5 Sunspot Models and Helioseismic Constraints

Since the advent of local helioseismology, the subsurface structure of sunspots (thermal, magnetic, and flows) is among the prime targets. Independent investigations of subsurface flow and field structure are of great interest, since we cannot infer from surface observations alone whether a sunspot is monolithic or better described by a cluster model. In this section we only highlight very briefly recent developments and refer to the reviews by Gizon and Birch (2005) and Gizon et al. (2010a) for details.

A major difficulty in applying helioseismic inversions to sunspots comes with the fact that sunspots are not small perturbations with respect to the surrounding quiet Sun – at least not for the upper most few Mm. This intrinsically limits the applicability of linear inversion methods most commonly used. In addition, the effect of magnetic field cannot be easily captured by a simple isotropic wave speed perturbation due to its directional character. Direct application of time distance helioseismology to sunspots revealed a two layer structure of sunspots. Interpreting travel time shifts as wave speed perturbation, Kosovichev et al. (2000) found a reduction of wave speed in the upper most 4 Mm and an enhancement of wave speed further down, extending in depth beyond 10 Mm. The relative amplitude of wave speed perturbation was found to be up to 5% translating to either a substantial temperature change of more than 2000 K or a magnetic field almost reaching 20 kG just a few Mm beneath the photosphere. A similar 2 layer pattern was also found by Zhao et al. (2001) with respect to subsurface flows: an inflow from about 1 – 4 Mm depth and an outflow beneath. These results were also summarized in a recent review by Kosovichev (2006). Complementary, also ring diagram analysis has been applied to active regions (Basu et al., 2004; Bogart et al., 2008), providing results that are qualitatively comparable to the previously mentioned time-distance inversions. However, the transition from reduced to enhanced wave speed is found at a depth of more than 7 Mm.

These results have been challenged recently by a long list of comparative studies summarized in Gizon et al. (2009, 2010b) and Moradi et al. (2010Jump To The Next Citation Point). Most of the observed travel time shifts can be attributed to the upper most 2 Mm, where relative perturbations are large with respect to the quiet Sun reference stratification. Independent support for a rather superficial thermal anomaly in sunspots comes also from the quasi-1D approach of Lindsey et al. (2010). Their method focuses on the frequency dependence of travel time shifts for waves that propagate almost vertical in sunspots (and are reflected in the umbra) to separate thermal and magnetic effects. A very promising new approach that has been taken by several groups in parallel is the forward modeling of wave propagation through stationary sunspot models that automatically addresses the complicated nature of the wave propagation in inclined magnetic field (Khomenko and Collados, 2006; Cameron et al., 2007a; Parchevsky and Kosovichev, 2007; Hanasoge, 2008; Shelyag et al., 2009). By modeling the propagation of f, p1, and p2-modes, Cameron et al. (2010) demonstrated that the observed helioseismic signatures of sunspots can be well captured by a rather shallow model. Evidence for a dominant influence from near surface effects was reported previously by Braun and Birch (2006) based on observations of the frequency dependence of p-mode travel times in sunspots. Also recent 3D simulations of sunspots (Rempel et al., 2009b,a) can be explored to infer helioseismic signatures. In these simulations the full spectrum (to the degree it is allowed by the finite size of the domain) of modes is naturally excited by convection and can be analyzed similar to observed modes by extracting artificial Dopplergrams on constant height or constant optical depth surfaces. As presented in Moradi et al. (2010) and Braun et al. (2011), fully compressible 3D radiative MHD models of sunspots predict a rather shallow wave speed perturbation and are in terms of travel times consistent with observations. Overall, there is now very strong evidence, both observationally and theoretically, that a detailed understanding of near surface effects will be essential for making progress in sunspot seismology.

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