In weakly dissipative media governed by the magnetohydrodynamics (MHD) equations, any efficient mechanism of energy dissipation requires the formation

of small scales. Using numerical simulations, we study the properties of Alfv´en waves propagating in a compressible inhomegeneous medium, with an inhomogeneity transverse to the direction of wave propagation. Two dynamical effects, energy pinching and phase mixing, are responsible for the small-scales formation, similarly to the incompressible case. Moreover, compressive perturbations, slow waves and a static entropy wave are generated; the former are subject to steepening and form shock waves, which efficiently dissipate their energy, regardless of the Reynolds number. Rough estimates show that the dissipation times are consistent with those required to dissipate Alfv´en waves of photospheric origin inside the solar corona

Malara, F.

Primavera, L.

Veltri, P.

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IL NUOVO CIMENTO VOL. 20 C, N. 6 Novembre-Dicembre 1997Dissipation of Alfve´n waves in compressibleinhomogeneous media()F. MALARA, L. PRIMAVERA and P. VELTRIDipartimento di Fisica, Universita` della Calabria - I-87030 Arcavacata di Rende (CS), Italy(ricevuto il 5 Dicembre 1996; approvato il 27 Febbraio 1997)Summary. — In weakly dissipative media governed by the magnetohydrodynamics(MHD) equations, any efficient mechanism of energy dissipation requires the forma-tion of small scales. Using numerical simulations, we study the properties of Alfve´nwaves propagating in a compressible inhomegeneous medium, with an inhomogeneitytransverse to the direction of wave propagation. Two dynamical effects, energy pinch-ing and phase mixing, are responsible for the small-scales formation, similarly to theincompressible case. Moreover, compressive perturbations, slow waves and a static en-tropy wave are generated; the former are subject to steepening and form shock waves,which efficiently dissipate their energy, regardless of the Reynolds number. Rough es-timates show that the dissipation times are consistent with those required to dissipateAlfve´n waves of photospheric origin inside the solar corona.PACS 96.60.Pb – Corona; coronal loops, streamers, and holes.PACS 52.35.Bj – Magnetohydrodynamic waves.PACS 01.30.Cc – Conference proceedings.1. – IntroductionThe dissipation of the energy of magnetohydrodynamic (MHD) waves represents oneof the mechanisms which have been proposed to explain the high temperature observed inthe plasma of the solar corona. The waves have a probable origin in the photospheric mo-tions and propagate along the magnetic field, which completely threads the corona. Thecorresponding energy flux [1] compares favorably with that required to heat the corona.However, in order to heat the plasma the wave energy must be significantly dissipatedbefore it leaves the system, i.e. due to the smallness of the dissipative coefficients in thecorona, the energy must be transferred to small scales. This can be achieved by the inter-action between the wave and the inhomogeneities of the coronal structures. This processhas been first studied analyzing the normal modes of inhomogeneous MHD structures.() Paper presented at the VII Cosmic Physics National Conference, Rimini, October 26-28, 1994.G Societa` Italiana di Fisica 903904 F. MALARA, L. PRIMAVERA and P. VELTRIIt has been found that these solutions develop small scales either localized in thin lay-ers where resonance conditions are satisfied (resonant modes, [2-12]) or across the wholeinhomogeneity region (resistive modes, [13, 14]).The study of normal modes gives useful indications on the dissipation mechanisms andon the associated time scales, but it leaves some important questions open: how are thesemodes formed? How long does it take to form the small-scale structures? The answers tothese questions require the study of the dynamical evolution of an initial wave in its propa-gation in a nonuniform medium. Using an asymptotic analysis Lee and Roberts [15] foundtwo main dynamical effects responsible for small-scale formation: i) energy pinching: theenergy of the wave concentrates at the resonance locations; ii) phase mixing [16], dueto inhomogeneities of the Alfve´n speed, transfers the disturbance energy to increasinglysmall-scale structures. Malara et al. [17], using numerical simulations of the MHD in-compressible equations showed that the both mechanisms are at work, but phase mixingor energy pinching dominates, according to the disturbance wavelength. In the formercase the small-scale formation time ssis roughly proportional to S 1=3 [4], S being theReynolds number. However, due to the large values of S, ssis too large to achieve anefficient dissipation within the corona.In the present paper we will discuss, using a numerical simulation code, the propertiesof Alfve´n waves propagation in a compressible medium, in oblique propagation. The aim isto elucidate the main differences in the mechanism of small-scale formation with respectto the incompressible case [17], and to see whether the coupling between compressibleand incompressible modes can speed up the dissipation. Compressive fluctuations un-dergo steepening and can give origin to shocks; this process corresponds to the formationof infinitely small lengths in a finite time (catastrophe), and the dissipation time is inde-pendent of S. This process is of interest in a medium where S is very large, like the solarcorona.2. – Numerical modelThe basic equations of our model are the compressible, dissipative, MHD equations,which can be written in the following form, using dimensionless variables:@@+r  (v) = 0;(1)@v@+ (v  r)v =  1r(T ) +1(r b) b+1Sr2v;(2)@b@= r (v  b) +1Sr2b;(3)@T@+(v  r)T+( 1)T (r  v)= 11Sr2T+1S@vi@xj@vi@xj+1S(r b)2;(4)where  is the density normalized to a characteristic value 0; b is the magnetic fieldnormalized to a value B0; v is the velocity normalized to the Alfve´n velocity cA0=DISSIPATION OF ALFVE´N WAVES IN COMPRESSIBLE INHOMOGENEOUS MEDIA 905B0=(40)1=2; T is the temperature normalized to mpc2A0=kB(with  the mean molec-ular weight, mpthe proton mass, and kBthe Boltzmann constant). The space variablesand the time  are normalized to a characteristic length a and to a=cA0, respectively. Thequantities S= acA0= and S= 4acA0=(c2) represent, respectively, the kinetic andmagnetic Reynolds numbers, while S= [=(mp)]S, and  is the adiabatic index.Equations (1)-(4) are solved in a rectangular spatial domain D = [ l;+l]  [0; Rl].The parameter l gives a measure of the domain width in units of the shear length a, whileR determines the aspect ratio of the domain.Since we will describe waves which essentially propagate along the y-direction, weimpose periodic boundary conditions at y = 0 and y = Rl, and free-slip boundary condi-tions at x = l (vanishing normal derivatives).The initial condition is represented by the superposition of an (ideal) equilibrium struc-ture and a perturbation. The equilibrium structure is defined byeq= 1 ; veq= 0 ;(5)beq=1 +tanhx xcosh2l(cos ey+ sin ez) ;(6)Teq(x) =peqeq jbeq(x)j22eq;(7)where  is the angle between beqand the propagation direction y;  measures the am-plitude of the inhomogeneity (which is in the x-direction). The temperature profile Teq(x)satisfies the total (magnetic + kinetic) pressure equilibrium. The total pressure peqisdetermined by peq= b2eq(l)=2; the temperature Teqis positive in the whole domain, pro-vided that  > 1. The parameter  determines the plasma :(x) =c2s(x)c2A(x)=2 jbeq(l)j2jbeq(x)j2  1!;(8)where cs(x) =pTeq(x) is the local sound velocity and cA(x) = beq(x) is the local Alfve´nspeed. Large values of  correspond to large average  values.Since waves in the corona of photospheric origin are believed to be essentiallyAlfve´nic [18,19], we have modeled such disturbances by superimposing on the above equi-librium structure the following perturbation:v(x; y) = b(x; y) = r [A1Re [(1=) exp[iy]]ez] ;(9)where A1is the amplitude of the perturbation and  is the wave number. This form cor-responds to a plane wave front initially propagating along the y-direction, with v and bpolarized along the x-axis.Equations (1)-(4) have been numerically solved using a 2 12-D pseudospectral code [20].At the initial time the equilibrium structure is homogeneous along the y-direction, whilethe perturbation has a vanishing space average along the same direction. Then weconsider any variable f as the superposition of two contributions: the average in they-direction f0, which we ascribe to the equilibrium structure, and the fluctuating partf = f   f0.906 F. MALARA, L. PRIMAVERA and P. VELTRI0 20 40 60 80 10001234w()  106Fig. 1. – Power dissipated on fluctuations as a function of the time for: S = 2000,  = 1:01 (thickline); S = 104,  = 1:01 (dashed line); S = 2000,  = 3:01 (thin line).3. – Numerical resultsIn the first run we used the following values for the parameters of the model:  = 1,corresponding to a wavelength in the y-direction y= 2, i.e. of the same order as theinhomogeneity scale length; the propagation angle is  = =4; the initial normalizedperturbation energy is "( = 0) = 2:510 5 (corresponding to a small amplitude pertur-bation); we chose  = 1:01 and  = 0:25, corresponding to  ranging between 8  10 3and 1:49; the Reynolds numbers are S= S= S=(   1) = 2000; the domain sizein the x-direction, determined by l, has been chosen sufficiently large (l = 20) to avoidthat the boundary conditions affect the time evolution; finally, the domain length in they-direction has been chosen equal to y, in order to have the maximum spatial resolutionin that direction (this corresponds to R = 2=l).This configuration, in the case of homogeneous equilibrium structure, would corre-spond to an Alfve´n monochromatic wave in oblique propagation, the wave vector beingparallel to the y-axis. The interaction of such a disturbance with the equilibrium inhomo-geneity generates both a modulation in the x-direction and an energy transfer to the yand z components, thus destroying the initial Alfve´nic character of the disturbance. Infig. 1 we plotted the fluctuation dissipated power, which is defined byw()=1E(0)(0)ZD(1S"@vi@xj2 @v0i@xj2#+1S[(r b)2 (r b0)2])dx dy;(10)where E(0)(0) is the initial equilibrium energy. It is seen that w() increases up to a maxi-mum and then it decreases, roughly exponentially, in time. This indicates that an effectivegeneration of small scales takes place; the time sswhen w() is maximum represents thesmall-scale formation time, and it gives a measure of the time necessary to dissipate arelevant part of the fluctuation energy. The dissipation time hom= S=, which the sameperturbation propagating in a homogeneous structure would have taken to be dissipated,is much longer than ss. Increasing the Reynolds numbers by a factor 5 (fig. 1), we verifiedDISSIPATION OF ALFVE´N WAVES IN COMPRESSIBLE INHOMOGENEOUS MEDIA 9070 1 2 3 4 5 6y-0.4-0.20.00.20.4  = 5 = 15 = 20 = 10Fig. 2. – Profiles of the density fluctuation as function of y, in the large-amplitude run, at x = 0.that ss/ S0:32, which is the scaling law typical of phase mixing [4]. Then, similarly to theincompressible case [17], for wavelengths of the order of the inhomogeneity length, phasemixing dominates in the small-scale formation process. However, increasing the value of (fig. 1), we verified that larger values of  correspond to larger ss, i.e. the dissipationprocess is faster in the more compressible case.Since small scales form in the x-direction, the effective wave vector is quasi-perpendicular to B0. Then, the direction perpendicular to B0and to the x-axis (z0) cor-respond to the Alfve´nic polarization, while the direction parallel to B0(y0) to the mag-netosonic polarization. The time evolution shows that the perturbation energy, initiallypolarized in the x-direction, in the inhomogeneity region is completely transferred tothe other directions (y0 and z0); these two polarizations evolve in different ways. They0-polarized fluctuations are formed by a superposition of two kinds of perturbations:1) A slow magnetosonic disturbance; we verified that the fluctuations  and By0 asso-ciated to such a disturbance are correlated as in a slow magnetosonic wave with quasi-perpendicular wave vector. Moreover, it oscillates with the local value of the cusp fre-quency !cusp= cAcs=[k(c2A+ c2s)1=2]; since both the Alfve´n velocity cAand the soundvelocity csvary across the inhomogeneity, the disturbance profile undergoes phase mix-ing which is responsible for small-scale formation in that polarization. The propagationdirection is the same as that of the initial wave. 2) A static entropy wave, in which  andT are anticorrelated, and the gas pressure fluctuation p is vanishing. This perturbationis due to the coupling between the initial wave and the entropy transverse modulationassociated to the equilibrium.The z0-polarized fluctuation is essentially Alfve´nic: Bz0' vz0 . It oscillates at the localvalue of the Alfve´n frequency !A= cA=kand then it undergoes both energy pinchingand phase mixing. The time evolution is similar to that observed in the incompressiblecase; actually, due to its Alfve´nic character, this disturbance should not be affected by thecompressibility of the medium. Globally,  10%–15% of the energy of the initial Alfve´nicwave is converted into a compressive disturbance, while the remaining part keeps thenon-compressive character.We have performed one more run characterized by larger values of the initial ampli-908 F. MALARA, L. PRIMAVERA and P. VELTRItude (vx= bx' 0:29). In this case, nonlinear effects play an important role in the wavedissipation. The dissipated power, rescaled to take into account that the amplitude is dif-ferent with respect to the previous run, increases faster and its maximum value is largerby a factor  1:5. This maximum is reached before the corresponding time of the small-amplitude case. The increase of w() is due to the formation of compressive shocks in thesimulation domain. In fig. 2 the profiles of  as functions of y at different times in the in-homogeneity region are shown. The small scales associated to the shock front contributeto dissipate the initial wave energy. This shock belongs to the slow mode, the density fluc-tuation being anticorrelated with By0 ; its formation follows the steepening of the slowmagnetosonic perturbation which form in the inhomogeneity region. The shock normal isstrongly oblique with respect to the y-direction as a consequence of the dependence on xof both cAand cs.4. – Discussion and conclusionsIn this paper we studied the formation of small scales in the propagation of an ini-tial Alfve´nic disturbance in a compressible medium along an inhomogeneous equilibriumstructure, in the case where the inhomogeneity direction is perpendicular both to the ini-tial propagation direction and to the background magnetic field. The main differencesbetween the compressible and incompressible [17] simulations can be summarized as fol-lows:i) In both cases small scales are efficiently formed by the same dynamical effects (phasemixing and, to a lower extent, energy pinching). However, the efficiency of these processesincreases in the compressible case, i.e. decreasing .ii) In the compressible case the interaction between the Alfve´nic disturbance and theinhomogeneous equilibrium structure generates slow and entropy waves. When the dis-turbance amplitude is sufficiently high the compressible perturbations steepen and thenform shock waves.As a consequence, the propagation of an Alfve´nic perturbation in a compressible in-homogeneous medium could be more efficient in producing small scales than the corre-sponding propagation in an incompressible medium, in that in the former case a sort ofcatastrophe can be observed, due to the production and steepening of compressible fluc-tuations.Now let us assume a wave period of say 10 s and an Alfve´n velocity cA 103 km/s, aninhomegeneity length a  1:6 103 km (corresponding to  = 1). We have found for thetypical time of small-scale formation ss 55 a=cAfor S = 104. Using the scaling ss/S0:31, and assuming S  108, we obtain a dissipation length lss cAss 1180 a  1:9106 km, i.e. small scales are formed by phase mixing and the wave energy is efficientlydissipated when the wave propagates less than 3Rabove the photosphere.For a fluctuating velocity to Alfve´n velocity ratio of about 0:29 (corresponding to veloc-ity fluctuations of the order of 290 km/s) we have found that the shock wave is formed ata time sh' ss 20–25 a=cA. The shock formation time is proportional to the perturba-tion amplitude, which, for our simulation, is larger than the velocity fluctuations observedin the corona by at least a factor 6. Multipling by the same factor sh, in coronal conditionsthe shock formation length is lsh' cAsh' 120–150 a  1:9–2:4 105 km, i.e. of the or-der of or smaller than one third Rand thus much smaller than the length for small-scaleformation lssdue to phase mixing. The energy associated with these shock waves can beas high as 10%–15% of the initial perturbation energy.DISSIPATION OF ALFVE´N WAVES IN COMPRESSIBLE INHOMOGENEOUS MEDIA 909  This work is part of a research program, which is financially supported by the Mini-stero dell’Universita` e della Ricerca Scientifica, the Consiglio Nazionale delle Ricerche(CNR) (contract 94.00802.CT02), and the Agenzia Spaziale Italiana (ASI) (contract 1994RS 164).REFERENCES[1] HOLLWEG J., in Solar Wind Five, edited by M. NEUGEBAUER (NASA CP-2280), 1983, p. 5.[2] KAPPRAFF J. M. and TATARONIS J. A., J. Plasma Phys., 18 (1977) 209.[3] MOK Y. and EINAUDI G., J. Plasma Phys., 33 (1985) 199.[4] STEINOLFSON R. S., Astrophys. J., 295 (1985) 213.[5] BERTIN G., EINAUDI G. and PEGORARO F., Comment Plasma Phys. Control. Fusion, 10(1986) 173.[6] EINAUDI G. and MOK Y., Astrophys. J., 319 (1987) 520.[7] DAVILA J. M., Astrophys. J., 317 (1987) 514.[8] HOLLWEG J., Astrophys. J., 312 (1987) 880.[9] HOLLWEG J., Astrophys. J., 320 (1987) 875.[10] DAVILA J. M., in Mechanisms of Chromospheric and Coronal Heating, edited by P.ULMSCHNEIDER, E. PRIEST and R. ROSNER (Springer-Verlag, New York) 1991, p. 464.[11] STEINOLFSON R. S. and DAVILA J. M., Astrophys. J., 415 (1993) 354.[12] CALIFANO F., CHIUDERI C. and EINAUDI G., Phys. Plasmas, 1 (1994) 43.[13] CALIFANO F., CHIUDERI C. and EINAUDI G., Astrophys. J., 365 (1990) 757.[14] CALIFANO F., CHIUDERI C. and EINAUDI G., Astrophys. J., 390 (1992) 560.[15] LEE E. M. and ROBERTS B., Astrophys. J., 301 (1986) 430.[16] HEYVAERTS J. and PRIEST E. R., Astron. Astrophys., 117 (1983) 220.[17] MALARA F., VELTRI P., CHIUDERI C. and EINAUDI G., Astrophys. J., 396 (1992) 297.[18] NARAIN U. and ULMSCHNEIDER P., Space Sci. Rev., 54 (1990) 377.[19] VELLI M., GRAPPIN R. and MANGENEY A., Geophys. Astrophys. Fluid Dyn., 62 (1991) 101.[20] MALARA F., VELTRI P. and CARBONE V., Phys. Fluids B, 4 (1992) 3070.

Dissipation of Alfven waves in compressible inhomogeneous media

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Dissipation of Alfven waves in compressible inhomogeneous media

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