Propagation and Dissipation of Slow Magneto-Acoustic Waves in Coronal Loops

Transcription

1 Reearch Journal of Recent Science ISSN 77- Vol. 1(), 34-41, Feb. (1) Re.J.Recent Sci. Propagation and Diipation of Slow Magneto-Acoutic Wave in Coronal Loop Abtract Kumar Pradeep 1, Singh Bhupendra, Chauhan Rajmani 1, and Kumar Anil 3 1 Department of Phyic, K G K College, Moradabad 44 1, Uttar Pradeh, INDIA Department of Phyic, Hindu College, Moradabad 44 1, Uttar Pradeh, INDIA 3 Dept. of Applied Science, Vivehwarya Intitute of Engg. and Tech., Dadri, G. B. Nagar, UP, INDIA Available online at: (Received 8 th January 1, revied 14 th January 1, accepted 16 th January 1) We tudy the patial damping of low magnetoacoutic wave in homogeneou, iothermal, and unbounded coronal plama permeated by a uniform magnetic field, with phyical propertie akin to thoe of coronal loop. Taking into account an energy equation with optically thin radiative loe, thermal conduction, and heating we obtain a fourth-order polynomial in the wavenumber k, which repreent the diperion relation for low and thermal MHD wave. The fourth order diperion relation ha been olved numerically for different loop parameter. It i found that damping length of low-mode wave exhibit varying behavior depending upon the phyical parameter of the loop. We found that for olar coronal loop, the dominant wave damping mechanim i compreive vicoity and thermal conduction with le ignificant contribution by radiation. For any conidered period, low wave have much horter damping length in hot coronal loop than that in cool loop and alo low wave damped very quickly in hot and long coronal loop. Keyword: Sun: Corona, MHD low wave, ocillation, coronal loop, heating. Introduction Recent obervational evidence from variou ground- and pace baed olar miion have confirmed that the Sun outer atmophere i highly dynamic. The obervation of Yohkoh, SoHO, TRACE, Hinode/EIS, STEREO and SDO/AIA revel that there i lot of example of mall amplitude wave and ocillation in different coronal tructure. Thee wave and ocillatory activitie of the olar corona are mainly in the form of low magneto-acoutic wave. Such wave mode are predominantly longitudinal and affect the coronal plama by perturbing the denity, temperature and the component of the velocity parallel to the magnetic field. Longitudinal ocillation may be in the form of either tanding or propagating low wave. Propagating intenity ocillation were firt oberved in coronal hole high above the limb uing the obervation of Ultra Violet Coronal Spectrometer (UVCS) on board SoHO 1. Similar propagating intenity ocillation (compreive wave) were alo oberved in olar polar plume with Extreme Imaging Telecope (EIT) on board SoHO with period of about 1 15 minute. Thee propagating ocillation were interpreted a the low magnetoacoutic wave 3-4 due to their propagating peed ( 1 km -1 ) that are cloe to the ound peed in the corona. There have alo been many obervation of coronal loop that are conitent with wave-guiding of the low mode along the loop tructure. Thee are oberved a low amplitude intenity and velocity ocillation located at the bae of quiecent coronal loop ytem 5-6. The damping of low magneto-acoutic wave in homogeneou medium ha been invetigated by taking into account thermal conduction and compreive vicoity 7. They found that the incluion of thermal conduction reult in thermal mode, which i purely decaying in cae of tanding wave, but i ocillatory and decaying in the cae of driven wave. The excitation and damping of tanding low-mode ocillation and wave ha alo been tudied by everal reearcher 8-1. They found that low tanding wave are attenuated within a few wave period. The main objective of thi paper i to invetigate the patial damping of low magneto-acoutic wave in coronal loop due to three poible damping mechanim, namely compreive vicoity, thermal conduction and radiation. The paper i organized a follow. Theoretical conideration and the baic equation governing plama motion are given in ection, a general diperion relation for the complex wavenumber a a function of frequency i alo derived in thi ection. In Section 3, the diperion relation i olved numerically and dicued to examine how wave damping depend upon compreive vicoity, thermal conduction and radiation. Finally, concluion are drawn in ection 4. Methodology Phyical Model and Governing Equation: We conider the propagation of low MHD wave in homogeneou, magnetically tructured and compreible plama. We retrict our attention to a motion along the uniform background magnetic field, which i directed along the z-axi. The baic International Science Congre Aociation 34

2 Reearch Journal of Recent Science ISSN 77- Vol. 1(), 34-41, Feb. (1) one-dimenional MHD equation governing the plama motion in coronal loop are the equation of continuity, equation of momentum, energy equation, and equation of tate 11. Conidering mall perturbation from the equilibrium in the form B ( r, t ) = B + B 1 ( r, t ), p ( r, t ) = p + p 1 ( r, t ), (1) ρ ( r, t ) = ρ + ρ ( r, t ), T ( r, t ) = T + T ( r, t ), v ( r, t ) = v ( r, t ) we linearize the baic equation (1) (4) of 11, to obtain the wave equation ( γ 1) κt 3 ρχ T 4η γγ ( 1) κt ( αγ )( 1) t γp z γp 3γ p γp t z vz c 4η 4η z T ( 1) γ ρχ + + 3γ p t 3γ p p t γγ ( 1) κt ρχt = + t γp p v ( γ 1) z z t () Now, ince the medium i unbounded and uniform we perform a Fourier analyi auming v( z, t) = v exp i( ωt kz) (3) ( ) and obtain the following diperion relation for low mode in term of wavenumber a function of frequency under the combined effect of compreive vicoity, thermal conduction and radiation a i( γ 1) κt ( ) 4η γγ 1κ T 4 c c ω k γp 3γp γp 4η γγ ( 1) κt 4η αγ ( 1) ρχ T ρχ T cω i cω i ω c ω i( γ )( 1) c k 3γp γp 3γp p γp ρχα T 3 i ( γ 1) ω+ ω=. p (4) Equation (4) can be made dimenionle by uing the nondimenional diipative ratio ϵ, d and r defined by 11-1 ε γ R 4 η µ 1 T 3 γ p L 3 γ p L = c = ( γ 1 ) κ ( 1 ) 1 T γ T d = = = γ c o n d γ p L c γ R γ p L µ ( γ 1) ρ χ T r = = L. γ p c r a d (5) Here p, T, R, and µ are the equilibrium preure, temperature, ga contant and mean molecular weight / 1 1 repectively; η = 1 T k g m i the compreive vicoity; / 1 1 i the coefficient of κ = 1 T W m d e g thermal conduction parallel to ambient magnetic field; = L p i the thermal conduction time and ( γ 1 ) κ T c o n d γ p i the radiative time, where χ and α are = ( ) r a d γ 1 ρ χ T (6) (7) the piecewie continuou function depend on the temperature 13. We have given the value of thee parameter and time for variou loop temperature and loop length in table For the conidered loop temperature and loop length, the radiative time i much larger than other diipative time except for the cooler loop, where it i horter than the other time, implying that at low temperature radiation may be the dominant diipative mechanim. In term of dimenionle quantitie, equation (4) can be rewritten a ( ) ( ) id d k i i d r i r k i r γε ω 4 ω+ εω + γ ω + αγε ω+ ( α) αγ ω + ω 3 = (8) Reult and Dicuion A we are intereted in the patial damping of low wave in coronal loop, we conider to be real and eek complex olution of the wavenumber k expreed a k = kr + iki. A number of theoretical and numerical tudie how that thermal conduction can play an important role in the quick damping of low magneto-acoutic wave in coronal loop 15, 7, 11. However, it ha been pointed out that the oberved damping of coronal loop ocillation oberved by trace can be explained via vicou or reitive diipation of wave energy if the vicoity or reitivity of the medium i enhanced compared with claical value 16. Therefore, it become utmot important to tudy the effect of different diipative mechanim on the damping of low wave in coronal loop and to identify the dominant diipative mechanim. Effect of Compreive Vicoity: According to 16 the oberved damping of coronal loop ocillation indicate the preence of trong diipation of the wave energy. Thu diipation via vicoity or reitivity lead to the heating of coronal loop. Therefore, in thi ection we invetigate the effect of compreive vicoity on the damping of low wave in coronal loop. So, in the abence of thermal conduction and radiation ( κ =, χ= ), equation become 4 η v z c c = v z 3 γ p t z t z (9) Uing equation 3 in equation 9, we obtain 1/ 1 εω k= ω ω 1 i, 1 iεω + (1) v exp iωt ikz = exp iω( t z)exp εω z. Thu, or, ( ) ( ) compreive vicoity will caue the amplitude of the velocity perturbation to decreae a exp( εω z ) i.e. in the abence of thermal conduction and radiation the motion i purely damped which i alo clear from figure 1a, where the dimenionle damping length of low wave i plotted againt dimenionale vicou coefficient parameter ' ε '. International Science Congre Aociation 35

3 Reearch Journal of Recent Science ISSN 77- Vol. 1(), 34-41, Feb. (1) Figure-1 Variation of normalized damping length L d a function of (a) compreive vicoity parameter ε (b) time period for contant temperature (c) time period for contant loop half length and (d) loop half length Figure 1 how the behaviour of damping length of low mode wave a a function of vicou parameter, time period (for contant temperature and contant loop half length) and loop half length. Damping length i the ditnace over which the amplitude of the wave drop to 1/e time of it original amplitude. Figure 1a how the variation of normalized damping length of low wave with vicou coefficient (ϵ) for different conidered period. The damping length of low wave i maximum for ε = and decreae from it maximum value to a minimum value for certain value of ε. Numerically, the minimum value of the damping length i found to be.193l forε =.1,.386L for ε =.4,.58L forε =.36 and.773l for ε =.48 for wave of period 5, 1, 15 and repectively. The damping length then increae lowely for uffeciently large vicou coefficient. A the wave period increae, minima of the curve hift toward the right correponding to the long period wave. Thi hifting of minima implie that long period wave would travel a longer ditance a compared to hort period wave before damping. Table 1 how that for different coronal loop parameter oberved by TRACE and SUMER the vicou coefficient in majority of cae remain very mall hence low wave have maximum damping length. We therefore, conclude that in uch coronal loop the damping effect of vicoity i very mall. However, for hot and hort coronal loop oberved by SUMER compreive vicoity have a ignificant effect on the damping of low magnetoacoutic wave. Figure 1b depict the behaviour of damping length with time period for different loop temperature i.e., for cool and hot coronal loop at contant loop half length. In cool coronal 6 loop ( T 1 K ) irrepective of vicou parameter it i found that hort period wave ( P <.5 ) damp very quickly wherea long period wave ( P >.5 ) travel undamped along the length of loop. However, in the cae of hot coronal 6 loop ( T 4 1 K ), damping length of low wave how a appericiable decreae a we go from long to hort period wave. Thi behaviour of damping length of low wave in hot coronal loop can be eaily undertood with the help of equation 5 a the vicou coefficient parameter ϵ varie with the cube of the loop temperature T and inverely with the loop half-length L for fixed loop preure p. Thu, it i clear that wave damping will low down at low temperature and large loop half length wherea in hort and hot coronal loop compreive vicoity will caue trong damping of 3/ low wave due to ε T L. Thee reult are in agreement with the imulation reult of 14. They found that the amplitude of ocillation of coronal loop increae with decreaing loop temperature and increaing loop length and the wave are diipated by the combined effect of vicoity and thermal conduction at higher temperature, wherea wave diipation i governed by radiation at lower temperature. International Science Congre Aociation 36

4 Reearch Journal of Recent Science ISSN 77- Vol. 1(), 34-41, Feb. (1) For different loop half lentgh, we have tudied the behaviour of damping length a function of wave period (figure 1c). The damping length of hort period wave i much horter than that of long period wave that lead to the wave energy of thee hort period wave will be diipated due to vicou diipation. It mean that the diipation of mechanical energy of low wave via compreive vicoity can balance the energy loe in coronal loop. Figure 1d further give the information about the dependence of damping length of low wave on the loop half length and temperature. On increaing the loop temperature vicou coefficient increae, leading to the decrement in damping length wherea the increment in loop half length act to balance out the effect of increaing temperature (equation 5). Accordingly, in hot and long coronal loop, damping length of low wave become almot contant which i evident from figure 1d. However, for hort and hot coronal loop oberved by SUMER, low wave damp quickly a the damping length decreae from maxima and ultimately attain a low contant value. Thu the overall behaviour of damping length of low wave i that it decreae from it maximum value and after howing a dip, attain a contant value. Table-1 Value of non-dimenional diipative parameter and characteritic time a function of loop half length and loop temperature. The dimenionle parameter ε, d and r repectively, the vicou ratio, the thermal ratio and radiation ratio; i the ound travel time, vic L (Mm) T = 1 MK T = MK T = 4 MK T = 6 MK T = 8 MK T = 1 MK i the vicou time, cond ε d r (min).883e-3.44e-3.1e-3.147e-3.11e-3.77e-.353e-.177e-.118e-.883e-3.565e-1.8e-1.141e-1.94e-.77e e-1.477e-1.318e-1.38e e-1.565e e-3.13e-.48e-.371e-.495e-.547e-4.19e-3.19e-3.38e-3.438e-3.484e-5.968e-5.193e-4.9e-4.387e-4.117e-5.34e-5.468e-5.73e-5.937e-5 4.8e e e-6.56e-6 3.4e e e e e e-6 i the heat-conduction time and rad vic (min) e e i the heat-radiation time cond (min) rad (min) International Science Congre Aociation 37

5 Reearch Journal of Recent Science ISSN 77- Vol. 1(), 34-41, Feb. (1) Effect of Thermal Conduction: In the abence of vicoity and radiation ( η=, χ= ), equation reduce to ( γ 1) κ T v ( 1) T v c γ γ κ = t γ p z z t γ p z t (11) Uing equation 3 in equation 11, we obtain 4 3 dk + ( iω γ dω ) k iω = (1) where all quantitie are dimenionle. The diperion relation (1) ha been olved numerically for different loop parameter and the reult are preented in figure. Figure a how the behaviour of damping length of low wave a a function of thermal conduction ratio for the exact olution given by equation 1. Thi figure clearly point out that all low wave have maximum damping length for mall value of d and decreae harply with d until wave attain a minimum damping length. Similar reult for low wave in coronal loop are alo obtained by 7. Numerically the minimum value of damping length i found to be.33l,.66l,.98l and 1.31L for 5, 1, 15 and period wave repectively at low thermal conduction ratio and thereafter damping length increae for ufficiently large d. It i alo evident from figure 1a and a that the effect of compreive vicoity on wave damping i more pronounced than that of thermal conduction. Figure b and c how the variation of damping length with time period for different loop temperature (i.e., for cool and hot coronal loop) and loop half length. For hot and cool coronal loop oberved by SUMER and TRACE repectively, the damping length of low wave irrepective of thermal ratio parameter how the ame pattern of variation with time period a in the cae of vicou damping. For fixed loop temperature, figure 1b and b clearly indicate that low wave in loop diipate energy trongly via compreive vicoity a compared to thermal conduction. However, for fixed loop half length, it can be een clear form figure 1c and c that the damping length of low wave due to compreive vicoity i larger than that due to thermal conduction. Thi imply implie that thermal conduction i the dominant wave damping mechanim a compared to compreive vicoity. Thi contradictory behaviour of low wave for fixed loop half length and loop temperature can be eaily undertood from the data given in table 1 and equation (5) - (6). Figure- Variation of normalized damping length L d a function of (a) thermal conduction parameter d (b) time period for contant temperature (c) time period for contant loop half length and (d) loop half length International Science Congre Aociation 38

6 Reearch Journal of Recent Science ISSN 77- Vol. 1(), 34-41, Feb. (1) The information about the damping length of low wave in relation to the loop half length and temperature i preented in figure d. On increaing the loop temperature thermal ratio increae that lead to an increae in damping length wherea the increment in loop half length act to balance out the effect of increaing temperature (equation 6). Accordingly, in cool and long coronal loop, damping length of low wave become almot contant which i evident from figure d. However, for hort and hot coronal loop uch a oberved by SUMER, low wave are undamped becaue they have larger damping hength in compariion with loop length. Thu, the low wave in cool coronal loop will diipate energy via thermal conduction however, compreive vicoity can diipate wave energy in both cool and hot coronal loop. Effect of Heat Radiation: In thi ection we invetigate the damping of low wave due to heat radiation and compare it with the wave damping due to compreive vicoity and thermal radiation. Whenα i negligible and diipative mechanim uch a compreive vicoity and thermal conduction are abent, from equation, we get 3 ρ χt v v ( γ 1) c = (13) 3 t γ p z t On uing equation 3 in equation 13, we obtain ir k ω 1 o, v exp iω( t z)exp( rz) (14) ω Figure-3 Variation of normalized damping length L d a function of (a) thermal conduction parameter d (b) time period for contant temperature (c) time period for contant loop half length and (d) loop half length International Science Congre Aociation 39

7 Reearch Journal of Recent Science ISSN 77- Vol. 1(), 34-41, Feb. (1) Thu, imilar to vicoity and thermal conduction effect radiation will alo caue the amplitude of velocity perturbation to decreae a exp(-rz). The behaviour of damping length of low wave with radiation coefficient for the exact numerical olution given by equation 13 ha been hown in figure 3. In figure 3a, we have hown the variation of damping length with radiation coefficient for low wave of different period. Thi figure how that the pattern of variation of damping length with radiation coefficient i ame for low wave of different period. Damping length of low wave decreae with radiation coefficient, a expected from the analytical expreion equation 14. From equation 7 we can ee that the value of dimenionle radiation parameter r i very mall for different coronal loop. For uch a mall value of r, the effect of radiation on wave damping would be hardly noticeable and therefore, in the preent invetigation we are uing ome large value of radaition parameter. Form figure 3 it can be een that initially for mall value of radiation coefficient r, low wave of any conidered period have large damping length. Thu, the wave damping due to optically thin radiation i not very effective for very mall value of radiation coefficient. For large value of radiation coefficient only hort period wave ( P 5 ) how meaningful damping wherea long period wave travel undamped in the loop. Concluion In thi paper, we have invetigated the individual effect of compreive vicoity, thermal conduction and heat radiation on the patial damping of low magneto-acoutic wave in coronal loop oberved by SUMER and TRACE. For it, we have numerically olved the diperion relation to tudy the damping length of low wave in a medium having phyical propertie akin to thoe of olar coronal loop. The main concluion that can be extracted from our tudy are: i) We find that for the typical obervational olar parameter of coronal loop, the dominant wave damping mechanim i compreive vicoity and thermal conduction, with le ignificant contribution by radiation. ii) Slow wave have minimum damping length due to compreive vicoity and if the oberved damping length of low wave in coronal loop i horter than the minimum damping length predicted by compreive vicoity, then the low MHD wave mut be damped by other damping mechanim. iii) For any conidered period, low wave have much horter damping length in hot coronal loop than that in cool loop and alo low wave damp very quickly in hot and long coronal loop. iv) Since the diipative ratio ε and d depend on the 3/ temperature and loop half length a T L, o in hort and hot coronal loop diipation i enhanced due to vicoity and thermal conduction. However, according to Hildner (1974), the cooling exponent α = 1 i for temperature range.8 T 1 MK, o it follow from Eq. (11) that radiative 5/ ratio r depend on temperature and loop half length a L T. Thi imply implie that radiative damping become the dominent damping mechanim in cool and long coronal loop. Our reult clearly how that at lower temperature, wave damping i almot goverend by radiative cooling wherea at higher temperature the wave damping i dominent due to vicoity and thermal conduction. Acknowledgement One of the author P K i thankful to the Department of Science & Technology, New Delhi for the financial upport in the form of reearch project under Fat Track Scheme for Young Scientit. B S acknowledge CSIR, New Delhi for providing financial upport under CSIR, NET-SRF reearch award. Reference 1. Ofman L. Romoli M. Poletto G., Noci G. and Kohl J. L., Ultraviolet coronagraph pectrometer obervation of denity fluctuation in the olar wind, Atrophyical J., 491, L111-L114 (1997). De Foret C.E. and Gurman, J.B., Obervation of Quai-periodic Compreive Wave in Solar Polar Plume, Atrophyical J., 51, L17-L (1998) 3. Marika J.T., Characteritic of olar flare doppler-hift ocillation oberved with the bragg crytal pectrometer on yohkoh, Atrophyical J., 639, (6) 4. Ogrodowczyk R. Murawki K. and Solanki S.K., Slow magnetoacoutic tanding wave in a curved olar coronal lab, Atron. Atrophy., 495, (9) 5. Marh M.S. Ireland, J. and Kucera, T., Bayeian analyi of olar ocillation, Atrophyical J., 681, (8) 6. Wang T.J. Ofman L. Davila J.M. and Marika J.T., Hinode/EIS obervation of propagating low-frequency low magnetoacoutic wave in fan-like coronal loop, Atron. Atrophy.,, 53 L-L8 (9) 7. De Moortel I. and Hood A.W., The damping of low MHD wave in olar coronal magnetic field, Atron. Atrophy., 48, (3) 8. Mc Ewan M.P. and De Moortel I., Longitudinal intenity ocillation oberved with TRACE: evidence of fine-cale tructure, Atron. Atrophy., 448, (6) International Science Congre Aociation 4

8 Reearch Journal of Recent Science ISSN 77- Vol. 1(), 34-41, Feb. (1) 9. Ogrodowczyk R. and Murawki K., Numerical imulation of low magnetoonic tanding wave in a traight olar coronal lab, Atron. Atrophy.,, 467, (7) 1. Kumar N. Kumar P. Singh S. and Kumar A., Damping of Slow Magnetoacoutic Wave in an Inhomogeneou Coronal Plama, J. Atrophyic and Atronomy, 9, (8) 11. De Moortel I. and Hood A.W., The damping of low MHD wave in olar coronal magnetic field - II. The effect of gravitational tratification and field line divergence, Atron. Atrophy., 415, (4) 1. Macnamara C.K. and Robert B., Effect of thermal conduction and compreive vicoity on the period ratio of the low mode, Atron. Atrophy., 515, A41 (1) 13. Hildner E., The formation of olar quiecent prominence by condenation, Solar Phy. 35, (1974) 14. Sigalotti L.D.G. Guerra G.A. and Mendoza-Briceño C. A., Propagation and Damping of a Localized Impulive Longitudinal Perturbation in Coronal Loop, Solar Phy., 4, (9) 15. Ofman L. and Wang. T., Hot coronal loop ocillation oberved by umer: low magnetoonic wave damping by thermal conduction, Atrophyical J., 58, L85-L88 () 16. Nakariakov V.M., Ofman L., DeLuca E.E., Robert B. and Davila J.M., TRACE Obervation of Damped Coronal Loop Ocillation: Implication for Coronal Heating, Science, 85, (1999) International Science Congre Aociation 41