Dispersion relations, rays and ray splitting in magnetohelioseismology

Abstract

Local helioseismology seeks to probe the near surface regions of the Sun, and in particular of active regions. These are distinguished by their strong magnetic fields, yet current local techniques do not take proper account of this. Here, we first derive appropriate gravito-magneto-acoustic dispersion relations, and then use these to examine how acoustic rays entering regions of strong field split into fast and slow components, and the subsequent fates of each. Specifically, two types of transmission point, where wave energy can transfer from the fast to slow branch (or vice versa) are identified; one close to the equipartition level where the sound and Alfvén speeds coincide, and one higher up near the acoustic cutoff turning point. This second type only exists for rays of low frequency or low l though. In accord with recent studies of fast-to-slow mode conversion from the perspective of p-modes, magnetic field inclination is found to have significant consequences for wave splitting.

Oscillations and Waves in Sunspots

“…The coloring of each ray indicates the amount of energy it contains after each transformation. By comparing the three panels, it can be seen that the most efficient conversion between fast and slow modes happens when the field inclination is 30 ∘ because the direction of propagation of the fast-mode ray is aligned with the magnetic field (i.e., the attack angle is zero) Cally, 2006;Schunker and Cally, 2006). The most efficient secondary conversion from fast magnetic mode to the slow magnetic mode happens on the second passage of the fast mode through the = 1 layer on its return produced after the refraction and reflection (see Section 7.1), for the −30 ∘ inclined field, again due to the almost perfect alignment between the ray and the field direction.…”

“…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):…”

“…In the works of Barnes and Cally (2001);Cally (2006) and Moradi and Cally (2008), the effects of the acoustic cut-off frequency were incorporated in the eikonal solution for waves in an isothermal atmosphere with constant inclined magnetic field. The following three-dimensional dispersion relation is obtained (Moradi and Cally, 2008):…”

Energy exchange and wave action conservation for magnetohydrodynamic (MHD) waves in a general, slowly varying medium

“…This is the first term in an asymptotic series. At this level of approximation the three modes -fast, slow, and transverse Alfvén wave -remain distinct, with no coupling or mode conversion studied, for example, by Cally (2006) or Cally and Andries (2010). Coupling processes take place near caustic surfaces when rays approach each other and need a higher order of approximation for proper treatment.…”