Issue 
A&A
Volume 521, October 2010



Article Number  A34  
Number of page(s)  6  
Section  The Sun  
DOI  https://doi.org/10.1051/00046361/201014367  
Published online  18 October 2010 
Numerical simulations of the attenuation of the fundamental slow magnetoacoustic standing mode in a gravitationally stratified solar coronal arcade
P. Konkol^{1}  K. Murawski^{1}  D. Lee^{2}  K. Weide^{2}
1  Group of Astrophysics and Gravity Theory, Institute of Physics, UMCS, ul. Radziszewskiego 10, 20031 Lublin, Poland
2 
ASC/Flash Center, The University of Chicago, 5640 S. Ellis Ave, Chicago, IL 60637, USA
Received 5 March 2010 / Accepted 2 June 2010
Abstract
Aims. We aim to explore the influence of thermal conduction
on the attenuation of the fundamental standing slow magnetoacoustic
mode in a twodimensional (2D) potential arcade that is embedded in a
gravitationally stratified solar corona.
Methods. We numerically solve the timedependent
magnetohydrodynamic equations to find the spatial and temporal
signatures of the mode.
Results. We find that this mode is strongly attenuated on a timescale of about 6 waveperiods.
Conclusions. The effect of nonideal plasma such as thermal
conduction is to enhance the attenuation of slow standing waves. The
numerical results are similar to previous observational data and
theoretical findings for the onedimensional plasma.
Key words: magnetohydrodynamics (MHD)  Sun: corona  Sun: oscillations
1 Introduction
Observational findings by SUMER SOHO/EIT and TRACE/EUV have revealed that solar coronal plasma is able to sustain (among other waves) both propagating (e.g., De Moortel et al. 2002a,b; Ofman et al. 1997; Murawski & Zaqarashvili 2010) and standing (e.g., Wang et al. 2002) slow magnetoacoustic (slow henceforth for slow magnetoacoustic) waves. Because slow waves are restricted to propagate along magnetic field lines, they are a valuable tool for probing the coronal plasma and providing information within the framework of coronal seismology (e.g., Roberts 2000; Nakariakov & Verwichte 2005; Ballai 2007; Andries et al. 2009).
Slow waves are found to be strongly attenuated over few waveperiods (Wang et al. 2003). Several mechanisms of attenuation of these waves were proposed: wave leakage into the internal layers of the solar atmosphere (Ofman 2002; Van Doorsselaere et al. 2004; Ogrodowczyk & Murawski 2007), lateral wave leakage as a result of the curvature of magnetic field lines (Roberts 2000; Ogrodowczyk et al. 2007), phase mixing (e.g., Ofman & Aschwanden 2002), resonant absorption (Ruderman & Roberts 2002) and nonideal magnetohydrodynamic (MHD) effects (Roberts 2000; Nakariakov et al. 2000; De Moortel & Hood 2003; Verwichte et al. 2008).
The intensive numerical investigations of slow standing waves in a straight magnetic field topology showed that without large amplitude slow mode oscillations attenuate rapidly due to shock dissipation (Haynes et al. 2008). Fully nonlinear 1D simulations of slow mode oscillations in the presence of thermal conduction were performed by De Moortel & Hood (2003) and Verwichte et al. (2008), who showed that thermal conduction is an important damping mechanism. In particular, De Moortel & Hood (2003) discussed in detail the influence of thermal conduction on slow waves. Their numerical results for driven waves revealed that for typical solar coronal conditions thermal conduction appears to be the dominant waveattenuation mechanism. Selwa et al. (2005) and Ogrodowczyk & Murawski (2007) showed that pressure pulses triggered near a footpoint excite the fundamental mode. The evolution of standing slow waves in a curved magnetic field topology was discussed by Selwa & Murawski (2006), Selwa et al. (2007a,b 2009), Ogrodowczyk et al. (2007) and Selwa & Ofman (2009) to find that loop curvature leads to the reduction of excitation and attenuation times of slow standing modes in a gravityfree arcade loop.
A study of slow wave propagation in gravitationally stratified media is rare. A theoretical 1D model of propagating slow waves in coronal loops by Nakariakov et al. (2000) includes the effects of stratification, nonlinearity, viscosity, resistivity, and thermal conduction. De Moortel & Hood (2004) investigated the effect of gravitational stratification in a straight magnetic field topology as a mechanism of attenuation of propagating slow waves. In another approach, MendozaBriceno et al. (2005) found that in a 1D gravitationally stratified atmosphere the attenuation time of slow standing waves was reduced by 1020% in comparison to a gravityfree case. Selwa & Ofman (2009) explored slow standing waves in a cold curved, gravitationally stratified loop.
Despite the significant achievements attained in the above mentioned papers there is still a lack of realistic modeling which takes a simultaneous presence of curved magnetic field and gravity into consideration. Although slow propagating waves are an interesting subject to study, we postpone this problem for future studies and limit ourself to slow standing waves. The aim of this paper is to study the influence of thermal conduction on the attenuation of slow standing waves in a gravitationally stratified solar coronal arcade. This way we generalize the model of Verwichte et al. (2008) into a 2D curved geometry and the model of Selwa & Ofman (2009) by inclusion of the photosphere layer, which results in wave reflection. Selwa & Ofman adopted linetying boundary conditions, which mimic action of the chromosphere/photosphere layer. We also extend the model of De Moortel & Hood (2003), in which damping of slow waves by thermal conduction was studied in the frame of a 1D homogeneous case. We extend their model to a curved magnetic field geometry and 2D case.
This paper is organized as follows. The numerical model is described in Sect. 2. The numerical results are presented in Sect. 3. We concluded with a short summary of the main results in Sect. 4.
2 The numerical model
We performed numerical simulations in a magnetically structured atmosphere. Henceforth, we neglect radiation and plasma heating, viscosity, and resistivity, but take into account isotropic thermal conduction, which is important for the damping of slow waves (Ofman 2002). As a consequence we use the following MHD equations to describe the coronal plasma:
with
Here p_{*} is total (gas plus magnetic) pressure, is total (kinetic plus internal and magnetic) energy density, is internal energy density, is the mass density, is the flow velocity, p is the gas pressure, is a constant gravity, is the magnetic field that is normalized as , T is plasma temperature, is a coefficient of thermal conductivity, is the adiabatic index, is Boltzman`s constant and denotes the mean mass. Although heat conduction is strongly prohibited across the field lines and prone to be along the field lines, we use isotropic conduction in this paper for simplicity reasons; anisotropic conduction effects will be studied in the future.
Equations (1)(5) are solved numerically using the new unsplit staggered mesh (USM) MHD solver (Lee & Deane 2009) in FLASH (Fryxell et al. 2000; Dubey et al. 2009). the FLASH architecture is described in Dubey et al. (2009).
The USM solver implements a highorder Godunov scheme in a directionally unsplit formulation, satisfying the condition to machine accuracy, using a constrained transport approach (Evans & Hawley 1988) on a uniform grid as well as the adaptive mesh refinement (AMR) grid. The USM solver can adopt various types of algorithms that provide first, second, and third order accurate Riemann reconstructions, different choices for slope limiters and Riemann solvers. In FLASH, the AMR grid is adopted, using the PARAMESH package (MacNeice et al. 1999).
We use a secondorder MUSCLHancock method (Toro 1997) with monotonized central slope limiter and the HLLD Riemann solver (Miyoshi & Kusano 2005). We set the simulation box as and impose fixedintime boundary conditions for all plasma quantities in the x and ydirections, while all plasma quantities remain invariant along the zdirection. In our studies we use the AMR grid with a minimum (maximum) level of refinement set to 4 (6), where each block contains interior cells. The refinement strategy is based on controlling numerical errors in mass density. This results in an excellent resolution of steep spatial profiles and greatly reduces numerical diffusion at these locations.
2.1 Initial setup
In this section, we detail the initial setup used in this paper. The solar corona is modeled as a low mass density, highly magnetized plasma overlaying a dense photosphere/chromosphere.
2.1.1 The structure of the solar atmosphere
We assume that the solar atmosphere is settled in a twodimensional
and still (
)
environment. Thus,
we assume that
the pressure gradient force is balanced by the gravity, that is
and the magnetic field is forcefree
Here the subscript corresponds to a background quantity.
Using the ideal gas law and the ycomponent of the hydrostatic pressure
balance indicated by Eq. (10), we express the background
gas pressure and mass density as
Here
is the pressure scaleheight, and denotes the gas pressure at the reference level that is chosen at Mm.
We adopt a smoothed stepfunction profile for the plasma temperature
where , and denote the chromospheric/photospheric temperature. The symbol corresponds to the temperature of the solar corona that is separated from the bottom layer at Mm by the transition region of its width km. Background gas pressure and mass density profiles that result from Eqs. (12) and (13) with the use of Eq. (15) are displayed in Fig. 1. Note that both and experience steep gradients at the transition region.
Figure 1: Vertical profiles of a background gas pressure (top panel) and a background mass density (bottom). 

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We adopt the magnetic field model was originally described by Priest (1982).
In this model, we assume that Eq. (11) is satisfied by a currentfree magnetic field
so that
Here A denotes the magnetic flux function
The background magnetic field components are then given by
in addition to . Here is the magnetic field at and the magnetic scale height is , with L=25 Mm. Magnetic field lines corresponding to Eq. (18) are illustrated by solid lines in Fig. 2. Note that we do not directly use Eq. (18) in order to get and . We instead take numerical derivatives of A(x,y), according to Eq. (16), to initialize and . This is important because a simple approach of using the analytical forms defined in Eq. (18) will not give the condition numerically.
Figure 2: Initial setup of the system. Magnetic lines are represented by solid lines. The mass density is displayed by a color bar. 

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Figure 3: Vertical profile of plasma . 

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The choice of magnetic field results in the plasma ,
The plasma is displayed in Fig. 3. Note that in the solar photosphere, it declines to a value of 0.025 at the transition region and later grows with altitude, reaching a value of at y=10 Mm.
2.1.2 Perturbations
Aim to study impulsively excited slow oscillations in the coronal arcade
that is described above. We found that these oscillations are triggered
most effectively by a simultaneous launching of initial pulses in gas pressure
and mass density
with
Here and are amplitudes of the initial pulses, w is their width and x_{0} and y_{0} denote their spatial positions. Unless otherwise stated we choose and hold fixed , , Mm, Mm, and . Additionally, these pulses are launched between two magnetic field lines determined by and as
Unless otherwise stated we set , and . Note that perturbations described by Eqs. (19) and (20) can be considered as a physical process in a coronal plasma where a sudden blast wave produced by flares can affect it.
Figure 3 shows vertical profile of plasma . As a result of this background and pulse the resulting standing wave will move along the magnetic line s with the mean value of .
3 Numerical results
The initial pulses of Eqs. (19) and (20) trigger magnetosonic waves. Fast waves propagate essentially isotropically and after a while they leave the simulation region. As slow waves are guided along magnetic field lines, they remain trapped in the system after experiencing reflections from the dense plasma regions.
Figure 4 displays a velocity component that is parallel to the magnetic field line, (solid line), and a mass density (dashed line) at t=3550 s. Here s denotes the coordinate along the magnetic field line which corresponds to the magnetic flux function , with s=0 denoting the arcade apex. Note that and consist a typical structure for the standing slow mode (e.g., Nakariakov & Verwichte 2005). The asymmetry in these profiles results from the strong initial pulse, which significantly perturbes the background state.
Figure 5 shows the timesignatures that are drawn by collecting wave signals in and at the detection point ( Mm, Mm). These timesignatures reveal oscillations that decay with time and exhibit the quarter waveperiod phase shift between and , which is characteristic for standing slow waves (Nakariakov & Verwichte 2005; Ogrodowczyk et al. 2009; Selwa et al. 2005).
Figure 4: Velocity (solid line) and mass density (dashed line) profiles along the magnetic line. 

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Figure 5: Timesignatures of a parallel velocity V_{} (solid line) and perturbed mass density (dashed line), collected at the detection point Mm, Mm. 

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We estimate the waveperiod P and attenuation time
of the
oscillation by fitting the following attenuated sine function:
to the timesignatures of Fig. 5. In this case, the waveperiod of the fundamental slow mode can be estimated as
where Mm is the length of the magnetic field line, which corresponds to the flux function and Mm s^{1} is the average sound speed along this line.
Figure 6: Attenuation time (upper panel) and waveperiod P (lower panel) vs. . Solid lines (dots) correspond to the analytical data of De Moortel & Hood (2003) (the numerical solutions of the full set of MHD equations). 

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Figure 7: Ratio of attenuation time and waveperiod P vs. the product of the normalized therma conductivity and a wavenumber k for the De Moortel & Hood (2003) (upper panel) and numerical solutions of the full set of MHD equations (lower panel). 

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Figure 8: Ratio of the attenuation time and period P vs. . 

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Figure 9: Ratio of attenuation time and waveperiod P vs. initial pulse strength . 

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Figure 6 illustrates the attenuation time (top panel) and waveperiod (bottom panel).
The solid lines result from the dispersion relation of
De Moortel & Hood (2003).
From their dispersion relation for a real value of k we obtain a complex wavefrequency
.
Hence we get a wave period
and
attenuation time
The index corresponds to the theoretical data of De Moortel & Hood (2003). Figure 7 compares ratio (upper panel) and taken from numerical simulations of full MHD equations with more complex background state (lower panel). In both cases there is some minimum located at about the same place.
Note that De Moortel & Hood (2003) deal with the homogeneous plasma and a straight magnetic field case, while we treat a more complex system in which the fundamental mode is attenuated by thermal conduction as well as by the curved magnetic field (Ogrodowczyk & Murawski 2007; Ogrodowczyk et al. 2009).
In numerical solutions waves are attenuated much stronger than in the theoretical model of De Moortel & Hood (2003). This is mainly due to the curved magnetic field, gravity, and nonideal reflections from the denser part of the atmosphere. Those effects are always present, so we have some finite attenuation even for .
The shorter waveperiod in the numerical simulations results from the action of gravity.
For a strictly vertical oscillation the dispersion relation can be written as
where is the cuttoff frequency. Hence we infer that in the presence of gravity the wave frequency (P) is higher (lower) than in the gravityfree case.
In order to show that these effects are present we ran a few cases for different values of the sound speed, . Note that according to Eq. (14) a higher value of corresponds to higher , which in turn results in a less inhomogeneous medium. Figure 8 illustrates a dependence of on . As grows with we infer that for a larger waves are less attenuated, which partially explains difference between our results of Fig. 6 for and those of De Moortel & Hood (2003).
Figure 9 illustrates a dependence of the ratio of attenuation time and waveperiod P on the initial pulse strength . According to our expectations declines with ; for a higher value of nonlinear effects become more important, which leads to wave steepening. As a result thermal conduction is more effective for steep profiles, which explains the fall, off of with (Verwichte et al. 2008).
4 Summary
We developed a twodimensional model of a coronal arcade to explore the attenuation of the fundamental slow magnetoacoustic standing mode in the presence of gravity and thermal conduction.
Our findings can be summarized as follows: the fundamental mode is excited impulsively by localized pulses in gas pressure and mass density that are initially launched in two nearby regions close to the chromosphere/photosphere. The obtained values of attenuation time depart from the analytical data for a homogeneous plasma studied by De Moortel & Hood (2003). This departure results from the adopted model, which implements wave energy leakage as a consequence of curved magnetic field lines and a presence of dense chromosphere/photosphere layer. However, what remains similar in both cases is the existence of some minimal value of damping time for a given wavelength andthermal conductivity. This minimum remains at about the same place.
AcknowledgementsP.K.'s and K.M.'s work was supported by a grant from the State Committee for Scientific Research Republic of Poland, with MNiI grant for years 20072010. The software used in this work was in part developed by the DOEsupported ASC/Alliance Center for Astrophysical Thermonuclear Flashes at the University of Chicago.
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All Figures
Figure 1: Vertical profiles of a background gas pressure (top panel) and a background mass density (bottom). 

Open with DEXTER  
In the text 
Figure 2: Initial setup of the system. Magnetic lines are represented by solid lines. The mass density is displayed by a color bar. 

Open with DEXTER  
In the text 
Figure 3: Vertical profile of plasma . 

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In the text 
Figure 4: Velocity (solid line) and mass density (dashed line) profiles along the magnetic line. 

Open with DEXTER  
In the text 
Figure 5: Timesignatures of a parallel velocity V_{} (solid line) and perturbed mass density (dashed line), collected at the detection point Mm, Mm. 

Open with DEXTER  
In the text 
Figure 6: Attenuation time (upper panel) and waveperiod P (lower panel) vs. . Solid lines (dots) correspond to the analytical data of De Moortel & Hood (2003) (the numerical solutions of the full set of MHD equations). 

Open with DEXTER  
In the text 
Figure 7: Ratio of attenuation time and waveperiod P vs. the product of the normalized therma conductivity and a wavenumber k for the De Moortel & Hood (2003) (upper panel) and numerical solutions of the full set of MHD equations (lower panel). 

Open with DEXTER  
In the text 
Figure 8: Ratio of the attenuation time and period P vs. . 

Open with DEXTER  
In the text 
Figure 9: Ratio of attenuation time and waveperiod P vs. initial pulse strength . 

Open with DEXTER  
In the text 
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