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7/27/2019 Anil Asi2011
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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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