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DAMPING OF SLOW MHD WAVES

IN FLOWING PLASMA

Anil Kumar1, Nagendra Kumar2, Pradeep Kumar3 and

Anuj Bhardwaj1

1. Department of Applied Sicences, V.I.E.T., Dadri, G.B.

Nagar, Uttar pradesh, India

2. Department of Mathematics, M.M.H. College,

Ghaziabad,Uttar pradesh, India

3. Department of Physics, Hindu College, Moradabad, Uttar

pradesh, India

Introduction

Recent observational findings by space born satellite

such as SOHO and TRACE have revealed that solar

coronal plasma support both propagating (De Moortel et

al. 2002a,b; Murawski & Zaqarashvili 2010) and stand-

ing slow magnetoacoustic waves(Wang et al.2002). Slow

waves are an important tool for diagnosing the coronal

plasma because they propagate along magnetic field lines.

Slow waves are believed to arise in sunspot (Bogdan,

2000) in the form of 3 minute oscillations, in the foot-

point of coronal loop (De Moortel, 2002 a-d) in the form

of both 3 and 5 minute oscillations. Slow waves may occur

both as standing and propagating waves. The propagat-

ing waves have been detected in sunspot, plumes and in

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loop footpoint regions, while standing slow waves have

been found in hot coronal loops.The effect of steady mass flow on the oscillatory modes

of magnetic structures has been studied theoretically by

several researchers. Nakariakov and Roberts (1995) stud-

ied the effect of steady flow on coronal and photospheric

structures in Cartesian geometry. Terra Homem et al.

(2003) extended the previous work to cylindrical geom-

etry. Ground and space born satellite observations haveconfirmed that the solar and space plasmas are always

dynamic showing steady flow (Gabriel et al. 2003; Kri-

jger et al. 2002). Therefore all the theoretical models

should include the presence of an equilibrium flow. Flow

breaks the symmetry between parallel and anti-parallel

wave propagation to flow direction. It has been seen that

slow modes can only propagate parallel to flow directionThe theoretical investigation of damping of slow mode

has been discussed in several papers. Several damping

mechanisms has been put forward to explain the attenua-

tion. De Moortel and Hood (2003) discussed the damping

of slow magnetoacoustc waves in a homogeneous medium

by taking into account thermal conduction and compres-

sive viscosity. Carbonell et al. (2004) has studied thedamping in homogeneous unbounded plasma by consid-

ering thermal conduction and radiative losses as damp-

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ing mechanism. The case of an isolated slab has been

addressed in Terradas et al. (2005) and that of a slabembedded in solar corona by Soler et al. (2007, 2008) con-

sidering non adiabatic effects. Kumar and Kumar (2006)

have studied the damping of MHD waves taking into ac-

count thermal conduction and compressive viscosity as

damping mechanism. Recently Carbonell et al. (2008)

discussed the combined effect of non-adiabatic mecha-

nism and steady flow on the damping of slow modes.In the present paper, we investigate the properties of

uncoupled slow magnetoacoustic waves from the point

of view of boundary driven oscillations by taking into

account the combined effect of steady flow and thermal

conduction on the time damping of slow waves.

2 Basic Equations

Basic equations describing the plasma motion in a isothre-

mal atmosphere of Sun are the standard ideal MHD equa-

tions and are as follows:

∂ρ

∂t + ∇ · (ρv) = 0, (1)

ρ

∂

∂t + v · ∇

v = −∇ p + 1

µ(∇ × B) × B + ρg (2)

∂ B

∂t = ∇ × (v × B), (3)

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Dp

Dt −

γp

ρ

Dρ

Dt − (γ − 1)∇ · (κ∇T ) = 0, (4)

p = RρT

µ̃ . (5)

where DDt

= ∂ ∂t

+ v · ∇ is the material derivative for

time variation following the motion; ρ, p, v and T repre-

sent density, pressure, velocity and temperature respec-

tively, R is the gs constant, µ is the mean molecular

weight, γ is the ratio of specific heats, We take thermal

conduction to act purely along the z-axis setting κ =

10−11T 5/2wm−1K −1 as thermal conduction is strongly

suppresed across a magnetic field (Spitzer 1962). Since

in this chapter we only consider the slow MHD oscilla-

tions so we restrict our attention to the motions along

background magnetic field directed along z-axis. There-

fore the system of MHD equations redeuces to 1D form

as∂ρ

∂t = − ∂

∂z (ρv), (6)

ρ∂v

∂t + ρv∂v

∂z = −∂p

∂z , (7)

∂p

∂t + v∂p

∂t = −γp∂v

∂z + (γ − 1)

∂

∂z (κ

)∂T

∂z , (8)

p = RρT

µ̃ . (9)

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Now we conider small deviations of physical quantities of

the medium from the equilibrium values as

ρ = ρ0 + ρ1

p = p0 + p1

v = v0 + v1

where the subscript ‘0’ and ‘1’ refer to the equilibrium

and perturbed quantities.After linearizing equations (6)–(9), we non-dimensionlize

these equations using the equilibrium values of pressure

and density so that v1 = cs v̄1 where c2s = γp0ρ0

is the

sound speed. Length and time are non-dimensionlized

as L = csτ , where L and τ are typical length and time

scale.

The resulting system of equations contains the dimen-sionlize thermal ratio, defined in De Moortel et al. (2002b)

as

d = (γ − 1)κT 0ρ0γ 2P 20 τ

= 1

γ

τ sτ cond

which is the ratio of the sound travel time τ s and the

thermal conduction timescale tcond.

Dropping bars from dimensionless quantities, the lin-

earized MHD equations are given by

∂ρ1∂t

= −∂v1∂z

− v0∂ρ1∂z , (10)

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∂v1

∂z = −

1

γ

∂p1

∂z − v0

∂v1

∂z , (11)∂p1∂t

= −γ ∂v1∂z

− v0∂p1∂z

+ γd∂ 2T 1∂z 2 , (12)

p1 = ρ1 + T 1. (13)

3 Numerical Scheme

We use MacCormack method to solve the system of PDE.

MacCormack method is a variation of the Lax-Wendroff

scheme that can be expressed as a Predictor-Corrector

scheme. A Predictor-Corrector scheme is a two-step method

that involves simultaneously solving coupled equations

that describe that PDE we wish to solve in real space

and in predictor space. The scheme uses half steps and

estimates the values at the half timestep and half spatial

step in predictor space. This is called the predictor step.

These values are then used to compute the values at the

whole time-step and whole spatial-step in real space. For

our problem, we use a forward difference for the spatial

step in the predictor stage and a backwards difference for

the spatial step in the corrector stage (although one can

swap these).

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4 Boundary conditions and Numerical simulation

In order to proceed further with numerical scheme, we

have to apply boundary conditions. As we are studying

the propagation of waves from the upper boundary, we

impose the following conditions:

v1(z, 0) = 0

p1(z, 0) = 0ρ1(z, 0) = 0

v1(z max, t) = f (t)

∂p1∂z

= −f (t)

v1(z min, t) = 0

∂p1∂z (z

min, t) = 0

where f (t) = sinωt, where ω is the driving freequency

which we choose as 2π. We run the simulation for 0 ≤

z ≤ 5 using 1001 steps in spatially dimensions.

Figure 1 shows the cross-section of perturbed veloc-

ity as a function of z at constant time. The dashed line

shows the cross section of perturbed velocity for idealplasma i.e. in the absence of thermal conduction and

steady flow, while the solid line shows the cross-section

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of perturbed velocity in the presence of thermal conduc-

tion, we have taken the value of thermal ratio d=0.25. Itis clear from solid line that the presence of thermal con-

duction decreases the amplitude of perturbed velocity. At

z=2.0, the amplitude of the perturbed velocity decreases

approximately to 50 percent to the initial amplitude at

z=5.0.

Figure 2 to 5 depict the cross section of perturbed ve-

locity in the presence of steady flow having different mag-nitude and thermal conduction. it is observed from these

figures when flow is parallel to propagation of slow MHD

waves it increases the damping length and when flow is

in anti parallel direction it decrease the damping length.

The damping length increases and decreases with the in-

crease in the magnitude of flow.

Therefore we conclude from our results of numericalsimulation that slow waves are strongly damped due to

thermal conduction effects in the presence of steady flow.

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0 0.5 1 1.5 2 2.5 3 3.5 4 4.5−1.5

−1

−0.5

0

0.5

1

1.5

z

v 1

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0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5−1.5

−1

−0.5

0

0.5

1

1.5

z

v 1

Fig2. Cross−ection of perturbed velocity when d=0.25 and v0=0.2

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0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5−1.5

−1

−0.5

0

0.5

1

1.5

v 1

Fig3. Cross−section of pertubed velocity when d=0.25 and v

0

=0.5

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0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5−1.5

−1

−0.5

0

0.5

1

1.5

v 1

Fig4. Cross−section of pertubed velocity when d=0.25 and v0

=0.2

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0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5−1.5

−1

−0.5

0

0.5

1

1.5

v 1

Fig5. Cross−section of perturbed velocity when d=0.25 and v0

=0.5

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