Fig.4: Typical values of the collisional and viscous damping time ratios for propagating a) Alfvén and fast magnetoacoustic waves and b) acoustic wave calculated from Eqs.(11), (12) for the prominence plasma with B0 =10 G and nn/n = 1.

Of interest for prominences is also the transverse (kpropagation of MHD waves with respect to the background magnetic field. Eqs. (4),(6),(9),(10)

(13)

(14)

Fig.5: Typical values of the collisional and viscous damping time ratios for pro-pagating a) fast magnetoacoustic and b) acoustic wave calculated from Eqs. (13), (14) for the prominence plasma with B0 =10 G and nn/n = 1.

Generalized Ohm’s law and magnetic induction equations

For the self-consistent description of collisional damping of MHD waves the generalized Ohm’s law

(15)

where , , ,

should be included into the modelling set of equations.

Application of the generalized Ohm’s law changes the form of the magnetic induction equation and the Joule heating term (E* j) in the energy equation. In the easiest case of a cold strongly magnetized plasma and relatively slow processes the induction equation is

(16)

where and are the Coulomb and Cowling coefficients of mag-netic diffusion. The induction equation in the partially ionized plasma contains an additional term, which plays a role of an anisotropic magnetic diffusion resulting from the strong dissipation of the transverse electric current due to the ion-neutral collisions.

Similar to the traditional magnetic Reynolds number , a new partially

ionized plasma magnetic Reynolds number can be introduced.

In the partially ionized plasma of the solar chromosphere with C and Rem ~ 105-107

. Thus, traditional diffusion term can be neg-

lected, whereas the anisotropic magnetic diffusion is important, and controls the dy-namics of plasma.

Conclusion• Collisional damping of MHD waves in the partially ionized plasmas of the low solar atmosphere and prominences is always stronger than their viscous damping.

• For a self-consistent description of MHD wave collisional and viscous damping in the partially ionized plasmas, besides of inclusion of appropriate terms into momentum and energy equations, the kinetic pressures pk, k=i,e,n in the generalized Ohm’s law should be replaced by generalized pressures, , k=i,e,n containing the viscous stress tensors .

• The expressions used here are valid only if damping decrements (linear approxi- mation). Therefore our analysis is correct only for waves with f = fc . De- pending on plasma parameters, dissipation mechanism and particular MHD mode, the critical frequency fc varies from ~ 0.1 Hz till 1011 Hz. For more details see Khodachen- ko et al., 2004.

• The damping time of a wave measured in wave periods , can be expressed as

.

ReferencesBraginskii, S.I., Transport processes in a plasma, in: Reviews of plasma physics, Vol.1, Consultants Bureau, New York, 1965.DePontieu, B., Haerendel, G., A&A, 338, 729, 1998.Khodachenko, M.L., Arber, T.D., Rucker, H.O., Hanslmeier, A., Collisional and viscous dam- ping of MHD waves in partially ionized plasmas of the solar atmosphere, A&A, (submitted)Nakariakov, V.M., Verwichte, E., Berghmans, D., Robbrecht, E., A&A, 362, 1151, 2000.Ofman, L., ApJ, 568, L135, 2002.Piddington, J.H., MNRAS, 116, 314, 1956.Vernazza, J.E., Avrett, E.H., Loeser, R., ApJ Suppl, 45, 635, 1981.

Abstract

A comparative study of the efficiency of MHD wave damping in solar plasmas due to collisional and viscous energy dissipation mechanisms is presented here. The damping rates are taken from Braginskii (1965) and applied to the VAL C model of the quiet Sun (Vernazza et al., 1981). These estimations show which of the mechanisms are dominant in which regions. In general the correct description of MHD wave damping requires the consideration of all energy dissipation mechanisms via the inclusion of the appropriate terms in the generalized Ohm's law, the momentum, energy and induction equations. Specific forms of the generalized Ohm's Law and induction equation are presented that are suitable for regions of the solar atmosphere which are partially ionised.

Introduction

Magnetohydrodynamic (MHD) waves are widely considered as a possible source of heating for various parts of the outer solar atmosphere. The heating effect of MHD waves is connected with a certain dissipation mechanism which converts the energy of damped MHD waves into the energy of the background plasma. Among the main energy dissipation mechanisms which play an important role under the solar conditions, are collisional dissipation (resistivity) and viscosity. The presence of neutral atoms in the partially ionized plasmas of the solar photosphere, chromosphere and prominences enhances the efficiency of both these energy dissipation mechanisms.

In previous publications collisional and viscous damping of MHD waves in partially ionized solar plasmas has been always considered separately. One group of researchers assumes the friction between ions and the neutral fraction is the dominant dissipation mechanism (Piddington, 1956; DePontieu & Haerendel, 1998). Another group of authors takes into account only the viscosity effects (Nakariakov et al., 2000; Ofman, 2002). Because of different physical nature of the above two mechanisms of MHD wave dissipation, in each particular case a certain reasoning is needed why one of the mechanisms is neglected and other is not. In the general case, the correct description of MHD wave damping requires the consideration of both dissipation mecha-nisms via the inclusion of the appropriate terms in the generalized Ohm's Law, as well the momentum and energy equations.

While the frictional dissipation (electrical resistivity) can usually be safely neglected as compared to viscosity effects in the fully ionized solar corona, the relation and mutual role of both these dissipation mechanisms for the damping of different types of MHD waves in the lower (partially ionized) solar atmosphere requires a special comparative study. Here we give some elementary estimations of the frictional and viscous damping of MHD waves which are needed to conclude on which of two dissipation mechanisms is dominant in different regions of the solar atmosphere.

MHD waves damping in the linear approximation

In this section we summarize the results from Braginskii (1965) on the linear damping rates of MHD waves, obtained from the calculation of the energy decay time using the local heating rates Qfrict, Qvisc, etc. These results are applied in later sections. The decay of wave amplitude is described by the complex frequency iso that the energy in a particular wave mode will decay as e- t where = (2)-1 is a characteristic wave damping time and is the logarithmic damping decrement.

A. The case of fully ionized plasma

Alfvén wave (A.w.) :

(1)

(2)

Fast magnetoacoustic (or magnetosonic) wave (f.ms.w.) :

(3)

(4)

Acoustic (or sound) wave (s.w.) :

(5)

(6)

Since in the fully ionized plasma , we use no-

tation Joulein place of frict. and are components of electr.conduc-tivity relative the magnetic field and 0, 1, 2 are viscosity coefficients

B. The case of partially ionized plasma

In the partially ionized plasma the expression for the frictional heating besides of the Joule heating contains and extra term arising due to the plasma - neutral gas collisional friction (Braginskii, 1965):

(7)

where , . Here

and are relative densities, and

, k = e,i ; l = i,n are effective collisional

frequencies.

For the Alfvén wave G=0, while for the fast magnetoacoustic wave G is small as compared to the first term in brackets in (7). Thus the fricti-onal heating term for these modes in the partially ionized plasma has a form of the Joule heating:

, where is well known Cowling

conductivity. In a strong enough magnetic field C .

Thus, the expressions (1) and (3) can be re-written for the damping decrements in a partially ionized plasma as the following:

Alfvén wave (A.w.) :

(8)

Fast magnetoacoustic (or magnetosonic) wave (f.ms.w.) :

(9)

For Acoustic wave (s.w.) both terms in brackets in (7) are important, and the frictional damping decrement in the case mi = mn is

(10)

Note that throughout this poster variables with a tilde relate to partial-ly ionized plasmas.

Application to the Sun

From (1), (3) and (8), (9) follows that parallel propagating ( k) Alfven and fast magnetoacoustic waves are equally damped due to col-lisional (Joule) dissi-pation. But because of C ( see Fig. 1 ) this damping is much higher in the partially ionized plasma.

Parallel propagating acoustic waves are not damped due to friction in a fully ionized co- rona (5), but are damped in partially ionized solar photo- spheric, chromospheric and prominence plasmas (10).

Viscous damping of MHD waves ((2), (4), (6)) in diffe-

rent parts of the solar atmo- sphere is defined by chang- ing with height plasma densi ty and viscosity coefficients.

Fig.1: Variation of C with height for the quiet Sun model (Vernazza et al. 1981) for different B0 : 1) 10 G; 2) 100 G; 3) 1000 G.

A. MHD wave damping in Photosphere/Chromosphere

Alfvén & Fast magnetoacoustic waves (case k):

Eqs. (2),(4),(8),(9) (11)

Weakly ionized photosphere ( , ): Eq. (11) gives

, where ,

Fig.2: Variation of with height in the low solar atmosphere

(Vernazza et al. 1981) for different B0 : 1) 5G; 2) 10G; 3) 100G; 4) 1000G.

Acoustic waves (case k) -- Important for non-magnetized regions in the

low solar atmosphere:

Eqs. (10),(6) = (12)

Fig. 3: Defined by Eq(12) variation of

with

height in the solar chromosphere (Ver nazza et al.1981).

B. MHD wave damping in Prominences

Prominence material is an important example of the solar partially ionized plasmas: T=(6...10)103 K; n=(1...50)10 cm -3; nn/n = 0.05...1, B0 ~10 G.

Eqs. (11), (12) can be used for calculation of the Alfvén, fast magnetoacoustic and acoustic waves collisional and viscous damping time ratios.

On the mechanisms of MHD wave dampingin the partially ionized solar plasmas

M. L. Khodachenko 1, H. O. Rucker 1,T. D. Arber 2 and A. Hanslmeier 3

(1) Space Research Institute, Austrian Academy of Sciences, Schmiedlstr.6, A-8042 Graz, Austria(2) Dept. of Physics, University of Warwick, Coventry, CV4 7AL, United Kingdom

(3) IGAM, Karl-Franzens University of Graz, Universitätsplatz 5, A-8010 Graz, Austria

D3.5-0045-04

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