Three-Dimensional MHD Wave Propagation and Conversion to Alfvén Waves near the Solar Surface. I. Direct Numerical Solution

Abstract

The efficacy of fast/slow MHD mode conversion in the surface layers of sunspots has been demonstrated over recent years using a number of modelling techniques, including ray theory, perturbation theory, differential eigensystem analysis, and direct numerical simulation. These show that significant energy may be transferred between the fast and slow modes in the neighbourhood of the equipartition layer where the Alfvén and sound speeds coincide. However, most of the models so far have been two dimensional. In three dimensions the Alfvén wave may couple to the magnetoacoustic waves with important implications for energy loss from helioseismic modes and for oscillations in the atmosphere above the spot. In this paper, we carry out a numerical "scattering experiment", placing an acoustic driver 4 Mm below the solar surface and monitoring the acoustic and Alfvénic wave energy flux high in an isothermal atmosphere placed above it. These calculations indeed show that energy conversion to upward travelling Alfvén waves can be substantial, in many cases exceeding loss to slow (acoustic) waves. Typically, at penumbral magnetic field strengths, the strongest Alfvén fluxes are produced when the field is inclined 30• - 40• from the vertical, with the vertical plane of wave propagation offset from the vertical plane containing field lines by some 60• -80• .

Oscillations and Waves in Sunspots

“…As mentioned above, the bases of the MHD wave conversion theory in a stratified solar atmosphere were developed by Zhugzhda andDzhalilov (1982, 1984a,c) and Cally (2001). Later on this theory was improved and extended to the case of inclined magnetic fields, by means of the ray theory and eikonal approximation (e.g., Cally, 2005Cally, , 2006Schunker and Cally, 2006;Cally and Goossens, 2008).…”

“…The ray theory shows that the direction and the efficiency of the mode conversion depends, among other parameters, on the wave frequency and the attacking angle between the wave vector ⃗ and the magnetic field (e.g., Cally, 2005Cally, , 2006Schunker and Cally, 2006;Cally and Goossens, 2008). In the two-dimensional case, an approximate formula for the fast-to-slow mode (acoustic to acoustic) transmission coefficient was derived by Cally (2005) for high-frequency waves (above the acoustic cut-off):…”

Helioseismology of Sunspots: Confronting Observations with Three-Dimensional MHD Simulations of Wave Propagation

“…This aspect of the problem was followed up by Cally, Crouch, and Braun (2003) and Crouch et al (2005), who placed constraints on the strength of the sunspot's magnetic field. More recently, attention has been placed on upward propagating magnetoacoustic waves and their possible observational signatures (Schunker and Cally, 2006;Khomenko and Collados, 2006;Cally and Goossens, 2008). The use of direct numerical simulations in helioseismology, with or without magnetic fields, has also undergone rapid development.…”

Helioseismic probing of the subsurface structure of sunspots

“…The bottom layer is a polytrope that represents the deep solar interior. In the polytrope and separately in the isothermal slab semi-analytic solutions for the linearized MHD equations are developed (e.g., Crouch & Cally, 2003, Cally & Goossens, 2008. Physical boundary conditions Figure 1.…”