Author name code: wilson-peter ADS astronomy entries on 2022-09-14 =author:"Wilson, P.R." OR =author:"Wilson, Peter R." ------------------------------------------------------------------------ Title: Solar and Stellar Activity Cycles Authors: Wilson, Peter R. Bibcode: 2005ssa..book.....W Altcode: How do you predict the parameters of future solar cycles? What is the role of dynamo theory in the cyclic activity of the Sun and similar stars? And what are the implications of chaos theory for stellar cycles? This book answers these questions and offers a timely review of studies in the cyclic activity of the Sun and other stars. This authoritative reference shows the importance of reliable predictions of the parameters of future solar cycles, and carefully explains the methods currently used to determine these (with special reference to the maximum of cycle 22). Some of the latest research into solar cycles is clearly presented; this includes helioseismology, observations of the extended activity cycle and the polar fields reversal, and contributions from dynamo theory and chaos theory. For graduate students and researchers, this monograph provides a much-needed synthesis of our understanding of activity cycles in the Sun and other stars. Title: Solar and Stellar Activity Cycles Authors: Wilson, Peter R. Bibcode: 2005ssac.book.....W Altcode: 1. Introduction; 2. Historical survey; 3. The structure of the Sun and the phenomena of activity; 4. The equations of magnetohydrodynamics and magnetostatics; 5. The one-dimensional configuration of the cycle; 6. Heuristic models of the solar cycle; 7. Stellar activity and activity cycles; 8. The two-dimensional representation of the extended activity cycle; 9. The origin of the large-scale fields; 10. The reversals of the polar magnetic fields; 11. The role of dynamo theory in cyclic activity; 12. Helioseismology and the solar cycle; 13. Chaos and the cycle; 14. Forecasting the cycle; 15. Summary and conclusion. Title: The Mechanism involved in the Reversals of the Sun's Polar Magnetic Fields Authors: Durrant, C. J.; Turner, J. P. R.; Wilson, P. R. Bibcode: 2004SoPh..222..345D Altcode: Models of the polarity reversals of the Sun's polar magnetic fields based on the surface transport of flux are discussed and are tested using observations of the polar fields during Cycle 23 obtained by the National Solar Observatory at Kitt Peak. We have extended earlier measurements of the net radial flux polewards of ±60° and confirm that, despite fluctuations of ∼20%, there is a steady decline in the old polarity polar flux which begins shortly after sunspot minimum (although not at the same time in each hemisphere), crosses the zero level near sunspot maximum, and increases, with reversed polarity during the remainder of the cycle. We have also measured the net transport of the radial field by both meridional flow and diffusion across several latitude zones at various phases of the Cycle. We can confirm that there was a net transport of leader flux across the solar equator during Cycle 23 and have used statistical tests to show that it began during the rising phase of this cycle rather than after sunspot maximum. This may explain the early decrease of the mean polar flux after sunspot minimum. We also found an outward flow of net flux across latitudes ±60° which is consistent with the onset of the decline of the old polarity flux. Thus the polar polarity reversals during Cycle 23 are not inconsistent with the surface flux-transport models but the large empirical values required for the magnetic diffusivity require further investigation. Title: Observations and Simulations of the Polar Field Reversals in Cycle 23 Authors: Durrant, C. J.; Wilson, P. R. Bibcode: 2003SoPh..214...23D Altcode: We have used observations obtained by the National Solar Observatory at Kitt Peak to study the reversals of the polar magnetic fields in Cycle 23. We have compared them with corresponding data obtained by the Mt. Wilson Observatory, when these are available, testing both data sets against the locations of Hα filaments. Because of the unreliability of the data at extreme latitudes and because the apparent time of reversal varies with the degree of smoothing applied to the data, it is difficult to determine precise reversal time in each hemisphere from direct observations. However, we show that it is possible to obtain a better-defined and more precise reversal time using polar maps derived from simulations of the synoptic fields. These indirect values, however, depend critically on the diffusivity used in the simulations. We applied various tests to confirm an empirical value for the diffusivity parameter of about 600 km2 s−1 and hence determined empirical reversal times of CR 1976 in the northern hemisphere and CR 1981 in the south. Title: Bipolar Magnetic Fields Emerging at High Latitudes Authors: Durrant, C. J.; Turner, J.; Wilson, P. R. Bibcode: 2002SoPh..211..103D Altcode: Three examples of the unusual development of high-latitude large-scale features during cycle 23 are described. These features are found in synoptic plots constructed using data obtained at both the NSOKP and the MWO Observatories. Several properties of these features cannot be reproduced in direct simulations using a modified form of the flux-transport equation appropriate for synoptic fields and it is inferred that their evolution is not due solely to the advection and diffusion of decaying active region fields. The analysis shows that one feature may be related to a high-latitude bipolar region which emerged in an earlier rotation. By imposing the locations of Hα filaments on enlargements of the NSOKP daily magnetograms, we can identify the location of the other features and study their structure at high resolution. This suggests that they are related to the emergence of small magnetic knots at high latitudes. By repeating the simulations including overlays of non-random patterns of bipoles emerging at appropriate times during the simulations, it is possible to study the effects of different patterns and to reproduce some of the qualitative properties of these features not present in the direct simulations. These results support Stenflo's contention that `quite minute deviations from a random distribution (in the emergence of small-scale fields) would suffice for these fields to have global effects'. Title: High-resolution Studies of the Polar Magnetic Fields during Cycle 23 Authors: Varsik, J.; Durrant, C. J.; Turner, J.; Wilson, P. R. Bibcode: 2002SoPh..205..231V Altcode: High-resolution mosaics of the solar polar magnetic fields have been constructed using individual magnetograms obtained with the video magnetograph of the Big Bear Solar Observatory, and the properties of these mosaics are demonstrated in this paper. The mosaics show selected regions of the polar fields on several days during the rising phase of Cycle 23, and are related to the global polar fields (i) by superposing the mosaic for a given day on to a full-disk SOHO-MDI magnetogram obtained on the same day, (ii) by plotting the mosaics in polar projection and using these to identify the approximate regions reported by the mosaics on the NSOKP polar synoptic plots, and (iii) by imposing the locations of the Hα filaments on to the mosaics in order to infer the neutral lines of the large-scale fields. We have studied the fine structure of the large-scale unipolar fields near the poles and, in particular, have constructed histograms of the magnetic field intensities within particular regions of the mosaics and, in this way, have estimated the ratios of the number of magnetic knots of opposite polarities within the unipolar plumes. We have also generated enlargements of the polar regions of the NSOKP daily magnetograms. These and statistical studies have shown that on days for which the BBSO mosaics are not available, the NSOKP enlargements may be used to study the high-resolution polar fields. Time-series of mosaics obtained over four-hour periods on September 6 and November 18 show that considerable evolution in the structure of existing flux knots and the formation of several new knots has taken place during these periods. Title: The Evolution of Trailing Plumes from Active Regions Authors: Durrant, C. J.; Kress, J. M.; Wilson, P. R. Bibcode: 2001SoPh..201...57D Altcode: We have studied the evolution of several high-latitude flux `plumes', i.e., unipolar regions, trailing from active regions which emerged near sunspot maximum in cycle 23. The observed patterns are compared with simulations using a simple flux transport equation based on the observed flux for an earlier Carrington rotation. In addition to the long recognized poleward migration and diffusion of flux from active regions, it is found that the evolution of the trailing plumes may be influenced by flux which emerges above latitude 35° over areas of all scales. We describe two cases in which the emerging flux appears in the form of bipolar flux patterns which are not obviously related to sunspots. Further, we find instances in which the observed surface flux decreases or spreads at rates which cannot be explained solely in terms of diffusion using the normally accepted rates. Thus in several cases the poleward migration of flux cannot be described in terms of passive transport by advection and diffusion as considered here, and further investigation of the processes that contribute to the evolution of the polar fields is required. Title: Simulations of the Polar Field Reversals during Cycle 22 Authors: Kress, J. M.; Wilson, P. R. Bibcode: 2000SoPh..194....1K Altcode: The revised Mount Wilson synoptic magnetic data for the period September 1987 through March 1996 are used as the basis of numerical simulations of the evolution of both the northern and southern polar magnetic fields during the reversal and declining phases of cycle 22. The simulations are based on numerical solutions of the flux-transport equation which involve, as parameters, the maximum meridional flow speed, v0, and the supergranule diffusivity, κ. By matching characteristics of the observed and simulated fields, such as the observed reversal times, the evolution of the net flux above 60 °, and the migration of the polar crown, empirical values of these parameters, i.e., v0=11 m s−1,κ=600 km2 s−1, may be determined. Further, the observed decrease in the mean net flux above 60 ° during the late declining phase of cycle 22 can be simulated only by increasing the diffusivity to 900 km2 s−1. However, direct observations of the supergranule velocities yield values of the diffusivity of order 200 km2 s−1, and we show that the inclusion of a pattern of emerging bipoles in the simulations can increase the diffusion of these fields and that, together with a more realistic value of the diffusivity, it is possible to reproduce qualitatively the features of the observed polar field reversals. Title: Observations of the Polar Magnetic Fields During the Polarity Reversals of Cycle 22 Authors: Snodgrass, H. B.; Kress, J. M.; Wilson, P. R. Bibcode: 2000SoPh..191....1S Altcode: The Mount Wilson synoptic magnetic data for the period September 1987 through March 1996 are completely revised and used to provide polar plots of the solar magnetic fields for both hemispheres. This period, from Carrington rotations 1793 to 1906, covers the reversals of the polar magnetic fields in cycle 22. Comparison of our plots with the presently available Hα filtergrams for this period shows that the polarity boundaries are consistent in these two data sets where they overlap. The Mount Wilson plots show that the polar field reversals involve a complex sequence of events. Although the details differ slightly, the basic patterns are similar in each hemisphere. First the old polarity becomes isolated at the pole, then shortly thereafter, the isolation is broken, and the polar field includes unipolar regions of both polarities. The old polarity then reclaims the polar region, but when the isolation of this field is established for a second time, it declines in both area and strength. We take the reversal to be complete when the old polarity field is no longer observed in the Mount Wilson plots. With this criterion we find that the polar field reversal is completed in the north by CR 1836, i.e., by December 1990, and in the south by CR 1853, i.e., March 1992. Title: The evolution of isolated active regions Authors: Kress, J. M.; Wilson, P. R. Bibcode: 1999SoPh..189..147K Altcode: The decay of several active regions which emerged early in cycle 22 has been studied using daily magnetograms and synoptic plots obtained at the Vacuum Telescope at the National Solar Observatory, Kitt Peak. The observed patterns are compared with simulations using the flux transport equation and some discrepancies are noted. For one region it is shown that, by including the emergence of a non-random pattern of small magnetic bipoles during the decay, the correspondence between the observed and simulated patterns may be improved. Title: High-Resolution Studies of the Solar Polar Magnetic Fields Authors: Varsik, J. R.; Wilson, P. R.; Li, Y. Bibcode: 1999SoPh..184..223V Altcode: We present high-resolution studies of the solar polar magnetic fields near sunspot maximum in 1989 and towards sunspot minimum in 1995. We show that, in 1989, the polar latitudes were covered by several unipolar regions of both polarities. In 1995, however, after the polar field reversal was complete, each pole exhibited only one dominant polarity region. Title: Curious Magnetic Changes in a Quiet Region Authors: Zirin, H.; Wilson, P. R.; Li, Y. Bibcode: 1998SoPh..179..269Z Altcode: We describe the evolution of weak magnetic fields in a quiet region observed at the Big Bear Solar Observatory on 1 October 1996. We observed puzzling changes in which one polarity changed without corresponding increases or decreases in the other. In the rest of the same field, no special changes were observed, and a search of nearby days revealed no similar changes. We do not wish to call Maxwell's laws into question, we simply state that there are surprising effects that we cannot understand with current models. Title: The Internal Solar Rotation Rate Inferred from Combined GONG and LOWL Data Authors: Li, Y.; Wilson, P. R. Bibcode: 1998ApJ...499..504L Altcode: Frequency splittings derived from the first 4 months of GONG data and from the 2 yr average of the LOWL data in the form of the Clebsch-Gordon coefficients are analyzed. The results show that the equatorial rotation rate is essentially ``flat'' (i.e., independent of radius) for 0.2 < R/R⊙ < 0.6. They also indicate an increase in the rotation rate at high latitudes (of ~15% at latitude 60° for R/R⊙ < 0.4), but it is emphasized that this result is a property of the data sets and should not be regarded as a property of the Sun without independent confirmation. Title: Is the Solar Rotation Uniform Below the Surface? Authors: Wilson, P. R.; Li, Y. Bibcode: 1998ESASP.418..857W Altcode: 1998soho....6..857W For want of reliable data, most analyses of solar frequency splittings (e.g. Thompson et al. 1996) have assumed that the tachocline layer connects the differentially rotating convection zone to a uniformly rotating interior. Yet there is some indication in the first four months GONG data of an increase in the angular velocity at high latitude below r = 0.6 (Wilson et al. 1997, and a stronger signal in the two year LOWL data (Charbonneau et al. 1998, Li and Wilson 1998) suggesting that there may be an increase of ~15% at latitude 60-circ below r = 0.4. There is also an indication in the SOHO data of a localized increase at r = 0.6 and latitude ~60-circ (Schou, private communication). The evidence from these diverse sources is assessed. Title: From "down under" Authors: Wilson, Peter R. Bibcode: 1998ASSL..222..136W Altcode: 1998ream.conf..136W No abstract at ADS Title: The internal rotation rate inferred from LOWL and GONG data Authors: Li, Y.; Wilson, P. R. Bibcode: 1998IAUS..185..181L Altcode: A slightly modified forward method (Wilson et al., 1995, 1996) is applied to the 2-year-average LOWL frequency splitting data recently provided by Jesper Schou and Steven Tomczyk. In comparison with GONG data, this data set contains less multiplets at small nu l, but more at large nu l which probes greater depth into the solar interior. Two approaches are to be described and the internal rotation profiles will be shown in this poster. One approach uses a combined data set of the GONG 4-month-average data and the 2-year-average LOWL data. For the second, multiplets with nu l > 54.6 are selected from the LOWL data (for reasons to be given in the paper). We use the rotation rate previously obtained from the 4-month-average GONG data as the value for depth from the surface to 0.75Rodot. Below this level we use the selected LOWL data to constrain the rotation rate. In both cases, the surface rotation rates are chosen equal to the measured plasma rotation rates by Snodgrass (1984). Title: Coronal Holes and the Polar Field Reversals Authors: Fox, P.; McIntosh, P.; Wilson, P. R. Bibcode: 1998SoPh..177..375F Altcode: A description of the reversal of the solar north polar magnetic field during cycle 22 is provided using polar projections which combine the large-scale magnetic fields as inferred from Hα synoptic charts and coronal holes mapped from Hei λ1083 nm spectroheliograms. These plots are supported by polar plots of the magnetic fields derived from synoptic magnetic field data from the Mount Wilson Observatory. The coronal holes showed some unexpected evolutionary patterns in relation to the polarity reversals, and these patterns appear to be coordinated with changes in the global patterns of coronal holes and the heliospheric current sheet, suggesting that the polar reversal originates from global processes rather than from local magnetic flux annihilation. Similar patterns have been observed in the reversal of the southern polar magnetic field in cycle 22 and in both hemispheres in cycle 21. The consequences of these findings for the solar dynamo process are discussed. Title: The Rotational Structure of the Region below the Solar Convection Zone Authors: Wilson, P. R.; Burtonclay, D.; Li, Y. Bibcode: 1997ApJ...489..395W Altcode: Frequency splittings derived from the first 4 month string of GONG data in the form of the Clebsch-Gordon coefficients are analyzed with particular emphasis on the region near the base of and just below the convection zone. It is found that the greatest changes in angular velocity occur across the region 0.60 < r = R/R⊙ < 0.72. At equatorial latitudes, the angular velocity increases (outward) from 416 nHz at r = 0.645 to 455 nHz at r = 0.71, with a maximum angular velocity gradient dω(λ)/dr of order 2.2 × 10-3 (in units of nHz km-1) at r = 0.68 +/- 0.01. At latitude 30°, there are some fluctuations in the angular velocity between r = 0.70 and r = 0.75, but below r = 0.70 the angular velocity is relatively uniform. At latitude 45°, the angular velocity decreases from 449 nHz at r = 0.645 to 415 nHz at r = 0.71, with a maximum (negative) gradient of magnitude 1.9 × 10-3 again at r = 0.68 +/- 0.01; while at latitude 60° the angular velocity decreases from 440 nHz at r = 0.67 to 383 nHz at r = 0.775, with a maximum negative gradient of order 2.4 × 10-3 at r = 0.70 +/- 0.01. Models exhibiting a discontinuous shear between the convection zone and a uniformly rotating radiative region are not strongly supported by these data. A model with a uniformly rotating core below r = 0.58 and with some latitudinal structure in the range 0.58 < r < 0.70 is found to be consistent with the data. Title: New IRIS constraints on the solar core rotation. Authors: Gizon, L.; Fossat, E.; Lazrek, M.; Cacciani, A.; Ehgamberdiev, S.; Gelly, B.; Grec, G.; Hoeksema, J. T.; Khalikov, S.; Palle, P. L.; Pantel, A.; Regulo, C.; Schmider, F. -X.; Wilson, P. R. Bibcode: 1997A&A...317L..71G Altcode: Four time series of IRIS data (4 to 6 months) have been used to obtain improved measurements of the low degree (l=1,2,3) rotational splitting frequencies. Assuming that the rotation law is known in the outer layers of the Sun, we investigate the implications of IRIS splittings for the central regions. Both a one-shell and a two-shell rotation model have been considered in the solar core. A core rotating slightly faster than the outer radiative envelope provides the best fit to the data. Some evidence for the reliability of the observations is shown by the visibility of differential rotation in the l=3 multiplets. Title: On the Inference of the Solar Internal Rotation Profile from Frequency-splitting Data Authors: Wilson, P. R.; Burtonclay, D.; Li, Y. Bibcode: 1996ApJ...470..621W Altcode: From the earliest helioseismology data it was inferred that the internal angular velocity of the Sun is invariant across the convection zone (i.e., it mimics the surface differential rotation). This result caused some concern to theoreticians since many dynamo and dynamical models of the convection zone require that the angular velocity be approximately constant on cylinders concentric about the rotation axis.

Stark and others have argued that in order to test models of the angular velocity against frequency- splitting data the uncertainties in these data must be magnified, and it is shown here that within these uncertainties it is indeed difficult to exclude some models in which the angular velocity is independent of radius across a region including the convection zone and some depths below it. Further, Gough and his colleagues have recently claimed that the currently available data are not inconsistent with some models for which the angular velocity is constant on cylinders within the Sun's convection zone. Thus, inferences from frequency-splitting data regarding the internal angular velocity of the Sun would seem to be somewhat uncertain.

In this paper, these uncertainties are discussed and an alternative approach is proposed in which a forward method is used to find the simplest model for the angular velocity (i.e., with the least positional variations) consistent with the data, including the quoted uncertainties. While it is not claimed that such a model represents the true angular velocity, its features may be said to "characterize" the essential properties of a particular data set. Title: Some aspects of the interpretation of frequency splitting data Authors: Wilson, P. R. Bibcode: 1996BASI...24..251W Altcode: No abstract at ADS Title: Book reviews Authors: Humphreys, R. M.; Kemp, S.; Savonije, G.; van der Hucht, K. A.; van der Kruit, P. C.; Miley, G.; Bumba, V.; van Nieuwkoop, J.; van Hoolst, T.; Cox, A.; Rutten, R. J.; Kleczek, J.; de Jager, Cornelis; Jerzykiewicz, M.; Zwaan, C.; Poedts, S.; Sakai, Jun-Ichi; Pecker, J. -C.; Heikkila, W.; de Jong, T.; Wilson, P. R.; Müller, E. A.; Hoyng, P.; Icke, V.; Shore, S. N.; Achterberg, A.; Lucchin, F.; Butcher, H.; Ne'Eman, Y.; Heidmann, J.; Belton, M. J. S.; de Graauw, Th.; Waters, L. B. F. M.; Pacini, F.; Hultqvist, B.; Akasofu, S. -I.; Vial, J. -C.; Schatzman, E.; van der Laan, H.; Cole, K. D.; Vanbeveren, D.; Southwood, D.; van der Klis, M.; Katgert, Peter Bibcode: 1996SSRv...76..339H Altcode: No abstract at ADS Title: Differential Rotation and Dynamics of the Solar Interior Authors: Thompson, M. J.; Toomre, J.; Anderson, E. R.; Antia, H. M.; Berthomieu, G.; Burtonclay, D.; Chitre, S. M.; Christensen-Dalsgaard, J.; Corbard, T.; De Rosa, M.; Genovese, C. R.; Gough, D. O.; Haber, D. A.; Harvey, J. W.; Hill, F.; Howe, R.; Korzennik, S. G.; Kosovichev, A. G.; Leibacher, J. W.; Pijpers, F. P.; Provost, J.; Rhodes, E. J., Jr.; Schou, J.; Sekii, T.; Stark, P. B.; Wilson, P. R. Bibcode: 1996Sci...272.1300T Altcode: Splitting of the sun's global oscillation frequencies by large-scale flows can be used to investigate how rotation varies with radius and latitude within the solar interior. The nearly uninterrupted observations by the Global Oscillation Network Group (GONG) yield oscillation power spectra with high duty cycles and high signal-to-noise ratios. Frequency splittings derived from GONG observations confirm that the variation of rotation rate with latitude seen at the surface carries through much of the convection zone, at the base of which is an adjustment layer leading to latitudinally independent rotation at greater depths. A distinctive shear layer just below the surface is discernible at low to mid-latitudes. Title: High-Resolution Studies of the Solar Polar Magnetic Fields Authors: Varsik, J. R.; Wilson, P. R.; Li, Y. Bibcode: 1996AAS...188.3504V Altcode: 1996BAAS...28..871V Most observations of the polar magnetic fields of the Sun have been at relatively low resolution. High-resolution studies of the polar fields offer new insight into their evolution. We show that near sunspot maximum in 1989, the polar regions are covered with several unipolar regions, each region containing magnetic knots of both polarities. These knots have average lifetimes greater than 7 hours but less than 24 hours. In 1995, after the polar field reversal was complete, each pole exhibits one dominant polarity, and the dominant polarity knots are in a ratio of 5:1 to the opposite polarity. By measuring the displacement of magnetic knots over a 7 hour period a rotation rate can be determined which is consistent with the Snodgrass (1982) relation determined by cross correlations of Mt. Wilson magnetograms. Most knots are not seen in magnetic bipoles either when the overall polar fields are mixed or when one polarity is dominant. Title: Calculations of the Solar Internal Angular Velocity for 1986--1990 Authors: Wilson, P. R.; Burtonclay, D.; Li, Y. Bibcode: 1996ApJ...457..440W Altcode: Frequency splitting data obtained at the Big Bear Solar Observatory for the years 1986, 1988, 1989, and 1990 are analyzed by a modified forward calculation in order to find the simplest solutions which embody the essential features of those data. The solutions are used to identify the properties of the Sun's internal angular velocity that are common to all years and to study possible changes from year to year.

The common properties include a maximum in the angular velocity at all latitudes just below the surface and a shear zone (negative outward), at high latitudes near the base of the convection zone. Surprisingly, there is no indication of a shear zone at intermediate latitudes, and at low latitudes there is only a weak suggestion of a shear in 1989 and 1990. While there is some evidence of other changes in the internal angular velocity from year to year, they do not appear to be qualitatively significant. Title: Book-Review - Solar and Stellar Activity Cycles Authors: Wilson, P. R.; Collier Cameron, A. Bibcode: 1995Obs...115..334W Altcode: No abstract at ADS Title: The Crisis in Helioseismology Authors: Wilson, P. R.; Burtonclay, D.; Li, Y. Bibcode: 1995ESASP.376b.295W Altcode: 1995help.confP.295W; 1995soho....2..295W No abstract at ADS Title: Calculations of the Solar Internal Rotation Coefficients, 1986-1990 Authors: Wilson, P. R.; Burtonclay, D. Bibcode: 1995ASPC...76...78W Altcode: 1995gong.conf...78W No abstract at ADS Title: Forward Analysis of the BBSO Frequency-splitting Coefficients for 1986 Authors: Wilson, P. R.; Burtonclay, D. Bibcode: 1995ApJ...438..445W Altcode: Frequency-splitting data obtained at the Big Bear Solar Observatory (BBSO) for 1986 are collated by a method which permits qualitative inferences regarding the internal rotation profile of the Sun. The data are analyzed by a recursive forward approach which is first tested by a 'hare and hounds' experiment. The method is then applied to the Big Bear data yields a solution for the Sun's internal rotation profile for 1986. Title: The Reversal of the Solar Polar Magnetic Fields - Part Five Authors: Wilson, P. R.; Giovannis, J. Bibcode: 1994SoPh..155...29W Altcode: The Mount Wilson synoptic magnetic data from CRs 1815 to 1866 are used to describe the reversal of the solar polar magnetic fields during the period May 1989-March 1993. These are compared with simulations based on the observed fields for CR 1815 using the flux transport equation. Simulations including the emergence of small bipoles with preferred poleward orientations are also described. It is shown that, while the former can provide a qualitative account of the evolution of the southern fields between CRs 1815 and 1860, only the latter can describe the evolution of the northern fields between CRs 1815 and 1865. Title: Books-Received - Solar and Stellar Activity Cycles Authors: Wilson, P. R. Bibcode: 1994JBAA..104..240W Altcode: No abstract at ADS Title: Books-Received - Solar and Stellar Activity Cycles Authors: Wilson, P. R. Bibcode: 1994Sci...265.1902W Altcode: No abstract at ADS Title: Solar and stellar activity cycles. Authors: Wilson, P. R. Bibcode: 1994CAS....24.....W Altcode: This book offers a unique survey of our knowledge of cyclic activity for the Sun and extends and links this to cycles in other stars. It provides a valuable review of our basic knowledge of the Sun, of solar activity cycles and of stellar activity cycles in general. It shows the importance of reliable predictions of future solar cycles, and carefully explains the methods currently used to determine the details of these (with special reference to the maximum of cycle 22). Some of the latest research into solar cycles is clearly presented, including helioseismology, observations of the extended activity cycle and the polar field reversals and theoretical contributions from dynamo theory and chaos theory. Title: Solar and stellar activity cycles Authors: Wilson, Peter R. Bibcode: 1994ssac.book.....W Altcode: 1994QB526.C9W55.... No abstract at ADS Title: Real and Virtual Unipolar Regions Authors: Snodgrass, H. B.; Wilson, P. R. Bibcode: 1993SoPh..148..179S Altcode: Difficulties in relating magnetograph measurements to the actual solar magnetic field are discussed. After a brief review both of problems inherent in the nature of the measurements and of sources of instrumental error, we show that field measurements taken within the photosphere can map out large-scale regions of a single magnetic polarity even though these regions contain no footpoints of large-scale magnetic structures, but instead only aggregates of small, unresolved bipoles. This may occur wherever the density of unresolved bipoles has a preferred orientation and a spatial variation along the direction of that orientation. We call these regionsvirtual unipolar regions, as they are not connected to regions of opposite polarity by field loops or lines passing through the corona. Investigation of these regions shows that they can arise at widely separated locations, and that they may evolve into real unipolar magnetic regions which are connected to the chromospheric and coronal fields. These results can explain a number of puzzling aspects of magnetograph observations of the solar background magnetic field. Title: Forward Calculations of Libbrecht's Frequency Splitting Coefficients Authors: Wilson, P. R. Bibcode: 1993ASPC...42..209W Altcode: 1993gong.conf..209W No abstract at ADS Title: The Reversal of the Solar Polar Magnetic Fields - Part Four Authors: Murray, N.; Wilson, P. R. Bibcode: 1992SoPh..142..221M Altcode: It is a basic feature of the Babcock-Leighton model of the solar cycle that the polar field reversal is due to the diffusive decay and poleward drift of the active region fields. The flux from follower regions moves preferentially polewards in each hemisphere, where it cancels with, and then replaces, the previously existing polar fields. A number of workers have attempted to model this process by numerical solutions of the flux transport equation, which include the surface effects of supergranule diffusion, differential rotation and meridional flow, with conflicting results. Title: Helioseismology Data and the Solar Dynamo Authors: Wilson, P. R. Bibcode: 1992ApJ...399..294W Altcode: Some of the difficulties raised by recent helioseismology data for dynamo wave models of the solar cycle are addressed. These data have been interpreted as indicating that the latitudinal variations of angular velocity across the convection zone are approximately independent of depth. Theoretical frequency splittings are calculated for several types of model of the sun's internal rotation profile, and it is confirmed that models in which the rotation rate is constant on concentric cylinders are incompatible with these data, but only if parameters for the surface rotation rate are permitted to depart from those derived from surface spectroscopic data. Another class of model, in which the rotation rate profiles are not independent of depth, is investigated, and these models are shown to accommodate the helioseismology data at least as well as the radial surfaces model. The implications for dynamo models and for the dynamics of the convection zone are considered. Title: The Reversal of the Solar Polar Magnetic Fields - Part Three Authors: Wilson, P. R. Bibcode: 1992SoPh..138...11W Altcode: Observations of the first major active regions and large-scale magnetic field patterns of Cycle 22 are presented. These show that, following the emergence of a trans-equatorial pattern, or cell, of positive flux related to old cycle activity, the first new cycle active regions of the longitude range emerged across the neutral lines of this cell, which continued to grow and expand across the equator for several rotations. The development of a parallel trans-equatorial band of flux of opposite (negative) polarity and the emergence of both new and old cycle active regions across a neutral line of this cell are also described. Title: Comment on the Discussion of the Extended Activity Cycle Authors: Wilson, P. R. Bibcode: 1992ASPC...27..376W Altcode: 1992socy.work..376W No abstract at ADS Title: The Reversal of the North Polar Field Authors: Wilson, P. R. Bibcode: 1992ASPC...27..116W Altcode: 1992socy.work..116W No abstract at ADS Title: Theoretical Implications of the Extended Activity Cycle Authors: Wilson, P. R. Bibcode: 1992ASPC...27..368W Altcode: 1992socy.work..368W No abstract at ADS Title: The Reversal of the Solar Polar Magnetic Fields - Part Two Authors: Wilson, P. R.; McIntosh, P. S. Bibcode: 1991SoPh..136..221W Altcode: Observations of the first large-scale patterns of magnetic fields near the sunspot minimum of 1986 (the start of cycle 22) are presented using synoptic magnetic data provided by the National Solar Observatory and contour maps constructed from data provided by the Mount Wilson Solar Observatory. The latter are compared with simulated contour maps derived from numerical solutions of the flux transport equation using data from particular Carrington rotations as initial conditions. Title: The Reversal of the Solar Polar Magnetic Fields - Part One Authors: Wilson, P. R.; McIntosh, P. S.; Snodgrass, H. B. Bibcode: 1990SoPh..127....1W Altcode: Some theoretical difficulties confronting the current model of the polar magnetic reversal by cancellation with the flux remnants of decaying active regions are discussed. It is shown that the flux transport equation does not adequately describe the essential physical consequences of the transport of large-scale fields, linked to deep subsurface toroids, over distances comparable with the solar radius. The possibility that subsurface reconnections may release these fields to form U-loops is discussed but it is shown that, in this event, the loops will quickly rise to the surface. Mechanisms whereby the flux may escape through the surface are considered. Title: An Extended Activity Cycle Picture of the Sun's Polar Magnetic Fields Authors: Snodgrass, H. B.; Wilson, P. R. Bibcode: 1990BAAS...22Q.855S Altcode: No abstract at ADS Title: The Origin of the Large-Scale Fields Authors: Wilson, P. R.; McIntosh, P. S. Bibcode: 1990BAAS...22..872W Altcode: No abstract at ADS Title: Solar Cycle Workshop - 3RD Meeting / Sydney, Australia / 1989JAN Authors: Wilson, P. R.; Donnelly, R. F.; Wilkinson, P. Bibcode: 1990SoPh..125..159W Altcode: 1990SoPh..125..157W The contributions to the third meeting of the Solar Cycle Workshop are briefly summarized. The topics discussed at the meeting included (i) predictions and precursors, (ii) large and small-scale magnetic fields, (iii) photospheric velocity fields, (iv) coronal phenomena, (v) the Sun as a star, (vi) limb temperature measurements and helioseismic data, (vii) theoretical modelling of the cycle, (viii) cyclic activity in stars, and (ix) the interpretation of the Elatina Sandstone Layers. Title: The buoyancy of U-loops Authors: Wilson, P. R. Bibcode: 1989PASA....8...32W Altcode: 1989PASAu...8...32W The problem of the surface migration of solar magnetic fields is discussed. It is shown that, unless some form of reconnection takes place below the surface, magnetic tension should effectively prevent such migration and the possibility of U-loop formation and transport is considered. As a first step, the buoyancy of a magnetic 'U-loop' within an isothermal atmosphere is considered. It is shown that such a configuration may form by the reconnection of the arms of an Omega-loop associated with an active region, and is likely to be buoyant initially and rise through the region. However, as it rises, the net buoyancy decreases and, provided that the temperature within the loop is even marginally less than that outside, it will reach a subsurface configuration in which the net buoyancy is zero. The significance of this result for solar magnetic fields is then assessed. Title: The Solar Dynamo and the Convective Rolls Authors: Wilson, P. R. Bibcode: 1988SoPh..117..217W Altcode: The most sophisticated attempts to model the convection zone have yielded results in which the angular velocity increases outwards and the largest scales of convection take the form of `banana cells' aligned with the rotation axis. However, not only does the sign of the angular velocity gradient present problems for dynamo theory, but attempts to detect banana type cells have so far been unsuccessful. Although by no means conclusive, current tracer, spectropic, and radiative data all tend to support models of azimuthal rolls encircling the axis as the fundamental mode. Title: Solar Cycle Workshop Authors: Wilson, P. R. Bibcode: 1988SoPh..117..205W Altcode: The presentations and discussions which took place during the second meeting of the Solar Cycle Workshop are summarized under the headings: sunspot minimum, the extended cycle, the large-scale photospheric motions, the large-scale magnetic fields and the polar reversal, the small-scale fields, global cyclic phenomena and the fundamental processes. The progress achieved so far is assessed and the directions for future observational and theoretical work are suggested. Title: The extended solar activity cycle Authors: Wilson, P. R.; Altrocki, R. C.; Harvey, K. L.; Martin, S. F.; Snodgrass, H. B. Bibcode: 1988Natur.333..748W Altcode: The solar cycle has been defined in terms of a sequential periodic variation in sunspot numbers, the period being the interval between successive minima, currently averaging 11.2 years. But a number of observations have indicated that the activity cycle may begin at higher latitudes before the emergence of the first sunspots of the new cycle. Here we report results from sunspot cycle 21 concerning the ephemeral active regions, the coronal green-line emission and the torsional oscillation signal, which confirm the earlier suggestions. In particular, we report the appearance of a high-latitude population of ephemeral active regions in the declin-ing phase of sunspot cycle 21, with orientations that tend to favour those for cycle 22 rather than 21. Taken together, these data indicate that sunspot activity is simply the main phase of a more extended cycle that begins at high latitudes before the maximum of a given sunspot cycle and progresses towards the equator during the next 18-22 yr, merging with the conventional 'butterfly diagram' (the plot of the latitudes of emerging sunspots against time) as it enters sunspot latitudes. We suggest that this extended cycle may be understood in the perspective of a model of giant convective rolls that generate dynamo waves propagating from pole to equator. Title: Solar torsional oscillations as a signature of giant cells Authors: Snodgrass, H. B.; Wilson, P. R. Bibcode: 1987Natur.328..696S Altcode: Although the existence of giant cells1 as the fundamental mode of solar convection has long been proposed on theoretical grounds, attempts to detect them observationally have been unsuccessful. During one search, using Mount Wilson magnetograph data, Howard and LaBonte2,3 discovered a pattern of latitudinal velocity bands that move from the poles towards the equator in synchrony with the sunspot cycle, and they interpreted this pattern as a torsional wave or 'oscillation' with wavenumber k=2 hemisphere-1. Here we suggest that this signal is not in fact an oscillation but represents a modulation of the mean differential rotation caused by a system of giant convective rolls which start at the poles at 11-yr intervals and migrate to the equator in a period of 18-22 yr. Additional evidence for the presence of these rolls is found in the zero offsets in the Mount Wilson data4 and in latitude variations of the limb temperature5. Thus we argue that the fundamental mode of giant-cell convection in the sun takes the form of equatorward migrating azimuthal rolls. This differs from the 'banana cell' mode suggested by Gilman6, and from the poleward propagating rolls reported by Ribes et al.7. Title: Azimuthal Rolls and the Solar Cycle Authors: Snodgrass, H. B.; Wilson, P. R. Bibcode: 1987BAAS...19Q.935S Altcode: No abstract at ADS Title: Solar Cycle Workshop Authors: Wilson, P. R. Bibcode: 1987SoPh..110....1W Altcode: Clearly there is no concensus or agreement at present about the nature and mechanism of the solar cycle or, indeed, about many of its observed features. However, by highlighting these areas of agreement and disagreement through the presentations and discussions during this meeting, it is hoped that the work of the Workshop Groups will be directed to resolving at least some of these questions at or before the next meeting (planned for August, 1987). In particular, it is hoped that Group V (The Sun as a Star) will be able to contribute through studies of the sun in relation to stellar cycles and activity. Title: Solar Rotation and the Giant Cells Authors: Wilson, P. R. Bibcode: 1987SoPh..110...59W Altcode: Departures from the mean solar differential rotation rate as a function of latitude, longitude, and epoch of the solar cycle, together with variations in the rotation rate as determined by spectroscopic and tracer measurements are reviewed. It is shown that, if giant convection cells do exist as predicted, real variations in the subsurface rotation rate should occur and that this may be responsible for the observed surface anomalies. Title: Solar Cycle Workshop. Proceedings of the 2nd meeting, held at Stanford Sierra Lodge, Lake Tahoe, California, 10 - 14 May 1987. Authors: Wilson, P. R. Bibcode: 1987scwp.book.....W Altcode: No abstract at ADS Title: Solar Cycle Workshop. Proceedings of the first meeting held at Big Bear Solar Observatory, Pasadena, CA, USA, 17 - 20 August, 1986. Authors: Wilson, P. R. Bibcode: 1987SoPh..110.....W Altcode: No abstract at ADS Title: A New Definition for the Solar Cycle Authors: Wilson, P. R. Bibcode: 1986BAAS...18.1010W Altcode: No abstract at ADS Title: The Generation of Magnetic Fields in Photospheric Layers Authors: Wilson, P. R. Bibcode: 1986SoPh..106....1W Altcode: Recent observations concerning the growth and decay of photospheric magnetic flux present a challenge to the conventional picture of the photosphere as a passive medium through which flux tubes emerge inertly. Rather, they suggest the possibility that interactions between the magnetic flux and the photospheric velocity fields may give rise to changes in the observed surface flux. Title: Flux changes in small magnetic regions. II - Further observations and analysis Authors: Simon, G. W.; Wilson, P. R. Bibcode: 1985ApJ...295..241S Altcode: Further examples of flux changes are described in which the flux associated with small magnetic knots of dimension 2-3 arcsec is observed to increase or decrease by 50-100 percent during periods of order 20-40 minutes. Within the same regions both positive and negative flux knots have been studied. In one case the positive and negative flux fluctuations appeared to be almost out of phase, but in another they were roughly in phase. From measurements of the velocity field within each region, an estimate of the vertical velocity shear has been calculated, and correlations of order 0.5 between the fluctuations in this shear and in both the positive and negative flux counts have been obtained in several, but not all cases. The possibility that these apparent flux changes are due to instrumental or seeing effects, changes in the line equivalent width, or vertical velocity gradients is discussed, but it is concluded that none of these effects provides a tenable alternative explanation of the observed changes. It is therefore suggested that these changes may provide examples of the generation of nonpotential magnetic fields (i.e., electric currents) within the photospheric layers, as has been proposed by several authors from time to time. Title: A new model for flux emergence and the evolution of sunspots and the large-scale fields Authors: McIntosh, P. S.; Wilson, P. R. Bibcode: 1985SoPh...97...59M Altcode: Existing models for the evolution of sunspots and sunspot groups, describing the subsurface structure of the magnetic fields and their interactions with the convective motions, are briefly reviewed. It is shown that they are generally unable to account for the most recent data concerning the relationship between the large-scale solar magnetic field structures and the magnetic fields of active regions. In particular, it is shown that the former do not arise directly from the decay of the latter, as required by the Babcock model and all other models based on it. Other observations which are not adequately explained by current models are also cited. Title: The possibility of a photospheric dynamo Authors: Wilson, P. R. Bibcode: 1985AuJPh..38..911W Altcode: Several recent observations are discussed which suggest that the current model for the emergence and decay of photospheric flux does not provide a complete account of all the processes involved. An elementary two-dimensional dynamo is discussed and it is shown how this may be adapted to photospheric conditions in order to provide a plausible kinematic account of these observations. Title: Photospheric flux changes and the MHD approximation Authors: Wilson, P. R. Bibcode: 1984PASA....5..500W Altcode: 1984PASAu...5..500W It is noted that observations of changes in photospheric magnetic flux present a challenge to the conventional picture of passive emerging flux tubes. This problem is assessed through an examination of the MHD approximation; both the pressure gradient term and the thermoelectric term are considered. It is concluded that, although they may provide seed fields, the two source terms, of themselves, seem unable to explain the observed rates of flux growth. However, observations suggest that regions of rapid growth and decay of flux are connected with velocity shears; it is thus necessary to consider the possibility that these nonaxisymmetric motions may give rise to dynamo action within these local regions. Title: Flux changes in small magnetic regions Authors: Wilson, P. R.; Simon, G. W. Bibcode: 1983ApJ...273..805W Altcode: High spatial and temporal resolution observations of continuum, velocity, and magnetic fields in active regions of the sun have been obtained which reveal rapid changes in the total magnetic flux associated with surface magnetic features. Attention is given to two cases. In the first, during February 1980, the total positive flux associated with a magnetic feature increased by about 50 percent in about 90 min without observable negative flux change. The topology of the feature developed from a single intense core to one exhibiting several discrete cores, all of like polarity. The second case, during July 1981, involved large fluctuations in the flux associated with a small negative feature over 20-min time intervals. Calculations of the correlation between velocity gradients and magnetic field gradients show values in the 0.4-0.7 range at times of good seeing, reaching a peak when the flux is greatest. Title: Free and forced oscillations of a flux tube Authors: Wilson, P. R. Bibcode: 1981ApJ...251..756W Altcode: The free oscillations of a magnetic flux tube of finite dimension, embedded in a nonmagnetic plasma, which may be in uniform motion relative to the tube, are studied for models in which energy may propagate laterally away from the tube. Although solutions obtained using the thin tube approximation yield both real and complex frequencies, the numerical solutions for effectively thick tubes all exhibit real frequencies and imaginary external wavenumbers. Thus the waves propagate parallel to e flux tube with constant amplitude and are evanescent in the transverse direction.

The same general formulation permits the study of the oscillations set up by an oscillatory disturbance applied at a point on the axis of the tube. Here it is found that for weak or moderately intense tubes the wave amplitudes decrease along the tube away from the disturbance as wave energy leaks away from the tube in the transverse direction. Intense tubes, however, tend to focus the wave energy along the tube, and the wave amplitude increases away from the disturbance. It is suggested that this effect may be relevant for photospheric flux tubes. Title: Flux Changes in Small Magnetic Regions Authors: Wilson, P. R.; Simon, G. W. Bibcode: 1981BAAS...13..882W Altcode: No abstract at ADS Title: Faculae, filigree and calcium bright points Authors: Wilson, P. R. Bibcode: 1981SoPh...69....9W Altcode: Simultaneous observations of fine structure photospheric features at several wavelengths are described. The observations, which include regions near disk center and at the limb, were obtained using a narrow band calcium filter and selected wavelengths in Mg b1, the red wing of Hα and the Hα continuum using the Universal Birefringent Filter of the Sacramento Peak Observatory. Title: Theories of sunspot structure and evolution Authors: Wilson, P. R. Bibcode: 1981phss.conf...83W Altcode: Various attempts to model all or part of the known phenomena associated with sunspots are reviewed. In summary, sunspots appear and grow as groups within active regions. Simple prototypes are the leader L and follower F spots, with the F spots decaying within a week, like a group, while L spots first form a moat-like structure and then last for several solar rotations. The magnetic structure stability of a sunspot is discussed, together with the reasons for the relative coolness, i.e., 3500-4000 K. A necessity is mentioned for understanding the subsurface structure of the spots, although the phenomena may in fact be shallow. Attention is given to the features and deficiencies of three sunspot models. It is suggested that improvements in modeling may be obtained from studies of the irradiance variability data, the growth and decay rates, time-dependent studies of the simultaneous magnetic and velocity fields and brightness patterns in the region around sunspots, and observations by the solar orbiting telescope of the ultrafine structure of the umbrae and penumbrae. Title: The general dispersion relation for the vibration modes of magnetic flux tubes Authors: Wilson, P. R. Bibcode: 1980A&A....87..121W Altcode: The general dispersion relation for wave modes in a magnetic flux tube embedded in a compressible but unstratified atmosphere is obtained and is solved numerically for several different sets of atmospheric parameters. Approximate analytic solutions for the phase velocity are also recovered in the thin tube approximation and these are compared with the numerical solutions. The nature of the different modes is discussed and particular attention is given to the interface or long wavelength mode, and to the 'taut wire' mode. Title: The interaction of acoustic waves with flux tubes Authors: Wilson, P. R. Bibcode: 1980ApJ...237.1008W Altcode: The interaction between a magnetic flux tube and an acoustic wave front propagating in the nonmagnetic region in which it is embedded is investigated by expressing the incident wave as a Fourier Bessel series. Using the velocity and pressure balance conditions at the interface, one may determine the amplitudes of the reflected and transmitted waves for each Bessel component. The component of the energy flux parallel to the field and averaged over azimuth and cycle is determined at the interface for both internal and external waves. It is found that when the component of the phase velocity of the incident wave along the tube approaches AC(i)/square root of (A-squared plus C/i/-squared) (A is the Alfven speed and C(i) the sound speed), the wave flux within the tube is concentrated at the interface. The implications of this for wave propagation in the solar atmosphere are briefly discussed. Title: Wave propagation in a magnetic flux sheath in the presence of a velocity field. Authors: Wilson, P. R. Bibcode: 1979ApJ...230..194W Altcode: Hydromagnetic-wave propagation in a plane-parallel magnetic flux sheath embedded in a compressible plasma is studied in the case where a uniform velocity field exists in either the sheath or the external plasma. The problem is formulated in the flux-sheath geometry partly for the mathematical convenience of the Cartesian coordinate system, and the formulation is generalized to include perturbation velocities parallel to the flux sheath and perpendicular to the magnetic field. Variations in the perturbation amplitudes in the indicated direction are considered in order to elucidate possible wave modes along supergranule boundaries. Attention is given to a strictly two-dimensional case, the thin-flux-tube approach, and the three-dimensional case. The effect of the velocity field on the stability of the system is evaluated. Title: Hydromagnetic wave modes in magnetic flux tubes. Authors: Wilson, P. R. Bibcode: 1979A&A....71....9W Altcode: The vibration modes and stability of a discrete magnetic flux element (a cylindrical flux tube or a two-dimensional flux sheath) embedded in an infinite compressible but unstratified plasma are analyzed. It is shown that the dispersion relation for a thin flux tube takes two distinct forms, depending on whether the oscillation modes are symmetrical or asymmetrical, that the m = 0 mode in the cylindrical geometry is analogous to the pulsation mode of a thin flux sheath, and that the axis of the flux tube has a finite velocity for the m = 1 mode, which is analogous to the vibration mode of a taut wire. Comparison of the results for the thin flux tube and thin flux sheath indicates that the taut-wire mode of the sheath corresponds to the m = 1 mode of the tube. The relevance of this analysis to solar magnetic fields is discussed. Title: Wave modes in magnetic elements in a stratified atmosphere. Authors: Wilson, P. R. Bibcode: 1978ApJ...225.1058W Altcode: The vibration modes of a magnetic flux element embedded in a compressible, gravitationally stratified atmosphere are considered. A local dispersion relation is first derived for a force-free field which decreases exponentially with height, and this is applied to a magnetic flux sheath lying within the local region. The sheath is embedded in an isothermal, gravitationally stratified plasma for which the standard dispersion relation applies globally. The velocity and pressure balance conditions across the interface restrict the possible vibration modes of the system. For the pulsation or varicose mode, a modified acoustic mode and a modified fast mode are obtained. The latter has a cutoff frequency which is greater than the acoustic cutoff and suggests that waves of shorter periods may occur in the presence of magnetic elements. For the sinuous or taut wire mode, complex solutions for the dispersion equation are obtained. For one of these the real part is large and negative, indicating that, within the limitations of a linear analysis, waves propagate with increasing amplitude. The possible significance of this is briefly discussed. Title: Wave modes in a magnetic flux sheath. Authors: Wilson, P. R. Bibcode: 1978ApJ...221..672W Altcode: The symmetric wave modes of a two-dimensional magnetic flux sheath embedded in a compressible isothermal but nongravitational atmosphere are investigated under more general conditions than those considered earlier by Cram and Wilson. Both two-dimensional 'taut wire' and pulsation modes are considered, and explicit solutions are obtained when the thickness of the flux sheath is small compared with the transverse scale size of the perturbations. In particular, it is found that the taut wire or Alfven-type mode may be unstable if the density within the sheath is less than that outside. The solutions for the pulsation mode are compared with those obtained by Defouw under similar conditions. Agreement is obtained only in the limit as the sound speed in the nonmagnetic region tends to infinity. It is shown that Defouw's analysis, which neglects all transverse variations in the perturbations, is unsuitable to the problem and that the approximation which he makes eliminates the interaction between the magnetic and nonmagnetic plasmas. Title: Structure within a magnetic flux tube. Authors: Wilson, P. R. Bibcode: 1978PASA....3..225W Altcode: 1978PASAu...3..225W X-ray observations of the solar corona show the existence of hot flux loops within which both the pressure and temperature appear to be in excess of that outside the loops. This paper discusses the structure within such flux tubes in terms of standard models, the enhancement of small-scale fields and approximate models of large radius. Title: The stability of a magnetic flux element in a horizontally stratified compressible plasma. Authors: Wilson, P. R. Bibcode: 1977SoPh...55...35W Altcode: The configuration of a magnetic flux element in a static, compressible, gravitationally stratified plasma is considered. Under isothermal conditions an exact force-free solution is given for a two-dimensional cartesian flux sheath but for an axi-symmetric element, i.e. a flux tube, approximate solutions, applicable only to thin flux tubes, are obtained. Title: Hydromagnetic Waves in Magnetic Flux Tubes Authors: Wilson, P. R. Bibcode: 1977PASA....3..173W Altcode: 1977PASAu...3..173W No abstract at ADS Title: The elementary theory of a twisted flux tube. II. Stability. Authors: Wilson, P. R. Bibcode: 1977ApJ...214..917W Altcode: Previously developed models for magnetic flux tubes are used to calculate the total potential energy of a tube embedded in an isothermal horizontally stratified compressible atmosphere. By comparing this with the total energy of a system of n flux tubes having the same total flux as the original tube, a necessary condition is obtained that the system is consistent with the adiabatic subdivision of a single untwisted flux tube. It is shown that under nonadiabatic conditions such as exist in the solar photosphere, the subdivision process must be endothermic (i.e., external energy is required) if the temperature within the original tube is significantly less than its surroundings, but exothermic if the temperatures are comparable. Thus it is conjectured that magnetic structures are less susceptible to subdivision if they are significantly cooler than their surroundings. While a twisted flux tube cannot subdivide simply like an untwisted tube, it may fray into several spiral filaments; it is shown that the cooling required to prevent this is actually greater than for an equivalent untwisted tube. It is shown that several features of the growth and decay of sunspots may be explained in terms of this conjecture. Title: The elementary theory of twisted flux tubes. I. Equilibrium configurations. Authors: Wilson, P. R. Bibcode: 1977ApJ...214..611W Altcode: An approximate parametric treatment of the equilibrium of a twisted, nonisothermal flux tube is given in which third and higher order terms in the Taylor expansion of the field are neglected. Families of nonisothermal, nonforce-free solutions are obtained. The equilibrium and pressure boundary conditions are satisfied correct to second-order terms, while the zero divergence condition is satisfied exactly. Models of untwisted flux tubes or of isothermal force-free tubes are obtained as particular cases of the general formulation. A comparison is made with Parker's analysis of twisted flux tubes, and it is shown that Parker's analysis is valid only for small angles of twist. The possibility of field enhancement by cooling and twisting is considered in terms of the models obtained. Title: Alfvén waves and Meyer's sunspot model. Authors: Wilson, P. R. Bibcode: 1975MNRAS.172..535W Altcode: The recent account of the growth and decay of a sunspot by Meyer, Schmidt, Weiss and Wilson is compared with models exhibiting a large emission of Alfven flux. It is shown that a composite model which involves both modified convection and Alfven waves within the sub-surface flux rope can overcome some of the difficulties faced by either type of model independently. It is shown that interactions between convective motions and the flux tubes may give rise to twists in these tubes. The significance of these twists in relation to the decay of the spot magnetic field and the appearance of the moving magnetic features in the moat is discussed. An approximate distribution of the radiative, convective and hydromagnetic energy flux in the neighborhood of a stable l-spot is suggested on the basis of the composite model. Title: Alfvén Waves and the Sunspot Phenomenon Authors: Wilson, P. R. Bibcode: 1975SoPh...42..333W Altcode: Parker's explanation of the sunspot phenomenon in terms of the enhanced emission of Alfvén waves (solar vulcanology) is shown to be compatible with observation only if ∼ 90% of the waves propagate downwards. Further difficulties arise if the region of cooling by Alfvén wave generation is restricted to a depth of 2 Mm. However, it is shown that, if Alfvén wave generation is included in a recent model proposed by Meyer, Schmidt, Weiss and Wilson, these difficulties may be resolved. The problem of the sharp umbra and penumbra boundaries is discussed and it is shown that features of this combined model are relevant to the flare phenomenon. Title: Hydromagnetic Waves in Structured Magnetic Fields Authors: Cram, L. E.; Wilson, P. R. Bibcode: 1975SoPh...41..313C Altcode: Although the inhomogeneous nature of solar magnetic fields is now well established, most theoretical analyses of hydromagnetic wave propagation assume infinite homogeneous fields. Here we reformulate the hydromagnetic wave problem for magnetic fields which vary in one direction perpendicular to the field. The permitted modes of small amplitude hydromagnetic oscillations are considered, first in the case of a single interface between semi-infinite magnetic and non-magnetic compressible regions, and secondly for a magnetic flux sheath of given thickness imbedded in a nonmagnetic region. It is shown that, for small values of R (the ratio of the Alfvén to the sound speed), an acoustic or p-mode wave front passes through the flux sheath with only minor deformation. However, for large R, the transmitted acoustic wave is attenuated and, depending upon the thickness of the flux sheath and the angle of incidence, a hydromagnetic wave may be effectively trapped and guided along the flux sheath. Title: The growth and decay of sunspots. Authors: Meyer, F.; Schmidt, H. U.; Weiss, N. O.; Wilson, P. R. Bibcode: 1974MNRAS.169...35M Altcode: Sunspots are formed between supergranules, at junctions in the chromospheric network. Many spots disappear rapidly. However, some large spots enter a phase of slow decay: an annular cell develops, centered on the spot and with a systematic outward velocity directed from the penumbra toward the nearest faculae. This flow forms a moat around the spot swept clear of magnetic field except for magnetic features which migrate from the penumbra to the surrounding network. This phase may persist for several months before the spot is finally destroyed. Throughout this period the sunspot area, and hence its flux, decrease at a constant rate. The evolution of a sunspot is related to subphotospheric convection, with particular emphasis on a model of the slow decay phase which explains the steady decay of magnetic flux. Title: The Cooling of a Sunspot. IV: Reply to M. H. Gokhale Authors: Wilson, P. R. Bibcode: 1974SoPh...37..483W Altcode: 1974SoPh...37..438W Some criticisms made by Gokhale (1974) on the present author's (1972) work on the cooling of a sunspot are met. It is agreed that axisymmetric models of magnetic and velocity fields are inadequate with respect to the zero divergence conditions, and that nonaxisymmetric models must be considered even if only in qualitative form. It is pointed out that the failure of Wilson's (1972) axisymmetric models to satisfy the zero divergence condition at all points where r is below a critical value does not invalidate some of the other conclusions. Calculations are carried showing the existence of a region where the zero divergence condition is satisfied to first order. It is agreed also that the power converted to magnetic form during field amplification is not a likely explanation of the residual energy deficit associated with sunspots. Title: The Cooling of a Sunspot IV: Reply to D. J. Mullan Authors: Wilson, P. R. Bibcode: 1974SoPh...35..111W Altcode: This paper considers the recent criticism by Mullan (1973) of sunspot models and the cooling mechanism which I have proposed in Papers I, II and III of this series. The discussion of the cooling produced by an idealized flow cycle has been extended to include vertical temperature gradients which are consistent with a convectively unstable atmosphere. This leads to an expression for Mullan's parameter f (the ratio in which estimates of the energy flux based on an idealized Carnot cycle should be reduced) which is appropriate to this situation. It is shown that, for a cycle similar to that of Paper III, f = 0.82, while for one which has a vertical extent of order 5 Mm, f= 0.4. Hence the energy flux which, in principle, can be transported away from a sunspot by such a cycle is conservatively estimated to be 1.1 × 1029 erg s−1 compared with a typical sunspot energy deficit of 2.2 × 1029 erg s−1. Other criticisms relating to the magnetic field amplification and the `cool one' model are discussed. It is concluded that the essential features of these models remain valid and that the modifications suggested by Mullan's criticism greatly increase their applicability to the sunspot problem. Title: The Stability of a Velocity Shear in the Presence of an Inhomogeneous Magnetic Field Authors: Wilson, P. R. Bibcode: 1974Ap&SS..26..363W Altcode: The stability of a velocity shear in the presence of a parallel but non-uniform magnetic field is considered in general terms. Two special cases are then investigated; (i) the well known case of a plane interface at which a discontinuity in the magnetic field coincides with the velocity shear; (ii) an axially symmetric flow in which discontinuities in the magnetic and velocity fields occur at a cylindrical surface whose axis is parallel to the flow. In the first case the flow is stabilized if the rms Alfvén velocity of the magnetic field exceeds the shear velocity; a result consistent with that obtained by other writers. In the second case it is shown that the discontinuity in the magnetic field increases the stability of the system. The significance of this result for the stability of the flux ropes associated with sunspots in the solar convection zone is considered. Title: A Theoretical Model for the Convection of Magnetic Flux in and Near Sunspots Authors: Meyer, F.; Schmidt, H. U.; Weiss, N. O.; Wilson, P. R. Bibcode: 1974IAUS...56..235M Altcode: No abstract at ADS Title: The Stability of a Magnetic Flux Rope and its Relation to Sunspots, Faculae and Flares Authors: Wilson, P. R. Bibcode: 1974IAUS...56..301W Altcode: No abstract at ADS Title: The Cooling of a Sunspot. III: Recent Observations Authors: Wilson, P. R. Bibcode: 1973SoPh...32..435W Altcode: The coordinates of the cooling cycle described in Paper I are re-defined in order to provide an account in which the part played by the cycle in cooling the sunspot is separated from the role of the supergranule cells in transporting energy away from it. Title: Erratum: "The cooling of a sunspot. II" [Sol. Phys., Vol. 27, p. 363 - 372 (1972)]. Authors: Wilson, P. R. Bibcode: 1973SoPh...30..280W Altcode: No abstract at ADS Title: The Cooling of a Sunspot. I: A Carnot Cycle and the Hydromagnetic Interactions Authors: Wilson, P. R. Bibcode: 1972SoPh...27..354W Altcode: A mechanism is proposed to explain the cooling of a sunspot in terms of the detailed interactions between the magnetic field and the convective motions. The mechanism provides that an axially symmetric concentration of magnetic field deforms the normal supergranule cell pattern below the sunspot into a radial outflow of plasma over a region of diameter ∼ 60 Mm. Title: The Cooling of a Sunspot. II: Convection Zone Models and the Magnetic Power Supply Authors: Wilson, P. R. Bibcode: 1972SoPh...27..363W Altcode: In order to discuss the detailed interactions between the magnetic and velocity fields below a sunspot, several models of the convection zone are considered. It is suggested that giant convection cells establish a pattern of large superadiabatic gradients near the top and bottom of the zone analogous with the temperature gradients in laboratory convection. These gradients in turn establish their own eddies or cells with dimensions typical of the local scale height. Those at the top correspond to the photospheric granules while overshoot motions from the bottom of the layer may be related to the surface supergranule motions. Title: Giant Cells and the Solar Cycle Authors: Wilson, P. R. Bibcode: 1972PASA....2..144W Altcode: 1972PASAu...2..144W Despite the success of the Babcock-Leighton model in explaining some phenomena of the solar cycle, e.g. Hale’s polarity law and Maunder’s butterfly diagram, it has recently encountered a number of observational and theoretical difficulties. The observational difficulties are centred around the different behaviour of bi-polar magnetic regions (BMRs) and uni-polar magnetic regions (UMRs). Babcock explains BMRs as the surface eruptions of toroidal fields wound by the differential rotation and it is clear that in their subsequent evolution and decay they are obviously influenced by this effect. However, UMRs display a sector structure in which the boundaries between regions of opposite polarities appear to be quite unaffected by the differential rotation but closely related to the interplanetary field sectors. Another feature of solar activity unexplained by the Babcock theory is the tendency of sunspots, flares, etc. to reappear at the same latitude during a cycle. Warwick has observed that proton flares show a preference for particular latitudes which may extend over several cycles. Again Bumba et al. have observed that new cycle fields tend to appear in longitude regions where old cycle fields are still visible. Title: Mean Values in Inhomogeneous Atmospheres Authors: Wilson, P. R.; Williams, N. V. Bibcode: 1972SoPh...26...30W Altcode: A brief summary of observations of inhomogeneities in the solar atmosphere and progress in the theoretical analysis of two-dimensional model atmospheres is given. In particular, it is asserted that reliable reference models of the mean temperature, pressure, etc. may be derived only by averaging over the horizontal coordinates of a two- or three-dimensional model. In discussion of a wide variety of cases including strong chromospheric lines, weak photospheric lines, and continuum radiation, it is shown that although the derivation of mean values by first averaging over the fluctuations in the data and then applying a one-dimension analysis is an obvious first approximation, it may lead to errors of a gross qualitative nature. Thus the recent deductions by several authors of very small temperature gradients in the upper regions of sunspots are shown to be subject to considerable doubt. Title: High Resolution Spectroscopy of the Disk Chromosphere. II. Time Sequence Observations of Ca II H and K Emissions Authors: Wilson, P. R.; Rees, D. E.; Beckers, J. M.; Brown, D. R. Bibcode: 1972SoPh...25...86W Altcode: Two independent sets of high resolution time series spectra of the CaII H and K emission obtained at the Solar Tower and at the Big Dome of the Sacramento Peak Observatory on September 11th, 1971 are reported. The evolutionary behaviour of the emission first reported by Wilson and Evans is confirmed but the detail of the evolution is found to be more complex. In one case, a doubly peaked feature showing some K3 emission evolves into a single K2 (red) peak with no K3 emission. Coincidentally, a neighbouring doubly peaked feature evolves to a very strong blue peak. In an entirely independent sequence a doubly peaked feature evolves into a single red peak. The K2 emission then fades completely although the continuum threads are still strong. Finally a strong K2 blue peak appears. These developments are confirmed by intensity profiles obtained from the spectra. Title: Alfvén Waves in Umbral Flux Tubes Authors: Wilson, P. R. Bibcode: 1972SoPh...22..434W Altcode: Savage has suggested that an energy flux of 2 × 1010 erg cm−2 s−1 passes through the umbra of a sunspot in the form of hydromagnetic waves. In this paper some of the consequences of this flux are considered. It is first shown that it is not inconsistent with the energy requirements for the heating of umbral dots and for solar wind storms, assuming in the latter case that the flux tubes emerging from about one tenth of the area of a large spot are open-ended. Title: Facular Models and the Sunspot Energy Deficit Authors: Wilson, P. R. Bibcode: 1971SoPh...21..101W Altcode: The problem of the energy deficit in a sunspot is shown to be critically related to the depth of a given sunspot model. Recent facular models are discussed and a new model is derived from recent data using a two-dimensional radiative transfer analysis. The excess non-radiative energy required by this and other models is evaluated and it is shown that in some models this may account for a considerable fraction of the sunspot energy deficit. For these models the Alfvén energy travelling along the closed flux loops from the sunspot is insufficient to supply the requirements of the faculae and it is suggested that excess energy flux from below the faculae is also required. These results provide further support for `deep' as opposed to `shallow' sunspot models. Title: Center Limb Observations of Inhomogeneities in the Solar Atmosphere. III: Time Dependent Fine Structure of the Ca II Emission Authors: Wilson, P. R.; Evans, C. D. Bibcode: 1971SoPh...18...29W Altcode: High resolution Ca II K-line spectra obtained at Kitt Peak National Observatory in September 1970 at a dispersion of 20 mm/Å and a spatial resolution approaching 1 arc sec are described. Title: Center Limb Observations of Inhomogeneities in the Solar Atmosphere. II: The Na D and Na 5688 Doublets and the MgI 4571 Line Authors: Cannon, C. J.; Wilson, P. R. Bibcode: 1971SoPh...17..288C Altcode: Center-limb observations of line-center intensity fluctuations in the Na D and Na 5688 doublets and the Mg 4571 line are described. For small scale structures the rms distributions for the photospheric lines show maxima of 20% at a heliocentric angle of 40° (Mg4571) and 9% at 20° (Na 5688). The rms values for Na D range between 10 and 13% but show no significant maximum. Values for large scale structures are somewhat less. Title: Sunspot Magnetic Fields and Umbral Dots Authors: Wilson, P. R. Bibcode: 1971IAUS...43..475W Altcode: No abstract at ADS Title: A three-component model for the formation of the chromospheric Ca ii K line Authors: Wilson, P. R. Bibcode: 1970SoPh...15..139W Altcode: A recent two-component model for the formation of the CaII K line in the solar chromosphere put forward by Beebe and Johnson is discussed. Although this model is a great advance on existing one-component models, it is pointed out that observations require a minimum of three components in order to understand the formation of the K2 peaks. Title: Center-Limb Observations of Inhomogeneities in the Solar Atmosphere. I: The Mg b Lines Authors: Cannon, C. J.; Wilson, P. R. Bibcode: 1970SoPh...14...29C Altcode: Center-limb observations of line-center intensity and velocity fluctuations in the Magnesium b lines are described. Autocorrelation and power spectral analyses indicate small scale brightness structures having periodicities of 3000 km and 8000 km and large scale structures of 22000 km. Corresponding velocity structures are 6000 km and 30000 km. Title: On the Properties of Umbral Dots Authors: Wilson, P. R. Bibcode: 1969SoPh...10..404W Altcode: On the basis of a three-dimensional radiative transfer analysis of several models it is shown that bright structures in sunspot umbrae which have horizontal diameters of 300 km or less cannot extend more than 300 km down into the umbra. Thus, such models are inconsistent with the hypothesis that the bright features are due to convection from the deep regions of the umbra. No such restrictions can be applied if the surface diameter is of order 500 km, but a model of this type is shown to be inconsistent with the available data. Thus a convective explanation of these bright features appears to be ruled out. Title: Structure of a Sunspot. V: What is the Wilson Effect? Authors: Wilson, P. R.; McIntosh, P. S. Bibcode: 1969SoPh...10..370W Altcode: From enlargements of patrol photographs of the disk passage of the sunspot of July 20 - August 2, 1966, intensity profiles across the spot are obtained at several positions near the disk-center and at each limb. It is found that these profiles show asymmetric features near each limb (increasingly sharp limb-side penumbra and poorly resolved disk-side penumbra) which are similar to those reported in Paper III of this series. It is suggested that these profile asymmetries are the essential feature of the center-limb variations in the appearance of a sunspot which have become known as the Wilson effect. Title: The Mean Temperature Gradient in the Umbra Authors: Wilson, P. R. Bibcode: 1969SoPh....9..391W Altcode: No abstract at ADS Title: Temperature Fluctuations in the Solar Photosphere. II: The Mean Limb-Darkening and the Second Maximum Authors: Wilson, P. R. Bibcode: 1969SoPh....9..303W Altcode: In paper I of this series it was shown that Edmonds' center-limb rms intensity fluctuation data provided strong evidence for the existence of a maximum in the horizontal temperature fluctuation near 250 km (optical depth 0.7). The data also gave a much less reliable indication of a second temperature fluctuation maximum approximately 100 km below this level. Two models, model 1 exhibiting a single temperature fluctuation maximum and model 2 which has two temperature fluctuation maxima, were put forward as worthy of further investigation. Title: On Granulation Models Authors: Wilson, P. R. Bibcode: 1969SoPh....8...20W Altcode: No abstract at ADS Title: Temperature Fluctuations and Convection in the Solar Photosphere Authors: Wilson, P. R. Bibcode: 1969BAAS....1R.296W Altcode: No abstract at ADS Title: Temperature Fluctuations and Convective Modes in the Solar Photosphere Authors: Wilson, P. R. Bibcode: 1969PASA....1..195W Altcode: 1969PASAu...1..195W Recently I have carried out a two-dimensional radiative-transfer analysis of the data given by Edmonds on centrelimb relative rms intensity fluctuations in terms of horizontal temperature fluctuations in the solar photosphere. Edmonds himself attempted an analysis of his data and found a ΔT rms which increased monotonically from about 300°K at unit mean optical depth (i.e. at a given physical depth) to 500°K at optical depth 3. Title: Temperature Fluctuations in the Solar Photosphere Authors: Wilson, P. R. Bibcode: 1969SoPh....6..364W Altcode: The general problem of interpreting granulation data, in particular Edmonds' r.m.s. intensity fluctuation distribution against heliocentric angle θ, is discussed. Title: On the Solution of the Transfer Equation for Continuum Radiation in Inhomogeneous Atmospheres Authors: Wilson, P. R. Bibcode: 1969ApJ...155..715W Altcode: A successful method of solving the equation of radiative transfer for a spectral line formed in a two- dimensionally varying atmosphere is adapted to continuum radiation. An important modification to the method, necessitated by the large horizontal intensity fluctuations found near the boundaries of large solar granules and sunspots, is described Title: The Structure of a Sunspot. IV: A Two-Dimensional Radiative Transfer Analysis of Center-Limb Intensity Profiles Authors: Wilson, P. R. Bibcode: 1968SoPh....5..338W Altcode: Typical intensity profiles across a sunspot at several heliocentric angles are selected from recent observations of the Wilson Effect. In addition, the profile of the mean intensity at the surface of the spot is inferred from these observed profiles. Title: The Structure of a Sunspot. III: Observations of the Wilson Effect Authors: Wilson, P. R.; Cannon, C. J. Bibcode: 1968SoPh....4....3W Altcode: White-light photographs of a fairly regular sunspot have been obtained for all but one day of its passage across the disk. From microphotometer tracings across these photographs, intensity profiles across the spot have been obtained at several heliocentric angles, θ. Title: The Structure of a Sunspot. II: The Magnetohydrodynamics of the Penumbra Authors: Wilson, P. R. Bibcode: 1968SoPh....3..454W Altcode: It is shown that horizontal radial flow in the penumbra is not inconsistent with a vertical component of the penumbral magnetic field. The magnetic amplification produced by the stretching of the magnetic-flux tubes in this velocity field yields a resultant field whose typical inclination to the horizontal is about 20°. It is shown that the thermal convection forces can balance the magnetic forces restraining this flow and that the magnetic energy produced by the stretching of the flux tubes is of the same order of magnitude as the sunspot-energy deficit. Finally, a mechanism is suggested for dispersing this magnetic energy over a large region of the solar surface. Title: Spectral-Line Formation in Inhomogeneous Atmospheres Authors: Wilson, P. R. Bibcode: 1968ApJ...151.1029W Altcode: A method is presented for solving the three-dimensional Eddington approximation of the radiative transfer equation for the spectral line formed by a two-level atom in an atmosphere exhibiting horizontal fluctuations in its physical properties. The method is demonstrated in two simple cases in which (i) the electron temperature is assumed to fluctuate sinusoidally in one direction parallel with the surface, and (ii) similar fluctuations are postulated for the density. Criteria for the stability of the solution and for assessing the accuracy of the results are discussed. In case (i) the opacity is arbitrarily assumed to be constant, since accurate methods are available for the solution of problems of this type, and the accurate and approximate solutions are compared. It is found that, although the source function fluctuations below the surface are comparable in each case, the computed emergent intensity fluctuations are much greater in case (i) than in case (ii) where the opacity fluctuations greatly reduce the effect of the source function fluctuations Title: On the Three-Dimensional Form of the Eddington Approximation Authors: Wilson, P. R. Bibcode: 1968ApJ...151.1019W Altcode: Derivations of the three-dimensional form of the Eddington approximation by Giovanelli and by Unno and Spiegel are examined and shown to be equivalent. They are compared with an earlier form obtained by Chandrasekhar for stellar interiors and shown to be superior near the surface. A third derivation is given which permits one to obtain a mathematically consistent boundary condition for zero incident flux on a plane surface This is similar to Krook's boundary condition for the one-dimensional problem. This derivation also permits the investigation of the accuracy of this approximation in a three-dimen- sional atmosphere. While in general agreement with Unno and Spiegel's claims for the value of the ap- proximation in astrophysical investigation, this paper disagrees with their general conclusions for opti- cally thin atmospheres and discusses several examples where errors are found. Finally, the importance of the three-dimensional approximation is emphasized in applications to atmospheres in which both in- tensity and opacity fluctuate generally Title: The Structure of a Sunspot. I: The Birth and Development of a Sunspot Authors: Wilson, P. R. Bibcode: 1968SoPh....3..243W Altcode: A new model for the structure of a sunspot is put forward. The features of the model are (i) the deep inhibition of convection by magnetic fields, (ii) the formation of a `cool cone' above the region of inhibition of convective transfer by the energy diverted around this region, and (iii) the development of the penumbra by the interaction of strong magnetic field with thermal forces in a region where the opacity prevents the transport of energy by radiation alone. A clear distinction is made between a pore, which results from the inhibition of deep convection across an area considerably greater than that of the pore, and isolated penumbral filaments, which result from strong local fields in the surface regions. Title: A Radiative Transfer Analysis of Wilson Effect Observations. Authors: Wilson, P. R. Bibcode: 1968AJS....73R..84W Altcode: Typical intensity profiles across a sunspot are selected from recent observations of the Wilson effect (previously reported at the Philadelphia meeting of the AAS and to appear in Solar Physics) at the heliocentric angles 00, 580, 710. In addition, at 0=54044 (cos 0 = 1/~) a mean profile (i.e. asymmetries are averaged out) is inferred from observed profiles at 0 = 580 and 520 since, within the limits of the Eddington approximation, the emergent intensity at 0 = cos-1 (1/W3) is equal to the mean intensity at the surface of a plane parallel atmosphere. Starting from these data the transfer equation is solved for the two-dimensional source function distribution within the sunspot for several models of the opacity distribution. It is found that only for an opacity model in which unit optical depth in the umbra occurs at least 700 km below unit optical depth in the mean photosphere do we obtain increases in the limb-side apparent penumbral width and decreases in the apparent umbral diameter which are qualitatively similar to the observed behavior but much smaller in magnitude. However, when some further assumptions are made regarding the release of convected heat in the penumbra near the umbra-penumbra boundary, it is possible to reproduce all the features of the observed profiles. Although no assumption is made about the shape of the umbra below the surface, these solutions clearly show that at a depth of 700 km below unit optical depth in the penumbra the diameter of the umbral region, which is 10 000 km. at the surface, has increased to about 11 500 km. Thus the shape of the umbral region below the surface is part of an inverted cone of semivertical angle about 450 Title: Recent Observations of the Wilson Effect in a Sunspot. Authors: Wilson, P. R. Bibcode: 1968AJS....73R..40W Altcode: White light photographs of a fairly regular sunspot have been obtained for all but one day of its passage across the disk. From microphotometer tracings across these photographs, intensity profiles across the spot have been obtained at several heliocentric angles. Apparent sunspot, umbral, and penumbral widths, have been obtained from these profiles and an examination of these reveals that the well-known Wilson effect (Wilson, A., Phil. Trans. 64, 1, 1774) is more complex than an earlier account by Loughhead and Bray (Australian J. Phys. 11, 177,1958) indicates. In particular, it appears that the apparent increase in the width of the limb-side penumbra as the spot approaches the limb is accompanied by an almost complementary decrease in the apparent umbral diameter. Further, these changes in apparent widths occur only at heliocentric angles greater than about 600. A model for the structure of a sunspot, which qualitatively explains these observations, is suggested. Its particular feature is the conical shape of the umbral region below the solar surface. Unlike most existing models, however, the vertex of the cone is upward so that the abnormally cool region at a given physical depth below the surface covers a large area than at the surface. Title: The Interpretation of Limb Intensity Profiles Authors: Wilson, P. R. Bibcode: 1966ApJ...146..241W Altcode: Limb intensity profiles at the center of Ha (i.e.,intensity distribution against distance perpendicular to the limb and to the line of sight of the observer) are calculated for several simple models of the absorption coefficient at the line center. It is found that an abrupt change of gradient, which appears in many observed profiles, also appears in the calculated profiles, and in the latter this corresponds to a total tangential optical thickness of 2.6 + 0.6 for a wide variety of models. Thus one relation between the parameters of the model may be obtained directly from observations. Title: On the Height of Formation of H-Alpha in the Solar Chromosphere Authors: White, O. R.; Wilson, P. R. Bibcode: 1966ApJ...146..250W Altcode: Several features of Ha limb spectra, including the line emission beyond the continuum limb, are analyzed to yield information about the height of formation of Ha in the solar chromosphere. It is concluded that de Jager's suggested value for the height of unit optical depth at the line center (5000 km) is too high. For the chromospheric features in these spectra the height of formation lies in a range from 1500 km (above a sunspot) to 2700 km. This is consistent with the limb profile analysis which yields a range from 1600 to 3000 km for a hypothetical mean chromosphere. Title: A Note on Deinzer's "theory of Sunspots" Authors: Wilson, P. R. Bibcode: 1966ApJ...143..993W Altcode: No abstract at ADS Title: On Photospheric Mean Values. Authors: Wilson, P. R. Bibcode: 1965ApJ...142.1195W Altcode: It is shown that, because the physical processes in the solar photosphere are non-linear, calculations of mean values of the source function must take into account not only the limb-darkening data but also the magnitude of the observed instensity fluctuations. An approximate iterative method is given for calculating mutually consistent mean values and horizontal fluctuations for the source function A discussion of physical considerations indicates that results obtained for model atmospheres may well apply to the solar atmosphere. Mean values of the source function against depth are obtained that are on the average 8 per cent higher than existing mean values. Mean values at fixed optical depth are also calculated, and the role of the optical depth coordinate in a fluctuating atmosphere is discussed. Title: The Wilson Effect and Sunspot Structure. Authors: Wilson, P. R. Bibcode: 1965ApJ...142..773W Altcode: A method is described for interpreting sunspot-intensity profiles at different heliocentric angles in terms of the attenuation coefficient distribution in the spot. Since the currently available data on sunspot profiles at various angles are qualitative rather than quantitative, the method cannot be applied in detail at present. However, it is hoped that the availability of such a method will encourage observers in the admittedly difficult task of obtaining such data. In order to demonstrate what may be done at present, a simplified form of the method is applied to Loughhead and Bray's photographs of a sunspot as it passed across the disk of the Sun from limb to limb. It is shown that the mean attenuation coefficient in the umbra of this spot is between 1 and 2 orders of magnitude less than in the surrounding photosphere. A difference of 1 order of magnitude is shown to be consistent with the assumption that the umbra of the spot is in radiative equilibrium. Title: Astrophysics: Summary and Discussion. (Book Reviews: Sunspots) Authors: Wilson, P. R. Bibcode: 1965Sci...149...46W Altcode: 1965Sci...149...46B No abstract at ADS Title: Photospheric Structure and RMS Fluctuation Data. Authors: Wilson, P. R. Bibcode: 1964ApJ...140.1148W Altcode: A new model approach to the interpretation of photospheric fluctuation data is developed by calculating the attenuation coefficient corresponding to any given value of the source function using the assumption of LTE and the absorption properties of the negative hydrogen ion and the neutral hydrogen atom. Application of this method to Edmonds' recent rms fluctuation data shows it to be inconsistent with a variety of simple columnar models of photospheric structure. Various photospheric models are examined to determine the features of model photospheric structures which affect the shape of calculated rms fluctuation distributions. Finally a qualitative interpretation is suggested for the appearance of the large-scale granulation pattern toward the limb, as observed by Edmonds. Title: On Methods of Interpreting Solar Fluctuation Data. Authors: Wilson, P. R. Bibcode: 1964ApJ...139..929W Altcode: Existing methods of interpreting solar fluctuation data are compared, and some major difficulties as[ated with the "empirical approach" of , Edmonds, and others are shown to be more readily dled by the model approach. Particular reference is made to the interpretation of the maximum in onds' distribution of rms intensity fluctuations against heliocentric angle. Difficulties in the existmodel method are also mentioned and a new approach is suggested. Title: The Wilson effect and sunspot structure Authors: Wilson, P. R. Bibcode: 1964susp.conf..146W Altcode: No abstract at ADS Title: An Interpretation of Edmonds' Granulation Data. Authors: Wilson, P. R. Bibcode: 1963ApJ...137..606W Altcode: Edmonds' recent data on granulation contrast is interpreted in terms of solutions of the equation of transfer in non-uniform media. It is shown, first, that his results cannot be readily explained in terms of atmospheres which are in radiative equilibrium, or even of non-equilibrium atmospheres in which emission varies only with depth. However, using a simple model representing emission from convective cells, it is possible to reproduce an essential feature of his results and to make some progress toward a photospheric model based on his observations. Title: Three-dimensional solutions of the transfer equation Authors: Wilson, P. R. Bibcode: 1963MNRAS.126..393W Altcode: Methods of solving the equation of radiative transfer are obtained sultable for media whose diffLtsing and absorbing properties vary generally in three dimensions. The existing two-dimensional solutions are then deduced as special cases. As an illustration of the second and more important method, a model problem is solved. Title: Radiative transfer in media exhibiting large horizontal variations Authors: Wilson, P. R. Bibcode: 1962MNRAS.124..383W Altcode: A method of solution of the equation of transfer is established in media where the emitted radiation exhibits large horizontal variations. Attenuation coefficient distributions are calculated for two given intensity distributions, one of which is relevant to the solar photosphere, and the interpretation of data concerning intensity variations across photospheric granules is discussed. The general form of the solution is also obtained for an axially symmetric medium such as an idealized sunspot, and the need for data regarding the directional intensity near a sunspot is stressed. Title: Spectral line profiles in emitting layers Authors: Wilson, P. R. Bibcode: 1962MNRAS.124..231W Altcode: Line emission profiles are calculated for a finite uniform layer in which scattering is completely non-coherent, using a power series solution of the equation of transfer in Eddington's approximate form. The results obtained are in satisfactory agreement with those obtained by Giovanelli using an independent method. The extension of the method to non-uniform media is also discussed. Title: The application of the equation of transfer to the interpretation of solar granulation Authors: Wilson, P. R. Bibcode: 1962MNRAS.123..287W Altcode: of The interpretation of solar granulation is discussed in terms of the equation radiative transfer. This is solved for non-uniform two-dimensional media in the absence of radiative equilibrium. The solution is obtained as a functional relation for the attenuation coefficient K in terms of the total intensity J and the emission function E, and estimates of some of the parameters of simple models are made by fitting mean values of J and K to tabulated values of Sykes (4) and Plaskett (3). Typical data concerning the horizontal variations in intensity at normal emergence are discussed in terms of the remaining parameters of the models, and the need is indicated for similar information at other angles of emergence. Title: On Solving the Equation of Radiative Transfer for Conservative Non-uniform Media Authors: Wilson, P. R. Bibcode: 1961AuJPh..14...57W Altcode: No abstract at ADS Title: The Constant Flux Problem in Non-uniform Exponential Media Authors: Wilson, P. R. Bibcode: 1960AuJPh..13..461W Altcode: No abstract at ADS