Local Helioseismology
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Abstract
We review the current status of local helioseismology, covering both theoretical and observational 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 dependence of these flows, perturbations associated with regions of magnetic activity, and solar supergranulation.
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 helioseismology^{1} 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.
2 Observations of Solar Oscillations
2.1 Data for local helioseismology
The fundamental data of modern helioseismology are high-resolution Doppler images of the Sun’s surface. Local helioseismology started with observations of acoustic absorption by sunspots using data from the Kitt Peak vacuum telescope (Braun et al., 1987, 1988). Observations obtained by Hill (1988, 1989) at the Sacramento Peak vacuum tower telescope demonstrated that local spectra of solar oscillations provide measurable information about internal horizontal flows. Continuous data from the 1988 south pole expedition lead to direct measurements of local travel times of acoustic waves (Duvall Jr et al., 1993). Today, the development of local helioseismology is fueled by high-quality data from space and ground based networks. The main datasets are provided by the Taiwan Oscillation Network (TON), the Global Oscillation Network Group (GONG), and the Michelson Doppler Imager (MDI) aboard the ESA/NASA SOHO spacecraft in a halo orbit around the Li Sun-Earth Lagrange point.
The TON consists of six identical telescopes at appropriate longitude around the globe. The data are series of 1080 × 1080 full-disk Ca^{+} K-line intensity images recorded at a rate of one image per minute. A description of the TON project is given by Chou et al. (1995). The TON data may be requested by contacting the Principal Investigator, Dr. Dean-Yi Chou (chou@phys.nthu.edu.tw).
The GONG is an international network of six extremely sensitive and stable solar velocity imagers that provide nearly continuous observations of solar oscillations (Leibacher, 1999). The GONG instruments, which are Michelson-interferometer-based Fourier tachometers, observe the Ni I 6768 Å line. In addition to Doppler and intensity images every minute, GONG provides full-disk magnetograms nominally every 20 minutes. The system became operational in October 1995, and will operate for at least an eleven-year solar cycle. The observation duty cycle has averaged about 90%. The original instruments used 256 × 256 pixel CCD cameras, which where replaced in 2001 by 1024 × 1024 square-pixel cameras. The GONG data products can be accessed at the project’s website (GONG, 2002).
The MDI has provided line-of-sight Doppler velocity images since 1996 with an excellent duty cycle (Scherrer et al., 1995). MDI Dopplergrams are obtained by combining 4 filtergrams on the wings and core of the Ni 6788 Å absorption line, formed just above the photosphere. Dopplergrams are available at a one minute cadence. MDI operates under several observing modes. The Dynamics Program runs for 2 to 3 months each year and provides 1024 × 1024 full-disk Doppler images; the plate scale is 2″ per pixel, or 0.12 heliographic degrees (1.45 Mm at disk center). The Structure Program provides continuous coverage: full-disk images are binned onboard into a set of about 20,000 regions of roughly similar projected areas on the Sun to make use of the narrow telemetry channel. The Structure Program data are used to measure mode frequencies up to spherical harmonics degrees of 250. MDI can also operate in High-Resolution mode by zooming on a 11′ square field of the Sun with a plate scale of 0.625″ per pixel and a diffraction-limited resolution of 1.25″. MDI data can be accessed at the project’s website (MDI, 1997).
2.2 Properties of solar oscillations
The five-minute solar oscillations were first discovered by Leighton et al. (1962) and interpreted as standing acoustic waves by Ulrich (1970) and Leibacher and Stein (1971). Deubner (1975) then confirmed that the power in the oscillations is concentrated at discrete frequencies for any given horizontal wavenumber, as predicted by Ulrich’s theory. The driving mechanism of solar oscillations is believed to be near-surface turbulent convection (Goldreich and Keeley, 1977). Solar and stellar oscillations are discussed in details by Cox (1980), Gough (1993), Unno et al. (1989), and Christensen-Dalsgaard (2002). Particularly useful are the lecture notes of J. Christensen-Dalsgaard (Christensen-Dalsgaard, 2003).
3 Models of Solar Oscillations
In order to understand local helioseismology it is crucial to understand wave propagation through generic solar models, including models with local inhomogeneities. In Section 3.1 we review the equations of motion for linear waves moving through non-magnetic backgrounds. The sources of excitation of solar oscillations are characterized in Section 3.2. In Section 3.3 we discuss methods for computing the Green’s functions for solar models. We describe the zero-order problem in Section 3.4 and the effects of weak steady perturbations in Section 3.5. Numerical tests of the Born approximation are described in Section 3.6. Some effects of magnetic fields are briefly reviewed in Section 3.7.
Throughout this section we will address two general classes of models: general models and plane-parallel models. We will not explicitly consider the case of spherically symmetric models. In general we will use the symbol r to denote position in three dimensions. For plane-parallel models we decompose the position vector as r = (x, z), where x is a two-dimensional horizontal vector and z is the height coordinate. The plane-parallel approximation is valid for very high degree modes, which are routinely used in local helioseismology. Plane-parallel models are often assumed for the interpretation of data collected over a small patch of the Sun: images may be remapped onto a grid which is, locally, approximately Cartesian.
3.1 Linear waves
3.2 Wave excitation
It is generally agreed that solar oscillations are excited by near surface turbulent convection (see, e.g., Goldreich et al., 1994). The two main approaches to modeling wave excitation have been numerical convection simulations and analytical models based on mixing length convection models.
Goldreich et al. (1994) used a mixing length model of solar convection to compute the energy input rates for modes with angular degree less than 60. The model energy input rates were very similar to the observed rates. In the Goldreich et al. (1994) model the main source of wave excitation was entropy fluctuations. Numerical simulations of near-surface turbulent convection have also been able to explain the observed frequency dependence of the energy input and damping rates (see, e.g., Stein and Nordlund, 2001). In the Stein and Nordlund (2001) model, the main source of wave excitation is Reynold’s stresses (turbulent pressure) near the boundaries of granules. Samadi et al. (2003a) compared wave excitation in a 3d numerical simulation and 1d mixing length based models. The numerical simulation gave about five times more energy input into the p-modes than did the mixing length model. In the numerical simulations of Samadi et al. (2003a), excitation by entropy fluctuations dominates over excitation by Reynold’s stresses. Samadi et al. (2003b) used a 3d numerical simulation to study the covariance function of the near-surface turbulent velocity and found that the temporal covariance was not Gaussian. As we will discuss in Section 3.4, this covariance is important for computing the power spectrum of solar oscillations.
As both the numerical convection simulations and the analytical convection models become more developed, it seems likely that they will converge and produce a definitive answer as to the source of solar oscillations.
3.3 Response to an impulsive source
There are three approaches to solving Equation (11). In the case of very simple problems it is sometimes possible to solve analytically for the Green’s functions in the Fourier domain (see, e.g., Gizon and Birch, 2002). Another, more general approach, is direct numerical solution (Section 3.3.1). An efficient approximate solution is normal mode summation (Section 3.3.2).
3.3.1 Direct solution in plane-parallel models
3.3.2 Normal-mode summation approximation
3.3.3 Green’s functions for the observable
3.4 The zero-order problem
The zero-order problem is to solve the driven equations of motion, Equation (7), when there are no perturbations to the background model. By the background model we mean a description of the background state together with specifications for wave damping and the statistics of wave excitation. As discussed in Section 3.1 we assume that the background state is a steady, wave-free, non-magnetic background solar model. The statistics of wave excitation which are needed to compute the wavefield covariance are described by the source covariance matrix M_{ij} (see Section 3.2). Wave damping can be described either physically (e.g., Balmforth, 1992) or phenomenologically (e.g., Gizon and Birch, 2002).
Notice that we have not attempted to include various corrections to the power spectrum (although it could in principle be done). For example, we have neglected the effects of the temporal and spatial window functions. Having observations over a finite spatial area or a finite amount of time will lead to smearing and lack of resolution in the k and ω domains. Also, we have ignored sphericity. Analyzing solar data as if the Sun were flat, which is common in local-helioseismology, leads to distortions in power spectra due to use of FFTs on data that have been projected onto a plane. Line-of-sight projection effects have also been ignored (local helioseismology typically uses the Doppler velocity as input data). For example, waves moving toward disk center are more visible than waves moving in the perpendicular direction; this results in anisotropic power spectra.
3.5 Effects of small steady perturbations
In order to do linear inversions of helioseismic data, it is necessary to first solve the linear forward problem. The linear forward problem is to compute the first-order effect of small perturbations to the solar model. By first-order we mean first order in the strength of the perturbations. Essentially all inversions that have been done have assumed that the perturbations to the solar model are time-independent over the time duration during which the observations are made.
There have been numerous efforts to approximate the kernel functions K for time-distance travel times. These efforts will be described in Section 4.3.5. The kernel functions for ring diagrams have typically been approximated as constant within the area the ring is measured over, with a depth dependence derived from normal mode theory (see Section 4.2.3).
The linear forward problem for normal-mode frequencies is quite well known (see the upcoming Living Reviews paper on global helioseismology, Thompson (2005)). For example, Gough and Thompson (1990), Goldreich et al. (1991), and Dziembowski and Goode (2004) applied first-order perturbation theory to estimate the effect of magnetic fields on normal mode frequencies. This work may be helpful for computing the effects of magnetic fields on local helioseismic measurements.
An alternative to the Born approximation is the Rytov approximation. In the Rytov approximation one computes the first order correction to the phase of the wavefield rather than the correction to the wavefield itself. For applications in the context of helioseismology, see Brüggen (2000) and Jensen and Pijpers (2003). The Born and Rytov approximations have been compared by Keller (1969).
3.6 Tests of Born approximation for sound speed and flow perturbations
The range of validity of the Born approximation is an important question for local helioseismology, as the linear forward problem proceeds naturally in the Born approximation.
3.7 Strong perturbations: magnetic tubes and sunspots
The interaction of solar waves with photospheric magnetic fields has been studied extensively in recent years. Yet, applications in the context of local helioseismology have proved difficult. The source of this difficulty is that magnetic perturbations are not small near the solar surface (e.g., Crouch and Cally, 2004): the first Born approximation cannot be applied there. Only deeper inside the Sun, can magnetic effects be treated as small perturbations. In this paragraph, we review theoretical results regarding wave interaction with magnetic flux tubes and sunspots.
Near the photosphere magnetic fields appear to be clumped into intense flux tubes with a typical field strength of order 1 kG and diameters of about 100 km. It is well known (see, e.g., Hollweg, 1990) that flux tubes support various modes of oscillation: the Alfvén modes (twisting motion), the sausage modes (propagating change in the cross-section of the tube), and the kink modes (bending of the tube). The interaction of acoustic waves with thin flux tubes, i.e., tubes with diameters much less than the wavelengths, has been studied extensively (see, e.g., Ryutova and Priest, 1993; Bogdan and Zweibel, 1985; Bogdan et al., 1996; Tirry, 2000). There is general agreement that scattering of acoustic waves by flux tubes contributes to the observed damping rates and frequency shifts (Rajaguru et al., 2001; Komm et al., 2002).
Finsterle et al. (2004b) have detected the interaction of high-frequency acoustic waves with the canopy magnetic field in the solar chromosphere. These exciting results were obtained by making observations of solar oscillations at different heights in the atmosphere (Finsterle et al., 2004a). We note that there have been theoretical studies of the effect of the magnetic canopy on acoustic modes (see, e.g., Campbell and Roberts, 1989; Goldreich et al., 1991).
Thomas et al. (1982) first predicted that solar oscillations could be used to probe the internal structure of sunspots. Sunspots are known to absorb incident p-mode energy (Braun et al., 1987), introduce phase shifts between the incident and scattered waves (Braun et al., 1992; Duvall Jr et al., 1996; Lindsey and Braun, 2004), and cause mode mixing (Braun, 1995). Effects that have been suggested to cause phase shifts are the Wilson depression (Braun and Lindsey, 2000), flows (see, e.g., Duvall Jr et al., 1996; Kosovichev, 1996), inhomogeneous absorption (Woodard, 1997), temperature/density/wave-speed anomalies (see, e.g., Kosovichev, 1996; Brüggen and Spruit, 2000; Tong et al., 2003), and the direct effect of magnetic fields. Bogdan et al. (1998) and Cally et al. (2003) have produced models of the coupling of ambient p and f modes with various magnetic waves in the sunspot that explain some aspects of the data, and demonstrate that magneto-atmospheric waves can not be ignored in local helioseismology. Yet, according to Bogdan (2000), ‘ignorance triumphs over knowledge’. The main difficulties are nonlinear aspects of wave propagation, radiative transfer in magnetised plasmas, and the relationship between velocity measurements in sunspots and real fluid motions. The theoretical study of oscillations in sunspots is an entire field of research and is crucial for the interpretation of local helioseismic measurements in and around sunspots. We refer the reader to the reviews by Bogdan and Braun (1995) and Bogdan (2000) for further details and references.
4 Methods of Local Helioseismology
4.1 Fourier-Hankel spectral method
4.1.1 Wavefield decomposition
4.1.2 Absorption coefficient
4.1.3 Phase shifts
Braun et al. (1992) and Braun (1995) were successful at measuring the relative phase shift between outgoing and ingoing waves. Phase shifts measurements require a lot of care. Braun et al. (1992) pointed out that the phases of A_{m} and B_{m} represent averages over a resolution element with size (ΔL, Δν). For example, Braun (1995) has ΔL = 20 and Δν = 4 µHz.
4.1.4 Mode mixing
4.2 Ring-diagram analysis
4.2.1 Local power spectra
Ring-diagram analysis is a powerful tool to infer the speed and direction of horizontal flows below the solar surface by observing the Doppler shifts of ambient acoustic waves from power spectra of solar oscillations computed over patches of the solar surface (typically 15° × 15°). Thus ring analysis is a generalization of global helioseismology applied to local areas on the Sun (as opposed to half of the Sun).
4.2.2 Measurement procedure
Other fitting techniques are described by Patrón et al. (1997) and Basu et al. (1999). Essentially all ring-diagram fitting has been done assuming symmetric profiles for the ridges in the power spectrum. Basu and Antia (1999) concluded that including line asymmetry in the model power spectrum improves the fits, but does not substantially change the inferred flows.
4.2.3 Depth inversions
The goal of inverting ring fit parameters is to determine the sound speed, density, and mass flows in the region underneath the tile over which the local power spectrum was computed. Generally, the assumption is that the sound speed, density, and flows are only functions of depth within the region covered by a particular tile. The forward problem for ring diagrams has traditionally been done by analogy with global mode helioseismology.
4.3 Time-distance helioseismology
The purpose of time-distance helioseismology (Duvall Jr et al., 1993) is to measure and interpret the travel times of solar waves between any two locations on the solar surface. A travel time anomaly contains the seismic signature of buried inhomogeneities within the proximity of the ray path that connects two surface locations. An inverse problem must then be solved to infer the local structure and dynamics of the solar interior (see, e.g., Jensen, 2003, and references therein).
4.3.1 Fourier filtering
The first operation in time-distance helioseismology is to track Doppler images at a constant angular velocity to remove the main component of solar rotation, as is done in ring-diagram analysis (see Section 4.2.1). Postel’s azimuthal equidistant projection is often used in time-distance helioseismology. The resulting data cube is Fourier transformed to obtain Φ(k, ω).
A filtering procedure is then applied to the data (Duvall Jr et al., 1997). First, frequencies below 1.5 mHz are filtered out in order to remove granulation and supergranulation noise. The data are further filtered to select parts of the wave propagation diagram.
Parameters of Fourier filters used in p-mode time-distance helioseismology. The Gaussian phase-speed filters are defined by \({\rm exp}{{[-(\omega/k - v_i)}^{2}}/(2 \delta v_{i}^{2})]\). Courtesy of T.L. Duvall and S. Couvidat.
i | distance range (Mm) | v_{i} (km s^{−1}) | δv_{i} (km s^{−1}) |
---|---|---|---|
1 | 3.7–8.7 | 12.8 | 2.6 |
2 | 6.2–11.2 | 14.9 | 2.6 |
3 | 8.7–14.5 | 17.5 | 2.6 |
4 | 14.5–19.4 | 25.8 | 3.9 |
5 | 19.4–29.3 | 35.5 | 5.3 |
6 | 26.0–35.1 | 39.7 | 3.0 |
7 | 31.8–41.7 | 43.3 | 3.2 |
8 | 38.4–47.5 | 47.7 | 3.6 |
9 | 44.2–54.1 | 52.3 | 4.5 |
10 | 50.8–59.9 | 57.2 | 3.8 |
11 | 56.6–66.7 | 61.1 | 3.4 |
A separate filtering procedure is applied for surface gravity waves, which are used to probe the near surface layer. In this case the filter function F (k, w) is 1 if \(\vert\omega\pm\sqrt{gk}\vert < kU_{\rm cut}\) and 0 otherwise. The parameter U_{cut} controls the width of the region around the f-mode dispersion relation ω^{2} = gk. A reasonable choice is U_{cut} = 1 km s^{−1}. This value allows for large Doppler frequency shifts introduced by flows, and does not let the p_{1} ridge through.
There is some freedom in the selection of the Fourier filters. For example one may construct filters that depend explicitly on the direction of the wavevector k, and not simply on the wavenumber (see, e.g., Giles, 1999).
4.3.2 Cross-covariance functions
The travel times of the wave packets are measured from the (first-bounce) cross-covariance function. Local inhomogeneities in the Sun will affect travel times differently depending on the type of perturbation. For example, temperature perturbations and flow perturbations have very different signatures. Given two points x_{1} and x_{2} on the solar surface the travel time perturbation due to a temperature anomaly is, in general, independent of the direction of propagation between x_{1} and x_{2}. However, a flow with a component directed along the direction x_{1} → x_{2} will break the symmetry in travel time for waves propagating in opposite directions: Waves move faster along the flow than against it. Magnetic fields introduce a wave speed anisotropy and will have yet another travel time signature (this has not been detected yet).
4.3.3 Travel time measurements
This definition has a number of useful properties. First, it is very robust with respect to noise. The fit reduces to a simple sum that can always be evaluated whatever the level of noise. Second, it is linear in the cross-covariance. As a consequence, averaging various travel time measurements is equivalent to measuring a travel time on the average cross-covariance. This is unlike previous definitions of travel time that involve non-linear fitting procedures. Third, the probability density functions of τ_{+} and τ_{−} are unimodal Gaussian distributions. This means, in particular, that it makes sense to associate an error to a travel time measurement. The Born sensitivity kernels discussed in Section 4.3.5 were derived according to this definition of travel times.
4.3.4 Noise estimation
In global helioseismology, it is well understood that the precision of the measurement of the pulsation frequencies is affected by realization noise resulting from the stochastic nature of the excitation of solar oscillations (see, e.g., Woodard, 1984; Duvall and Harvey, 1986; Libbrecht, 1992; Schou, 1992). It is important to study these properties since the presence of noise affects the interpretation of travel time data. In particular, the correlations in the travel times must be taken into account in the inversion procedure.
An interesting approach, pioneered by Jensen et al. (2003a), consists of estimating the noise directly from the data by measuring the rms travel time within a quiet Sun region. The underlying assumptions are that the fluctuations in the travel times are dominated by noise, not by ‘real’ solar signals, and that the travel times measured at different locations can be seen as different realizations of the same random process. By ‘real’ solar signals we mean travel time perturbations due to inhomogeneities in the solar interior that are slowly varying over the time of the observations. Jensen et al. (2003a) studied the correlation between the center-to-annulus travel times as a function of the distance between the central points, at fixed annulus radius.
Gizon and Birch (2004) derived a simple model for the full covariance matrix of the travel time measurements. This model depends only on the expectation value of the filtered power spectrum and assumes that solar oscillations are stationary and homogeneous on the solar surface. The validity of the model is confirmed through comparison with MDI measurements in a quiet Sun region. Gizon and Birch (2004) showed that the correlation length of the noise in the travel times is about half the dominant wavelength of the filtered power spectrum. The signal-to-noise ratio in quiet-Sun travel time maps increases roughly like the square root of the observation time and is maximum for a distance near half the length scale of supergranulation.
4.3.5 Travel time sensitivity kernels
Notice that kernels can be computed for quantities other than travel times. For example, in geophysics Dahlen and Baig (2002) computed the first-order sensitivity of the amplitude of the observed waveform to a small local change in sound speed. In particular it might be helpful to compute kernels for the mean frequency and amplitude of the cross-correlations, with the aim of using these quantities to help constrain inversions.
4.3.5.1 Ray approximation
Ray theory is based on the assumption that the perturbations to the model are smooth and that the wavepacket frequency bandwith is very large. Bogdan (1997) showed that the energy density of a realistic wavepacket was substantial away from the ray path. This result strongly suggested that perturbations located away from the ray path could have substantial effects on travel times. It is now well known that ray theory fails when applied to perturbations that are smaller that the first Fresnel zone (see, e.g., Hung et al., 2001; Birch et al., 2001).
4.3.5.2 Finite-wavelength kernels in the single-source approximation
The three approximations that have been used to treat small scale perturbations for time-distance helioseismology are the Born approximation, the Rytov approximation, and the Fresnel zone approximation. Here we describe some results that have been obtained using these methods in the single-source approximation. The single-source approximation, which was motived by the “Claerbout Conjecture” (Rickett and Claerbout, 1999) and by analogy with previous ray-theory based work, says that the one-way travel time perturbation can be found by looking at the time-shift of a wavepacket created at one observation location and then observed at the other. Gizon and Birch (2002) showed that the single-source approximation ignores a potentially important scattering process; this issue is discussed in detail in the paragraph on distributed source models.
The Rytov approximation gives a first order correction to the phase of the wavefield, instead of a first order correction to the wavefield itself as one obtains from the Born approximation. Jensen and Pijpers (2003) used the Rytov approximation to write travel time kernels for the effects of sound-speed perturbations. The results were qualitatively similar to the Born approximation results of Birch and Kosovichev (2000). No comparison has been made, in the helioseismology literature, of the ranges of validity of the Rytov and Born approximations.
4.3.5.3 Distributed source models
4.3.6 Inversions of travel times
The inverse problem is to determine the perturbations δq_{α}(r) to the solar model that are consistent with a particular set of observed travel times δτ_{i}. For the inverse problem, the kernel functions and the noise covariance are in general assumed to be known.
The minimization of χ^{2}, Equation (66), has been done using either the LSQR algorithm (Paige and Saunders, 1982), the multi-channel deconvolution (MCD; Jacobsen et al., 1999), or via singular value decomposition (Hughes and Thompson, 2003). The inputs to either of these methods are the travel time kernels, which result from the linear forward problem (see Section 3.5), the regularization operator, the choice of regularization parameter, and in general also the covariance matrix of the noise.
The central question about inversions is the degree to which any particular inversion method is able to retrieve the true Sun, e.g., the actual sound-speed variations and flows in the interior, from a set of observed travel times. The degree to which any inversion method succeeds will presumably depend on a number of things: (i) the accuracy of the forward model, (ii) the noise level in the travel times, (iii) the depths and spatial scales of the real variations in the Sun, (iv) the number and type of travel times that are available as input to the inversion, and (v) the accuracy of the travel time noise covariance matrix. The two main approaches that have been taken to study inversion methods are either to invert artificial data or to invert real data for perturbations that are relatively well known from global helioseismology or from surface measurements. It is also useful to compare the results of different inversion methods applied to the same set of observed travel times.
There have been a number of efforts to validate inversions by the generation and inversion of artificial data. These tests are useful as they provide an intuitive understanding of various regularization schemes, the choice of regularization parameter, the effects of limited sets of input travel times, the effects of incorrect assumptions regarding the noise covariance, the potential resolution of any particular inversion, and the cross-talk between different model parameters. A drawback to testing inversions with artificial data is that one always wonders about the realism of the artificial data. Hopefully, in the not distant future quite realistic data will be available for the purpose of testing inversion schemes.
Couvidat et al. (2004) compared inversions done using ray-approximation kernels with inversions done with Fresnel-zone kernels. The results were quite similar for depths between the surface and the lower turning point of the deepest rays used in the inversion. Inversions done with the ray kernels cannot retrieve sound speed perturbations below the lower turning point of the deepest ray, while the Fresnel-zone kernels extend below the depth of the deepest ray. The results of Couvidat et al. (2004), as well as those of Giles (1999), suggest that even though the ray approximation overestimates travel times for small scale perturbations this effect does not seriously corrupt the large scale features found by inversion.
4.4 Helioseismic holography
Helioseismic holography was introduced in detail by Lindsey and Braun (1990), although the basic concept was first suggested by Roddier (1975). For a recent theoretical introduction to helioseismic holography see Lindsey and Braun (2000a). The central idea in helioseismic holography is that the wavefield, e.g., the line-of-sight Doppler velocity observed at the solar surface, can be used to make an estimate of the wavefield at any location in the solar interior at any instant in time. In this sense, holography is much like seismic migration, a technique in geophysics that has been in use since the 1940’s (see, e.g., Hagedoorn, 1954, and references therein). In migration, the wave equation is used to determine the wavefield in the interior of the earth at past times, given the wavefield observed at the surface (see, e.g., Claerbout, 1985).
In Section 4.4.1 we introduce the basic computations of helioseismic holography, the ingression and egression. In Section 4.4.2 we review the various calculations that have been used to obtain the Green’s functions used in holography. In Section 4.4.3 we introduce the notion of the local control correlation. Acoustic power measurements are described in Section 4.4.4. Phase-sensitive holography is introduced in Section 4.4.5. Section 4.4.6 describes the technique of far-side imaging. In Section 4.4.7 we describe acoustic imaging (Chang et al., 1997), a technique closely related to holography.
4.4.1 Ingression and egression
The ingression and egression are attempts to the answer the following question: Given the wavefield observed at the solar surface, what is the best estimate of the wavefield at some point in the solar interior assuming that the observed wavefield resulted entirely from waves diverging from that point (for the egression) or waves converging towards that point (for the ingression)?
The development of holography has, historically, been motived by analogy with optics (e.g., Lindsey and Braun, 2000a) and as a result the holography literature often employs optical terminology. The target point in the solar interior at which we attempt to estimate the wavefield is termed the “focus point”. In practice, only data in a restricted area on the surface above the focus point is used to compute the egression and ingression. This region is called the “pupil”. The choice of pupil geometry depends on the particular application and will be described when we discuss particular applications of holography.
An intuitive way to interpret Equation (70) is to view the Green’s functions as propagators. In this way, the egression can be seen as the result of using the anti-causal Green’s function \(G_{+}^{{\rm holo}}\) to propagate the surface wavefield backwards in time. Likewise, the ingression can be seen as the result of using the causal Green’s function \(G_{-}^{{\rm holo}}\) to propagate the observed surface wavefield forwards in time.
The egression and ingression are the essential quantities in helioseismic holography. There are many particular ways in which these quantities can be combined to learn about the solar interior. The main techniques are control-correlations (Section 4.4.3), acoustic power holography (Section 4.4.4), phase-sensitive holography (Section 4.4.5), and far-side imaging (Section 4.4.6) which is a special case of phase-sensitive holography. Of central importance to all of these methods is the choice of the holography Green’s function \(G_{\pm}^{{\rm holo}}(r,\omega)\), which we now describe.
4.4.2 Holography Green’s functions
The two approaches that have been used to construct the holography Green’s functions, \(G_{\pm}^{{\rm holo}}\), involve ray theory and wave theory.
4.4.2.1 Ray theory
4.4.2.2 Wave theory
4.4.3 Local control correlations
If the egression/ingressions were perfect reconstructions of the observed signal, then the phases of the C_{±} would both, on average, be zero in the quiet Sun. In practice, this is not the case (Lindsey and Braun, 2004). The phases of the holography Green’s functions are typically corrected so that the phase of the average quiet Sun local control correlation is zero (e.g., Lindsey and Braun, 2000a). We refer to this correction as the empirical dispersion correction.
4.4.4 Acoustic power holography
4.4.5 Phase-sensitivity holography
4.4.6 Far-side imaging
4.4.7 Acoustic imaging
Acoustic imaging was first introduced by Chang et al. (1997); for a recent review see Chou et al. (2003). We have included acoustic imaging in the section on helioseismic holography as the definitions and philosophical motivation for the two techniques are quite similar.
In some sense, acoustic imaging is a special case of holography. Here, the pupil is an annulus with inner radius Δ_{1} and outer radius Δ_{2} given by the time-distance relation Δ_{j} = Δ(τ_{j}, z). The equivalent holography Green’s function is given by Wδ_{D}(t − τ). The signal Ψ_{out} corresponds to the egression, i.e., is the signal reconstructed from the observations of waves diverging from the focal point, while corresponds to the ingression, the signal estimated from the waves seen converging towards the focus position.
The forward problem that has received the most attention in acoustic imaging is the dependence of travel times on changes in the sound speed. Chou and Sun (2001) used the ray approximation to estimate the sensitivity of acoustic imaging phase travel times to changes in sound speed. The results showed that in general the horizontal resolution is greater than the vertical resolution and that the resolution decreases with increasing focus depth.
The inverse problem of determining the sound speed from a given set of travel times has been studied in the ray approximation as well. Sun and Chou (2002) tested ray-theory RLS inversions of phase-time measurements on artificial data. These tests showed that the RLS inversions were, for good choices of the regularization parameter, capable of reconstructing the model used to generate the artificial data. Inversions of travel times for sound speed were also discussed in detail by Chou and Sun (2001).
4.5 Direct modeling
Woodard (2002) introduced the idea of estimating subsurface flows from direct inversion of the correlations seen in the wavefield in the Fourier domain. The central notion is that for horizontally homogeneous steady models with no flow Fourier components of the physical wavefield are uncorrelated. Departures from horizontal homogeneity or time-dependence in general introduce correlations into the wavefield. Thus observations of the correlations in the Fourier domain can be used to estimate flows in the interior. Woodard (2002) gave a practical demonstration of the ability of the technique to recover near-surface flows from the f-mode part of the spectrum.
4.5.1 Forward problem
4.5.2 An example calculation
Woodard and Fan (2005) make the further approximation that oscillations of different radial orders are excited incoherently. This is certainly not the case in reality, as evidenced by the asymmetry in the power spectrum. It is not clear what effect this approximation will have on the final result, as most of the power is near the resonances, where line-asymmetry is not important.
Notice that we also need to compute the term \(E[\Phi(k,\omega)\delta\Phi^{\ast}(k^{\prime},\omega^{\prime})]\) as the final result that we wish to obtain is the first order change in \(E[\Phi(k,\omega)\Phi^{\ast}(k^{\prime},\omega^{\prime})]\) introduced by a small change in the model. This second term can be computed in exactly the same manner as the \(E[\delta\Phi(k,\omega)\Phi^{\ast}(k^{\prime},\omega^{\prime})]\) term, which we have already done.
4.5.3 Inverse problem
So far we have only addressed the forward problem. The other half of the direct-modeling approach is the inverse problem, that is inferring the flows in the Sun from the Fourier domain correlations seen in the data. The inverse problem in direct modeling has been solved as a linear least-squares problem (Woodard, 2002; Woodard and Fan, 2005). A crucial observation is that a flow with horizontal wavenumber q only couples modes with wavenumbers k and k′ when ∥k − k′∥ = q. This is a result of the assumption that the background model is plane-parallel and translation invariant. As a result the different wavenumber components of the flow can be inferred one at a time, in the spirit of the MCD inversion scheme (Jacobsen et al., 1999). The same argument applies in the time domain, and flows can be inferred one frequency component at a time.
5 Scientific Results from Local Helioseismology
5.1 Global scales
Flows in the upper convection zone have been measured by local helioseismology on a wide variety of scales, including differential rotation and meridional circulation, local flows around complexes of magnetic activity and sunspots, and convective flows. Here we mostly discuss the longitudinal averages of flows measured by local helioseismology, and their solar-cycle variations. The focus is on temporal variations that are slow compared to the typical lifetime of active regions. By computing longitudinal averages of rotation and meridional circulation over a few solar rotation periods it is possible to filter out small-scale flows and to reach a sensitivity of the order of 1 m s^{−1} near the surface. The evolution of flows through cycle 23 reveals connections between mass motions in the solar interior and the large-scale characteristics of the magnetic cycle.
5.1.1 Rotation and torsional oscillations
Howard and Labonte (1980) discovered small (±10 m s^{−1}) latitudinal variations in the surface Doppler rotation profile that propagate toward the equator with the sunspot latitudes. These variations, known as torsional oscillations, have since been confirmed by global helioseismology (Kosovichev and Schou, 1997; Schou, 1999). Inversions of global-mode frequency splittings have shown that torsional oscillations persist at a depth over a large fraction of the convection zone (Howe et al., 2000a), especially at high latitudes where they extend down to the bottom of the convection zone (Vorontsov et al., 2002). At low latitudes, the data support the idea that the torsional oscillation pattern originates inside the convection zone and propagates both upward and equatorward (Vorontsov et al., 2002; Basu and Antia, 2003). Above 45° latitude, bands of faster and slower rotation appear to move toward the poles (Antia and Basu, 2001; Schou, 2003a) suggesting a connection with the poleward migration of high-latitude magnetic activity seen at the surface (see, e.g., Callebaut and Makarov, 1992).
Schüssler (1981) and Yoshimura (1981) suggested that the torsional oscillations may be driven by the Lorentz force due to a migrating dynamo wave. Recent numerical calculations by Covas et al. (2000) seem to be consistent with this hypothesis; their calculations also reproduce the poleward propagating torsional oscillations at high latitudes. Other explanations attribute the torsional oscillations to the feedback of the smaller-scale magnetic fields on the angular momentum transport mechanisms responsible for differential rotation, e.g., changes in the Reynolds/Maxwell stresses (Küker et al., 1996; Kitchatinov et al., 1999) or the local suppression of turbulent viscosity by active regions (Petrovay and Forgács-Dajka, 2002). An alternative explanation by Spruit (2003) suggests that zonal flows may be driven by temperature perturbations near the surface due to the magnetic field: Local flows around regions of enhanced magnetic activity act through the Coriolis force to balance horizontal pressure gradients. The model of Spruit (2003) naturally predicts a solar-cycle variation in the meridional flows. The reader is referred to the review paper of Shibahashi (2004) for other possible explanations.
5.1.2 Meridional flow and its variations
Surface Doppler measurements have shown the existence of a meridional flow from the equator to the poles with an amplitude of about 20 m s^{−1} (e.g. Hathaway, 1996). Meridional circulation is an important ingredient of solar dynamo models where magnetic flux is transported by a deep equatorward flow (flux-transport dynamo models). In such models, the solar-cycle period is largely determined by the amplitude of the meridional flow near the base of the convection zone (Dikpati and Charbonneau, 1999). Hathaway et al. (2003) claimed that the butterfly diagram is evidence for flux-transport dynamo models, and estimated a 1.2 m s^{−1} flow at the bottom of the convection zone from the drift of sunspots toward the equator. However, Schüssler and Schmitt (2004) argue that the butterfly diagram is equally well reproduced by a conventional dynamo model with migrating dynamo waves without transport of magnetic flux by a flow. The depth of penetration of the meridional flow is another parameter in some dynamo models (see, e.g., Nandy and Choudhuri, 2002). Knowing meridional circulation is also important to understand its feedback on differential rotation (Rekowski and Rüdiger, 1998; Gilman, 2000).
5.1.3 Vertical flows
5.1.4 Search for variability at the tachocline
Howe et al. (2000b) discovered temporal changes in the rotational velocity near the bottom of the convection zone with a period of approximately 1.3 yr and a peak amplitude at 0.72R_{⊙}. The angular velocity variations at 0.72R_{⊙} and 0.63R_{⊙} are anticorrelated (the signal seemed much weaker near the second half of 2001 but it has come back since). These results are somewhat controversial since they have not been confirmed by others (see, e.g., Basu and Antia, 2001; Vorontsov et al., 2002). However, using wavelet analysis over cycles 21–23, Boberg et al. (2002) found signals with a 1–2 yr period in the solar mean magnetic field and Knaack et al. (2005) detected transient 1.3 yr oscillations in the unsigned photospheric magnetic flux, which may be related to the large-scale magnetic surges towards the poles. The 1.3 yr periodicity is also seen in observations of the interplanetary magnetic field and geomagnetic activity (Lockwood, 2001). Gough (2000) suggested that a 500 G radial magnetic field across the tachocline may lead to exchange of angular momentum with the right time scale, while Covas et al. (2001) suggested a highly non-linear mechanism that does not require different physical time scales for torsional and tachocline oscillations.
A different approach by Birch (2002) and Duvall Jr (2003) was to search for longitudinal structures near the bottom of the convection zone. Raw travel times measured between surface points separated by 45° were observed to be correlated from day to day, indicating a real solar signal (Duvall Jr, 2003). However, it is not known whether this signal is due to surface effects or deep perturbations.
It should be mentioned that global seismology may also be used to detect a toroidal magnetic field at the level of a fraction of a megagauss, if such a field were confined to a thin layer near the base of the convective envelope (e.g., Basu, 1997; Dziembowski and Goode, 2004). For example, Basu (1997) gave an upper limit of 0.3 MG near solar minimum.
It is fair to say that local helioseismological techniques have encountered difficulties in probing the tachocline. In order to reach a depth of 200 Mm, it is necessary to consider ray paths that connect surface points separated by almost 45°. All local methods thus suffer from foreshortening near the limb. Also, the plane-wave approximation is obviously not appropriate for probing the deep convection zone. Certain methods, like ring-diagram analysis, need to be adapted to take this into account (see, e.g., Haber et al., 1995).
5.2 Active regions and sunspots
5.2.1 Ordered flows near complexes of magnetic activity
The observed flows around active regions could be caused by the magnetic field. Indeed, the model of Spruit (2003) mentioned earlier predicts a surface inflow toward regions of enhanced magnetic field. An alternative explanation by Yoshimura (1971) suggests that the longitudinal ordering of solar magnetic fields could be due to the existence of large convective patterns that favor the formation of active regions at particular sites on the solar surface. We note that far away from active regions, complex flows like meanders, jets, and vortices are seen in the synoptic maps (Figure 39); Toomre (2002) suggested that they may be related to the largest scales of deep convection.
5.2.2 Effect on longitudinal averages of large-scale flows
An important question is to know whether the local flows that surround active regions contribute significantly to the solar-cycle variations of the longitudinal averages of rotation and meridional circulation in the upper layers of the convection zone (as discussed in Section 5.1.1 and Section 5.1.2). In other words, are the time-varying components of rotation and meridional circulation global phenomena or are they modulated in longitude by the presence of active regions?
5.2.3 Sunspot flows
Here we consider flows in the immediate vicinity of sunspots, which should not be confused with the flows discussed in the previous two sections. Duvall Jr et al. (1996) used the technique of time-distance helioseismology to measure the travel time difference between incoming and outgoing p-mode wavepackets around sunspots. They suggested that the observations are consistent with the existence of a downflow below sunspots with a velocity of about 2 km s^{−1}. It was estimated that the downflow persists down to a depth of 2 Mm below the surface. Kosovichev (1996) performed a 3D inversion of travel time measurements. The inversion results are consistent with 1 km s^{−1} downflows that can reach depths of about 25 Mm.
Lindsey et al. (1996) employed “knife-edge” diagnostics to estimate horizontal flows around sunspots. In knife-edge diagnostics the Fourier transform of the data is multiplied by some filter F(k, ω), and then transformed back to the space-time domain. The square of the absolute value of the signal is then averaged over time. The filter is constructed to let through the part of the signal that has been Doppler shifted by a flow in a particular direction. An estimate of the x component, for example, of the flow is then formed by subtracting the local amplitudes that correspond to waves that have been Doppler shifted in the \(+\hat{x}\) and \(-\hat{x}\) directions. Using this technique, Lindsey et al. (1996) were able to detect a horizontal outflow from the center of sunspots at depths of about 15 Mm. The amplitude of this flow reaches a maximum of about 180 m s^{−1} at a distance of 40 Mm from the center of a sunspot.
The Hankel analysis also has the capability of detecting flows below sunspots by interpreting the phase shifts between inward and outward traveling acoustic waves. Braun et al. (1996) applied this technique to south pole data from 1988 and 1991, and reported phase shifts that are consistent with the Doppler effect of a radial outflow from sunspots. The mean horizontal outflow appears to increase with depth and reaches 200 m s^{−1} at a depth of 20 Mm. A more detailed analysis by Sun et al. (1997) using TON data demonstrates that there is a positive frequency shift between outgoing and ingoing waves which corresponds to a 40–80 m s^{−1} radial outflow from sunspots. These results exhibit properties similar to those reported by Lindsey et al. (1996).
Almost all local helioseismology results seem to indicate that there exists a radial outflow around sunspots down to at least 10 Mm, which relates to the moat flow seen at the surface. An exception is the inflow measured with p-mode time-distance helioseismology at a depth of 0–3 Mm. It is especially puzzling that the f-mode and p-mode results are inconsistent. Regarding vertical flows, it is difficult to understand how both downflows and horizontal outflows can coexist just below the sunspot. We caution that many complications introduced by the magnetic field are simply ignored by all local techniques. For example, the effect of the magnetic field on excitation and damping mechanisms is known to introduce a small travel time difference between incoming and outgoing waves (Woodard, 1997; Gizon and Birch, 2002).
5.2.4 Sinks and sources of acoustic waves
Spruit and Bogdan (1992) suggested that the partial conversion of the incoming acoustic waves into slow magnetoacoustic waves that propagate downward, channelled by the magnetic field, may explain the observations. Cally and Bogdan (1993), Bogdan (1997), and Cally et al. (2003) showed using numerical models that this mechanism is indeed capable of producing strong absorption coefficients, as defined by Braun. The best agreement is obtained when the magnetic field is inclined to the vertical, to simulate a spreading magnetic field with height (Crouch and Cally, 2003; Cally et al., 2003). In these models the maximum absorption is near 30° inclination.
We note that acoustic absorption by sunspots may also be studied with local power maps from ring-diagram analysis (e.g., Rajaguru et al., 2001; Howe et al., 2004), and by comparing cross-correlation amplitudes for incoming and outgoing waves in time-distance helioseismology (Duvall, 2002, private communication).
5.2.5 Phase shifts and wave-speed perturbations
Kosovichev et al. (2000) studied the emergence in time of an active region. Their results show that the wave-speed perturbations rise very fast across the upper 18 Mm of the convection zone. An analysis for 2 hr time steps suggests that the emerging magnetic flux travels the upper 10 Mm in less than 2 hr, implying a minimum speed of 1.3 km s^{−1}. Jensen et al. (2001) inferred the wave-speed structure beneath the same emerging active region using an inversion technique based on Fresnel-zone sensitivity kernels, as opposed to ray-based kernels. Their results are similar to those of Kosovichev et al. (2000).
Using local mode frequency differences between active and quiet Sun regions measured with the ring-diagram technique (see, e.g., Hindman et al., 2000; Rajaguru et al., 2001), Basu et al. (2004) performed structure inversions to estimate the sound speed and adiabatic index in active regions. The results are for regions with horizontal size 15° and 7-day averages. An attempt was made to remove the uncertainty in the modelisation of the surface layers (Section 4.2.3): inversions do not provide information for depths less than 1 Mm. The uncertainties in modeling the surface layers were ‘removed’ in this process (Section 4.2.3), such that no reliable information at depths of less than 1 Mm could be extracted.
The wave-speed anomalies below sunspots could be caused by a variety of physical effects, for example thermal and magnetic perturbations. It has not been possible to disentangle these effects yet. The low sound-speed regions just below the surface have been attributed to a smaller temperature (Kosovichev et al., 2000; Basu et al., 2004). On the other hand, the higher wave speeds measured at a depth of 10 Mm below sunspots are unlikely to be due only to the direct effect of the magnetic field (or it would imply very large field strengths of a several tens of kG). The likely cause is a combination of magnetic and structural/thermal effects (Brüggen and Spruit, 2000; Basu et al., 2004). Barnes and Cally (2001) and Cally et al. (2003), however, have questioned the interpretation of travel time anomalies in terms of linear perturbations to the wave speed. Their numerical simulations of wave propagation through a model sunspot have been able to reproduce the phase shifts measured by Hankel analysis without the need for a thermal perturbation. This stresses the need for a proper solution of the forward problem of time-distance helioseismology in sunspots.
Lindsey and Braun (2003) argued that strong magnetic fields near the photosphere introduce large phase shifts in waves passing upwards into the photosphere of active regions and termed this effect the “showerglass” effect. They argue that the showerglass makes measurements of local variations in the subsurface more difficult, and thus attempts should be made to correct for this effect before inverting for subphotospheric structure and flows. Measurements of the showerglass were made by Lindsey and Braun (2003), and in more detail by Lindsey and Braun (2005a). In both cases the local egression and ingression control correlations (Section 4.4.3) were computed in active regions using MDI data. Maps of the phase and amplitude of the control correlations showed a clear relationship with the line-of-sight magnetograms, suggesting that the surface magnetic field was altering the amplitudes and phases of the waves used to compute the control correlation. Lindsey and Braun (2005b) used the measurements of the phase shifts of Lindsey and Braun (2005a) to correct the data before doing phase-sensitive holography (Section 4.4.5). They suggest that with the effect of near-surface magnetic field removed there is no clear evidence for sound-speed perturbations at depths greater than 5 Mm below a sunspot and that the effect of strong photospheric magnetic field on local helioseismic measurements should be studied further.
5.2.6 Far-side imaging
The sign of the travel time perturbation associated with active regions seen by far-side imaging is consistent with increased wave speeds or shorter paths traveled by the waves. This is consistent with the results of phase-sensitive holography of plage regions on the visible disk (see, e.g., Braun and Lindsey, 2000) and may be related to the observed solar cycle variation of normal mode frequencies (see, e.g., Lindsey and Braun, 2000b, and references therein).
5.2.7 Excitation of waves by flares
Later, Donea et al. (1999) used acoustic power holography (see Section 4.4.4) of SOHO/MDI data to investigate the same flare as was studied by Kosovichev and Zharkova (1998). The acoustic power holography showed that the wave source was strongest around 3 mHz, though the signal-to-noise ratio was higher around 6 mHz. The signal in the 6 mHz band appeared about 4 min after the signal in the 3 mHz band. Donea and Lindsey (2004) found significant acoustic power signatures associated with two other flares and showed a possible connection between the fine-scale spatial structure of the acoustic power maps and motions of the footpoints of the flaring loops. We note that detection of flare-induced acoustic waves using ring-diagram analysis has been claimed by Ambastha et al. (2003).
The basic mechanism by which a flare excites helioseismic waves is not entirely clear. Wolff (1972) suggested that the energy released in the flare heats the atmosphere, the expansion of which in turn causes a downward propagating wave. Zharkova and Kosovichev (1998) argue, on the basis of the momentum needed to explain the observed Bastille Day flare wave amplitude, that the shock wave created by chromospheric evaporation may not be the complete explanation and suggested that perhaps momentum transfer by electron or proton beams impacting directly on the photosphere may be important. Zharkova and Kosovichev (2001) argued that electron beams can carry sufficient momentum to explain the observed amplitude of the helioseismic wave caused by the Bastille Day flare.
A separate issue is the mechanism that leads to the initiation of solar flares. The contribution of local helioseismology to this problem has been rather poor so far. We note that Dzifcakova et al. (2003) have searched for a relationship between the evolution of subsurface flows and flaring activity.
5.3 Supergranulation
5.3.1 Paradigms
Granulation, with a typical size of 1.5 Mm, is well understood as a convective phenomenon and can be studied with realistic numerical simulations (see, e.g., Stein and Nordlund, 2000). Supergranules, however, have remained puzzling since their detection by Hart (1954, 1956). In a classic paper, Leighton et al. (1962) reported “large cells of horizontally moving material distributed roughly uniformly over the entire solar surface” with a characteristic scale of 30 Mm that are outlined by the chromospheric network; they suggested that the cells are a “surface manifestation of a supergranulation pattern of convective currents”. Unfortunately, there is no accepted theory that explains why solar convection should favor a 30 Mm scale. The solar convection zone is so highly turbulent and stratified that numerical modeling at supergranular scales has remained elusive (see, e.g., Rincon, 2004, and references therein).
Simon and Leighton (1964) presumed that a convective instability due to the recombination of ionized Helium is at the origin of the distinct supergranular scale. Forty years later, this hypothesis has not been proved to be right or wrong. The depth of the supergranulation layer is not known either. We note that Parker (1973) believes that the effect of stratification on convection may imply that supergranules are a deep phenomenon, with depths in excess of their horizontal diameters. Antia and Chitre (1993) investigated the stability of linear convective modes in the solar convection zone and found that a few convective modes dominate the power spectrum, one of which was identified as supergranulation. Using 1D numerical caculations, Ploner et al. (2000) suggested that the scale of supergranulation may be due to the interaction and merging of individual granular plumes (see also Rast, 2003). A somewhat related model was proposed by Rieutord et al. (2000, 2001) whereby supergranulation is the result of a non-linear large-scale instability of the granular flow, triggered by exploding granules. In the two previous models, supergranulation is not a proper scale of thermal convection. Regarding the influence of supergranular flows on magnetic fields, it is well established that a stationary cellular flow tends to expel the magnetic field from the regions of fluid motion and concentrate the magnetic flux into ropes at the cell boundaries (Parker, 1963; Galloway et al., 1977; Galloway and Weiss, 1981).
On the observational side, the original work of Leighton and coworkers has been refined. A variety of methods have been used to characterize the distribution of the cell sizes. A characteristic scale can be obtained from the spatial autocorrelation function (see, e.g., Hart, 1956; Simon and Leighton, 1964; Duvall Jr, 1980), the spatial Fourier spectrum (see, e.g., Hathaway, 1992; Beck, 1997), and segmentation or tessellation algorithms (Hagenaar et al., 1997). Although definitions vary, average cell sizes are in the range 15–30 Mm. It is unclear whether there is a variation of cell sizes with latitude: Rimmele and Schroeter (1989) and Komm et al. (1993a) report a possible decrease with latitude, Berrilli et al. (1999) an increase, and Beck (1997) no significant variation. The typical lifetime of the supergranular/chromospheric network is found to be in the range 1–2 d (e.g Rogers, 1970; Worden and Simon, 1976; Duvall Jr, 1980; Wang and Zirin, 1989). The rms horizontal velocity of supergranular flows is known to be about 300 m s^{−1} (see, e.g., Hathaway et al., 2000). The vertical component of the flows, however, has been extremely difficult to measure (see, e.g., Giovanelli, 1980) or infer (November, 1989). Indeed, Miller et al. (1984) caution that Doppler velocity measurements at the cell boundaries may be polluted by the network field. Estimates of the rms vertical flow are provided by Chou et al. (1991) and Hathaway et al. (2002) who find speeds of about 30 m s^{−1} (the topology of the vertical flows is largely unknown). Even more difficult to measure are the related temperature fluctuations. Observers have searched for the thermal signature of a convective process, i.e., rising hot material at the cell centers and sinking cool material at the cell boundaries. Unfortunately, answers vary too widely (see Lin and Kuhn, 1992, and references therein).
Estimates of the pattern rotation rate of supergranulation can be obtained by correlation tracking techniques. Duvall Jr (1980) and Snodgrass and Ulrich (1990) used Doppler scans separated in time by Δt = 24 hr. The results indicate that the equatorial rotation of the supergranulation pattern is faster than the spectroscopic rate (Snodgrass and Ulrich, 1990) by about 4% and faster than the magnetic features (Komm et al., 1993b) by about 2%. Duvall Jr (1980) also considered same-day scans with Δt = 6 hr and found a slightly smaller pattern rotation rate than for Δt = 24 hr.
It was suggested by Foukal (1972) that the difference between the rotation of magnetic features and the spectroscopic rate may be due to magnetic structures being rooted in deeper, more rapidly rotating layers. Supergranules must also sense increase rotation in the shear layer below the surface. However, Beck (2000) remarks that the pattern rotation of supergranulation measured from correlation tracking with Δt = 24 hr is significantly faster than the rotation of the solar plasma measured by helioseismology at any depth in the interior. It is especialy puzzling that the rotation of the magnetic network is significantly less than that of the supergranular pattern, since small magnetic elements are believed to be advected by supergranular flows. Hathaway (1982) suggested that a faster supergranular rate may be a direct consequence of the interaction of convection and rotation.
As explained in the following sections, local heliososeismology has become a powerful tool to study the structure and evolution of supergranular flows. In fact, helioseismological measurements have transformed our knowledge of the dynamics of supergranulation.
5.3.2 Horizontal flows and vertical structure
In order to resolve structures at supergranular scales, Duvall Jr et al. (1996) measured p-mode travel times at distances shorter than 10 Mm over an 8.5 hr time interval. Directional information was obtained by cross-correlating a point on the surface with surrounding quadrants centered on the four cardinal directions. In a first order approximation, the south-north and east-west travel time differences were converted into an apparent horizontal vector flow field without inversion. Duvall Jr et al. (1997) found that the line-of-sight projection of the inferred flow field is highly correlated with the mean MDI Dopplergram (correlation coefficient 0.74). This comparison was initialy used as a validation of the time-distance technique.
Three dimensional inversions of quiet-sun travel times have been presented by Duvall Jr et al. (1997), Kosovichev and Duvall Jr (1997), and Zhao and Kosovichev (2003a). Tests of the ray-based inversion procedure of Zhao and Kosovichev (2003a) show that the horizontal components of the velocity can be infered with some confidence in the upper 5 Mm (and perhaps down to 10 Mm). The small vertical flows, on the other hand, cannot be inferred reliably near the surface due to significant “cross-talk” with the horizontal divergence signal.
Near-surface horizontal flows have been measured at a depth of 1 Mm with f-mode time-distance helioseismology (Duvall Jr and Gizon, 2000). In this case travel time differences are directly sensitive to horizontal flows because f modes propagate horizontaly. The f-mode time-distance technique gives results that are comparable to correlation tracking of granulation (De Rosa et al., 2000). Shown in Figure 22 is an inversion of f-mode travel times that employs 2D Born kernels (Gizon et al., 2000). The correlation coefficient between the estimated line-of-sight velocity and the surface Doppler image is about 0.7.
The holographic technique also measures flows at supergranular scales. As shown earlier in Figure 45, supergranules are easy to identify as regions of outflows for a 3 Mm focus depth (24 hr time average).
Local heliososeismology opens prospects for mapping the structure of supergranular flows below the surface. A major goal is to answer the long-standing question of how deep supergranular flows persist below the surface.
We note that inversion results heavily rely on the assumed travel time sensitivity kernels. While most inversions use ray-based kernels, Jensen et al. (2000) and Birch and Kosovichev (2000) showed that finite-wavelength effects must be taken into account. It may be that wave-based kernels could yield depth inversion results that are qualitatively different from the ones presented above.
It would appear that local helioseismology has not yet provided a definitive answer regarding the depth of supergranulation.
5.3.3 Rotation-induced vorticity
Since the typical lifetime of supergranules is significantly less than the solar rotation period, the influence of rotation on supergranular convection is expected to be small. Solar rotation effects in supergranules were illustrated by Hathaway (1982) in non-linear numerical simulations. The Coriolis force causes divergent and convergent horizontal flows to be associated with vertical components of vorticity of opposite signs (on average). In the northern hemisphere, cells rotate clockwise where the horizontal divergence is positive, while they rotate counterclockwise in the convergent flow towards the sinks. The sense of circulation is reversed in the southern hemisphere.
5.3.4 Pattern evolution
As mentioned above, the pattern rotation rate of supergranulation derived from correlation tracking of Doppler scans appears to be significantly faster than the spectroscopic rate (for a 24 hr time lag). Correlation tracking algorithms can also be applied to images of supergranular flows derived from local helioseismology, such as the horizontal divergence signal. Unlike raw Doppler images, the divergence signal has uniform sensitivity across the solar disk and is subject to few systematic errors.
Earlier observations of solar convection assumed that supergranulation can be characterized by an autocorrelation function that exhibits a simple exponential decay in time (Harvey, 1985; Kuhn et al., 2000). It is now obvious that supergranulation does not follow such a simple model. The oscillation period of the correlation function, of the order of 6 d, suggests an underlying long-range order. As shown below, these puzzling observations are easier to describe in Fourier space (Section 5.3.5).
5.3.5 Traveling-wave convection
The 3D power spectrum of the divergence signal was studied first by Gizon et al. (2003). The same analysis was extended by Gizon and Duvall Jr (2004) to cover the period from 1996 to 2002. They considered series of MDI full-disk Dopplergrams from the Dynamics campaigns (two to three months each year). Dopplergrams were tracked at the Carrington angular velocity to remove the main component of rotation. Every 12 hr, a 120^{°} × 120^{°} map of the horizontal divergence of the near-surface flows was obtained using f-mode time-distance helioseismology. At a given target latitude λ a longitudinal section of the data was extracted, 10° wide in latitude. Gizon and Duvall Jr (2004) rearranged the data in a frame of reference with angular velocity Ω_{mag}(λ) = 14.43° − 1.77° sin^{2} λ − 2.58° sin^{4} λ d^{−1} (rotation rate of small magnetic features; Komm et al., 1993b). This choice of reference is convenient, although arbitrary. The position vector is denoted by x = (x, y) in the neighborhood of latitude λ, where coordinate x is prograde and y is northward, with a spatial sampling of 2.92 Mm in both coordinates.
The lifetime of supergranules is about 2.3 d at kR_{⊙} = 115, i.e., a somewhat larger value than other estimates derived from the decay of the autocorrelation function. This is because the lifetime is not strictly given the decay of the correlation function at short time lags, but by the decay of the envelope of the correlation function, which oscillates. The lifetime decreases by about 20% at active latitudes (Gizon and Duvall Jr, 2004).
As mentioned earlier, estimates of supergranulation rotation obtained by tracking the Doppler pattern are systematically found to be higher than the rotation of the magnetic network (for large time lags, see panel (a) of Figure 63). This apparent superrotation of the pattern can now be understood as the result of the waves being predominantly prograde. The east-west motion of the pattern is effectively a power-weighted average of the true rotation and the non-advective phase speed u_{w} = ω_{0}/k ∼ 65 m s^{−1} for kR_{⊙} = 120. Similarly, the excess of wave power toward the equator is reflected in the equatorward meridional motion of the pattern at large time lags (see panel (b) of Figure 63) even though the advective flow is poleward. Correlation tracking measurements must take into account the fact that the pattern propagates like a modulated traveling wave.
It was a source of concern that the wavelike properties of supergranulation had not been seen previously in surface Doppler shifts. Schou (2003b) showed that the same phenomenon can be observed directly using MDI Dopplergrams, thereby confirming the observations of Gizon and Duvall Jr (2003). In addition to confirming those results, Schou (2003b) was able to extend the measurements for the rotation and meridional velocities, U_{x} and U_{y}, beyond ±70° latitude. We note that, earlier, Beck and Schou (2000) had also used a spectral method to estimate the equatorial rotation of supergranulation from surface Doppler images; this method, however, was incorrect as it did not take account of the full complexity of the power spectrum.
All the evidence shows that supergranulation displays a high level of organization in space and time. Although no serious explanation has been proposed yet, it would seem that supergranulation is an example of traveling-wave convection. The prograde excess of wave power is perhaps due to the influence of rotation (or rotational shear) that breaks the east-west symmetry, allowing for new instabilities to propagate (see, e.g., Busse, 2003, 2004). We note that convection in oblique magnetic fields also exhibits solutions that take the form of traveling waves (Hurlburt et al., 1996). Future work should focus on measuring the evolution of the pattern at different depth in the interior and the phase relationship between the different Fourier components. More than forty years after its discovery, supergranulation remains a complete mystery.
Acknowledgements
We thank D.C. Braun, D.-Y. Chou, S. Couvidat, T.L. Duvall, D. Haber, S. Hughes, R. Komm, A.G. Kosovichev, C. Lindsey, P.H. Scherrer, and J. Zhao for their encouragements, figures, and comments. We are grateful to M.J. Thompson and an anonymous referee for suggesting improvements. SOHO is an international collaboration between the European Space Agency and the National Aeronautics and Space Administration. The GONG project is managed by the National Solar Observatory and is supported by AURA and the National Science Foundation. This work was supported in part by NASA grants NAG5-13261 and NAG5-12452.