1 Outline

Helioseismology is a powerful tool to study the interior of the Sun from surface observations of naturally-excited internal acoustic and surface-gravity waves. Helioseismological studies based on the interpretation of the eigenfrequencies of the resonant modes of oscillations have yielded many exciting results about the internal structure and dynamics of the Sun (see, e.g., Christensen-Dalsgaard, 2002

). For example, a major achievement has been the inference of the large-scale rotation as a function of depth and unsigned latitude (see, e.g., Thompson et al., 2003 ). The angular velocity inside the Sun is now known to be larger at the equator than at the poles throughout the convection zone, while the radiative interior rotates nearly uniformly. The layer of radial shear at the bottom of the convection zone, known as the tachocline, is commonly believed to be the seat of the solar dynamo (see, e.g., Gilman, 2000

). The current research focuses on small temporal variations connected to the solar cycle that are likely to be related to the magnetic dynamo.

With global-mode helioseismology, however, it is not possible to detect longitudinal variations or flows in meridional planes. To complement global helioseismology, techniques of local helioseismology¹ are being developed to probe local perturbations in the solar interior or on its surface (see review by Duvall Jr, 1998 ). The goal of local helioseismology is to interpret the full wave field observed at the surface, not just the eigenmode frequencies. Local helioseismology provides a three-dimensional view of the solar interior, which is important to understand large-scale flows, magnetic structures, and their interactions in the solar interior. Local helioseismology includes a number of different approaches that complement each other. This paper is an attempt to review all these techniques and their achievements. Not all methods of local helioseismology have reached the same degree of maturity. In Section 2 we give basic information about the data that are currently most commonly used for local helioseismology and about the properties of solar oscillations. In Section 3 we discuss equations of motion for solar oscillations, Green’s functions for the response of solar models to forcing, and basic perturbation methods and their range of validity. The main methods of local helioseismology, i.e., Fourier-Hankel decomposition, ring-diagram analysis, time-distance helioseismology, helioseismic holography, and direct modeling, are described in Section 4. In Section 5 we give a summary and discussion of the main results obtained using local helioseismology regarding global-scale features, active regions and sunspots, excitation of waves by flares, and supergranulation. Whenever possible, we discuss the physical implications of the observations.