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Chapter 2 Time-Distance Helioseismology Chapter 2 Time-Distance Helioseismology 2.1 Solar Oscillations Solar oscillations were rst observed in 1960 by Leighton et al. 1960; 1962, who saw ubiquitous vertical motions of the solar photoshere with p erio ds of ab out 5 minutes and amplitudes of ab out 1000 m=s. Ten years later, it was suggested that these os- cillations might b e manifestations of acoustic waves trapp ed b elow the photosphere Ulrich, 1970; Leibacher and Stein, 1971. Deubner 1975 later con rmed this hy- p othesis by showing that the observed relationship b etween the spatial and temp oral frequencies see gure 2.1 of the oscillations was close to that predicted by theory. This con rmation marked the birth of helioseismology as a to ol for probing the solar interior. The identi cation of the solar oscillations with waves trapp ed in a resonantcavity leads naturally to an analysis in terms of normal mo des. In section 2.2 I will brie y describ e this formalism and mention some of the earliest helioseismic results. Since this work is concerned with the large-scale dynamics of the Sun, I will describ e in slightly more detail the metho ds used to measure the Sun's internal rotation using a normal mo de analysis. After p ointing out some of the limitations of this metho d, section 2.3 will describ e the foundations of time-distance helioseismology, the metho d used in this work, and its application to the measurementof ows. 14 CHAPTER 2. TIME-DISTANCE HELIOSEISMOLOGY 15 8 6 (mHz) 4 π /2 ω 2 0 0.0 0.5 1.0 1.5 2.0 -1 kh (Mm ) Figure 2.1: The greyscale denotes the p ower sp ectrum computed from 8 hours of Dynamics Dopplergrams from MDI. Dark regions are areas of high p ower. The ver- tical axis is the temp oral cyclic frequency, and the horizontal axis is the horizontal wavenumb er. The ridges in the sp ectrum are regions of resonance which indicate the presence of normal mo des of oscillation. The horizontal wavenumb er determines the spherical harmonic degree l of the normal mo de see section 2.3.2. The lowest- frequency ridge is the fundamental mo de see section 4.2.3 2.2 Normal Mo des of Oscillation The Sun forms a resonantcavity for acoustic waves; since the temp erature increases with depth, waves which propagate inward from the surface are refracted toward the horizontal and eventually return to the surface. At the surface, they are re ected by the sudden decrease in density. The depth of p enetration dep ends on the horizontal phase sp eed of the waves. These oscillations are generally observed by measuring either intensity uctuations or Doppler motions of the Sun's surface. One way to represent the oscillations is as a CHAPTER 2. TIME-DISTANCE HELIOSEISMOLOGY 16 sum of standing waves or normal mo des, where the signal observed at a p ointr; ; at time t is given by X f r; ;;t= a r; ; expi[! t + ]: 2.1 nlm nlm nlm nlm nlm In this equation, the three integers n, l , and m identify each mo de and are commonly called the radial order, angular degree, and azimuthal order resp ectively.For each mo de, a is the mo de amplitude, ! is the eigenfrequency, and is the phase. nlm nlm nlm The spatial eigenfunction for each mo de is denoted by .For a spherically sym- nlm metric Sun, the eigenfunctions can b e separated into radial and angular comp onents: r; ;= r Y ; ; 2.2 nlm nl lm where Y is the spherical harmonic and the radial eigenfunction is denoted nowby lm r . nl The surface manifestation of solar oscillations can therefore b e decomp osed into a sum of spherical harmonics for each instant of observation. The result is that the power sp ectrum of the acoustic signal shows resonant p eaks at a particular set of temp oral frequencies for each pair n; l . These p eaks identify the eigenfrequencies ! of the normal mo des, which can then b e used as a diagnostic of the solar interior. nlm The basic pro cedure is to identify the eigenfrequencies for a set of normal mo des, and then to \invert" these measurements for some prop erty for example, the sound sp eed of the solar interior. In the earliest days of helioseismology, the frequencies were interpreted using simple mo dels and analytical formulae derived from asymp- totic analysis. For example, Deubner's measurements implied that the solar con- vection zone was signi cantly deep er than predicted by standard mo dels of the day Gough, 1976. Helioseismology also put an imp ortant constraint on the solar helium abundance Christensen-Dalsgaard and Gough, 1980. More quantitative results were later derived from sophisticated inverse metho ds. In a spherically symmetric Sun, all the mo des with the same n and l would have the same eigenfrequency ! , regardless of the value of m = fl; l +1; :::; l 1;lg. The nl CHAPTER 2. TIME-DISTANCE HELIOSEISMOLOGY 17 Sun, however, is not spherically symmetric, which causes this 2l + 1-degeneracy in the frequencies to b e broken. This splitting of the frequencies is the basis for helioseismic measurements of the solar internal rotation. 2.2.1 Measuring rotation with mo de frequencies The most obvious symmetry breaker is rotation ab out a xed axis. The Sun's rotation causes a splitting of the m-degeneracy in frequency which can b e describ ed by Z ! = m K r; r; rdrd 2.3 nlm nlm Here r; is the angular velo city in the solar interior, ! is the frequency di er- nlm ence b etween the rotating and non-rotating case, and the rotation kernel K r; nlm isaweighting function which describ es the sensitivity of the mo de n; l ; m to di er- ent regions of the solar interior. The kernels are determined by the structure of the eigenfuntions r; ; in equation 2.2. nlm In practice, the measured frequency splitting as a function of m and l is often 1 parametrized as an expansion in terms of Legendre p olynomials k max X m 0 k : 2.4 ! ! l; m ! l; 0 = l a P k lm l l k =0 In this expansion, the rotation contributes only to the o dd-ka-co ecients, whereas the even-ka-co ecients for each l arise from aspherical e ects which cannot distin- guish east from west Brown and Morrow, 1987. 2.2.2 Limitations of the global approach As noted in section 1.2.1, helioseismic measurements of the solar rotation have b ecome incredibly precise. However, there are several ways in which this approach to the problem is somewhat limited. 1 Other p ossibilites exist and are also used. CHAPTER 2. TIME-DISTANCE HELIOSEISMOLOGY 18 First, the approach is inherently global. This results in a high precision of mea- surement but is also an imp ortant limitation. In equation 2.3, the frequency splitting is shown to b e related to the internal angular velo city through the sensitivitykernels K . Because of the symmetry of the mo de eigenfunctions, all the kernels K nlm nlm are symmetric ab out the equatorial plane. Therefore one consequence of the global approach is that the frequency splitting is sensitive only to that part of whichis symmetric ab out the equatorial plane. Furthermore, since the spherical harmonics im dep end on longitude only through e , the kernels are also indep endent of longi- tude. Anyvariations in the angular velo city due to, for example, active regions, are therefore hidden. Second, until recently the resolution in latitude has b een somewhat limited. This resolution is basically limited by the value of k in equation 2.4. Signal-to-noise max considerations have meant that most early results from ground-based observations were limited to k = 5; the resulting rotation pro le had three indep endent p oints max in latitude. This limitation has b een largely removed with the advent of the GONG network and the SOHO spacecraft, as frequency splittings are now b eing extracted for larger values of k sometimes up to 35. This higher resolution is needed to max resolve small-scale structures like the torsional oscillation. Third, as noted in the previous section, all the spherically asymmetric e ects other than rotation | magnetic elds, structural asphericity, and meridional circulation, for example | cause a splitting of the normal mo de frequencies which app ears in the even a-co ecients of expansion 2.4. Thus, it is imp ossible to disentangle any one of these e ects from the others Zweib el and Gough, 1995. 2.3 Time-Distance Helioseismology Solar oscillations are essentially the only to ol astronomers have for lo oking into the interior of the Sun. Naturally, with the successes of helioseismology on global scales, solar physicists were intrigued by the p ossibilty of probing the solar interior on smaller scales. Some of the earliest work in this area involved measurements of the interaction between acoustic waves and sunsp ots Braun et al., 1988. More recently, there has CHAPTER 2. TIME-DISTANCE HELIOSEISMOLOGY 19 b een a great interest in using propagating waves to measure the meridional circula- tion and other ows. The lo cal approach hop es to b e complementary to the global approachbyovercoming some of the limitations outlined in section 2.2.2. The various lo cal metho ds all rely on the interaction of traveling acoustic waves with small p erturbations to the background state.
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