magnetic field with large excursions in latitude

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1 JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 102, NO. All, PAGES 24,175-24,181, NOVEMBER 1, 1997 Direct observational evidence for a heliospheric magnetic field with large excursions in latitude T. H. Zurbuchen, N. A. Schwadron, and L. A. Fisk Department of Atmospheric, Oceanic and Space Sciences, University of Michigan, Ann Arbor Abstract. Fisk has pointed out that the heliospheric magnetic field in fast solar wind, that is, at the higher heliographic latitudes, may undergo large excursions in heliographic latitudes, and thus that the field will deviate from the expected Archimedes spiral pattern. These excursions result from the interplay between the nonradial expansion of the solar wind in rigidly rotating coronal holes, and the differential rotation of the footpoints of the field lines anchored in the photosphere. In this paper magnetic field data from the Ulysses spacecraft in the southern solar hemisphere are analyzed and show that the field exhibits the systematic variations with latitude and longitude expected from the field configuration predicted by Fisk. We also show that variations about the predicted field should exhibit a unique periodicity of about 20 days at high latitudes and that the predicted periodicity is in fact observed. The observation conformation of the predicted field provides support for several other applications of this theory. 1. Introduction Fisk [1996] has pointed out that the interplay between the differential rotation of the photosphere of the Sun, and the nonradial expansion of the solar wind in the solar corona, can result in large excursions of the heliospheric magnetic field in latitude. This theory has important implications for the transport of energetic particles in the heliosphere and can account for the observations that energetic particles accelerated in or modulated by corotating interaction regions, which occur near the equatorial plane of the Sun, are seen at high heliographic latitudes [Sirenerr et al., 1995; McKibben et al., 1995]. The theory also has implications for the modulation of galactic cosmic rays [Fisk et al., 1996], and for the behavior of the magnetic field in the solar corona and the origin of the slow solar wind [Zurbuchen et al., 1996b]. In this paper we explore whether there is evidence in the observed heliospheric magnetic field for the field variations predicted by Fisk [1996]. The predicted field variations at any one location are quite small and are easily masked by the large Alfvenic fluctuations seen particularly at high latitudes [e.g., Smith et al., 1995], as well as by larger-scale variations in the solar corona and in th flow of the solar wind. Nonetheless, with careful analysis of the observed magnetic field data from the southern solar pass of Ulysses, when the coronal hole pattern of the Sun was quite stable, it is possible to find evidence that the predicted field configuration is present. This small systematic variation of the Copyright 1997 by the American Geophysical Union. Paper number 97JA /97/97JA ,175 predicted field over large ranges of latitude and radial distance is responsible for important effects in the heliosphere. In the next section we review the theory of œisk [1996] and point out tests which can be compared with observations. The analysis of the observed field is presented in section 3, and concluding remarks are provided in section Theory The basic concepts in the Fisk [1996] theory are straightforward. Field lines are assumed to be anchored in the photosphere, which differentially rotates. The field lines undergo a subsequent nonradial expansion in coronal holes, which rotate more rigidly, at near the equatorial rotation rate, and in general have an axis which is offset from the rotation axis of the Sun. A given field line, then, as its footpoint in the photosphere differentially rotates, will move through the coronal hole, experiencing different nonradial expansions. Due to this effect, the heliospheric footpoints on the solar wind source surface, where the solar wind flows radially outward into the heliosphere, experience large excursions in latitude and longitude. A given field line in the heliosphere will thus experience large excursions, particularly in heliographic latitude, and can provide direct magnetic connection between high and low latitudes. The basic geometry is sketched in Figure 1 for the northern hemisphere. The magnetic field, originating in the high-latitude photosphere, nonradially expands to the source surface, from where it is carried out by the solar wind. The magnetic field configuration is as- sumed to be centered on the axis M which is offset from

2 24,176 ZURBUCHEN ET AL- HELIOSPHERIC MAGNETIC FIELD Figure 1. A schematic illustration of the expansion of magnetic field lines from a polar coronal hole according to the model by Fisk [1996]. The differential rotation of photospheric magnetic flux elements is projected onto the solar wind source surface. For details refer to text. the rotation axis f by c. Consider the field line which originates from the heliographic pole. Due to the nonradial expansion it will be bent back and penetrates the source surface as the field line p which is offset from the rotation axis f by the angle t3. The actual size of t3 is determined by two parameters, namely, the angle c and the amount of overexpansion from photosphere to source surface [see Fisk, 1996]. All other field lines differentially rotate about the polar field line and nonradially expand onto the rigidly rotating source surface. Because of this differential rotation of the photosphere, the position of the field lines changes with time, and subsequently, the differential rotation in the photosphere is projected onto a circular pattern on the source surface. Also shown in Figure I are field lines which intersect the coronal hole boundary. Note that this boundary is projected onto the source surface and is represented by the solid/dashed line near the equator. The interplay of the footpoints with the boundary of the coronal hole is an interesting issue, which has implications for the origin of the slow solar wind, as is discussed by Zurbuchen et al. [1996b] and $chwadron et al. [1997]. Sketched in Figure 2 is the solar wind source surface, as seen from the perspective of the south solar pole, since we will consider Ulysses observations in the southern solar hemisphere. Again, the trajectories of the footpoints of the heliospheric magnetic field lines execute circular patterns on the source surface about the location where the field line from the southern heliographic pole penetrates the surface (at polar angle /3). From this perspective the direction of motion of the footpoints is of course opposite to those shown in Figure 1. The footpoint motions and their implications can readily be discussed by considering their projection onto the equatorial plane, as is shown in Figure 3. Again the projection is from the south solar pole. The trajectories of the footpoints on the solar wind source surface in the southern hemisphere are assumed to be circles about the point where the field line from the southern heliographic pole intersects the source surface. As can be seen by comparing Figure 3 with Figure 3 in Fisk [1996], footpoint trajectories which are circles on the source surface provide a reasonable approximation to the trajectories that result if the expansion of the magnetic field in the polar coronal hole is from a dipole near the solar surface to uniform field strength on the source surface. First we note that with this circular approximation to the footpoint trajectories an analytic expression for the magnetic field in the heliosphere can be readily determined. Assuming that the solar wind flows radially outwards with constant speed V, we find that the time stationary magnetic field lies in a frame which corrects both for the equatorial rotation of the Sun with rate and the differential rotation of the footpoints with rate cv. Two rotational frame transformations must be performed to obtain the time-stationary frame: one transformation about the Sun's rotation axis and one transformation about the field line iv. In the corotating reference frame, the coordinates of the streamline and s are related as follows: Figure 2. The southern hemisphere of the solar wind source surface drawn in the frame corotating with the equatorial rotation rate. The trajectories of the heliospheric magnetic field lines execute circular patterns moving in heliographic latitude and longitude.

3 ZURBUCHEN ET AL.: HELIOSPHERIC MAGNETIC FIELD 24,177 o The first test of the footpoint motion theory is then to look for this field configuration in the observed heliospheric magnetic field. In particular, the observed field should include a systematic component Bo, and an additional component of B, which (1) vary sinusoldally with longitude, (2) are 900 out of phase (sin versus cos), (3) have phases which are determined by the longitude of the solar magnetic pole, and (4) have amplitudes which depend on the differential rotation 180 ø ' 0 ø rate and the latitude fi, at which the polar magnetic field line crosses the source surface. Second, we note that although the footpoint trajectories are shown in Figures 2 and 3 as being featureless, in practice they should contain variations in field magnitude and distortions in direction, which move along the source surface at the speed of the footpoints, that is, at essentially the differential rotation rate. These variations and distortions will in turn generate features 270 ø in the hellospheric magnetic field seen by Ulysses. Con- Figure 3. High-latitude footpointrajectories pro- sider, then, the periodicity with which Ulysses observes jected onto the equator plane as seen in the frame coro- these features. In the corotating frame of Figure 3, rating with the equatorial rotation rate. Ulysses, as- Ulysses will execute counterclockwise motion along the sumed to be at a constant latitude of 0-20 ø, can ob- arc marked U, which for times of order a solar rotaserve features on this footpoint trajectory at the starred tion occur at essentially a single latitude. The feature location with a recurrence time of 20 days. For low lat- will move at the differential rotation rate along the arc itudes, footpointrajectories end when they encounter marked F. Thus if Ulysses observes a feature at the the coronal hole boundary, given by the solid black line. starred location in the bottom half of Figure 3, it can observe that same feature at the starred location in the top half of the figure, provided that the transit time of sin sin( s + f rs/v - c)o) the feature along its footpointrajectory (F) and the co sin cos sin 0s 8s transit time of Ulysses in its orbit (U) are the same. cos( s + f rs / V - o) + In Figure 4 we plot the transit time of a feature which moves along a footpoint trajectory between its V cos fi sin(, + fir /V - o) V' two points of intersection with the Ulysses trajectory. The transit time is plotted as a function of the az- We insist that imuthal angle 0 where the initial intersection between O0 /Or the footpointrajectory and the Ulysses trajectory oc- (2) Bo /B - sin O OqSs /Ors and V. B = 0 which leads to the following result Br Be B = 2 2 Boro = V ,co sin fi sin( 5 + f r/v - q3o) Boro 2 = Vr [co(cos fi sin 0 + sin fi cos 0 cos( 5 + f r/v - q3o)) - f sin 0] (3) 24 ', 22f ' ', Here B0 is the radial field strength on the source surface at r = r0. The angle b = b0 occurs in the plane 6 I defined by the rotation axis and the magnetic axis of the 4 Sun. is the polar angle where the field line p, from the heliographic pole, crosses the source surface. The angle fi can be calculated in the model by Fisk [1996] for a % [deg] given orientation c of the magnetic axis M and a given Figure 4. The transit time of Ulysses in its orbit and nonradial expansion. For the configuration discussed by the transit time of a feature on the source surface. The Fisk [1996], fi is of the order 30 ø. Note in comparison curves cross when Ulysses can observe a feature twice. that the standard Parker spiral for the heliospheric mag- In an autocorrelation analysis, such observations will netic field yields Bo = 0 and B = -Bo(f ro2/vr)sin 0. result in a peak around 20 days. 20."o

4 24,178 ZURBUCHEN ET AL.: HELIOSPHERIC MAGNETIC FIELD curs, as shown in Figure 3. The footpoint trajectories are assumed to be circles on the source surface and the motion along these trajectories is at the differential rotation rate, w radians per day. This value for w is obtained by extrapolating the differential rate observed by Shodgrass [1983] to high latitudes. We take the differential rotation rate as constant even though in this model the field lines observed by Ulysses at a given latitude will originate from a range of latitudes near the solar pole. Ulysses is assumed to be at 70 o south latitude, and the angle /3 is taken to be 30 ø, a value which provides a good fit to the observations of energetic particles at high latitudes [Fisk, 1996]. Where the two curves for the transit times intersect, Ulysses can observe the feature twice. The features in the field may be strong and more significanthan the underlying systematic field. We therefore expect a clear periodicity at high latitudes of approximately 20 days. Such a periodicity is unique to the footpoint motion theory; in a standard Parker theory for the hellospheric magnetic field, only periodicities of 34 days, the high-latitude solar rotation period, or perhaps 26 days for the coronal hole boundary, are expected. As can be seen in Figure 3, at lower latitudes (corre- sponding to polar angles greater than/3) the trajectories of the footpoints and the orbit of Ulysses are more coaligned. Thus features in the magnetic field of reasonably large spatial extent will be seen to move in the same direction as Ulysses and will be observed with the usual periodicity of approximately 34 days. Features moving along footpoint trajectories at low latitudes can produce a response in the magnetic field at higher lati- tudes. The solar wind source surface occurs where lat- eral stresses in the magnetic field become small, and the flow of the solar wind becomes radial. Therefore a localized increase in the field strength at low latitudes or a change in field direction can produce a response over ro a e rb r o.oo O. lo røb -o.35 " rbr i ' i l -o.4o "...,. Figure 6. Magnetic field measurements at -620 < 0 < -350 using the same approach as Figure 5. The shape of the coronal hole boundary clearly influences the magnetic field measured by Ulysses. Notice the clear signature around 1200, which can be related to observed photospheric structures. a broader region of the source surface in the process of reducing the lateral stresses. We expect that a 34 day periodicity may also be seen at higher latitudes, resulting from lower latitude features, although we expect it will be less significant than the 20 day periodicity. Finally, we note that the clearest signal of the field resulting from footpoint motions should occur at high lat- itudes. The simple footpoint trajectories shown in Figure 3 should be distorted near the coronal hole boundaries, which may be ragged and may have complicated extensions. Thus in the mid-latitude to low-latitude range we expect a much more complicated field configuration than the simple field given in (3), and we expect less pronounced periodicities. Hence the signal produced by the footpoint motions should be difficult to extract from the noise produced by local turbulence and solar wind variations. ro 0.00 o.o oo Figure 5. Magnetic field measurements at 0 ( -65 o binned by the Cartington longitude of their origin on the source surface. The error bars denote standard deviations from the mean in each bin. The measurements are compared with the field model described by (3), with two of the model parameters being determined by remote sensing observations of the southern coronal hole during the measurement period of Ulysses. Local deviations are probably caused by the structure of the coronal hole boundary at midlatitudes. 3. Observations We consider the heliospheric magnetic field observed by Ulysses in the southern solar hemisphere. The southern polar coronal hole was quite stable during the Ulysses pass over the south pole, and the heliospheric field in this hemisphere should reveal the effects of footpoint motions. In contrast, the coronal hole at the north pole was seen by Ulysses to evolve in time [e.g., Roelof et al., 1997], and the heliospheric field in this hemisphere is less likely to reveal easily the small systematic field component due to footpoint motions. The magnetic field data from Ulysses has been provided for general use by the Principal Investigator, A. Balogh [Balogh et al., 1992], through the Coordinated Heliospheric Observations Data of the NASA National Space Science Data Center. The data are provided in 1 min averages. Consider first the systematic variations in the magnetic field. Again, these systematic variations are small and are easily masked by the large-amplitude Alfven waves which occur particularly at high latitudes [e.g., Smith et al., 1995], as well as by fluctuations introduced

5 ZURBUCttEN ET AL.: tteliospheric MAGNETIC FIELD 24,179 by variations in the solar corona. It is important, then, to use an averaging technique which is designed to reveal the small underlying systematic variations; conversely, there are averaging techniques that are appropriate in other circumstances which will not be success- cated field pattern. ful. We use the following steps: 1. An exponential filter, exp(-t/r), is applied to the data with day, which will suppress high-frequency variations (e.g., Afvenic fluctuations) larger than 1/r and will not introduce any artificial periodicity. 2. The Carrington longitude from which the field originates on the source surface is determined using the actual solar wind velocity. 3. The magnetic field data from different solar rotations are binned by their Carrington longitude on the source surface. 4. The data are smoothed by performing a running average over in Carrington longitude. This last step reduces the impact of the fluctuations on the source surface that will give rise to the periodicities. We have also used boxcar averaging and other binning procedures without substantively changing the result. The results are shown in Figure 5 for latitudes above 66øS measured in 1994 from day 150 to 320. This latitude range was chosen because higher latitudes were significantly free of large-scale magnetic features as observed in midlatitude regions. The data are plotted as Boro/rBr with r0 denoting 1 AU, since from (3) we expect this ratio not to be a strong function of radial distance. The error bars are consistent with the standard deviation of the mean, = rr/x/, in each Carrington longitude bin. Here rr is the standard deviation determined from the filtered data points in each bin, and no is the number of statistically independent data points in each bin. However, in this range we do not expect that the trajectories of the footpoints will be the simple curves shown in Figure 3, but rather the influence of the distorted coronal hole boundaries will produce a more compli- The pattern is particularly distorted around Carrington longitude bc = 1000 to 160 ø. This very strong and highly localized deviation from our simple model should be rooted in a photospheric distortion and therefore observable in coronal hole polarity maps. Indeed, Wang e! al. [1996] report the occurrence of a localized excursion of the coronal hole boundary occurring in this latitude range. The effect of this photospheric structure can also be observed in variations in the solar wind speed [e.g., Zurbuchen et al., 1996a]. Consider next the periodicity in the magnetic field. Here we apply the following averaging procedure: 1. An exponential filter, exp(-t/,'), is applied to the data with r = 0.5 day, which will suppress variations with frequencies larger than 1/r and will not introduce any artificial periodicity. 2. A standard autocorrelation estimate is applied to the filtered data. Errors are estimated by assuming the time series is a Markov process with Gaussian statistics. A detailed description of both the filtering and the auto- correlation procedure is given by Priestley [1981]. Note that for the error analysis, time stationarity and Gaussian statistics must be assumed. In the time series of interest both assumptions are not strictly valid. Using artificial data and model calculations it can be shown that in our case the procedure as described above can be applied. Note also that a smaller,' value (0.5 day) has been used for the autocorrelation analysis than was used in determining the average field configuration (r = Also shown in Figure 5 are the predicted field compo- ' I '"! I I I I I nents from (3). The differential rotation rate is taken to be tad per day and/3 = 30 ø. The solar magnetic 0.8 pole, or equivalently, the b = b0 plane, is oriented such that the hellospheric current sheet has the observed tilt 15ø; the required Carrington longitude of the magnetic pole is 1000 [see, e.g., Phillips et al., 1995, Figure 3]. Note that all free parameters of the model, except for fi, are directly observed with methods other than the observation of the heliospheric magnetic field '" '. - The angle/3 has to be determined using the observed 0.8 fl 0.6 tilt of the current sheet and the overexpansion of the magnetic field from the photosphere to the source sur- 0.4 Bp face. The predicted field is then subjected to the same 0.2 averaging technique as the data. 0!?!.:..::.::: ii... i :.:::::-i!i '::.:: ::. ::.'.,,.'.:ii:' ::::::::::::::::::::: : :: "g:::-: i.i: ::-:::2 i :ø - ": g::'::::' qn,..'.,:?:::-! i '",:' ' There are, of course, deviations between the observed and predicted field. Nonetheless, the amplitudes and particularly the phases of the predicted field components, which are specified by (3) and the orientation of the solar magnetic pole, are in agreement with the time [day] observations. Figure 7. The autocorrelation functions for Bo and In Figure 6 the procedure is repeated for midlati- B for southern latitudes higher than 65 ø. The dashed tudes, from -350 to -620 measured from day 200 in area denotes a l rr estimation error of the autocorrela until day 122 in Here the fit between the tion procedure. Maxima are observed around t = 20 predictions of (3) and observations is less satisfactory. days and t = 34 days.

6 ß 24,180 ZURBUCHEN ET AL.' HELIOSPHERIC MAGNETIC FIELD 1 day). In order not to rule out the recurrent structures which actually cause the autocorrelation signal, we must take a value for ' smaller than I day. The results are shown in Figure 7 in the top panel for the polar component of the field B. There is a statistically significant (more than 2er) result at a periodicity of about 20 days, and a lesser peak at about 34 days, as expected. In the bottom panel of Figure 7, the results are shown for the azimuthal component of the field B. Here the peaks are less significant, a t approximately 18 and 36 days. The azimuthal field due to footpoint motions'is superimposed on the normal Parker field, which varies in latitude and radius, and thus, as observed by Ulysses, may introduce its own periodicities. The po- lar field, in contrast, is unique to the footpoint motion theory and may provide a clearer test of the expected periodicities. The autocorrelation procedure can be repeated for midlatitudes. It is shown in Figure 8 and does provide clear periodicities, as expected. It shows a significant maximum in the autocorrelation function around 25 days, the equatorial solar rotation period. In addition, there is a maximum between 30 and 40 days as expected. Although the coronal hole boundary and the connection of the field lines from low to high latitudes for energetic particles seemed to be more or less constant, small changes clearly occur during this timeperiod. Due to the drift of newly emerging flux in the photosphere the coronal hole boundary can change in time (see, e.g., discussion by Zurbuchcn ½t al. [1996a]). 4. Concluding Remarks The motion of footpoints across the solar wind source surface, as proposed by Fisk [1996], and the resulting hellospheric magnetic field, have many important consequences for the solar corona and the hellosphere. The footpoint motions imply a considerable transport of magnetic flux in the solar corona, which has implications for the dynamics of the corona and the origin of the solar wind [Zurbuchen et al., 1996b]. The large I I I I I time [days] Figure 8. The autocorrelation functions for Bs for the midlatitude range from-62 O -35. Maxima are observed around t - 25 days and t = 35 days. For a detailed description refer to text. latitude excursions of the heliospheric magnetic field will have a profound effect on the modulation of galactic cosmic rays [Fisk et al., 1996]. It is thus important to determine whether this footpoint motion theory is in fact correct. Although the effects of the systematic variations in the magnetic field resulting from footpoint motions are large, at any one location the field which causes these effects is quite small and difficult to discern among the large-amplitude fluctuations in the hellospheric field. Nonetheless, with careful analysis we have provided evidence that the principal tests of the predicted field, the amplitude and phases of the field components, and the expected periodicities, are in agreement with the observations particularly at highhellographic latitudes. At lower latitudes the agreement is not as significant. However, here we expect that the simple form of the theory, in which the footpoint trajectories have circular patterns, does not apply due to the influence of the ragged boundaries of coronal holes. There is no reason to believe that if the theory is valid at high latitudes, it will not also be valid at midlatitudes. The footpointrajectories at midlatitudes, however, will be more complex and difficult to discern in the observa- tions, but understanding them should reveal interesting aspects of the dynamics and field configurations in the corona. Acknowledgments. We re grateful to R. Jokipii nd N. R. Sheeley for helpful discussions. T.H.Z. was supported, in p rt, by the Swiss N tional Science Foundation. The work w s supported, in p.rt, by NASA/JPL contracts nd NASA gr nt NAG The editor th nks V. J. Pizzo nd D. Winterh lter for their ssist nce in evaluating this p per. References Balogh A., T. J. Beek, R. J. Forsyth, P. C. Hedgecock, R. J. M rqued nt, E. J. Smith, D. J. Southwood, and B. T. Tsurut ni, The m gnetic field investigation on the Ulysses mission: Instrumentation nd preliminary scientific esults, Astro. Astrophys. Suppl. Set., 92, 221, Fisk, L. A., Motion of the footpoints oi hellospheric m g- netic field lines t the Sun: Implications for recurrent energetic p rticle events t high heliogr phic l titudes, J. Geophys. Res., 101, 15547, Fisk, L. A., N. A. Schw dron, nd T. H. Zurbuchen, The consequences of heliospheric m gnetic field with l rge excursions in l titude for the transport of energetic p rticles ( bstr ct), Eos Tra s. AGU, 77 (46), F ll Meet. Suppl., F571, McKibben, R. B., J. J. Cormell, C. Lop te, J. A. Simpson, nd M. Zh ng, Cosmic r y modulation in the 3-D heliosphere, Space Sci. Rev., 7œ, 367, Phillips, J. L., B. E. Goldstein, J. T. Gosling, C. M. H mmond, J. T. Hoeksem, nd D. J. McCom s, Sources of shocks nd compressions in the high-l titude solar wind: Ulysses, Geophys. Res. Left., œœ, 3305, Priestley, M. B., Spectral A alysis a d Time Series, 106 pp., Academic Press, S n Diego, CMif., Roelof, E. C., G. M. Simnett, R. B. Decker, L. J. L nzerotti, C. G. M clenn n, T. P. Armstrong, nd R. E. Gold, Reappearance of recurrent low-energy p rticle events t Ulysses/HI-SCALE in the northern heliosphere, J. Geophys. Res., 10œ, 251, 1997.

7 ZURBUCHEN ET AL.: HELIOSPHERIC MAGNETIC FIELD 24,181 Schwadron, N. A., L. A. Fisk, and T. H. Zurbuchen, On the slow solar wind, A strophys. J., in press, Simnett, G. M., K. A. Sayle, S. J. Tappin, and E. C. Rodof, Corotating particl enhancements out of the ecliptic plane, Space Sci. Rev., 7œ, 327, Smith, E. J., M. Neugebauer, A. Balogh, S. J. Bame, R. P. Lepping, and B. T. Tsurutani, Space Sci. Rev., 73, 165, in Solar Wind Eight, Conf. Proc. 38œ, edited by D. Winterhalter et al., p.273, AIP Press, College Park, Md.,1996a. Zurbuchen, T. H., L. A. Fisk, and G. Gloeckler, On the slow solar wind (abstract), Eos Trans. A GU, 77 (46), Fall Meet. Suppl., F563, 1996b L. A. Fisk, N. A. Schwadron, and T. H. Zurbuchen, Space Physics Research Laboratory, University of Michi- Snodgrass, H. B., Magnetic rotation of the solar photo- gan, 2455 Hayward Street, Ann Arbor, MI sphere, Astrophys. J., œ70, 288, ( lafisk@umich.edu; nathanas@engin.umich.edu; Wang, Y.-M., S. H. Hawley, and N. R. Sheeley, The magnetic thomasz@engin.umich.edu) nature of coronal holes, Science, œ71,464, Zurbuchen, T. H., P. Bochsler, and R. von Steiger, Coronal (Received March 20, 1997; revised July 18, 1997; hole differential rotation rate observed with SWICS/Ulysses, accepted July 29, 1997.)