# Numerical Simulations

### Theoretical Modeling of Propagation of Magnetoacoustic Waves in Magnetic Regions Below Sunspots

We use two-dimensional numerical simulations and eikonal approximation to study properties of magnetohydrodynamic (MHD) waves traveling below the solar surface through the magnetic structure of sunspots. We consider a series of magnetostatic models of sunspots of different magnetic field strengths, from 10 Mm below the photosphere to the low chromosphere. The purpose of these studies is to quantify the effect of the magnetic field on local helioseismology measurements by modeling waves excited by subphotospheric sources. Time-distance propagation diagrams and wave travel times are calculated for models of various field strengths and compared to the nonmagnetic case. The results clearly indicate that the observed time-distance helioseismology signals in sunspot regions correspond to fast MHD waves. The slow MHD waves form a distinctly different pattern in the time-distance diagram, which has not been detected in observations. The numerical results are in good agreement with the solution in the short-wavelength (eikonal) approximation, providing its validation. The frequency dependence of the travel times is in good qualitative agreement with observations.

Wave paths of the fast mode waves launched from the same lower turning point below the photosphere propagating through the model sunspot with Bphot = 2.4 kG for different frequencies: 6.6 mHz (top), 4.5 mHz (middle), and 3 mHz (bottom). The white curves are the wave paths when the magnetic ﬁeld is taken into account. The red curves are the wave paths in the sunspot models with the same thermal properties, but setting B = 0. The black dots on the trajectory are separated from each other by 1 minute in time. The green curve marks the position of the v_A = c_S layer. The yellow curves mark the heights where the frequency of the wave is equal to the local acoustic cutoff frequency. The background gray image is acoustic speed c_s.

Khomenko, E., Kosovichev, A., Collados, M., Parchevsky, K., Olshevsky, V. 2009, ApJ, 694, 411

### Three-dimensional Radiative Transfer Modeling of the Polarization of the Sun's Continuous Spectrum

Polarized light provides the most reliable source of information at our disposal for diagnosing the physical properties of astrophysical plasmas, including the three-dimensional (3D) structure of the solar atmosphere. Here we formulate and solve the 3D radiative transfer problem of the linear polarization of the solar continuous radiation, which is principally produced by Rayleigh and Thomson scattering. Our approach takes into account not only the anisotropy of the solar continuum radiation but also the symmetry-breaking effects caused by the horizontal atmospheric inhomogeneities produced by the solar surface convection. We show that such symmetry-breaking effects do produce observable signatures in Q/I and U/I, even at the very center of the solar disk where we observe the forward scattering case, but their detection would require obtaining very high resolution linear polarization images of the solar surface. Without spatial and/or temporal resolution U/I ≈ 0 and the only observable quantity is Q/I, whose wavelength variation at a solar disk position close to the limb has been recently determined semi-empirically. Interestingly, our 3D radiative transfer modeling of the polarization of the Sun's continuous spectrum in a well-known 3D hydrodynamical model of the solar photosphere shows remarkable agreement with the semi-empirical determination, significantly better than that obtained via the use of one-dimensional (1D) atmospheric models. Although this result confirms that the above-mentioned 3D model was indeed a suitable choice for our Hanle-effect estimation of the substantial amount of "hidden" magnetic energy that is stored in the quiet solar photosphere, we have found however some small discrepancies whose origin may be due to uncertainties in the semi-empirical data and/or in the thermal and density structure of the 3D model. For this reason, we have paid some attention also to other (more familiar) observables, like the center-limb variation of the continuum intensity, which we have calculated taking into account the scattering contribution to the continuum source function. The overall agreement with the observed center-limb variation turns out to be impressive, but we find a hint that the model's temperature gradients in the continuum-forming layers could be slightly too steep, perhaps because all current simulations of solar surface convection and magnetoconvection compute the radiative flux divergence ignoring the fact that the effective polarizability is not completely negligible, especially in the downward-moving intergranular lane plasma.

Symbols show the variation with wavelength of the observed solar continuum intensity for some selected μ-values, normalized to the ensuing disk center value. The ﬁlled circles correspond to the observations of Pierce & Slaughter (1977), while the stars correspond to those of Neckel & Labs (1994). The solid lines show the results of our radiative transfer calculations in the 3D model.

Emergent Q/I (top panels) and U/I (bottom panels) at 4600 Å calculated for three LOSs in the 3D model and accounting for the diffraction limit effect of a 1 m telescope. The positive reference direction for Stokes Q lies along the vertical direction of the corresponding panel, which for the μ = 0.1 and μ = 0.5 cases coincides with the parallel to the limb of the chosen solar atmospheric model. Note that we have taken into account the projection effects by means of which the off-disk-center images appear contracted by a factor μ along the horizontal direction of the ﬁgure panels. Note also that the “surface distances” given in the plots measure the true separation between the points on the actual surface of the solar model. The solid-line contours in the μ = 1 panels delineate the (visible) upﬂowing granular regions.

Trujillo Bueno, J., Shchukina, N. 2009, ApJ, 694, 1364

### Numerical magnetohydrodynamics

We have developed a numerical magnetohydrodinamic code for the calculation of the response of a equilibrium magnetic atmosphere to an arbitrary perturbation. The code solves the 3D nonlinear MHD equations for perturbations, which are obtained by subtracting the equations of initial magnetohydrostatic equilibrium from the system of MHD equations. Spatial derivatives are discretized using fourth-order centered differences and the solution is advanced in time using a fourth-order Runge-Kutta scheme. It is stabilized by artificial diffusion terms and its parallel design is based on domain decomposition scheme. Extensive tests have been performed to prove the robustness of the code. The figure presents the results of the simulations of the waves excited by a harmonic driver in a 3D sunspot atmosphere.

This movie was created with the help of VAPOR visualization software.