Differential rotation and dynamics of the solar interior

Science

Volumn 272, Issue 5266, 1996, Pages 1300-1305

  Thompson, M J ^a   Toomre, J ^b   Anderson, E R ^c   Antia, H M ^d   Berthomieu, G ^e   Burtonclay, D ^f   Chitre, S M ^d   Christensen Dalsgaard, J ^g   Corbard, T ^e   DeRosa, M ^b   Genovese, C R ^h   Gough, D O ⁱ   Haber, D A ^b   Harvey, J W ^c   Hill, F ^c   Howe, R ^a   Korzennik, S G ^j   Kosovichev, A G ^k   Leibacher, J W ^c   Pijpers, F P ^g   Provost, J ^e   Rhodes Jr , E J ^l   Schou, J ^k   Sekii, T ⁱ   Stark, P B ^m   Wilson, P R ^f   more..

^a QUEEN MARY UNIVERSITY OF LONDON   (United Kingdom)

^b UNIVERSITY OF COLORADO   (United States)

^c NATIONAL SOLAR OBSERVATORY   (United States)

^d TATA INSTITUTE OF FUNDAMENTAL RESEARCH   (India)

^e UNIVERSITÉ CÔTE D'AZUR   (France)

^f UNIVERSITY OF SYDNEY   (Australia)

^g AARHUS UNIVERSITY   (Denmark)

^h CARNEGIE MELLON UNIVERSITY   (United States)

ⁱ UNIVERSITY OF CAMBRIDGE   (United Kingdom)

^j HARVARD SMITHSONIAN CENTER FOR ASTROPHYSICS   (United States)

^k STANFORD UNIVERSITY   (United States)

^l UNIVERSITY OF SOUTHERN CALIFORNIA   (United States)

^m UNIVERSITY OF CALIFORNIA   (United States)

more..

Author keywords

[No Author keywords available]

Indexed keywords

ADJUSTMENT LAYER; DIFFERENTIAL ROTATION; GLOBAL OSILLATION NETWORK GROUP; OSCILLATION FREQUENCY; ROTATION; SHEAR LAYER;

SUN;

EID: 0029750499     PISSN: 00368075     EISSN: None     Source Type: Journal    
DOI: 10.1126/science.272.5266.1300     Document Type: Article

References (115)

1. Rotational histories of low-mass stars like the sun are discussed in L. W. Hartmann and R. W. Noyes, Annu. Rev. Astron. Astrophys. 25, 271 (1987); M. H. Pinsonneault, S. D. Kawaler, S. Sofia, P. Demarque, Astrophys. J. 338, 424 (1989); B. Chaboyer, P. Demarque, M. H. Pinsonneault, ibid. 441, 865 (1995).
2. Rotational histories of low-mass stars like the sun are discussed in L. W. Hartmann and R. W. Noyes, Annu. Rev. Astron. Astrophys. 25, 271 (1987); M. H. Pinsonneault, S. D. Kawaler, S. Sofia, P. Demarque, Astrophys. J. 338, 424 (1989); B. Chaboyer, P. Demarque, M. H. Pinsonneault, ibid. 441, 865 (1995).
3. Rotational histories of low-mass stars like the sun are discussed in L. W. Hartmann and R. W. Noyes, Annu. Rev. Astron. Astrophys. 25, 271 (1987); M. H. Pinsonneault, S. D. Kawaler, S. Sofia, P. Demarque, Astrophys. J. 338, 424 (1989); B. Chaboyer, P. Demarque, M. H. Pinsonneault, ibid. 441, 865 (1995).
4. Possible dynamical processes in the solar core are discussed in D. O. Gough, in Solar-Terrestrial Relationships and the Earth Environment in the Last Millennium. G. Castagnoli-Cini, Ed. (Soc. Italiana di Fisica, Bologna, 1988), pp. 90-132; W. J. Merryfield, J. Toomre, D. O. Gough, Astrophys. J. 367, 658 (1991); S. Ghosal and E. A. Spiegel, Geophys. Astrophys. Fluid Dyn. 61, 161 (1991). Computation of modifications to the neutrino flux is considered in D. O. Gough, Ann, N.Y. Acad. Sci. 647, 119 (1991).
5. Possible dynamical processes in the solar core are discussed in D. O. Gough, in Solar-Terrestrial Relationships and the Earth Environment in the Last Millennium. G. Castagnoli-Cini, Ed. (Soc. Italiana di Fisica, Bologna, 1988), pp. 90-132; W. J. Merryfield, J. Toomre, D. O. Gough, Astrophys. J. 367, 658 (1991); S. Ghosal and E. A. Spiegel, Geophys. Astrophys. Fluid Dyn. 61, 161 (1991). Computation of modifications to the neutrino flux is considered in D. O. Gough, Ann, N.Y. Acad. Sci. 647, 119 (1991).
6. Possible dynamical processes in the solar core are discussed in D. O. Gough, in Solar-Terrestrial Relationships and the Earth Environment in the Last Millennium. G. Castagnoli-Cini, Ed. (Soc. Italiana di Fisica, Bologna, 1988), pp. 90-132; W. J. Merryfield, J. Toomre, D. O. Gough, Astrophys. J. 367, 658 (1991); S. Ghosal and E. A. Spiegel, Geophys. Astrophys. Fluid Dyn. 61, 161 (1991). Computation of modifications to the neutrino flux is considered in D. O. Gough, Ann, N.Y. Acad. Sci. 647, 119 (1991).
7. Possible dynamical processes in the solar core are discussed in D. O. Gough, in Solar-Terrestrial Relationships and the Earth Environment in the Last Millennium. G. Castagnoli-Cini, Ed. (Soc. Italiana di Fisica, Bologna, 1988), pp. 90-132; W. J. Merryfield, J. Toomre, D. O. Gough, Astrophys. J. 367, 658 (1991); S. Ghosal and E. A. Spiegel, Geophys. Astrophys. Fluid Dyn. 61, 161 (1991). Computation of modifications to the neutrino flux is considered in D. O. Gough, Ann, N.Y. Acad. Sci. 647, 119 (1991).
8. The average sidereal rotation rate Ω/2π from photospheric Doppler measurements is ∼452 nHz at the equator, 432 nHz at 30°, and 369 nHz at 60° [R. K. Ulrich et al., Sol. Phys. 117, 291 (1988)]. The rotation rate determined by tracking sunspots varies with the size of the spot group [R. Howard, P. A. Gilman, P. I. Gilman, Astrophys. J. 283, 373 (1984)] or with the age of the spots [E. Nesme-Ribes et al., Astron. Astrophys. 274, 563 (1993)]. Average sunspot rotation rates are 467 nHz and 465 nHz, respectively, near the equator, and 444 nHz and 450 nHz at 30°. See also E. H. Schröter, Sol. Phys. 100, 141 (1985).
9. The average sidereal rotation rate Ω/2π from photospheric Doppler measurements is ∼452 nHz at the equator, 432 nHz at 30°, and 369 nHz at 60° [R. K. Ulrich et al., Sol. Phys. 117, 291 (1988)]. The rotation rate determined by tracking sunspots varies with the size of the spot group [R. Howard, P. A. Gilman, P. I. Gilman, Astrophys. J. 283, 373 (1984)] or with the age of the spots [E. Nesme-Ribes et al., Astron. Astrophys. 274, 563 (1993)]. Average sunspot rotation rates are 467 nHz and 465 nHz, respectively, near the equator, and 444 nHz and 450 nHz at 30°. See also E. H. Schröter, Sol. Phys. 100, 141 (1985).
10. The average sidereal rotation rate Ω/2π from photospheric Doppler measurements is ∼452 nHz at the equator, 432 nHz at 30°, and 369 nHz at 60° [R. K. Ulrich et al., Sol. Phys. 117, 291 (1988)]. The rotation rate determined by tracking sunspots varies with the size of the spot group [R. Howard, P. A. Gilman, P. I. Gilman, Astrophys. J. 283, 373 (1984)] or with the age of the spots [E. Nesme-Ribes et al., Astron. Astrophys. 274, 563 (1993)]. Average sunspot rotation rates are 467 nHz and 465 nHz, respectively, near the equator, and 444 nHz and 450 nHz at 30°. See also E. H. Schröter, Sol. Phys. 100, 141 (1985).
11. The average sidereal rotation rate Ω/2π from photospheric Doppler measurements is ∼452 nHz at the equator, 432 nHz at 30°, and 369 nHz at 60° [R. K. Ulrich et al., Sol. Phys. 117, 291 (1988)]. The rotation rate determined by tracking sunspots varies with the size of the spot group [R. Howard, P. A. Gilman, P. I. Gilman, Astrophys. J. 283, 373 (1984)] or with the age of the spots [E. Nesme-Ribes et al., Astron. Astrophys. 274, 563 (1993)]. Average sunspot rotation rates are 467 nHz and 465 nHz, respectively, near the equator, and 444 nHz and 450 nHz at 30°. See also E. H. Schröter, Sol. Phys. 100, 141 (1985).
12. P. A. Gilman, in Physics of the Sun, P. A. Sturrock, T. E. Holzer, D. M. Mihalas, R. K. Ulrich, Eds. (Reidel. Dordrecht, Netherlands, 1986), vol. 1, pp. 95-160; S. K. Solanki, Space Sci. Rev. 63, 1 (1993); F. Cattaneo, in Solar Magnetic Fields, M. Schüssler and W. Schmidt, Eds. (Cambridge Univ. Press, Cambridge, 1993), pp. 261-275; P, R. Wilson, Solar and Stellar Activity Cycles (Cambridge Univ. Press, Cambridge, 1994).
13. P. A. Gilman, in Physics of the Sun, P. A. Sturrock, T. E. Holzer, D. M. Mihalas, R. K. Ulrich, Eds. (Reidel. Dordrecht, Netherlands, 1986), vol. 1, pp. 95-160; S. K. Solanki, Space Sci. Rev. 63, 1 (1993); F. Cattaneo, in Solar Magnetic Fields, M. Schüssler and W. Schmidt, Eds. (Cambridge Univ. Press, Cambridge, 1993), pp. 261-275; P, R. Wilson, Solar and Stellar Activity Cycles (Cambridge Univ. Press, Cambridge, 1994).
14. P. A. Gilman, in Physics of the Sun, P. A. Sturrock, T. E. Holzer, D. M. Mihalas, R. K. Ulrich, Eds. (Reidel. Dordrecht, Netherlands, 1986), vol. 1, pp. 95-160; S. K. Solanki, Space Sci. Rev. 63, 1 (1993); F. Cattaneo, in Solar Magnetic Fields, M. Schüssler and W. Schmidt, Eds. (Cambridge Univ. Press, Cambridge, 1993), pp. 261-275; P, R. Wilson, Solar and Stellar Activity Cycles (Cambridge Univ. Press, Cambridge, 1994).
15. P. A. Gilman, in Physics of the Sun, P. A. Sturrock, T. E. Holzer, D. M. Mihalas, R. K. Ulrich, Eds. (Reidel. Dordrecht, Netherlands, 1986), vol. 1, pp. 95-160; S. K. Solanki, Space Sci. Rev. 63, 1 (1993); F. Cattaneo, in Solar Magnetic Fields, M. Schüssler and W. Schmidt, Eds. (Cambridge Univ. Press, Cambridge, 1993), pp. 261-275; P, R. Wilson, Solar and Stellar Activity Cycles (Cambridge Univ. Press, Cambridge, 1994).
16. D. O. Gough and J. Toomre, Annu. Rev. Astron, Astrophys. 29, 627 (1991).
17. The effect of large-scale flows on mode frequencies and linewidths is discussed by E. M. Lavely and M. H. Ritzwoller, Astrophys. J. 403, 810 (1993).
18. Low-degree oscillations were studied through Doppler shifts in light from the entire solar disk (8) [G. Grec, E. Fossat, M. A. Pomerantz, Sol. Phys. 82, 55 (1983)] and through fluctuations in the limb-darkening function [H. Hill, R. J. Bos, P. R. Goode, Phys. Rev. Lett. 49, 1794 (1982)].
19. Low-degree oscillations were studied through Doppler shifts in light from the entire solar disk (8) [G. Grec, E. Fossat, M. A. Pomerantz, Sol. Phys. 82, 55 (1983)] and through fluctuations in the limb-darkening function [H. Hill, R. J. Bos, P. R. Goode, Phys. Rev. Lett. 49, 1794 (1982)].
20. A. Claverie et al., Nature 293, 443 (1981).
21. Sectoral modes were studied with a cylindrical lens that folded the solar image into the entrance slit of a spectrograph aligned with the solar equator [T. L. Duvall Jr. and J. W. Harvey, Nature 310, 19 (1984)], and splitting data were inverted (T. L. Duvall Jr. et al., ibid, p. 22).
22. Sectoral modes were studied with a cylindrical lens that folded the solar image into the entrance slit of a spectrograph aligned with the solar equator [T. L. Duvall Jr. and J. W. Harvey, Nature 310, 19 (1984)], and splitting data were inverted (T. L. Duvall Jr. et al., ibid, p. 22).
23. Frequency splittings for intermediate-degree modes were done with the Fourier Tachometer, the progenitor of the GONG instrument [T. M. Brown and C. A. Morrow, Astrophys. J. 314, L21 (1987)], and the data were inverted [(11); see also T. M. Brown, Nature 317, 591 (1985)].
24. Frequency splittings for intermediate-degree modes were done with the Fourier Tachometer, the progenitor of the GONG instrument [T. M. Brown and C. A. Morrow, Astrophys. J. 314, L21 (1987)], and the data were inverted [(11); see also T. M. Brown, Nature 317, 591 (1985)].
25. T. M. Brown et al., Astrophys. J. 343, 526 (1989).
26. Intermediate-degree frequencies and splittings from Big Bear Solar Observatory, using Doppler imaging with a tunable birefringent filter, were reported for a 1986 campaign [K. G. Libbrecht, Astrophys. J. 336, 1092 (1989)], with further campaigns during 1988-1990 [K. G. Libbrecht and M. F. Woodard, Nature 345, 779 (1990); M. F. Woodard and K. G. Libbrecht, Astrophys. J. 402, L77 (1993)], For discussions of rotation inversions, see J. Christensen-Dalsgaard and J. Schou, in Seismology of the Sun and Sun-Like Stars, E. J. Rolfe, Ed. (ESASP-286, ESA, Noordwijk, Netherlands, 1988), pp. 149-153; W. A. Dziembowski, P. R. Goode, K. G. Libbrecht, Astrophys. J. 337, L53 (1989); D. O. Gough et al., Astron. Soc. Pac. Conf. Ser. 40, 93 (1993); M. F. Woodard and K. G. Libbrecht, Science 260, 1778 (1993).
27. Intermediate-degree frequencies and splittings from Big Bear Solar Observatory, using Doppler imaging with a tunable birefringent filter, were reported for a 1986 campaign [K. G. Libbrecht, Astrophys. J. 336, 1092 (1989)], with further campaigns during 1988-1990 [K. G. Libbrecht and M. F. Woodard, Nature 345, 779 (1990); M. F. Woodard and K. G. Libbrecht, Astrophys. J. 402, L77 (1993)], For discussions of rotation inversions, see J. Christensen-Dalsgaard and J. Schou, in Seismology of the Sun and Sun-Like Stars, E. J. Rolfe, Ed. (ESASP-286, ESA, Noordwijk, Netherlands, 1988), pp. 149-153; W. A. Dziembowski, P. R. Goode, K. G. Libbrecht, Astrophys. J. 337, L53 (1989); D. O. Gough et al., Astron. Soc. Pac. Conf. Ser. 40, 93 (1993); M. F. Woodard and K. G. Libbrecht, Science 260, 1778 (1993).
28. Intermediate-degree frequencies and splittings from Big Bear Solar Observatory, using Doppler imaging with a tunable birefringent filter, were reported for a 1986 campaign [K. G. Libbrecht, Astrophys. J. 336, 1092 (1989)], with further campaigns during 1988-1990 [K. G. Libbrecht and M. F. Woodard, Nature 345, 779 (1990); M. F. Woodard and K. G. Libbrecht, Astrophys. J. 402, L77 (1993)], For discussions of rotation inversions, see J. Christensen-Dalsgaard and J. Schou, in Seismology of the Sun and Sun-Like Stars, E. J. Rolfe, Ed. (ESASP-286, ESA, Noordwijk, Netherlands, 1988), pp. 149-153; W. A. Dziembowski, P. R. Goode, K. G. Libbrecht, Astrophys. J. 337, L53 (1989); D. O. Gough et al., Astron. Soc. Pac. Conf. Ser. 40, 93 (1993); M. F. Woodard and K. G. Libbrecht, Science 260, 1778 (1993).
29. Intermediate-degree frequencies and splittings from Big Bear Solar Observatory, using Doppler imaging with a tunable birefringent filter, were reported for a 1986 campaign [K. G. Libbrecht, Astrophys. J. 336, 1092 (1989)], with further campaigns during 1988-1990 [K. G. Libbrecht and M. F. Woodard, Nature 345, 779 (1990); M. F. Woodard and K. G. Libbrecht, Astrophys. J. 402, L77 (1993)], For discussions of rotation inversions, see J. Christensen-Dalsgaard and J. Schou, in Seismology of the Sun and Sun-Like Stars, E. J. Rolfe, Ed. (ESASP-286, ESA, Noordwijk, Netherlands, 1988), pp. 149-153; W. A. Dziembowski, P. R. Goode, K. G. Libbrecht, Astrophys. J. 337, L53 (1989); D. O. Gough et al., Astron. Soc. Pac. Conf. Ser. 40, 93 (1993); M. F. Woodard and K. G. Libbrecht, Science 260, 1778 (1993).
30. Intermediate-degree frequencies and splittings from Big Bear Solar Observatory, using Doppler imaging with a tunable birefringent filter, were reported for a 1986 campaign [K. G. Libbrecht, Astrophys. J. 336, 1092 (1989)], with further campaigns during 1988-1990 [K. G. Libbrecht and M. F. Woodard, Nature 345, 779 (1990); M. F. Woodard and K. G. Libbrecht, Astrophys. J. 402, L77 (1993)], For discussions of rotation inversions, see J. Christensen-Dalsgaard and J. Schou, in Seismology of the Sun and Sun-Like Stars, E. J. Rolfe, Ed. (ESASP-286, ESA, Noordwijk, Netherlands, 1988), pp. 149-153; W. A. Dziembowski, P. R. Goode, K. G. Libbrecht, Astrophys. J. 337, L53 (1989); D. O. Gough et al., Astron. Soc. Pac. Conf. Ser. 40, 93 (1993); M. F. Woodard and K. G. Libbrecht, Science 260, 1778 (1993).
31. Intermediate-degree frequencies and splittings from Big Bear Solar Observatory, using Doppler imaging with a tunable birefringent filter, were reported for a 1986 campaign [K. G. Libbrecht, Astrophys. J. 336, 1092 (1989)], with further campaigns during 1988-1990 [K. G. Libbrecht and M. F. Woodard, Nature 345, 779 (1990); M. F. Woodard and K. G. Libbrecht, Astrophys. J. 402, L77 (1993)], For discussions of rotation inversions, see J. Christensen-Dalsgaard and J. Schou, in Seismology of the Sun and Sun-Like Stars, E. J. Rolfe, Ed. (ESASP-286, ESA, Noordwijk, Netherlands, 1988), pp. 149-153; W. A. Dziembowski, P. R. Goode, K. G. Libbrecht, Astrophys. J. 337, L53 (1989); D. O. Gough et al., Astron. Soc. Pac. Conf. Ser. 40, 93 (1993); M. F. Woodard and K. G. Libbrecht, Science 260, 1778 (1993).
32. Intermediate-degree frequencies and splittings from Big Bear Solar Observatory, using Doppler imaging with a tunable birefringent filter, were reported for a 1986 campaign [K. G. Libbrecht, Astrophys. J. 336, 1092 (1989)], with further campaigns during 1988-1990 [K. G. Libbrecht and M. F. Woodard, Nature 345, 779 (1990); M. F. Woodard and K. G. Libbrecht, Astrophys. J. 402, L77 (1993)], For discussions of rotation inversions, see J. Christensen-Dalsgaard and J. Schou, in Seismology of the Sun and Sun-Like Stars, E. J. Rolfe, Ed. (ESASP-286, ESA, Noordwijk, Netherlands, 1988), pp. 149-153; W. A. Dziembowski, P. R. Goode, K. G. Libbrecht, Astrophys. J. 337, L53 (1989); D. O. Gough et al., Astron. Soc. Pac. Conf. Ser. 40, 93 (1993); M. F. Woodard and K. G. Libbrecht, Science 260, 1778 (1993).
33. Intermediate- and high-degree mode parameters deduced from South Pole Ca II intensity observations were reported in T. L. Duvall Jr., J. W. Harvey, M. A. Pomerantz, Nature 321, 500 (1986); T. L. Duvall Jr. et al., Astrophys. J. 324, 1158 (1988); T. L. Duvall Jr., J. W. Harvey, S. M. Jefferies, M. A. Pomerantz, ibid. 373, 308 (1991); S. M. Jefferies et al., ibid. 377, 330 (1991); T. L. Duvall Jr. et al., ibid. 410, 829 (1993). The most recent South Pole instrumentation now serves as the High Degree Helioseismometer (HDH) [K. T. Bachmann, T. L. Duvall Jr., J. W. Harvey, F. Hill, ibid. 443, 837 (1995)].
34. Intermediate- and high-degree mode parameters deduced from South Pole Ca II intensity observations were reported in T. L. Duvall Jr., J. W. Harvey, M. A. Pomerantz, Nature 321, 500 (1986); T. L. Duvall Jr. et al., Astrophys. J. 324, 1158 (1988); T. L. Duvall Jr., J. W. Harvey, S. M. Jefferies, M. A. Pomerantz, ibid. 373, 308 (1991); S. M. Jefferies et al., ibid. 377, 330 (1991); T. L. Duvall Jr. et al., ibid. 410, 829 (1993). The most recent South Pole instrumentation now serves as the High Degree Helioseismometer (HDH) [K. T. Bachmann, T. L. Duvall Jr., J. W. Harvey, F. Hill, ibid. 443, 837 (1995)].
35. Intermediate- and high-degree mode parameters deduced from South Pole Ca II intensity observations were reported in T. L. Duvall Jr., J. W. Harvey, M. A. Pomerantz, Nature 321, 500 (1986); T. L. Duvall Jr. et al., Astrophys. J. 324, 1158 (1988); T. L. Duvall Jr., J. W. Harvey, S. M. Jefferies, M. A. Pomerantz, ibid. 373, 308 (1991); S. M. Jefferies et al., ibid. 377, 330 (1991); T. L. Duvall Jr. et al., ibid. 410, 829 (1993). The most recent South Pole instrumentation now serves as the High Degree Helioseismometer (HDH) [K. T. Bachmann, T. L. Duvall Jr., J. W. Harvey, F. Hill, ibid. 443, 837 (1995)].
36. Intermediate- and high-degree mode parameters deduced from South Pole Ca II intensity observations were reported in T. L. Duvall Jr., J. W. Harvey, M. A. Pomerantz, Nature 321, 500 (1986); T. L. Duvall Jr. et al., Astrophys. J. 324, 1158 (1988); T. L. Duvall Jr., J. W. Harvey, S. M. Jefferies, M. A. Pomerantz, ibid. 373, 308 (1991); S. M. Jefferies et al., ibid. 377, 330 (1991); T. L. Duvall Jr. et al., ibid. 410, 829 (1993). The most recent South Pole instrumentation now serves as the High Degree Helioseismometer (HDH) [K. T. Bachmann, T. L. Duvall Jr., J. W. Harvey, F. Hill, ibid. 443, 837 (1995)].
37. Intermediate- and high-degree mode parameters deduced from South Pole Ca II intensity observations were reported in T. L. Duvall Jr., J. W. Harvey, M. A. Pomerantz, Nature 321, 500 (1986); T. L. Duvall Jr. et al., Astrophys. J. 324, 1158 (1988); T. L. Duvall Jr., J. W. Harvey, S. M. Jefferies, M. A. Pomerantz, ibid. 373, 308 (1991); S. M. Jefferies et al., ibid. 377, 330 (1991); T. L. Duvall Jr. et al., ibid. 410, 829 (1993). The most recent South Pole instrumentation now serves as the High Degree Helioseismometer (HDH) [K. T. Bachmann, T. L. Duvall Jr., J. W. Harvey, F. Hill, ibid. 443, 837 (1995)].
38. Intermediate- and high-degree mode parameters deduced from South Pole Ca II intensity observations were reported in T. L. Duvall Jr., J. W. Harvey, M. A. Pomerantz, Nature 321, 500 (1986); T. L. Duvall Jr. et al., Astrophys. J. 324, 1158 (1988); T. L. Duvall Jr., J. W. Harvey, S. M. Jefferies, M. A. Pomerantz, ibid. 373, 308 (1991); S. M. Jefferies et al., ibid. 377, 330 (1991); T. L. Duvall Jr. et al., ibid. 410, 829 (1993). The most recent South Pole instrumentation now serves as the High Degree Helioseismometer (HDH) [K. T. Bachmann, T. L. Duvall Jr., J. W. Harvey, F. Hill, ibid. 443, 837 (1995)].
39. Extensive sets of intermediate- and high-degree frequencies and splittings observed from Mount Wilson with a Doppler magneto-optical analyzer have been reported [S. Tomczyk et al., in Seismology of the Sun and Sun-Like Stars, E. J. Rolfe, Ed. (ESA SP-286, ESA, Noordwijk, Netherlands, 1988), pp. 141-147], and the data have been inverted [S. G. Korzennik et al., ibid., pp. 117-124; E. J. Rhodes Jr. et al., Astrophys. J. 351, 687 (1990)]. For inferences from 1988 and 1990 campaigns, see S. G. Korzennik et al., in Proceedings of the Oji International Seminar Progress of Seismology of the Sun and Stars, Y. Osaki and H. Shibahashi, Eds. (Springer-Verlag, Berlin, 1990), pp. 341-348; P. R. Goode et al., Astrophys. J. 367, 649 (1991); S. G. Korzennik, A. Cacciani, E. J. Rhodes Jr., Astron. Soc. Pac. Conf. Ser. 42, 201 (1993); S. G. Korzennik et al., ibid. 76, 12 (1995).
40. Extensive sets of intermediate- and high-degree frequencies and splittings observed from Mount Wilson with a Doppler magneto-optical analyzer have been reported [S. Tomczyk et al., in Seismology of the Sun and Sun-Like Stars, E. J. Rolfe, Ed. (ESA SP-286, ESA, Noordwijk, Netherlands, 1988), pp. 141-147], and the data have been inverted [S. G. Korzennik et al., ibid., pp. 117-124; E. J. Rhodes Jr. et al., Astrophys. J. 351, 687 (1990)]. For inferences from 1988 and 1990 campaigns, see S. G. Korzennik et al., in Proceedings of the Oji International Seminar Progress of Seismology of the Sun and Stars, Y. Osaki and H. Shibahashi, Eds. (Springer-Verlag, Berlin, 1990), pp. 341-348; P. R. Goode et al., Astrophys. J. 367, 649 (1991); S. G. Korzennik, A. Cacciani, E. J. Rhodes Jr., Astron. Soc. Pac. Conf. Ser. 42, 201 (1993); S. G. Korzennik et al., ibid. 76, 12 (1995).
41. Extensive sets of intermediate- and high-degree frequencies and splittings observed from Mount Wilson with a Doppler magneto-optical analyzer have been reported [S. Tomczyk et al., in Seismology of the Sun and Sun-Like Stars, E. J. Rolfe, Ed. (ESA SP-286, ESA, Noordwijk, Netherlands, 1988), pp. 141-147], and the data have been inverted [S. G. Korzennik et al., ibid., pp. 117-124; E. J. Rhodes Jr. et al., Astrophys. J. 351, 687 (1990)]. For inferences from 1988 and 1990 campaigns, see S. G. Korzennik et al., in Proceedings of the Oji International Seminar Progress of Seismology of the Sun and Stars, Y. Osaki and H. Shibahashi, Eds. (Springer-Verlag, Berlin, 1990), pp. 341-348; P. R. Goode et al., Astrophys. J. 367, 649 (1991); S. G. Korzennik, A. Cacciani, E. J. Rhodes Jr., Astron. Soc. Pac. Conf. Ser. 42, 201 (1993); S. G. Korzennik et al., ibid. 76, 12 (1995).
42. Extensive sets of intermediate- and high-degree frequencies and splittings observed from Mount Wilson with a Doppler magneto-optical analyzer have been reported [S. Tomczyk et al., in Seismology of the Sun and Sun-Like Stars, E. J. Rolfe, Ed. (ESA SP-286, ESA, Noordwijk, Netherlands, 1988), pp. 141-147], and the data have been inverted [S. G. Korzennik et al., ibid., pp. 117-124; E. J. Rhodes Jr. et al., Astrophys. J. 351, 687 (1990)]. For inferences from 1988 and 1990 campaigns, see S. G. Korzennik et al., in Proceedings of the Oji International Seminar Progress of Seismology of the Sun and Stars, Y. Osaki and H. Shibahashi, Eds. (Springer-Verlag, Berlin, 1990), pp. 341-348; P. R. Goode et al., Astrophys. J. 367, 649 (1991); S. G. Korzennik, A. Cacciani, E. J. Rhodes Jr., Astron. Soc. Pac. Conf. Ser. 42, 201 (1993); S. G. Korzennik et al., ibid. 76, 12 (1995).
43. Extensive sets of intermediate- and high-degree frequencies and splittings observed from Mount Wilson with a Doppler magneto-optical analyzer have been reported [S. Tomczyk et al., in Seismology of the Sun and Sun-Like Stars, E. J. Rolfe, Ed. (ESA SP-286, ESA, Noordwijk, Netherlands, 1988), pp. 141-147], and the data have been inverted [S. G. Korzennik et al., ibid., pp. 117-124; E. J. Rhodes Jr. et al., Astrophys. J. 351, 687 (1990)]. For inferences from 1988 and 1990 campaigns, see S. G. Korzennik et al., in Proceedings of the Oji International Seminar Progress of Seismology of the Sun and Stars, Y. Osaki and H. Shibahashi, Eds. (Springer-Verlag, Berlin, 1990), pp. 341-348; P. R. Goode et al., Astrophys. J. 367, 649 (1991); S. G. Korzennik, A. Cacciani, E. J. Rhodes Jr., Astron. Soc. Pac. Conf. Ser. 42, 201 (1993); S. G. Korzennik et al., ibid. 76, 12 (1995).
44. Extensive sets of intermediate- and high-degree frequencies and splittings observed from Mount Wilson with a Doppler magneto-optical analyzer have been reported [S. Tomczyk et al., in Seismology of the Sun and Sun-Like Stars, E. J. Rolfe, Ed. (ESA SP-286, ESA, Noordwijk, Netherlands, 1988), pp. 141-147], and the data have been inverted [S. G. Korzennik et al., ibid., pp. 117-124; E. J. Rhodes Jr. et al., Astrophys. J. 351, 687 (1990)]. For inferences from 1988 and 1990 campaigns, see S. G. Korzennik et al., in Proceedings of the Oji International Seminar Progress of Seismology of the Sun and Stars, Y. Osaki and H. Shibahashi, Eds. (Springer-Verlag, Berlin, 1990), pp. 341-348; P. R. Goode et al., Astrophys. J. 367, 649 (1991); S. G. Korzennik, A. Cacciani, E. J. Rhodes Jr., Astron. Soc. Pac. Conf. Ser. 42, 201 (1993); S. G. Korzennik et al., ibid. 76, 12 (1995).
45. Extensive sets of intermediate- and high-degree frequencies and splittings observed from Mount Wilson with a Doppler magneto-optical analyzer have been reported [S. Tomczyk et al., in Seismology of the Sun and Sun-Like Stars, E. J. Rolfe, Ed. (ESA SP-286, ESA, Noordwijk, Netherlands, 1988), pp. 141-147], and the data have been inverted [S. G. Korzennik et al., ibid., pp. 117-124; E. J. Rhodes Jr. et al., Astrophys. J. 351, 687 (1990)]. For inferences from 1988 and 1990 campaigns, see S. G. Korzennik et al., in Proceedings of the Oji International Seminar Progress of Seismology of the Sun and Stars, Y. Osaki and H. Shibahashi, Eds. (Springer-Verlag, Berlin, 1990), pp. 341-348; P. R. Goode et al., Astrophys. J. 367, 649 (1991); S. G. Korzennik, A. Cacciani, E. J. Rhodes Jr., Astron. Soc. Pac. Conf. Ser. 42, 201 (1993); S. G. Korzennik et al., ibid. 76, 12 (1995).
46. The low- and intermediate-degree (LOWL) magnetooptical analyzer [S. Tomczyk et al., Sol. Phys. 159, 1 (1995)] has produced a long campaign of observations from Mauna Loa, Hawaii. For rotation inversions, see S. Tomczyk, J. Schou, M, J. Thompson, Astrophys. J. 448, L57 (1995).
47. The low- and intermediate-degree (LOWL) magnetooptical analyzer [S. Tomczyk et al., Sol. Phys. 159, 1 (1995)] has produced a long campaign of observations from Mauna Loa, Hawaii. For rotation inversions, see S. Tomczyk, J. Schou, M, J. Thompson, Astrophys. J. 448, L57 (1995).
48. Recent interpretations of low-l splittings have been reported from the International Research on the Interior of the Sun (IRIS) network [E. Fossat et al., Astron. Soc. Pac. Conf. Ser, 76, 24 (1995)], from the Birmingham Solar Oscillation Network (BiSON) (Y. Elsworth et al., ibid., p. 43) (17), and from the Interplanetary Helioseismology by Irradiance (IPHIR) measurements (T. Toutain, ibid., p. 34).
49. Recent interpretations of low-l splittings have been reported from the International Research on the Interior of the Sun (IRIS) network [E. Fossat et al., Astron. Soc. Pac. Conf. Ser, 76, 24 (1995)], from the Birmingham Solar Oscillation Network (BiSON) (Y. Elsworth et al., ibid., p. 43) (17), and from the Interplanetary Helioseismology by Irradiance (IPHIR) measurements (T. Toutain, ibid., p. 34).
50. Recent interpretations of low-l splittings have been reported from the International Research on the Interior of the Sun (IRIS) network [E. Fossat et al., Astron. Soc. Pac. Conf. Ser, 76, 24 (1995)], from the Birmingham Solar Oscillation Network (BiSON) (Y. Elsworth et al., ibid., p. 43) (17), and from the Interplanetary Helioseismology by Irradiance (IPHIR) measurements (T. Toutain, ibid., p. 34).
51. Y. Elsworth et al., Nature 376, 669 (1995).
52. W. A. Dziembowski and P. R. Goode, Astron. Soc. Pac. Conf. Ser. 42, 225 (1993).
53. A. Jiménez et al., Astrophys. J. 435, 874 (1994).
54. T. Toutain and A. G. Kosovichev, Astron. Astrophys. 284, 265 (1994); T. Appourchaux et al., in Fourth SOHO Workshop Helioseismology, T. Hoeksema, V. Domingo, B. Fleck, B. Battrick, Eds. (ESA SP-376, ESA, Noordwijk, Netherlands, 1995), vol. 2, pp. 265-270; S. Tomczyk, J. Schou, M. J. Thompson, Bull. Astron. Soc. India, in press.
55. T. Toutain and A. G. Kosovichev, Astron. Astrophys. 284, 265 (1994); T. Appourchaux et al., in Fourth SOHO Workshop Helioseismology, T. Hoeksema, V. Domingo, B. Fleck, B. Battrick, Eds. (ESA SP-376, ESA, Noordwijk, Netherlands, 1995), vol. 2, pp. 265-270; S. Tomczyk, J. Schou, M. J. Thompson, Bull. Astron. Soc. India, in press.
56. T. Toutain and A. G. Kosovichev, Astron. Astrophys. 284, 265 (1994); T. Appourchaux et al., in Fourth SOHO Workshop Helioseismology, T. Hoeksema, V. Domingo, B. Fleck, B. Battrick, Eds. (ESA SP-376, ESA, Noordwijk, Netherlands, 1995), vol. 2, pp. 265-270; S. Tomczyk, J. Schou, M. J. Thompson, Bull. Astron. Soc. India, in press.
57. For numerical simulations of convection constrained by rotation within spherical shells, see G. A. Glatzrnaier, Astrophys. J. 291, 300 (1985); P. A. Gilman and J. Miller, Astrophys. J. (Suppl.) 61, 585 (1986); G. A. Glatzmaier, in The Internal Solar Angular Velocity, B. R. Durney and S. Sofia, Eds. (Reidel, Dordrecht, Netherlands, 1987), pp. 263-274, Modified mixing-length and anisotropic-diffusion models can also be used to estimate Reynolds stresses of rotating convection [G. Rüdiger, Geophys. Astrophys. Fluid Dyn. 16, 239 (1980); G. Belvedere, L. Paterno, M. Stix, Astron. Astrophys. 86, 40 (1980); D. H. Hathaway, Astrophys. J. 276, 316 (1984); B. R. Durney, ibid. 378, 378 (1991)].
58. For numerical simulations of convection constrained by rotation within spherical shells, see G. A. Glatzrnaier, Astrophys. J. 291, 300 (1985); P. A. Gilman and J. Miller, Astrophys. J. (Suppl.) 61, 585 (1986); G. A. Glatzmaier, in The Internal Solar Angular Velocity, B. R. Durney and S. Sofia, Eds. (Reidel, Dordrecht, Netherlands, 1987), pp. 263-274, Modified mixing-length and anisotropic-diffusion models can also be used to estimate Reynolds stresses of rotating convection [G. Rüdiger, Geophys. Astrophys. Fluid Dyn. 16, 239 (1980); G. Belvedere, L. Paterno, M. Stix, Astron. Astrophys. 86, 40 (1980); D. H. Hathaway, Astrophys. J. 276, 316 (1984); B. R. Durney, ibid. 378, 378 (1991)].
59. For numerical simulations of convection constrained by rotation within spherical shells, see G. A. Glatzrnaier, Astrophys. J. 291, 300 (1985); P. A. Gilman and J. Miller, Astrophys. J. (Suppl.) 61, 585 (1986); G. A. Glatzmaier, in The Internal Solar Angular Velocity, B. R. Durney and S. Sofia, Eds. (Reidel, Dordrecht, Netherlands, 1987), pp. 263-274, Modified mixing-length and anisotropic-diffusion models can also be used to estimate Reynolds stresses of rotating convection [G. Rüdiger, Geophys. Astrophys. Fluid Dyn. 16, 239 (1980); G. Belvedere, L. Paterno, M. Stix, Astron. Astrophys. 86, 40 (1980); D. H. Hathaway, Astrophys. J. 276, 316 (1984); B. R. Durney, ibid. 378, 378 (1991)].
60. For numerical simulations of convection constrained by rotation within spherical shells, see G. A. Glatzrnaier, Astrophys. J. 291, 300 (1985); P. A. Gilman and J. Miller, Astrophys. J. (Suppl.) 61, 585 (1986); G. A. Glatzmaier, in The Internal Solar Angular Velocity, B. R. Durney and S. Sofia, Eds. (Reidel, Dordrecht, Netherlands, 1987), pp. 263-274, Modified mixing-length and anisotropic-diffusion models can also be used to estimate Reynolds stresses of rotating convection [G. Rüdiger, Geophys. Astrophys. Fluid Dyn. 16, 239 (1980); G. Belvedere, L. Paterno, M. Stix, Astron. Astrophys. 86, 40 (1980); D. H. Hathaway, Astrophys. J. 276, 316 (1984); B. R. Durney, ibid. 378, 378 (1991)].
61. For numerical simulations of convection constrained by rotation within spherical shells, see G. A. Glatzrnaier, Astrophys. J. 291, 300 (1985); P. A. Gilman and J. Miller, Astrophys. J. (Suppl.) 61, 585 (1986); G. A. Glatzmaier, in The Internal Solar Angular Velocity, B. R. Durney and S. Sofia, Eds. (Reidel, Dordrecht, Netherlands, 1987), pp. 263-274, Modified mixing-length and anisotropic-diffusion models can also be used to estimate Reynolds stresses of rotating convection [G. Rüdiger, Geophys. Astrophys. Fluid Dyn. 16, 239 (1980); G. Belvedere, L. Paterno, M. Stix, Astron. Astrophys. 86, 40 (1980); D. H. Hathaway, Astrophys. J. 276, 316 (1984); B. R. Durney, ibid. 378, 378 (1991)].
62. For numerical simulations of convection constrained by rotation within spherical shells, see G. A. Glatzrnaier, Astrophys. J. 291, 300 (1985); P. A. Gilman and J. Miller, Astrophys. J. (Suppl.) 61, 585 (1986); G. A. Glatzmaier, in The Internal Solar Angular Velocity, B. R. Durney and S. Sofia, Eds. (Reidel, Dordrecht, Netherlands, 1987), pp. 263-274, Modified mixing-length and anisotropic-diffusion models can also be used to estimate Reynolds stresses of rotating convection [G. Rüdiger, Geophys. Astrophys. Fluid Dyn. 16, 239 (1980); G. Belvedere, L. Paterno, M. Stix, Astron. Astrophys. 86, 40 (1980); D. H. Hathaway, Astrophys. J. 276, 316 (1984); B. R. Durney, ibid. 378, 378 (1991)].
63. For numerical simulations of convection constrained by rotation within spherical shells, see G. A. Glatzrnaier, Astrophys. J. 291, 300 (1985); P. A. Gilman and J. Miller, Astrophys. J. (Suppl.) 61, 585 (1986); G. A. Glatzmaier, in The Internal Solar Angular Velocity, B. R. Durney and S. Sofia, Eds. (Reidel, Dordrecht, Netherlands, 1987), pp. 263-274, Modified mixing-length and anisotropic-diffusion models can also be used to estimate Reynolds stresses of rotating convection [G. Rüdiger, Geophys. Astrophys. Fluid Dyn. 16, 239 (1980); G. Belvedere, L. Paterno, M. Stix, Astron. Astrophys. 86, 40 (1980); D. H. Hathaway, Astrophys. J. 276, 316 (1984); B. R. Durney, ibid. 378, 378 (1991)].
64. N. Brummell, F. Cattaneo, J. Toomre, Science 269, 1370 (1995); G. A. Glatzmaier and J. Toomre, Astron. Soc. Pac. Conf. Ser. 76, 200 (1995).
65. N. Brummell, F. Cattaneo, J. Toomre, Science 269, 1370 (1995); G. A. Glatzmaier and J. Toomre, Astron. Soc. Pac. Conf. Ser. 76, 200 (1995).
66. 5) (latitude 30°). Further details of the combining of data are given in (11) and in P. R. Wilson and D. Burtonclay, Astrophys. J. 438, 445 (1995).
67. 5) (latitude 30°). Further details of the combining of data are given in (11) and in P. R. Wilson and D. Burtonclay, Astrophys. J. 438, 445 (1995).
68. The combining of kernels as in OLA was introduced by G. Backus and F. Gilbert [Geophys. J. R. Astron. Soc. 16, 169 (1968); Philos. Trans. R Soc. London Ser. A 266, 123 (1970)]. Its use in helioseismology is discussed in D. O. Gough, Sol. Pbys. 100, 65 (1985); the particular formulation known as SOLA is discussed in F. P. Pijpers and M. J. Thompson, Astron. Astrophys. 262, L33 (1992).
69. The combining of kernels as in OLA was introduced by G. Backus and F. Gilbert [Geophys. J. R. Astron. Soc. 16, 169 (1968); Philos. Trans. R Soc. London Ser. A 266, 123 (1970)]. Its use in helioseismology is discussed in D. O. Gough, Sol. Pbys. 100, 65 (1985); the particular formulation known as SOLA is discussed in F. P. Pijpers and M. J. Thompson, Astron. Astrophys. 262, L33 (1992).
70. The combining of kernels as in OLA was introduced by G. Backus and F. Gilbert [Geophys. J. R. Astron. Soc. 16, 169 (1968); Philos. Trans. R Soc. London Ser. A 266, 123 (1970)]. Its use in helioseismology is discussed in D. O. Gough, Sol. Pbys. 100, 65 (1985); the particular formulation known as SOLA is discussed in F. P. Pijpers and M. J. Thompson, Astron. Astrophys. 262, L33 (1992).
71. The combining of kernels as in OLA was introduced by G. Backus and F. Gilbert [Geophys. J. R. Astron. Soc. 16, 169 (1968); Philos. Trans. R Soc. London Ser. A 266, 123 (1970)]. Its use in helioseismology is discussed in D. O. Gough, Sol. Pbys. 100, 65 (1985); the particular formulation known as SOLA is discussed in F. P. Pijpers and M. J. Thompson, Astron. Astrophys. 262, L33 (1992).
72. OLA is computationally expensive, requiring the inversion of a matrix, the order of which is equal to the number of data. The 1 ⊗ 1 inversion approach we use [T. Sekii, Mon. Not. R. Astron. Soc. 264, 1018 (1993); in Fourth SOHO Workshop Helioseismology, T. Hoeksema, V. Domingo, B. Fleck, B. Battrick, Eds. (ESA SP-376, ESA, Noordwijk, Netherlands, 1995), vol. 2, pp. 285-288] is much less expensive because it exploits the fact that the kernels are approximately the product of a function of radius and a function of latitude (Fig. 1).
73. OLA is computationally expensive, requiring the inversion of a matrix, the order of which is equal to the number of data. The 1 ⊗ 1 inversion approach we use [T. Sekii, Mon. Not. R. Astron. Soc. 264, 1018 (1993); in Fourth SOHO Workshop Helioseismology, T. Hoeksema, V. Domingo, B. Fleck, B. Battrick, Eds. (ESA SP-376, ESA, Noordwijk, Netherlands, 1995), vol. 2, pp. 285-288] is much less expensive because it exploits the fact that the kernels are approximately the product of a function of radius and a function of latitude (Fig. 1).
74. 2 misfit) between the data and the splittings predicted by the parameterized model, plus a function that penalizes solutions that are large or that vary on small length scales. The competing misfit and regularity of the solution are balanced by a tradeoff parameter [see I. J. D. Craig and J. C. Brown, Inverse Problems in Astronomy (Hilger, Bristol, UK, 1986)].
75. i (i = 1, 3, . . . , 17) averaged over 4 months, for 957 (n,l) multiplets in the ranges l = 3 to 178 and v = 1500 to 3500 μHz. Splittings were determined separately for each of GONG months 1, 2, 4, and 5, and these were then averaged with equal weights.
76. nl.
77. -5 so as to give a reasonable tradeoff between resolution and variance of the solution. More automated procedures for selecting values for the trade-off parameters, such as generalized crossvalidation, are available.
78. The formal 1σ error bars in Fig. 4 indicate how the solution at a point would vary given different realizations of the data noise. They do not necessarily provide a confidence interval that covers the true solution because of the finite resolution of the inversion. Specifically, in the RLS the choice of finite basis and the smoothing penalty term both introduce biases. It is possible to estimate the magnitude of the latter effect by computing the terms in an expansion about the estimated parameters, but we have not done so here.
79. T. L. Duvall Jr., Sol. Phys. 66, 213 (1980).
80. P. Foukal and J. R. Jokipp, Astrophys. J. 199, L71 (1975); P. Foukal, ibid. 218, 539 (1977); P. A. Gilman and P. V. Foukal, ibid. 229, 1179 (1979).
81. P. Foukal and J. R. Jokipp, Astrophys. J. 199, L71 (1975); P. Foukal, ibid. 218, 539 (1977); P. A. Gilman and P. V. Foukal, ibid. 229, 1179 (1979).
82. P. Foukal and J. R. Jokipp, Astrophys. J. 199, L71 (1975); P. Foukal, ibid. 218, 539 (1977); P. A. Gilman and P. V. Foukal, ibid. 229, 1179 (1979).
83. N. H. Brummell, X. Xie, J. Toomre, C. Baillie, Astron. Soc. Pac. Conf. Ser. 76, 192 (1995); J. Toomre and N. H. Brummell, in Fourth SOHO Workshop Helioseismology, T. Hoeksema, V. Domingo, B. Fleck, B. Battrick, Eds. (ESA SP-376, ESA, Noordwijk, Netherlands, 1995), vol. 1, pp. 47-62; N. H, Brummell, N. E. Hurlburt, J, Toomre, Astrophys. J., in press.
84. N. H. Brummell, X. Xie, J. Toomre, C. Baillie, Astron. Soc. Pac. Conf. Ser. 76, 192 (1995); J. Toomre and N. H. Brummell, in Fourth SOHO Workshop Helioseismology, T. Hoeksema, V. Domingo, B. Fleck, B. Battrick, Eds. (ESA SP-376, ESA, Noordwijk, Netherlands, 1995), vol. 1, pp. 47-62; N. H, Brummell, N. E. Hurlburt, J, Toomre, Astrophys. J., in press.
85. N. H. Brummell, X. Xie, J. Toomre, C. Baillie, Astron. Soc. Pac. Conf. Ser. 76, 192 (1995); J. Toomre and N. H. Brummell, in Fourth SOHO Workshop Helioseismology, T. Hoeksema, V. Domingo, B. Fleck, B. Battrick, Eds. (ESA SP-376, ESA, Noordwijk, Netherlands, 1995), vol. 1, pp. 47-62; N. H, Brummell, N. E. Hurlburt, J, Toomre, Astrophys. J., in press.
86. Time-distance methods based on studies of acoustic travel time along sets of ray paths may eventually enable mapping of localized structures in wave speed and flow velocities, as discussed for HDH data from the South Pole [T. L. Duvall Jr., S. M. Jefferies, J. W. Harvey, M. A. Pomerantz, Nature 362, 430 (1993); T. L. Duvall Jr., S. D'Silva, S. M. Jefferies, J. W. Harvey, J. Schou, ibid. 379, 235 (1996)] and with inversion of maps of travel times [A. G. Kosovichev, Astrophys. J. 461, L55 (1996)]. Ringdiagram analysis involves observing the wave fields while tracking localized regions on the sun to deduce the average horizontal flows and sound-speed perturbations beneath the region [F. Hill, Astrophys. J. 333, 996 (1988); Sol. Phys. 128, 321 (1990); Astron. Soc. Pac. Conf. Ser. 76, 484 (1995)). Data about the displacement of the rings evident in slices at fixed frequency in the power spectra have been inverted [J. Pátron et al., Astrophys. J. 455, 746 (1995)].
87. Time-distance methods based on studies of acoustic travel time along sets of ray paths may eventually enable mapping of localized structures in wave speed and flow velocities, as discussed for HDH data from the South Pole [T. L. Duvall Jr., S. M. Jefferies, J. W. Harvey, M. A. Pomerantz, Nature 362, 430 (1993); T. L. Duvall Jr., S. D'Silva, S. M. Jefferies, J. W. Harvey, J. Schou, ibid. 379, 235 (1996)] and with inversion of maps of travel times [A. G. Kosovichev, Astrophys. J. 461, L55 (1996)]. Ringdiagram analysis involves observing the wave fields while tracking localized regions on the sun to deduce the average horizontal flows and sound-speed perturbations beneath the region [F. Hill, Astrophys. J. 333, 996 (1988); Sol. Phys. 128, 321 (1990); Astron. Soc. Pac. Conf. Ser. 76, 484 (1995)). Data about the displacement of the rings evident in slices at fixed frequency in the power spectra have been inverted [J. Pátron et al., Astrophys. J. 455, 746 (1995)].
88. Time-distance methods based on studies of acoustic travel time along sets of ray paths may eventually enable mapping of localized structures in wave speed and flow velocities, as discussed for HDH data from the South Pole [T. L. Duvall Jr., S. M. Jefferies, J. W. Harvey, M. A. Pomerantz, Nature 362, 430 (1993); T. L. Duvall Jr., S. D'Silva, S. M. Jefferies, J. W. Harvey, J. Schou, ibid. 379, 235 (1996)] and with inversion of maps of travel times [A. G. Kosovichev, Astrophys. J. 461, L55 (1996)]. Ringdiagram analysis involves observing the wave fields while tracking localized regions on the sun to deduce the average horizontal flows and sound-speed perturbations beneath the region [F. Hill, Astrophys. J. 333, 996 (1988); Sol. Phys. 128, 321 (1990); Astron. Soc. Pac. Conf. Ser. 76, 484 (1995)). Data about the displacement of the rings evident in slices at fixed frequency in the power spectra have been inverted [J. Pátron et al., Astrophys. J. 455, 746 (1995)].
89. Time-distance methods based on studies of acoustic travel time along sets of ray paths may eventually enable mapping of localized structures in wave speed and flow velocities, as discussed for HDH data from the South Pole [T. L. Duvall Jr., S. M. Jefferies, J. W. Harvey, M. A. Pomerantz, Nature 362, 430 (1993); T. L. Duvall Jr., S. D'Silva, S. M. Jefferies, J. W. Harvey, J. Schou, ibid. 379, 235 (1996)] and with inversion of maps of travel times [A. G. Kosovichev, Astrophys. J. 461, L55 (1996)]. Ringdiagram analysis involves observing the wave fields while tracking localized regions on the sun to deduce the average horizontal flows and sound-speed perturbations beneath the region [F. Hill, Astrophys. J. 333, 996 (1988); Sol. Phys. 128, 321 (1990); Astron. Soc. Pac. Conf. Ser. 76, 484 (1995)). Data about the displacement of the rings evident in slices at fixed frequency in the power spectra have been inverted [J. Pátron et al., Astrophys. J. 455, 746 (1995)].
90. Time-distance methods based on studies of acoustic travel time along sets of ray paths may eventually enable mapping of localized structures in wave speed and flow velocities, as discussed for HDH data from the South Pole [T. L. Duvall Jr., S. M. Jefferies, J. W. Harvey, M. A. Pomerantz, Nature 362, 430 (1993); T. L. Duvall Jr., S. D'Silva, S. M. Jefferies, J. W. Harvey, J. Schou, ibid. 379, 235 (1996)] and with inversion of maps of travel times [A. G. Kosovichev, Astrophys. J. 461, L55 (1996)]. Ringdiagram analysis involves observing the wave fields while tracking localized regions on the sun to deduce the average horizontal flows and sound-speed perturbations beneath the region [F. Hill, Astrophys. J. 333, 996 (1988); Sol. Phys. 128, 321 (1990); Astron. Soc. Pac. Conf. Ser. 76, 484 (1995)). Data about the displacement of the rings evident in slices at fixed frequency in the power spectra have been inverted [J. Pátron et al., Astrophys. J. 455, 746 (1995)].
91. Time-distance methods based on studies of acoustic travel time along sets of ray paths may eventually enable mapping of localized structures in wave speed and flow velocities, as discussed for HDH data from the South Pole [T. L. Duvall Jr., S. M. Jefferies, J. W. Harvey, M. A. Pomerantz, Nature 362, 430 (1993); T. L. Duvall Jr., S. D'Silva, S. M. Jefferies, J. W. Harvey, J. Schou, ibid. 379, 235 (1996)] and with inversion of maps of travel times [A. G. Kosovichev, Astrophys. J. 461, L55 (1996)]. Ringdiagram analysis involves observing the wave fields while tracking localized regions on the sun to deduce the average horizontal flows and sound-speed perturbations beneath the region [F. Hill, Astrophys. J. 333, 996 (1988); Sol. Phys. 128, 321 (1990); Astron. Soc. Pac. Conf. Ser. 76, 484 (1995)). Data about the displacement of the rings evident in slices at fixed frequency in the power spectra have been inverted [J. Pátron et al., Astrophys. J. 455, 746 (1995)].
92. Time-distance methods based on studies of acoustic travel time along sets of ray paths may eventually enable mapping of localized structures in wave speed and flow velocities, as discussed for HDH data from the South Pole [T. L. Duvall Jr., S. M. Jefferies, J. W. Harvey, M. A. Pomerantz, Nature 362, 430 (1993); T. L. Duvall Jr., S. D'Silva, S. M. Jefferies, J. W. Harvey, J. Schou, ibid. 379, 235 (1996)] and with inversion of maps of travel times [A. G. Kosovichev, Astrophys. J. 461, L55 (1996)]. Ringdiagram analysis involves observing the wave fields while tracking localized regions on the sun to deduce the average horizontal flows and sound-speed perturbations beneath the region [F. Hill, Astrophys. J. 333, 996 (1988); Sol. Phys. 128, 321 (1990); Astron. Soc. Pac. Conf. Ser. 76, 484 (1995)). Data about the displacement of the rings evident in slices at fixed frequency in the power spectra have been inverted [J. Pátron et al., Astrophys. J. 455, 746 (1995)].
93. The transport of angular momentum within the convection zone and in an adjustment layer at its base is discussed in P. A. Gilman, C. A. Morrow, E. E. DeLuca, Astrophys. J. 338, 528 (1989). The properties of a turbulent "solar tachocline" to achieve a transition in the rotation rate of the convection zone to that of the deeper interior are discussed in E. A. Spiegel and J.-P. Zahn, Astron. Astrophys. 265, 106 (1992), assuming that the helioseismically inferred rotation profile of the convection zone must adjust rapidly in radius to the interior state while transferring little or no net torque. If shear instabilities in the adjustment layer lead to substantial turbulent diffusive processes in the horizontal that can balance the advection of angular momentum, then a thin transition region can result.
94. The transport of angular momentum within the convection zone and in an adjustment layer at its base is discussed in P. A. Gilman, C. A. Morrow, E. E. DeLuca, Astrophys. J. 338, 528 (1989). The properties of a turbulent "solar tachocline" to achieve a transition in the rotation rate of the convection zone to that of the deeper interior are discussed in E. A. Spiegel and J.-P. Zahn, Astron. Astrophys. 265, 106 (1992), assuming that the helioseismically inferred rotation profile of the convection zone must adjust rapidly in radius to the interior state while transferring little or no net torque. If shear instabilities in the adjustment layer lead to substantial turbulent diffusive processes in the horizontal that can balance the advection of angular momentum, then a thin transition region can result.
95. E. A. Spiegel and N. O. Weiss, Nature 287, 616 (1980);
96. R. Rosner, in Cool Stars, Stellar Systems, and the Sun, A. K. Dupree, Ed. (SAO SR-389, Smithsonian Astrophysical Observatory, Cambridge, 1980), pp. 79-96;
97. L. Golub, R. Rosner, G. S. Vaiana, N. O. Weiss, Astrophys. J. 243, 309 (1981);
98. G. A. Glatzmaier, Geophys. Astrophys. Fluid Dyn. 31, 137 (1985);
99. D. Schmitt, in The Cosmic Dynamo, F. Krause, K.-H. Rädler, G. Rüdiger, Eds. (Kluwer, Dordrecht, Netherlands, 1993), pp. 1-12.
100. The finite resolution smooths out even discontinuous transitions [see M. J. Thompson, Sol. Phys. 125, 1 (1990)]. Techniques more appropriate for detecting sharp features include looking directly for the signature of a discontinuity or jet stream [D. O. Gough, in Proceedings of the Oji International Seminar Progress of Seismology of the Sun and Stars, Y. Osaki and H. Shibahashi, Eds. (Springer-Verlag, Berlin, 1990), pp. 283-318] and designing averaging kernels that are sensitive to sharp gradients or local extrema [F. P. Pijpers and M. J. Thompson, Astron. Soc. Pac. Conf. Ser. 42, 241 (1993)].
101. The finite resolution smooths out even discontinuous transitions [see M. J. Thompson, Sol. Phys. 125, 1 (1990)]. Techniques more appropriate for detecting sharp features include looking directly for the signature of a discontinuity or jet stream [D. O. Gough, in Proceedings of the Oji International Seminar Progress of Seismology of the Sun and Stars, Y. Osaki and H. Shibahashi, Eds. (Springer-Verlag, Berlin, 1990), pp. 283-318] and designing averaging kernels that are sensitive to sharp gradients or local extrema [F. P. Pijpers and M. J. Thompson, Astron. Soc. Pac. Conf. Ser. 42, 241 (1993)].
102. The finite resolution smooths out even discontinuous transitions [see M. J. Thompson, Sol. Phys. 125, 1 (1990)]. Techniques more appropriate for detecting sharp features include looking directly for the signature of a discontinuity or jet stream [D. O. Gough, in Proceedings of the Oji International Seminar Progress of Seismology of the Sun and Stars, Y. Osaki and H. Shibahashi, Eds. (Springer-Verlag, Berlin, 1990), pp. 283-318] and designing averaging kernels that are sensitive to sharp gradients or local extrema [F. P. Pijpers and M. J. Thompson, Astron. Soc. Pac. Conf. Ser. 42, 241 (1993)].
103. R. Rosner and N. O. Weiss, Nature 317, 790 (1985); L. Mestel and N. O. Weiss, Mon. Not. R Astron. Soc. 226, 123 (1987); P. Charbonneau and K. B. MacGregor, Astrophys. J. 417, 762 (1993).
104. R. Rosner and N. O. Weiss, Nature 317, 790 (1985); L. Mestel and N. O. Weiss, Mon. Not. R Astron. Soc. 226, 123 (1987); P. Charbonneau and K. B. MacGregor, Astrophys. J. 417, 762 (1993).
105. R. Rosner and N. O. Weiss, Nature 317, 790 (1985); L. Mestel and N. O. Weiss, Mon. Not. R Astron. Soc. 226, 123 (1987); P. Charbonneau and K. B. MacGregor, Astrophys. J. 417, 762 (1993).
106. N. O. Weiss, Proc. R. Soc. London Ser. A 293, 310 (1966).
107. J.-P. Zahn, Astron. Astrophys. 288, 829 (1994).
108. E. Schatzman, Astrophys. Lett. 3, 139 (1969); Astron. Astrophys. 56, 211 (1977); J-P. Zahn, ibid. 265, 115 (1992).
109. E. Schatzman, Astrophys. Lett. 3, 139 (1969); Astron. Astrophys. 56, 211 (1977); J-P. Zahn, ibid. 265, 115 (1992).
110. E. Schatzman, Astrophys. Lett. 3, 139 (1969); Astron. Astrophys. 56, 211 (1977); J-P. Zahn, ibid. 265, 115 (1992).
111. θ = 0.10R at r = 0.91R, 30°. The FWHM in each direction is larger than these values by roughly a factor of 1.7.
112. J. Christensen-Dalsgaard, J. Schou, M. J. Thompson, J. Toomre, Astron. Soc. Pac. Conf. Ser. 76, 212 (1995); in Fourth SOHO Workshop Helioseismology, J. Hoeksema, V. Domingo, B. Fleck, B. Battrick, Eds. (ESA SP-376, ESA, Noordwijk, Netherlands, 1995), vol. 2, pp. 41-45.
113. J. Christensen-Dalsgaard, J. Schou, M. J. Thompson, J. Toomre, Astron. Soc. Pac. Conf. Ser. 76, 212 (1995); in Fourth SOHO Workshop Helioseismology, J. Hoeksema, V. Domingo, B. Fleck, B. Battrick, Eds. (ESA SP-376, ESA, Noordwijk, Netherlands, 1995), vol. 2, pp. 41-45.
114. J. Schou, J. Christensen-Dalsgaard, M. J. Thompson, Astrophys. J. 433, 389 (1994).
115. Supported by the UK Particle Physics and Astronomy Research Council, NSF, NASA, Danmarks Grundforskningsfond, CNRS, and the Institut du Développement et des Ressources en Informatique Scientifique, France. GONG is managed by NSO, a division of NOAO that is operated by the Association of Universities for Research in Astronomy under cooperative agreement with NSF.