viscous damping of alfvén surface waves in the solar atmosphere

4
ISSN 1063780X, Plasma Physics Reports, 2009, Vol. 35, No. 11, pp. 976–979. © Pleiades Publishing, Ltd., 2009. Published in Russian in Fizika Plazmy, 2009, Vol. 35, No. 11, pp. 1055–1058. 976 1 1. INTRODUCTION The observed inhomogeneous nature of the solar atmosphere has motivated many researchers to study the wave propagation in a magnetically structured atmosphere. In the solar atmosphere, discontinuities in magnetic field and density is observed on the edges of sun spots, flux tubes, prominences, coronal holes, etc., which may lead to the existence of surface waves. The magnetohydrodynamic (MHD) surface waves at a single interface in the incompressible medium have been studied by several authors [1–3]. Recently, Alfvén waves are detected in the solar corona [4] and followed by many researchers through Hinode observations. The presence of Alfvén waves in the solar chromosphere [5|, prominences [6] and solar xray jets [7] have been observed. The ubiquitous pres ence of Alfvén waves in the solar atmosphere has been revealed and accepted by the scientific community. Hence, I the study of Alfvén waves in the structured atmosphere namely the Alfvén surface waves may reveal many interesting phenomena occurring in the solar atmosphere. The investigation on viscous damping of MHD waves in the solar corona was started much earlier [8] and significant works have been done in the recent times [9, 10]. Ruderman has studied the viscous damping of surface MHD waves at a magnetic inter face with cold plasma approximation in the solar corona [11] and in the solar chromosphere [12] by including thermal conductivity as an additional damping mechanism. Viscosity plays a crucial role in the damping of waves over the other existing mecha nisms such as phase mixing, ohmic heating, resonance 1 The article is published in the original. absorption, etc. [13] and the viscous MHD spectra for cylindrical plasma has been analyzed and confirmed the viscosity as the dominant mechanism for coronal loop heating [14]. In this work, we study the viscous damping of Alfvén surface waves at a single magnetic interface separating two incompressible viscous plasma media and estimate the damping length in the solar atmosphere. 2. FORMULATION OF THE PROBLEM AND DISPERSION RELATION In the highly structured solar atmosphere, in order to describe the propagation of Alfvén surface waves, we consider a single magnetic interface separating two magnetized media. It is assumed that the magnetized media occupying above (x > 0) and below (x < 0) the interface (x = 0) are incompressible and viscous in nature with different magnitudes of magnetic field and density. In the MHD approximation, the linearized MHD equations for incompressible viscous plasma with den sity , viscosity , embedded with uniform mag netic field have been adopted from Uberoi and Somasundaram [15]. Applying a small perturbation of the form f (x, y, z, t) = and solving the linear MHD equations, the field components can easily be derived. Applying the boundary conditions that the normal and tangential component of the velocity, tan gential viscous stress and total pressure are continu ous, we obtain the dispersion relation for AS W as, ρ 01 ν 1 01 B [ ] + −ω ( )exp ( ) fx i ly kz t SPACE PLASMA Viscous Damping of Alfvén Surface Waves in the Solar Atmosphere 1 G. D. Rathinavelu a , M. Sivaraman a , and A. S. Narayanan b a Plasma Physics Group, Department of Physics, Gandhigram Rural University, Gandhigram 624 302, Tamil Nadu, India b Indian Institute of Astrophysics, Bangalore 560 034, India Received January 21, 2009; in final form, March 16, 2009 Abstract—The dispersion relation for the propagation of viscous Alfvén surface waves along viscous plasma plasma interface has been derived. Two modes of Alfvén surface waves are found to propagate with their char acteristics depend on the interface parameters like magnetic field, density ratio, viscosity, etc. The viscous damping of Alfvén surface waves has been studied in the astrophysical point of view. The damping length of Alfvén surface waves due to viscosity in the solar atmosphere has been estimated. PACS numbers: 95.30.Qd; 96.50.Tf DOI: 10.1134/S1063780X09110075

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Page 1: Viscous damping of Alfvén surface waves in the solar atmosphere

ISSN 1063�780X, Plasma Physics Reports, 2009, Vol. 35, No. 11, pp. 976–979. © Pleiades Publishing, Ltd., 2009.Published in Russian in Fizika Plazmy, 2009, Vol. 35, No. 11, pp. 1055–1058.

976

1 1. INTRODUCTION

The observed inhomogeneous nature of the solaratmosphere has motivated many researchers to studythe wave propagation in a magnetically structuredatmosphere. In the solar atmosphere, discontinuitiesin magnetic field and density is observed on the edgesof sun spots, flux tubes, prominences, coronal holes,etc., which may lead to the existence of surface waves.The magnetohydrodynamic (MHD) surface waves at asingle interface in the incompressible medium havebeen studied by several authors [1–3].

Recently, Alfvén waves are detected in the solarcorona [4] and followed by many researchers throughHinode observations. The presence of Alfvén waves inthe solar chromosphere [5|, prominences [6] and solarx�ray jets [7] have been observed. The ubiquitous pres�ence of Alfvén waves in the solar atmosphere has beenrevealed and accepted by the scientific community.Hence, I the study of Alfvén waves in the structuredatmosphere namely the Alfvén surface waves mayreveal many interesting phenomena occurring in thesolar atmosphere.

The investigation on viscous damping of MHDwaves in the solar corona was started much earlier [8]and significant works have been done in the recenttimes [9, 10]. Ruderman has studied the viscousdamping of surface MHD waves at a magnetic inter�face with cold plasma approximation in the solarcorona [11] and in the solar chromosphere [12] byincluding thermal conductivity as an additionaldamping mechanism. Viscosity plays a crucial role inthe damping of waves over the other existing mecha�nisms such as phase mixing, ohmic heating, resonance

1 The article is published in the original.

absorption, etc. [13] and the viscous MHD spectra forcylindrical plasma has been analyzed and confirmedthe viscosity as the dominant mechanism for coronalloop heating [14]. In this work, we study the viscousdamping of Alfvén surface waves at a single magneticinterface separating two incompressible viscousplasma media and estimate the damping length in thesolar atmosphere.

2. FORMULATION OF THE PROBLEM AND DISPERSION RELATION

In the highly structured solar atmosphere, in orderto describe the propagation of Alfvén surface waves,we consider a single magnetic interface separating twomagnetized media. It is assumed that the magnetizedmedia occupying above (x > 0) and below (x < 0) theinterface (x = 0) are incompressible and viscous innature with different magnitudes of magnetic field anddensity.

In the MHD approximation, the linearized MHDequations for incompressible viscous plasma with den�sity , viscosity , embedded with uniform mag�netic field have been adopted from Uberoi andSomasundaram [15].

Applying a small perturbation of the form f (x, y, z,t) = and solving the linearMHD equations, the field components can easily bederived. Applying the boundary conditions that thenormal and tangential component of the velocity, tan�gential viscous stress and total pressure are continu�ous, we obtain the dispersion relation for AS W as,

ρ01 ν1

01B

[ ]+ − ω( )exp ( )f x i ly kz t

SPACE PLASMA

Viscous Damping of Alfvén Surface Waves in the Solar Atmosphere1

G. D. Rathinavelua, M. Sivaramana, and A. S. Narayananb

a Plasma Physics Group, Department of Physics, Gandhigram Rural University, Gandhigram 624 302, Tamil Nadu, India

b Indian Institute of Astrophysics, Bangalore 560 034, IndiaReceived January 21, 2009; in final form, March 16, 2009

Abstract—The dispersion relation for the propagation of viscous Alfvén surface waves along viscous plasma�plasma interface has been derived. Two modes of Alfvén surface waves are found to propagate with their char�acteristics depend on the interface parameters like magnetic field, density ratio, viscosity, etc. The viscousdamping of Alfvén surface waves has been studied in the astrophysical point of view. The damping length ofAlfvén surface waves due to viscosity in the solar atmosphere has been estimated.

PACS numbers: 95.30.Qd; 96.50.Tf

DOI: 10.1134/S1063780X09110075

Page 2: Viscous damping of Alfvén surface waves in the solar atmosphere

PLASMA PHYSICS REPORTS Vol. 35 No. 11 2009

VISCOUS DAMPING OF ALFVÉN SURFACE WAVES 977

(1)

where

, ,

and , , and are,

respectively, the Alfvén velocity, magnetic field, den�sity and viscosities in media 1 and 2. Equation (1) isnormalized by introducing the following nondimen�sional interface parameters,

, ,

z = 1 + tan2θ,

where

, and

are the normalized phase velocity, viscosity in media 1and 2, magnetic filed ratio, and density ratio, respec�tively, and is the angle between the magnetic fieldand propagation wave vector.

Normalizing the Eq. (1) with , we get thedimensionless dispersion relation of Alfvén surfacewaves modified by viscosity as,

(2)

3. RESULTS AND DISCUSSION

Equation (2) is solved numerically by the softwareMATLAB 6.0 and drawn curves for the interface

parameters ; = 0.5 and 1.5; 0.5, 1.0,and 1.5; and ranging from 0 to 1.0. We have plotted

( )

( ) ( )( )

( )( )

( ) ( )( ) ( )( )

( )( )

( ) ( )( )

⎡ρ ω −⎣⎤ ⎡+ ρ ω − ρ ω − τ −⎦ ⎣

⎤+ ρ ω − τ − ⎦

+ ρ ω − ρ ω −

⎡− ω ν − ν ρ ω − τ −⎣⎤−ρ ω − τ − ⎦

+ ω ν − ν τ − τ − =

2 2 201 1

2 2 2 2 2 202 2 01 1 2

2 2 202 2 1

2 2 2 2 2 201 1 02 2

2 2 2 21 2 01 1 2

2 2 202 2 1

23 21 2 1 2

4

4

4 0,

A

A A

A

A A

A

A

i k V

k V k V K

k V K

i K k V k V

K k V K

k V K

i K K K

( )⎡ ⎤ρ

τ = − ω −⎢ ⎥ν ω⎣ ⎦

/1 22 2 2 201,2

1,2 1,21,2

Ai

K k V = +

2 2 2K l k

⎡ ⎤= ⎢ ⎥μρ⎣ ⎦

/1 2201,2

1,201,2

AVB

01,2B ρ01,2 ν1,2

( )⎡ ⎤−τ= = −⎢ ⎥

⎢ ⎥⎣ ⎦

/1 22

11

1

11

xT

K V z

( )⎡ ⎤η− ατ= = −⎢ ⎥α⎢ ⎥⎣ ⎦

/1 22 2

22 2

2

1x

TK V z

ω=

1

,A

xkV

ν ω=ρ

1,21,2 2

01,2 1,2

,A

VV

α =02

01

BB

ρη =

ρ

01

01

θ

1AV

( ) ( ) ( )( )

( )( ) ( ) ( )

( ) ( )( ) ( )( )

( ) ( )( )

⎡ ⎤ ⎡− + η − α − −⎣ ⎦ ⎣⎤+ η − α − + − η − α⎦

⎡ ⎤+ − α − − − η − α −⎣ ⎦

+ − α − − =

2 2 2 22

2 2 2 2 21

2 2 2 21 2 2 1

22 21 2 1 2

1 1 1

1 4 1

4 1 1 1

4 1 1 0.

x x x T

x T x x

i z V V x T x T

z V V T T

α = 0.2 η =2V

1V

the dependence of and (in units of ) on thenormalized viscosity for various magnitudes of (0.5, 1.0, and 1.5). The values of and can be cal�culated from the obtained values of x , wherek is the complex wavenumber .

Figures 1a–1c are plotted for and = 0.5with various magnitudes of . It can be seen from thefigures that two modes of Alfvén surface waves areobserved. It is interesting to note that in Figs. 1a–1c,for a typical critical value of ( ), there are a downhump and a raising hump, which is vice versa in thesecond mode. This critical value increases as increases. From Figs. 1a–1c, for = 0.5, 1.0, and1.5 are obtained as 0.1, 0.2, and 0.3, respectively.However, the second critical value appears for > 1which is less than (Fig. 1c).

From Figs. 2a–2c, we observe the same character�istics as that of Figs. 1a–1c. These graphs are drawn

for and η = 1.5 with various V2 values. Theamplitudes of the observed peaks are significantlyhigher than observed in Figs. 1a–1c. The critical val�ues of ( ) are same for Figs. 1a–1c and Figs. 2a–2c, except that the amplitude of the humps are foundto be different.

In this study, we have taken typical numerical val�ues in the solar corona, the field aligned viscosity

νρ⎯1 = [16], temperature T =105–106 K, number density n = 1015 m–3, Alfvénvelocity VA1 = 106 m/s and frequency ω = 1 s–1 [17].For the above values, the calculated value of normal�ized viscosity lies between 2.18 × 10–5 and 6.9 × 10–3.Damping length of ASW due to viscosity has been esti�mated at the critical values of V1 observed in Figs. 1aand 1c for different values of V2 when η = 0.5. At V1c =0.1 for V2 = 0.5, the first mode in Fig. 1a,

, . Hence, the

propagation wavelength m,

damping length m, the ratio can be calculated. Similarly for the

second mode, m, λi2 =

2π/ki2 = 9.38 × 107 m and , whichreveals that the first mode damps shortly since thedamping length is just twice that of propagation wave�length, whereas, the second mode gets damped slowlyas the damping length is 19 times that of propagationwavelength. Hence, first mode suffers heavy dampingwhile second mode experiences slow damping whichcan propagate outwards to longer distance. The samedamping and propagation wavelengths arc calculatedfor = 1.5 when η = 0.5 at V1c = 0.3 (Fig. 1c).

rk ik ω 1AV

1V 2V

rk ik= ω 1AkV

( )= +r ik k i k

α = 0.2 η

2V

1V 1cV

1V 2V

1cV 2V

1V 2V

1cV

α = 0.2

1V 1cV

− −

×/3 5 2 1 2 16.9 10 sT n m

= ω1 11.1949r Ak kV = ω1 11.591i Ak kV

λ = π = ×6

1 12 5.42 10r rk

λ = π = ×7

1 12 1.1 10i ikλ λ = ≈1 1 2.03 2i r

λ = π = ×6

2 22 4.87 10r rkλ λ = ≈2 2 19.26 19i r

2V

Page 3: Viscous damping of Alfvén surface waves in the solar atmosphere

978

PLASMA PHYSICS REPORTS Vol. 35 No. 11 2009

RATHINAVELU et al.

From Fig. 2a, for = 0.5 when η = 1.5 at V1c = 0.1,for the first mode λr1 = 1.5 × 105 m, λi1 =3.95 × 106 m,the ratio , for the second modeλr2 = 3.26 × 106 m, λi2 = 2.21 × 107 m, the ratio

. In this case, first mode can propa�gate to a distance more than second mode beforedamping since the damping length of first mode is

2V

λ λ = ≈1 1 26.33 26i r

λ λ = ≈2 2 7.02 7i r

higher (26 times of propagation wavelength) than sec�ond mode (seven times of propagation wavelength).

This change is observed due to the influence of ,because the already reported ratio of the dampinglength to the propagation wavelength at the viscousplasma�vacuum interface is 14 [15]. Hence the densityplays a major role in damping of surface waves. How�

η

3.0

2.5

2.0

1.5

1.0

0.5

0

kr1

ki1kr2

ki2

kr, ki

(а)

3.0

2.5

2.0

1.5

1.0

0.5

0

kr1

ki1

kr2

ki2

(b)

3.0

2.5

2.0

1.5

1.0

0.5

0

kr1

ki1

kr2

ki2

(c)

1.00.80.60.40.2V1

Fig. 2. Curves are drawn for , (in units of ω/VA1) as

functions of with , 1.5; and =(a) 0.5,(b) 1.0, and (c) 1.5.

rk ik

1V α = 0.2 η = 2V

1.8

1.6

1.4

1.2

1.0

0.8

0.6

0.4

0.2

0

kr, ki

kr1ki1kr2

ki2

1.8

1.6

1.4

1.2

1.0

0.8

0.6

0.4

0.2

0

kr1ki1kr2ki2

(а)

(b)

1.8

1.6

1.4

1.2

1.0

0.8

0.6

0.4

0.2

0

kr1ki1kr2ki2

(c)

1.00.80.60.40.2V1

Fig. 1. Curves are drawn for , (in units of ω/VA1) as

functions of with , = 0.5; and = (a) 0.5,(b) 1.0, and (c) 1.5.

rk ik

1V α = 0.2 η 2V

Page 4: Viscous damping of Alfvén surface waves in the solar atmosphere

PLASMA PHYSICS REPORTS Vol. 35 No. 11 2009

VISCOUS DAMPING OF ALFVÉN SURFACE WAVES 979

ever, when = 1.5, from Fig. 2c, at V1c = 0.3, for thefirst mode, λri = 3.29 × 106 m, λi1 = 3.95 × 106 m. Thisindicates that the damping length and the propagationwavelength are nearly equal ( );hence, it suffers a very heavy damping. In the secondmode, λr2 = 3.28 × 106 m, λi2 = 2.21 × 107 m and

. Therefore, the second mode prop�agates seven times that of the propagation wavelengthbefore being damped.

It is clear from the numerical calculation thatamong the two modes of ASW modified by viscosity,one of them is damped normally while the other isheavily damped. From Figs. 1a and 1c, first mode isheavily damped and the second mode is normallydamped. The calculation from Fig. 2a shows that thefirst mode suffers normal damping while the secondmode exhibits heavy damping. However, the firstmode in Fig. 2c experiences rapid damping since and are nearly equal, whereas the second modeshows normal damping.

Among the two modes observed, one mode dampsstronger and the other damps weaker. The momentumtransfer between the modes chooses the presence ofdifferent dissipation rate in modes. Since, the viscosityand density are the main cause of momentumexchange between two media through the surface [18].At the critical viscosity, the mode with high momen�tum damps weakly; where the mode with low momen�tum damps strongly. This critical viscosity changeswith respect to the density ratio. Hence, this Alfvénwave damping is very important in the aspect of coro�nal heating, since this is the only likely candidatewhich can carry the energy to a greater distance.

4. CONCLUSIONS

In this work, the viscous damping of ASW at a mag�netic interface in the solar atmosphere has been stud�ied. Two modes of ASW modified by viscosity areobserved. The damping length has been calculated inthe solar coronal situations. The results show that one

of the modes is normally damped and the other isheavily damped depending on the interface parame�

ters. Interestingly, for , = 1.5, and = 1.5,one of the modes suffer rapid damping since the prop�agation wavelength and damping length are almostequal.

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2V

λ λ = ≈1 1 1.2 1.0i r

λ λ = ≈2 2 6.74 7i r

λ 2r

λ 2i

α = 0.2 η 2V