Living Rev. Solar Phys., 2, (2005), 6 http://www.livingreviews.org/lrsp-2005-6 Local Helioseismology Laurent Gizon Stanford Solar Observatories Group W. W. Hansen Experimental Physics Laboratory 455 Via Palou, Stanford University Stanford, CA 94305-4085, U.S.A. and Max-Planck-Institut f¨urSonnensystemforschung Max-Planck-Str. 2 D-37191 Katlenburg-Lindau, Germany email: [email protected] http://www.mps.mpg.de/ Aaron C. Birch Colorado Research Associates A division of NorthWest Research Associates, Inc. 3380 Mitchell Lane Boulder, CO 80301-5410, U.S.A. email: [email protected] http://www.nwra.com/ Accepted on 13 May 2005 Published on 15 November 2005 Living Reviews in Solar Physics Published by the Max Planck Institute for Solar System Research Max-Planck-Str. 2, 37191 Katlenburg-Lindau, Germany ISSN 1614-4961 Abstract We review the current status of local helioseismology, covering both theoretical and observa- tional results. After a brief introduction to solar oscillations and wave propagation through in- homogeneous media, we describe the main techniques of local helioseismology: Fourier–Hankel decomposition, ring-diagram analysis, time-distance helioseismology, helioseismic holography, and direct modeling. We discuss local helioseismology of large-scale flows, the solar-cycle de- pendence of these flows, perturbations associated with regions of magnetic activity, and solar supergranulation. c Max Planck Society and the authors. Further information on copyright is given at http://solarphysics.livingreviews.org/About/copyright.html For permission to reproduce the article please contact [email protected]. How to cite this article Owing to the fact that a Living Reviews article can evolve over time, we recommend to cite the article as follows: Laurent Gizon and Aaron C. Birch, “Local Helioseismology”, Living Rev. Solar Phys., 2, (2005), 6. 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Contents 1 Outline 5 2 Observations of Solar Oscillations 6 2.1 Data for local helioseismology .............................. 6 2.2 Properties of solar oscillations .............................. 6 3 Models of Solar Oscillations 9 3.1 Linear waves ....................................... 9 3.2 Wave excitation ...................................... 10 3.3 Response to an impulsive source ............................ 10 3.3.1 Direct solution in plane-parallel models .................... 11 3.3.2 Normal-mode summation approximation .................... 11 3.3.3 Green’s functions for the observable ...................... 13 3.4 The zero-order problem ................................. 13 3.5 Effects of small steady perturbations .......................... 15 3.6 Tests of Born approximation for sound speed and flow perturbations . 17 3.7 Strong perturbations: magnetic tubes and sunspots . 17 4 Methods of Local Helioseismology 21 4.1 Fourier–Hankel spectral method ............................. 21 4.1.1 Wavefield decomposition ............................. 21 4.1.2 Absorption coefficient .............................. 22 4.1.3 Phase shifts .................................... 22 4.1.4 Mode mixing ................................... 24 4.2 Ring-diagram analysis .................................. 24 4.2.1 Local power spectra ............................... 24 4.2.2 Measurement procedure ............................. 26 4.2.3 Depth inversions ................................. 27 4.3 Time-distance helioseismology .............................. 29 4.3.1 Fourier filtering .................................. 29 4.3.2 Cross-covariance functions ............................ 30 4.3.3 Travel time measurements ............................ 31 4.3.4 Noise estimation ................................. 38 4.3.5 Travel time sensitivity kernels .......................... 38 4.3.6 Inversions of travel times ............................ 41 4.4 Helioseismic holography ................................. 44 4.4.1 Ingression and egression ............................. 47 4.4.2 Holography Green’s functions .......................... 48 4.4.3 Local control correlations ............................ 50 4.4.4 Acoustic power holography ........................... 50 4.4.5 Phase-sensitivity holography .......................... 51 4.4.6 Far-side imaging ................................. 52 4.4.7 Acoustic imaging ................................. 52 4.5 Direct modeling ...................................... 54 4.5.1 Forward problem ................................. 54 4.5.2 An example calculation ............................. 55 4.5.3 Inverse problem .................................. 56 5 Scientific Results from Local Helioseismology 57 5.1 Global scales ....................................... 57 5.1.1 Rotation and torsional oscillations ....................... 57 5.1.2 Meridional flow and its variations ........................ 61 5.1.3 Vertical flows ................................... 61 5.1.4 Search for variability at the tachocline ..................... 65 5.2 Active regions and sunspots ............................... 67 5.2.1 Ordered flows near complexes of magnetic activity . 67 5.2.2 Effect on longitudinal averages of large-scale flows . 69 5.2.3 Sunspot flows ................................... 71 5.2.4 Sinks and sources of acoustic waves ....................... 76 5.2.5 Phase shifts and wave-speed perturbations ................... 76 5.2.6 Far-side imaging ................................. 86 5.2.7 Excitation of waves by flares .......................... 86 5.3 Supergranulation ..................................... 88 5.3.1 Paradigms ..................................... 88 5.3.2 Horizontal flows and vertical structure ..................... 91 5.3.3 Rotation-induced vorticity ............................ 94 5.3.4 Pattern evolution ................................. 97 5.3.5 Traveling-wave convection ............................ 99 6 Acknowledgements 106 References 130 Local Helioseismology 5 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 helioseis- mology1 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