The Effect of the Heliospheric Current Sheet on Cosmic Ray Intensities at Solar Maximum' Two Alternative Hypotheses

Transcription

1 JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 91, NO. A3, PAGES , MARCH 1, 1986 The Effect of the Heliospheric Current Sheet on Cosmic Ray Intensities at Solar Maximum' Two Alternative Hypotheses BARRY T. THOMAS Computer Science Department, University of Bristol, Bristol, United Kingdom BRUCE E. GOLDSTEIN AND EDWARD J. SMITH Jet Propulsion Laboratory, California Institute of Technology, Pasadena There is now a growing awareness that solar cycle related changes in the large-scale structure of the heliosphericurrent sheet may play an important role in the modulation of galactic cosmic rays. To date, attention'has been focused on. the configuration of the current sheet at times near solar minimum when the current sheet structure is relatively simple. Previous analyses have explored the effect on cosmic ray intensities of a single current sheet which is tilted with respect to the heliographic equator under the assumption that the tilt of the current sheet is a minimum at solar minimum and increases as solar maximum approaches. This paper attempts to extend the previous analyses into the period near solar maximum. Two alternative hypotheses are explored: (1) that the tilt of the current sheet continues to increase as solar maximum approaches, finally becoming vertical and overturning, and (2) that the single sheet structure breaks down near solar maximum and the sun at this time sheds the poloidal flux of the previous cycle and develops a new field structure of the opposite polarity. It is found that both hypotheses lead to variations in cosmic ray intensity comparable to those actually observed over the solar cycle. INTRODUCTION It is now well established thathe heliospheric current sheet is not coincident with the heliographic equator but, in general, is tilted. This leads 'to a large-scale warp in the current sheet structure as it propagates away from the sun. It has been further demonstrated that there is a solar cycle variation in the angle of tilt and, therefore, in the latitudinal extent of the durrent sheet warp [Svalgaard and Wilcox, 1974; Thomas and Smith, 1981]. Recent analyses have demonstrated that the effect of this variable warp in the current sheet is to produce a solar cycle variation in the intensity of galacticosmic rays reaching 1 AU [Kota, 1979; Jokipii and Thomas, 1981; Kota and Jokipii, 1983]. These papers assumed that the latitudinal extent of the current sheet warp varied from approximately 10 ø near solar minimum up to 30 ø near solar maximum. The period very near solar maximum has not been studied, as it has been assumed that the simple single current sheet hypothesis is no longer valid during the period of field reversal. In this paper we attempt to extend the range of the previous analyses by considering two alternative possibilities for the current sheet structure very near solar maximum. The first hypothesis is a straightforward extrapolation of the previous studies in which it is assumed that the tilt of the current sheet continues to increase as solar maximum approaches, at which time it becomes vertical and overturns. This is the view propounded by a number of authors [Saito, 1975; Saito et al., 1977; Kaburaki and Yoshii, 1979; Swinson et al., 1981]. At first sight this view seems oversimplistic and is inconsistent with dynamo theories on the mechanism by which the solar field reverses [e.g., Babcock, 1961; Stix, 1976]. However, recent calculations of the probable current sheet configuration by potential field calculations do suggest that the effective current sheet tilt becomes very large near solar maximum [Hoeksema et al., 1983]. Furthermore, a recent Copyright 1986 by the American Geophysical Union. Paper number 5A /86/005 A study using the results of Hoeksema et al. indicates that the simple picture of a single highly inclined current sheet at this time is capable of organizing the observed cosmic ray intensities quite well [Smith and Thomas, this issue]. The second hypothesis is more complicated. The solar corona at solar maximum is very complex and suggests the existence of not one but Several heliospheric current sheets. If this is the case, then the simple picture of northern hemisphere flux migrating southward and vice versa is implausible. In this case we would be forced to believe that the highly inclined current sheets emerging from the potential field models are an artifact of the method used, and, indeed, the results reported by Hoeksema et al. [1983] often reveal the presence of closed unipolar regions, presumably separated from the background field by additional current sheets. The model we present here is more consistent with dynamo models for the solar field reversal and is based on the hypothesis that the sun reverses its polarity by shedding the poloidal field of the previous cycle and developing a new field of the opposite polarity. The model is one in which unipolar magnetic field regions representing the emergence of new solar flux develop on the sun near the polar regions. These regions are assumed to spread over the sun as solar maximum approaches, displacing the flux of the previous solar cycle. There is direct observational evidence for the existence of closed regions of anomalous polarity on the sun I-Hundhausen et al., 1981; Newkirk and Fisk, 1985]. That the sun is capable of producing sufficient new flux to entirely displace the field of the previous cycle is supported by the observations of Howard and Labonte [1981]. From a study of solar surface fields they show that large quantities of apparently new flux emerge in association with active regions and that the flux can spread rapidly to surrounding areas. A process in which the sun completely sheds the flux of the previous cycle will imply the existence of closed field lines in the inner heliosphere, for the periods very near solar maximum, which will greatly impede cosmic ray access to the inner heliosphere. The method used to investigate the effect of these alternative hypotheses is a three-dimensional numerical solution of the cosmic ray transport equation in simplified models of the

2 2890 THOMAS ET AL.'CosMIC RAYS interplanetary magnetic field. For the overturning current sheet we investigate the effect on cosmic ray intensities of a series of steady state heliospheres with a planar current sheet with different inclinations. The results are then used to approximate the intensity variations in a heliosphere in which the tilt of the current sheet varies continuously throughout the solar cycle. For the second hypothesis we use a highly simplified model for the heliosphere containing a region of closed field lines representing the newly emerged magnetic flux. THE OVERTURNING CURRENT SHEET In this section we assume that the increase in tilt angle of the heliospheric current sheet can be simply extended through solar maximum. For simplicity we assume that the tilt is zero (6) (i.e., parallel to the solar equator) at solar minimum, increases steadily toward 90 ø at solar maximum, and then diminishes where V. is here called the adiabatic focusing velocity, given by steadily toward zero again at the next minimum, but with the polarity now reversed. Above and below the current sheet the field is assumed to be a pure Archimedian spiral given by B r = AB o/r 2 B, = ABo sin O/(rV, ) (1) Bo= 0 where r, 0, b are spherical polar coordinates aligned with the sun's rotation axis, f the angular rotation velocity of the sun, and V, the solar wind velocity. A = _+ 1, which determines the field polarity. In one cycle, A is positive above the current sheet and negative below, and in the next cycle the polarities are reversed. The current sheet surface in the outflowing solar wind is given by cot 0 = -tan sin ( b - b0 + rfl/v, ) (2) where cz is the tilt angle of the current sheet and bo is a constant. A pictorial representation of this surface for intermediate tilt angle can be found in the work by Jokipii and Thomas [1981]. The tilt angle in this analysis is assumed to vary continuously between zero at solar minimum and 90 ø at solar maximum. This is given by = rrt/(11 years) (3) where t is the time measured from solar minimum. The technique used to calculated the cosmic ray intensity at 1 AU resulting from the above field structures is similar to the approximate solution of the cosmic ray transport equation given by Jokipii and Thomas [1981]. In this analysis, however, a further term is retained, leading to a considerable increase in accuracy. The basic transport equation for cosmic rays is the Fokker- Planck equation [e.g., Jokipii and Parker, 1970], given here in terms of the phase space density f(r, P)= j/p2, where j is the intensity and P the particle momentum' c f = O= c ( K o _ )-(V,., + Va.,) c f V. V, p c f (4) where K j is the symmetric part of the diffusion tensor (offdiagonal terms are zero). Va is the pitch angle averaged drift velocity which represents the divergence of the antisymmetric part of the diffusion tensor and can be determined from the field configuration by Pvc = v x (5) where v and q are the particle velocity and charge [e.g., lsenberg and dokipii, 1979]. In the work by dokipii and Thomas [1981] the transport equation was solved using the approximation that K 0 could be neglected and the equation reduced to first order, a technique first shown to be useful by dokipii and Kopriva [1979]. In this paper, however, a further first-order term is retained, leading to a significant increase in accuracy. Since the offdiagonal terms of the diffusion tensor are zero, the equation can be simply expanded as c - c f = 0 = K,, c9- c 2f 2 + (V.., - V,., - Va.,) - x c f + V. 3 V, e OP c f = (7) The fact that this term does indeed represent the effects of adiabatic focusing can be seen by considering the extreme cases. Focusing physically represents the tendency for a particle to maintain its magnetic moment between collisions and therefore retain part of the collimation effect that would exist in the absence of scattering. If the degree of scattering is large, the particles no longer see the mean field structure, and the focusing term should vanish. This is the situation in which K ii - K _, and the divergence of K j does indeed vanish under this assumption. If, on the other hand, Kii >> K _, then the particle is well aware of the field topology and is partially collimated with the field between collisions (to an extent which is regulated by the precise value of Kii ). Under the assumption that Kz can be neglected, the equation for V reduces to Vf = gliv. (KiiB/B) (8) where gll is a unit vector aligned with the field. If the further assumption is made that K ii is everywhere constant, the equation reduces to the familiar expression for the adiabatic focusing velocity developed by Earl [1976], i.e., Vf -- dllkii Vo (B/B) = dll KII (9) where 1/L---(1/B)aB/&, z being the distance measured along the field line. In this paper we do not make this last assumption but allow K II to increase in the outer heliosphere so that the unphysical problem of the mean free path becoming smaller than a particle gyroradius cannot arise. For simplicity, K ii is assumed to scale inversely with the field magnitude so that the mean free path and gyroradius have a fixed ratio. Under this assumption the focusing velocity reduces to Vf -'" 11KoBoV. (B/B 2) (10) where Bo and K o are the field magnitude and parallel diffusion coefficient at 1 AU. In order to solve (6) three dimensionally we now make the assumption that the second-order term can be neglected. The neglected term represents primarily the one-dimensional random walk of particles along the field line. Of course, the effects of perpendicular scattering are now totally ignored in this analysis. The remaining terms now reduce to of v. L of (V,., + l/a.,- V.,,) c9x, 3 = 0 (11)

3 THOMAS ET AL.' COSMIC RAYS 2891 We see that adiabatic focusing appears in the equation as a convective term in a similar way to drifts and solar wind convection. As has been pointed out in the earlier papers [Jokipii and Kopriva, 1979; Jokipii and Thomas, 1981], an equation of this form states that f is unchanged along its characteristics, given by dx i = (V, i + Va, i - Vf,i) dt dp P (V'Vw) 3 dt (12) where dt is a time increment. The characteristics are clearly the particle trajectories in phase space (which now retains the previously neglected term representing adiabatic focusing). This corresponds to the statement that within our approximations the pitch angle averaged motion of the particles obeys Liouville's theorem (i.e., the phase space density is conserved along the mean particle trajectory). The reason that the focusing velocity appears in the equation as a negative convection term is related to the fact that unlike drifts and solar wind convection, focusing is reversible in time. This can be seen more clearly in the extreme case of no scattering. In this case a distribution of particles released at 1 ^U will rapidly become collimated with the field and move outward through the hellosphere at a large fraction of the particle velocity. Conversely, particles released in the outer hellosphere will only reach 1 ^U if they start out moving almost parallel with the field (otherwise they will mirror long before they reach 1 AU), and therefore those that reach the inner hellosphere have also moved rapidly inward along the field lines. In applying Liouville's theorem directly to particle motion it is the history of particles reaching 1 AU which is relevant, and therefore it is the inward (or negative) aspect of foc. using which is important. To find the particle intensity from the above equations, one integrates backward in time (dt < 0) along the particles trajectory, keeping account of momentum changes, until one strikes the model boundary. The intensity j can then be obtained by 101 Unmodulated % o ø.'.' \ > 1 - % 1,.. \ & - 2 ;" %,,\ :' ',,\ z -2 (1) dokipii Qnd D vil 1981 X 10 -(2) V t=-vf.v d V w X ) vt = Vd*Vw + Full Tr jectocy Integn tion -3,,,,! l l,,,!,,,i,,,... lo E 2 0 ø KINETIC ENERGY (GeV) Fi.. S eral solutions to t e cosmic ray transport e uatio, i, si pl spiral elds. Cur e is a two-di e,sio,al full solutio, publis d by ]okipii 1 [1981]. Cur es 2 a d 3 are our t reedi e,sio,al solutio,s to t e sa e proble usi, t e approximations described i t e text. T e points represent solutio,s obtai,ed by a full trajectory i,te ratio, tec,i ue a ai described i t e text. k 1 10 ø 1 2 ' -3 lo c 2 Unmodulc ted Tilt ø9' Angle K// = 3 x1022r1/2 %N \,,,... l,, I,l,,I,,, f... I,,, f... i ø KINETIC ENERGY (GeV) Fig. 2. Approximate solutions to the transport equation for several tilt angles (radians) of the hellospheric current sheet. The boundary is at 25 AU. The northern hemisphere field is inward for these calculations, which is the situation when cosmic ray protons drift inward along the current sheet. setting j = fp2. The boundary condition at the inner boundary (solar surface) is simply f-0 (no solar contribution) and at the outer boundary is given by a power law in total energy: 1 f oc cc(mø2c4 q- p2c2)-1.8 (13) To avoid the additional complexity introduced in the earlier papers of dealing with the infinite drift velocities which can result from the first-order expressions (in which it was necessary to treat the current sheet as an additional boundary), we chose here to calculate the drift and focusing velocities numerically. This is achieved by deriving the divergences and curls from the variation in field magnitude over a threedimensional Cartesian grid with step size equal to one particle gyroradius. In this way the drift velocity never exceeds the particle velocity (for a flat sheet it is easy to show that the drift velocity is two-thirds the particle velocity--the value expected). That this technique is capable of giving reasonably accurate results is illustrated in Figure 1. The diagram displays a number of solutions at 1 AU for the transport equation in a simple Parker spiral field with a flat current sheet. Curve 1 is a solution published by Jokipii and Davila [1981] using a two-dimensional solution of the full transport equation, curve 2 is the solution for the same problem performed three dimensionally, using the technique outlined above, and curve 3 is the result using the earlier approximation in which adiabatic focusing is neglected. These results indicate both the reliability of this approximate technique and the improvement bbtained by retaining focusing. The points with error bars represent another three-dimensional technique for solving the transport equation which will be outlined in connection with our second hypothesis. Figure 2 illustrates the solutions obtained for the tilted current sheet model described in this paper. These results represent the case where the field is inward in the northern hemisphere, which is the situation in which the particles drift in along the current sheet. Curves are displayed for a flat current sheet (tilt equal to 0) and for three finite tilt angles (given in

4 ß TI OM^S ET AL.'CosMIC R^¾S '1 I i I I. I I i I I I I I I I I I I I I z z o z _! I I I!! I! I I I! I I I I I I I I i I I I I i I! I.4 I i! I I! I I I! linde YEAR${ Fig. 3. Intensity versus time plot for 2.5-GeV protons assuming the current sheet rotatesmoothly in latitude and overturns at solar maximum. Shown for comparison are neutron monitor data for the last two solar cycles. radians). For simplicity the outer boundary is taken as 25 AU, is a consequence of our assumption that the tilt of the current as this is sufficient to allow several wavelengths in the wavy sheet increase smoothly, whereas the evidence suggests that current sheeto be present. The profiles are averages of inten- large tilt angles would occur only very near solar maximum. It sities obtained at two longitudes, one when the earth is exactly is clear that our results do not rule out the possibility that the on the current sheet and the other when the earth is at its current sheet is very steeply inclined at this time and illustrate greatest displacement from the sheet. To avoid overlapping that the current sheet would still be a primary factor in modulation. curves, the results for large tilt angles are not shown as the intensity modulation actually diminishes at large angles (al- FLUX SHEDDING though only slightly). This result is not unreasonable, the current sheet structure is relatively simple when the sheet is As stated earlier, the concept of the solar field reversal being vertical and intermediate angles give rise to the greatest cur- accomplished by a complete overturning of the poloidal field rent sheet deformations. The effect on our calculations is that of the sun is inconsistent with dynamo theory [e.g., Babcock, the completely vertical current shee that is assumed to exist 1961; Stix, 1976]. In this section we propose an additional at solar maximum gives rise to slightly less modulation than model for the heliospheric field near solar maximum which is shortly before or after solar maximum. based on the hypothesis that the solar field reverses by shed- This is illustrated in Figure 3, which displays the intensity of ding the poloidal field of the previous cycle and developing a 2.5-MeV particles as a function of time covering two consecu- new field of the opposite polarity. Figure 4 is a schematic tive solar cycles. It is produced by assuming that the current representation of a mechanism by which this may be accomsheet tilt varies continuously throughout the solar cycle as plished. The three panels represent a time sequence. given by (3), and plotting the intensity appropriate for each tilt The top panel shows the interplanetary field (solid lines) as angle obtained from the static models. This is a reasonably it may appear above the solar corona at a time well before good approximation in this particular calculation, as the solar solar maximum. For simplicity we assume there is a single flat wind transit time to 25 AU is only 3 months and the tilt angle equatorial current sheet at this time. The circle represents not will not have changed greatly over this period. Also shown for the sun but the source surface of the solar wind, and thus all comparison the actual cosmic ray intensity measured by the field lines are open at this time. Mount Washington neutron monitor detectors plotted on the The middle panel illustrates the situation near solar maxisame time scale. Although differing in detail near solar maxi- mum with isolated regions of opposite polarity now existing in mum, the overall agreement is reasonably good, and the the two hemispheres, separated from the background field by peaked profile in one cycle and flat top of the next, features additional current sheets. The dashed lines represent new field that emerged in previous analyses, are again confirmed. The lines, associated with the developing current systems, which relatively broad reduction in intensity around solar maximum are drawn into the interplanetary medium by the outflowing

5 T ostns.t AL..' COSMIC RAYS 2893 solar wind. The hypothesis is that as the new regions of opposite polarity grow they push the old flux toward lower latitudes where it is ultimately shed from the sun. The bottom panel illustrates the situation shortly after solar maximum when the new polarity regions have spread completely over both hemispheres, establishing the new cycle, with the old flux now completely shed from the sun. If the bulk of the field reversal takes place on a time scale of one year, then the new flux will not have had time to convect to the hello- pause, and the field lines in the inner heliosphere at this time will therefore be closed. This model may be oversimplified, there may be more than one region of anomalous polarity which develops in each hemisphere, and the flux shedding process, if it occurs, will probably be spasmodic and patchy. There is direct observational evidence for the existence of regions of anomalous polarity [e.g., Hundhausen et al., 1981], although the observa- Fig. 5. Meridian plane projection of a simple model for the heliospheric field, shortly after solar maximum, containing the essential features of the bottom panel in Figure 4. Fig. 4. Schematic representation of a model for the solar field reversal. (Top) Meridian plane projections of field lines above the solar corona prior to solar maximum (the circle represents the interplanetary magnetic field source surface, not the sun). (Middle) Field geometry during the reversal period, with new current systems producing new magnetic flux (dashed lines). (Bottom) The situation shortly after the reversal, with the new flux having completely displaced the old. The radial distances in these diagrams have been greatly foreshortened. tions suggest that such regions first appear on the sun at mid-latitudes, rather than at the poles. Field line reconnection processes may also be expected to play a role in the reversal, which will further complicate the flux shedding process, but if isolated regions of opposite flux do develop on the sun it is not easy to see how these regions can reconnect with the old field from the opposite hemisphere. Topologically, however, there are only two possibilities for the solar reversal process. Either field lines migrate over the solar surface, as in our first hypothesis, or new flux emerges and displaces the old. If flux is, indeed, shed from the sun, then the cosmic ray response will be largely independent of the detailed geometry at the solar surface. It is clear that the magnetic field geometry in the hellosphere resulting from a process of this kind will be very complex. We now outline a greatly simplified model for the field geometry shortly after solar maximum, for which solutions of the cosmic ray transport equation may be obtained. The objective is simply to determine if the existence of a closed field line configuration in the inner heliosphere is capable of providing cosmic ray modulation of the required magnitude. Figure 5 displays the model schematically. The diagram shows only the meridian plane components of the field. Over most of the diagram the azimuthal component is dominant, and the field geometry is close to an Archimedian spiral, with a fiat current sheet in the equatorial plane. The outer boundary is taken to be spherical and is located at 100 AU (probably a more realistic distance than that used in the previous section). Inside 20 AU the field is a pure Archimedian spiral given by the equations used previously (equations(1)). A is chosen to give outward fields in the northern hemisphere and inward in the southern hemisphere. This region represents the new field after the reversal process is completed. Outside 80 AU the field equations are identical, but A is chosen to give inward fields in the northern hemisphere and outward fields in the southern hemisphere. This region represents field from the previous solar cycle which has now been completely shed from the sun but which has not yet convected to the model boundary.

6 2894 THOMAS ET AL.'CosMIC RAYS 1 10 ø Unmodulated \ K// = 3 xlo 22 R 1/2 Because of the important role played by scattering in assisting particles to cross the interface between the separate field regions, it is not possible to use the approximate solution to the transport equation described in the last section. We therefore use instead a technique involving full trajectory integration and the direct application of Liouville's theorem. We fully integrate the equation of motion of individual cosmic ray protons: dt - q E + - c (15) z Ld F- Z 2 1( 2 ß Celculeted Intensities 1. Parkerian Field \ 2. Closed Fields \ ø KINETIC ENERGY (GeV) Fig. 6. Results of a numerical calculation of cosmic ray intensities in a simple spiral (Parkerian) field and in the closed field geometry of Figure 5. Between 20 and 80 AU is the transition region representing the period during which the field reversal occurred. This corresponds to a reversal period of approximately eight months. A simple method of closing the internal and external field regions is given by the divergence free expressions: B, = ABo(r -- ro)/(raar) Be = ABofl sin O(r - ro)/(rv Ar ) (14) Bo = ABo(1 -cos O)/(rAr sin O) where r o is the interface between old and new flux located at 50 AU and Ar is the width of the transition region. This model actually corresponds to a field on the sun in which the radial component dies away linearly and builds up again in the opposite direction. Although not a representation of the true situation, it contains the essential features: separate regions of old and new flux, closed fields in the inner heliosphere, and a field geometry which is almost Parkerian (except very near the interface). The magnitude of the north-south field component as this structure convects past 1 AU would be less than 0.1 nt. The fact that the radial component diminishes over the reversal period may be unrealistic but will have the effect of providing easier access to the inner heliosphere and thus will weaken the overall modulation effect rather than exaggerate it. It is our purpose to compare the solution of the transport equation in this field configuration with that obtained in the pure Archimedian fields that will exist at periods well away from solar maximum. It is clear that particles will have easier access to the inner heliosphere in the simple spiral field configuration. One sees from (6) that the primary influences on particle motion in the heliosphere are drifts and focusing along the field lines. In spiral fields both the field lines and the drift patterns extend to the heliopause. In a closed field configuration, exactly the opposite is true. Not only are the field lines closed, but since particle drifts are divergence free, the drift patterns are also closed. Therefore particles gain no help from either process and can gain access to the inner heliosphere only by scattering perpendicular to the field, across the interface between one field region and the other. where E is the convection electric field. The effect of scattering by small-scale irregularities is represented by introducing small random angular perturbations in such a way that the desired diffusion coefficient is obtained. Perpendicular scattering is implicit in this method, with the particle scattering typically one gyroradius perpendicular to the field in each parallel mean free path. The basic technique is similar to that described in the last section. We integrate a given trajectory backward in time, keeping account of momentum changes, until the particle reaches the model boundary. By direct application of Liouville's theorem, each particle gives an independent estimate of the phase space density at 1 AU (at a given particle energy), and by averaging over a large number of individual estimates we obtain a representative value for the omnidirectional intensity at 1 AU. Full details of this method are given by Thomas and Gall [1984]. To demonstrate that this method gives reasonably accurate solutions, we have also included in Figure 1 results obtained by full trajectory integration for comparison with the other methods. Figure 6 displays the cosmic ray intensities obtained as a function of particle energy for simple Parker fields and for the closed field model outlined here. Again it is clear that the difference in the two curves is indeed comparable to the observed variation in cosmic ray intensity between solar minimum and maximum. DISCUSSION AND CONCLUSION The mechanism by which the sun reverses polarity is crucial to understanding the solar cycle modulation of galactic cosmic rays. In this paper we have explored the effect of two possibilities: first, that the sun reverses polarity by a simple and continuous increase in inclination of the heliospheric current sheet until it becomes vertical and overturns and, second, that the sun sheds the magnetic field of the previous solar cycle and develops a new field of the opposite polarity. It is now well known that the inclination of the current sheet does increase as solar maximum approaches, and present evidence suggests that it can be steeply inclined in the years surrounding solar maximum. Thus the results of the first half of this paper are likely to be relevant even if other mechanisms do exist very near maximum. If the current sheet does overturn at solar maximum, the large-scale structure of the heliosphere will be rather more complicated than the simple static models used in this analysis, since the continuously increasing inclination will induce a large meridional warp in the current sheet in addition to the equatorial warp induced by solar rotation. Thus our results will only be directly applicable if the current sheet overturns smoothly and relatively slowly. For a current sheet that overturns rapidly during solar maximum we begin to approach the situation analyzed in the second part of this paper, in which the inner heliosphere and outer heliosphere have different field polarities. The situation

7 THOMAS ET AL..' COSMIC RAYS 2895 will be different in a fundamentally important sense, however, as there will be no regions of closed flux. For the second hypothesis, that the sun sheds its flux, there is less direct evidence, but the hypothesis is certainly more consistent with dynamo theories. There is, however, general supporting evidence from coronal data that the situation at solar maximum is a good deal more complicated than a simple near-vertical current sheet. Isolated regions of anomalous polarity do appear to develop on the sun as solar maximum approaches and spread over the surface as the new cycle develops. Such a process is, of course, consistent with our flux shedding hypothesis. In an attempt to demonstrate that such a process is capable of producing significant modulation we have investigated only a very simple and rather unrealistic field configuration, but we are confident that we have established the general principle. Our primary objective in this paper has been to give a general idea of the importance of the large-scale structure of the heliospheric current sheet in producing cosmic ray modulation and to further demonstrate that its effects (whether there exists a single or multiple current sheets) may still be important throughout solar maximum. There are, of course, other large-scale magnetic structures in the heliosphere that are capable of producing modulation, and presently there is substantial interest in the effect of large solar flares at solar maximum. It does not appear to be clear at this time whether solar flares are capable of producing very long term reductions in cosmic ray intensities, or whether the apparent cascade effect of solar flares during the last approach to solar maximum [McDonald et al., 1981; Burlaga et al., 1984] was actually a series of standard Forbush decreasesuperimposed on an overall decline in cosmic ray intensity produced by different mechanisms, such as those we propose here. Acknowledgments. The impetus to perform part of this analysis was the result of involvement in the Workshop on Cosmic Ray Modulation organized by L. Burlaga, F. McDonald, G. Newkirk, and others. We have benefited greatly from the many presentations and discussions involving the other workshop participants. We would particularly like to thank J. R. Jokipii both for his contributions at the workshop and for many helpful discussions previously. 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Smith, Jet Propulsion Laboratory, Building 169, Room 506, 4800 Oak Grove Drive, Pasadena, CA B. T. Thomas, Computer Science Department, University of Bristol, University Walk, Bristol BS8 1TW, England. (Received April 29, 1985; revised November 1, 1985; accepted November 6, 1985.)