Numerical Simulation of Acoustic Wave Propagation in the Solar Sub ...

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the Solar Sub-Photosphere with Localized Magnetic Field. Concentration ... ifications are derived from a standard solar model and adjusted to maintain.
Solar-Stellar Dynamos as Revealed by Helio- and Asteroseismology c 2009 ASP Conference Series, Vol. 416, ! M. Dikpati, T. Arentoft, I. Gonz´ alez Hern´ andez, C. Lindsey and F. Hill, eds.

Numerical Simulation of Acoustic Wave Propagation in the Solar Sub-Photosphere with Localized Magnetic Field Concentration S. Shelyag, S. Zharkov, V. Fedun, R. Erd´elyi, and M. J. Thompson Solar Physics and upper-Atmosphere Research Center, Department of Applied Mathematics, University of Sheffield, Hicks Building, Hounsfield Rd., Sheffield, S7 3RH, United Kingdom Abstract. The results of numerical simulations of acoustic wave propagation and dispersion in the solar sub-photosphere with a localised magnetic field concentration are presented. The initial equilibrium density and pressure stratifications are derived from a standard solar model and adjusted to maintain magnetohydrostatic and convective stability. Acoustic waves are generated by a perturbation located at the height corresponding to the visible surface of the Sun. The time-distance diagram of the vertical velocity perturbation at the level corresponding to the visible solar surface shows that the magnetic field perturbs and scatters acoustic waves and absorbs the acoustic power of the wave packet.

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Introduction

The internal structure of sunspots is still not quite well known. Helioseismic techniques, which model the propagation of sound waves through the solar interior and the influence of internal solar inhomogeneities on sound wave propagation, might be of great help in analysing the invisible, sub-photospheric solar processes. The ability of forward numerical simulations to predict and reproduce various solar phenomena in helioseismology has been shown by e.g. Shelyag et al. (2006, 2007); Hanasoge et al. (2007); Parchevsky & Kosovichev (2007) and many others. Since magnetic fields are, perhaps, the most important property of the sunspots, it is of particular interest to analyse the influence on acoustic wave propagation of solar magnetic field concentrations, such as sunspots or solar active regions. The appearance of slow magnetoacoustic waves has been shown in forward MHD simulations in non-realistic polytropic models by Crouch & Cally (2003); Gordovskyy & Jain (2007). Ray-approximation simulations in a more realistic and applicable magnetised model have shown a similar behavour of the acoustic waves (Moradi & Cally 2008). Shelyag et al. (2007) have analysed the influence of sub-photospheric flows on acoustic wave propagation using foward modeling. Now it is timely to perform a full forward magneto-hydrodynamic simulation of a wave packet propagating through a non-uniform magnetic field region in the realistic solar sub-photosphere. In the next sections we show some preliminary results and a brief analysis of such a simulation. 167

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Shelyag et al. Simulation Model

The code developed by Shelyag et al. (2008) has been used to carry out the simulations. The code is designed to solve the ideal magneto-hydrodynamic equations in a multi-dimensional Cartesian domain for gravitationally-stratified media. Fourth-order central spatial and fourth-order Runge-Kutta time discretization schemes are used. Hyperviscosity and hyperresistivity are implemented to stabilize the numerical solution. Here we focus on 2D simulations. The 2D box is 180 Mm wide and 50 Mm deep, and has a resolution of 960x1000 grid points; the upper boundary of the domain being near the solar temperature minimum. The boundaries of the domain are open. The perturbation source is located 500 km below the upper boundary of the simulation box. The vertical velocity measurement level is near the temperature minimum. We use a self-similar magnetic field configuration, which can be obtained from the following set of equations (Schl¨ uter & Temesv´ary 1958; Sch¨ ussler & Rempel 2005; Cameron et al. 2008): Bx = −∂f /∂z · G (f ) ,

(1)

Bz = ∂f /∂x · G (f ) ,

(2)

f = x · B0z (z) ,

(3)

where B0z describes the decrease of the vertical component of magnetic field towards the top of the model, and G is a function which defines how the magnetic field opens up with height. The vector field, which is built in this manner, is divergence free by definition. Pressure and density are then recalculated numerically using the magnetohydrostatic equilibrium condition. The parameters are chosen in such a way that the vertical component of magnetic field with a strength of 1000 G at the top of the model mimicks the magnetic field of a weak sunspot. We introduce a perturbation source described by the expression: vz = sin(2πt/T0 ) exp(−((t − T1 )2 /σ12 )) exp(−((r − r0 )2 /σ02 )),

(4)

where T0 =300 s, T1 =600 s, σ1 =100 s, σ2 =0.1 Mm, r0 is the source location. The source is located in the top-middle of the computational box, allowing us to compare the waves propagating to the left (uninfluenced by the magnetic field) and to the right from the source location. 3

Time-Distance Analysis

We study the influence of the magnetic field on the acoustic response of the simulated solar sub-photosphere. The analysis shows that for this particular configuration of magnetic field, background model and spatial resolution, the influence the magnetic field exercises on the wave propagation is mainly caused by the temperature (and hence local sound-speed) change in the magnetised region. The temperature increase below the simulated sunspot causes negative phase shifts of the wave packets propagating through it. The phase shifts are observed in all of the bounces, however, the first bounce is only affected in the

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Figure 1. Power spectrum of the vertical velocity perturbation generated by the source. The p modes are visible up to high orders. Eigenmodes of the background model are overplotted by solid lines.

magnetic field region (in Figure 2, the phase shifts for the first bounce at 0-30 Mm and at 70-90 Mm are of the order of numerical noise). Higher-order bounces are all affected for > 30 Mm distance from the source. The internal structure of the acoustic perturbation can be revealed by making the ratio of acoustic power in the left and right parts of the simulation domain. Figure 3 shows a complicated structure of changing acoustic shadows created by the magnetic field. The acoustic rays are plotted over the power ratio image in the figure and correspond well to the acoustic shadows. Acknowledgments. This work was supported by a grant from the UK Science and Technology Facilities Council (STFC). RE acknowledges M. K´eray for patient encouragement. RE is also grateful to NSF, Hungary (OTKA, Ref.No. K67746). S. Zharkov acknowledges the support from the HELAS European Network. References Crouch, A. D., & Cally, P. S. 2003, Solar Phys., 214, 201 Cameron, R., Gizon, L., & Duvall, T. L., Jr. 2008, Solar Phys., 251, 291 Gordovskyy, M., & Jain, R. 2007, ApJ, 661, 586 Hanasoge, S. M., Duvall, T. L., Jr., & Couvidat, S. 2007, ApJ, 664, 1234 Moradi, H., & Cally, P. S. 2008, Solar Phys., 251, 309 Parchevsky, K. V., & Kosovichev, A. G. 2007, ApJ, 666, 547 Schl¨ uter, A., & Temesv´ary, S. 1958, Electromag Phen Cosm Phys, 6, 263 Sch¨ ussler, M., & Rempel, M. 2005, A&A, 441, 337 Shelyag, S., Erd´elyi, R., & Thompson, M. J. 2006, ApJ, 651, 576 Shelyag, S., Erd´elyi, R., & Thompson, M. J. 2007, A&A, 469, 1101 Shelyag, S., Fedun, V., & Erd´elyi, R. 2008, A&A, 486, 655

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Figure 2. Vertical speed difference image. The difference is computed between the points located at the same distance and opposite sides from the source. The first bounce is affected only locally by the magnetic field, however, the second and third bounces are also affected in the 60-80 Mm distance region.

Figure 3. Acoustic power ratio image for the magnetized region of the model. The ratio is computed between the points located at the same distance and opposite sides from the source. The image shows the regions of acoustic power decreased compared with the ambient non-magnetic medium. The acoustic rays are overplotted. The lower turning points of the acoustic rays calculated for the frequency f =4.8 mHz correspond well with the acoustic shadows.