(z)v2A(z)=const:and 0(z)v2c(z)=const:The object of the present paper is to study the absorptionof slow and fast magnetosonic waves by coupling to local res-onant slow magnetosonic waves .

9 The resonant slow waves arecontrolled by the local cusp frequency which is determined bythe wave vector and the local cusp speed. Hence in the presentcontext it is convenient to prescribe the variation of the squareof the local cusp velocityv2cv2c=v2Av2sv2A+v2s(2)as a function that is monotonous inside the nonuniform layer0 z Land is constant elsewhere:v2c(z)=8> <>:v22=const:;z Lv21 (v21 v22 zL n;0 z Lv21=const:z 0(3)The expressions for density, local Alfv en speed and localsound speed take the simple forms: 0= 00v21v2c(z);v2A= 1+2 v2c(z)v2s= 1+ 2 v2c(z);(4) Cade z et al.: Absorption of magnetosonic waves in presence1243wherev2c(z) is prescribed by Eqs (3). The dimensionless Alfv en,sound, and cusp speed as functions of coordinatezare shownin Fig. use this model for studying the resonant Absorption ofmagnetosonic waves in different structures in the solar atmo-sphere.)

10 The incoming wave is launched from the lower uniformlayerz>Lwith a prescribed frequency!and a prescribedwave vectork, and propagates towards the nonuniform the boundaryz=Lthe wave is partly reflected and partlytransmitted. The energy of the transmitted wave is partly ab-sorbed in the nonuniform plasma due to the resonant excitationof local waves . With our choice of the direction of thez-axis,the incident wave propagates in the negativez-direction and thereflected wave propagates in the positivez-direction. We do notwant to consider leaky waves and restrict our analysis to wavesthat are evanescent in the upper uniform Governing equations and solutionsThe driven waves are studied with the standard set of linearizedequation of ideal MHD:@ 1@t+r ( 0v1)=0; 0@v1@t= rp1+1 0(r B1) B0+1 0(r B0) B1;@B1@t=r (v1 B0)@p1@t+v1 rp0=v2s(@ 1@t+v1 r 0)(5)An index '0' denotes an equilibrium quantity, while an in-dex '1' denotes an Eulerian perturbation.