long-term cosmic-ray modulation in the heliosphere - IOPscience

The Astrophysical Journal, 603:744–752, 2004 March 10 # 2004. The American Astronomical Society. All rights reserved. Printed in U.S.A.

LONG-TERM COSMIC-RAY MODULATION IN THE HELIOSPHERE S. E. S. Ferreira and M. S. Potgieter Unit for Space Physics, School of Physics, Potchefstroom University for Christian Higher Education, 2520 Potchefstroom, South Africa; [email protected], [email protected] Received 2003 August 27; accepted 2003 November 24

ABSTRACT A time-dependent model based on a numerical solution of Parker’s transport equation is used to model the modulation of cosmic-ray protons, electrons, and helium for full 11 year and 22 year modulation cycles using a compound approach. This approach incorporates the concept of propagating diffusion barriers, increases in the heliospheric magnetic field as they propagate from the Sun throughout the heliosphere, time-dependent gradient, curvature, and current-sheet drifts, and other basic modulation mechanisms. The model results are compared with those of the observed 11 year and 22 year cycles for 1.2 GV electrons and 1.2 GV helium at Earth for the period of 1975–1998. The model solutions are also compared with the observed charge-sign–dependent modulation along the Ulysses trajectory for the period of 1990–1998. This compound approach to long-term modulation, especially charge-sign–dependent modulation, is found to be remarkably successful. It is shown that the model can account for the latitude dependence of cosmic-ray protons and electrons by assuming a large perpendicular diffusion in the polar direction. This approach contributes to an improved understanding of how diffusion and drifts vary from solar minimum to maximum modulation and what the time dependence of the heliospheric diffusion coefficients may be. It is found that less than 10% of the available drifts are needed at solar maximum, when the solar magnetic field reverses, to explain, e.g., the observed electron-to-proton ratio along the Ulysses trajectory. Subject headings: convection — cosmic rays — diffusion — ISM: magnetic fields — solar-terrestrial relations — solar wind

1. INTRODUCTION

to simulate, to the first order, a complete 11 year proton modulation cycle by including a combination of drifts and GMIRs in a comprehensive time-dependent model. The addition of GMIRs in such a model could explain the steplike appearance for the observed cosmic-ray intensities, which could not be obtained with a drift model with a time-dependent HCS alone. The degree to which GMIRs affect long-term modulation depends on their rate of occurrence, the size of the heliosphere, the speed with which they propagate, their spatial extent (especially their latitudinal extent), and the background diffusion coefficients they encounter. Drifts, on the other hand, seem to dominate the solar-minimum modulation periods so that during an 11 year cycle, there must be a transition from a period dominated by drifts to a period dominated by propagating structures. For reviews on long-term modulation modeling, see Potgieter (1993, 1995, 1997) and Potgieter, Burger, & Ferreira (2001). The simulations of le Roux & Potgieter (1995) were done for radial distances greater than 20 AU, allowing enough time for the merging of corotating structures to take place. Concerning the merging effect, Cane et al. (1999), Wibberenz & Cane (2000), and Wibberenz, Richardson, & Cane (2002) argued that the cosmic-ray step decreases observed at Earth could not be caused by GMIRs, because these decreases occurred before any GMIRs could form beyond 10–20 AU. Instead, they suggested that time-dependent global changes in the heliospheric magnetic field (HMF) strength (which roughly varies by a factor of 2 from solar minimum to maximum) might be responsible for long-term modulation. This scenario does not require GMIRs, which occur beyond 10–20 AU, before steplike modulation sets in at Earth. Modeling done by le Roux & Fichtner (1999) also showed that a series of only GMIRs, as they occurred during the

The record of the long-term solar modulation of cosmic rays in the heliosphere shows a clear 11 year modulation cycle (Fig. 1). This cycle is anticorrelated with solar activity, with maximum cosmic-ray intensity values for solar-minimum conditions and vice versa. Apart from the 11 year modulation cycle, a 22 year cycle is also evident, when some periods of maximum intensity show a plateau-like and some peaklike shapes. By modeling these observed intensities, it was shown by Potgieter & le Roux (1992) that a time-dependent modulation model including gradient, curvature, and current-sheet drift effects (Jokipii, Levy, & Hubbard 1977) could reproduce in general some of the modulation features visible in cosmicray observations for solar-minimum periods (Ko´ta & Jokipii 1983). For these calculations, the waviness of the heliospheric current sheet (HCS) was assumed to be the only timedependent parameter. However, le Roux & Potgieter (1995) subsequently showed that their model could not reproduce cosmic-ray observations during a phase of increased solar activity with the tilt of the HCS as the only time-dependent parameter. This was especially true when large step decreases in the observed cosmic-ray intensities occurred (e.g., Lockwood 1960; McDonald et al. 1981). These steps are very prominent during the increasing phase of solar activity, when the changes in the HCS are no longer primarily responsible for the modulation. To successfully model the cosmic-ray intensities during moderate to higher solar activity requires some form of propagating diffusion barriers (PDBs), as first introduced by Perko & Fisk (1983). The extreme forms of these diffusion barriers are called global merged interaction regions (GMIRs), as introduced by Burlaga, McDonald, & Ness (1993). It was illustrated by le Roux & Potgieter (1995) that it was possible 744

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also computed in order to study charge-sign–dependent modulation over a full 22 year cycle (see also Burger & Potgieter 1999; Potgieter et al. 2001). The purpose of this study is to establish how the heliospheric diffusion coefficients and drifts change from solar minimum to maximum modulation and the effects on cosmic-ray modulation. 2. MODEL AND PARAMETERS The model is based on the numerical solution of Parker’s (1965) time-dependent transport equation: @f ¼ ð V þ h vd iÞ G Hf þ H G ð Ks G Hf Þ @t 1 @f þ Q; þ ð H G VÞ 3 @ ln P

Fig. 1.—Model computations together with Hermanus NM count rates expressed as percentage values for the period of 1980–1992. Shaded areas indicate the time periods when the HMF polarity was not well defined.

1978–1987 modulation cycle, could not reproduce the observed level of cosmic-ray modulation. This could only be achieved by adding some global, long-term variation into, e.g., the diffusion coefficient (in association with the changing magnetic field strength of a factor of 2 from solar minimum to maximum) or the tilt of the HCS. The concept of Cane et al. (1999) was tested by Ferreira (2002) by changing all the diffusion coefficients in a fully time-dependent model to reflect the time-dependent changes in the measured HMF magnitude at Earth. These regions of diffusion changing with the HMF were propagated outward at the solar wind speed to form effective PDBs at solar maximum throughout the heliosphere. As shown in Figure 1, this approach can simulate an 11 year modulation cycle successfully for cosmic-ray observations at neutron monitor energies, e.g., greater than 10 GV. However, for rigidities less than 5 GV (not shown here; e.g., Ferreira 2002), it resulted in far less modulation than what was observed, so that a modified approach was needed. In this paper we present an approach, called the compound approach, that combines the effects of the global changes in the HMF magnitude with drifts, and therefore also time-dependent current-sheet ‘‘tilt angles,’’ that establishes a time dependence for the diffusion coefficients as a function of the HMF. The compound model was previously discussed by Potgieter & Ferreira (2001), Ferreira (2002), and Ferreira et al. (2003a) with the essence as discussed below. In this work the compound approach is applied to model the long-term modulation of 1.2 GV electron and helium intensities at Earth and the 2.5 GV electron and proton intensities along the Ulysses trajectory. The ratio of these intensities is

ð1Þ

where f ðr; P; tÞ is the cosmic-ray distribution function, P is the rigidity, r is the position, V is the solar wind velocity, and t is the time. Terms on the right-hand side represent convection, gradient, and curvature drifts, diffusion, adiabatic energy changes, and a source function. The symmetric tensor Ks consists of a parallel diffusion coefficient (Kk) and two perpendicular diffusion coefficients, in the radial direction (K?r) and the polar direction (K?). In equation (1) the pitch angle– averaged guiding center drift velocity for a near isotropic cosmic-ray distribution is given by hvd i ¼ H ðKA eB Þ, with eB ¼ B=Bm and Bm the magnitude of the modified average HMF (e.g., Jokipii & Ko´ta 1989; Potgieter 2000). Equation (1) was solved time-dependently, in a two-dimensional spherical spatial coordinate system, based on the numerical procedure of le Roux & Potgieter (1995) for both the so-called A > 0 (1970–1980 and 1990–2001) and A < 0 epochs (1980–1990 and >2001). In this model the effects of the HCS on cosmic-ray transport are simulated as described in, e.g., Hattingh & Burger (1995). It was shown in Ferreira, Potgieter, & Burger (1999) that there are no qualitative differences and insignificant quantitative differences between this approach and a three-dimensional approach including an actual wavy HCS. The time-dependent input parameters used for the model, such as the tilt angle of the HCS and the HMF strength, are taken from measurements and shown in Figure 2.1 There are two different models for calculating (Hoeksema 1992); the ‘‘classical’’ model uses a line-of-sight boundary condition, while the ‘‘new’’ model uses a radial boundary condition at the photosphere. Ferreira (2002) found that the with the smallest rate of change over a period of decreasing or increasing solar activity provides the best compatibility with cosmic-ray observations (see Ferreira & Potgieter 2003). Therefore, in this work, the tilt of the HCS corresponding to the new model is used for periods of increasing solar activity (1976.0–1979.9, 1987.4–1990.0, and 1995.5–2000.0), and corresponding to the classical model is used for periods of decreasing solar activity (1979.9–1987.4 and 1990.0–1995.5). The outer modulation boundary (heliopause) was assumed to be at 120 AU, where the local interstellar spectra were specified. For solar-minimum conditions, the solar wind speed V was assumed to change from 400 km s1 in the equatorial 1 The data for the middle and bottom panels of Fig. 2 were obtained from NSSDC COHOWeb (http://nssdc.gsfc.nasa.gov/cohoweb) and Wilcox Solar Observatory (http://sun.stanford.edu), respectively.

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Fig. 2.—Function f2 ðtÞ in eq. (2) (top), representing the time dependence of all the heliospheric diffusion coefficients for the period of 1975–2000. This function combines the effects of the observed global changes in the HMF magnitude (middle) with drifts, and therefore also with time-dependent current-sheet tilt angles (bottom), to establish the time dependence of the diffusion coefficients.

plane (polar angle ¼ 90 ) to a maximum of 800 km s1 for 60 and 120 (McComas et al. 2001), while for solar maximum, V ¼ 400 km s1 for all polar angles. The time dependence in the latitude dependence of the solar wind speed changes according to Langner, Potgieter, & Webber (2003), and the effects on the modulation of the cosmic rays, in particular low-energy electrons, are illustrated by Ferreira et al. (2003b). For the parallel diffusion coefficient, we assume a basic form: B0 n ; ð2Þ Kk ¼ f1 ðr; P; tÞf2 ðtÞ; f2 ðtÞ ¼ BðtÞ where f1 ðr; P; tÞ is a function determining the rigidity and spatial dependence as given by Ferreira et al. (2001), with ¼ 1:3 for A < 0 epochs and ¼ 1:0 for A > 0 epochs. These different diffusion coefficients for alternate solar cycles agree qualitatively with Reinecke, Moraal, & McDonald (1996) and Potgieter (2000), who found that the coefficients were different for consecutive solar-minimum periods. This might also indicate that a charge-sign–dependent mechanism such as magnetic helicity (e.g., Smith & Bieber 1993; Burger,

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Hattingh, & Bieber 1997) may affect diffusion. The function f2 ðtÞ, with B0 ¼ 5 nT, determines the dependence of all the diffusion coefficients on the HMF BðtÞ, which is time-dependent according to the measurements at 1 AU. The effects of magnetic turbulence are included in f2 ðtÞ. Potgieter & Ferreira (2001) illustrated that n ¼ 1 in equation (2) resulted in the best fit to the observed modulation at neutron monitor rigidities, indicating that the diffusion coefficients could change with time by a factor of 2, corresponding to changes in the HMF magnitude from solar minimum to maximum (Fig. 1). For lower rigidities, when the computed intensities were compared with observed 1.2 GV electron and helium intensities at Earth, this approach did not reproduce the observed modulation amplitudes over 11 years, not even with values of n 3:0. These authors concluded that n could not be a constant and had to change with time (and rigidity), depending on the level of solar activity. As a successful proxy of solar activity (from a drift point of view), the obvious choice was the time-varying . The compound approach for long-term cosmic-ray modulation was consequently proposed by assuming n ¼ = 0 , with 0 ¼ 11, which may vary with rigidity. (In the future we plan to report more on this aspect.) Using this form means that n is small (n ! 0) for minimum modulation but increases with increasing solar activity (n ¼ 2 5). The larger n is, the larger are the temporal changes in the diffusion coefficients when the magnetic field changes in the interaction regions, simulating essentially a series of PDBs of changing magnitude. In this work the same compound model is used, and in Figure 2 (top), the time dependence of the diffusion coefficients, represented by f2 ðtÞ, is shown for the period of 1975– 2000 (see also Ferreira et al. 2003b). Evidently, the diffusion coefficients are larger (a factor of 10 or more, depending on the rigidity) at solar minimum than at solar maximum (see also Cummings & Stone 2001) and are highly time-dependent. These time-dependent changes are propagated outward into the heliosphere at the solar wind speed, causing time-dependent diffusive barriers to move at 1 AU and beyond. Because these diffusive barriers play an important role only for intermediate and larger solar activity, to avoid numerical problems their propagation speed through the heliosphere is chosen to be the slow solar wind speed. However, these barriers may also eventually merge, but no merging was allowed for this work. Because we use the observed time-dependent changes in the HMF at 1 AU as a proxy for the rest of the heliosphere, f2 ðtÞ may differ when merging is allowed, e.g., when interaction regions are formed and merged in the middle of the heliosphere, resulting in a stronger HMF. This could lead to larger diffusion coefficients for solar maximum, as shown in Figure 2. In addition, the spatially two-dimensional nature of the model means that we average the cosmic-ray intensities over one solar rotation. Therefore, the effects of recurrent features of the solar wind speed, such as corotating interaction regions, on the modulation of high-energy cosmic rays were not considered. (For such treatments, see Ko´ta & Jokipii 1995; Kissmann et al. 2003). However, the effects of merged corotating interaction regions on the long-term modulation of cosmic rays was studied by Potgieter et al. (1993) and Potgieter & le Roux (1994). They confirmed that they contribute little to long-term modulation and that large PDBs are needed to simulate longterm cosmic-ray modulation successfully. Concerning perpendicular diffusion, no exact theory exists to adequately describe it (for a comprehensive discussion, see le Roux, Zank, & Ptuskin 1999). It has therefore become

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standard practice when using numerical modulation models to scale K? as Kk (e.g., Ko´ta & Jokipii, 1998; Burger, Potgieter, & Heber 2000). For theoretical motivation, see le Roux et al. (1999). For K?r and K?, we assume that

K?r

P ¼ 0:02 P0

FðÞ ¼ Aþ þ A

0:3

K? ¼ bFðÞ; Kk 1 tanh ð A 90 þ F Þ : Kk ;

ð3Þ

Here P0 ¼ 1 GV and b ¼ 0:03. This results in K?r =Kk 0:02 for P 1 GV and K? =Kk ¼ 0:03 (in the equatorial plane), as required by the simulations done by Giacalone & Jokipii (1999). Furthermore, Burger et al. (2000) illustrated that in order to produce the correct magnitude and rigidity dependence of the latitudinal cosmic-ray proton density gradient observed by Ulysses, enhanced latitudinal transport is required (see also Ko´ta & Jokipii 1995; Potgieter et al. 1997). This is accomplished by increasing K? toward the poles by a factor d ¼ 8 with respect to the value in the equatorial plane by assuming the function FðÞ in equation (3), where A ¼ ðd 1Þ=2, ¼ 1=8, A ¼ , and F ¼ 35 for 90, while for > 90, A ¼ 180 . A justification of this increase in K? toward the polar regions was given by, e.g., Burger et al. (2000). These arguments are based on Ulysses measurements that show the variance in the transverse and normal directions of the HMF to increase more than in the radial direction, resulting in larger diffusion in those directions. Furthermore, in a Fisk-type HMF (Fisk 1996), which is probably a more realistic HMF geometry for solar-minimum conditions, latitudinal transport is supposedly more effective than in a Parker field, and to account for this effect, K? needs to be enhanced toward the polar regions when a Parker HMF is used. The function FðÞ is shown in Figure 3c with d ¼ 8 and F ¼ 35 , and with d ¼ 6 and F ¼ 15 . The effects of these two scenarios on model computations are shown and discussed in Figure 3 and later in the text. Because both perpendicular diffusion coefficients scale as Kk, their values change with time, as shown in Figure 2. The drift coefficients change with time as follows: Kdrift ðPÞ f2 ðtÞ; 3Bm Dfak P2 ; Kdrift ðPÞ ¼ P Dfak P2 þ 1 K A ¼ ð K A Þ0

Fig. 3.—(a) Computed 2.5 GV e=p along the Ulysses trajectory (solid line) and at Earth (dotted line) in comparison with the 2.5 GV e=p observations from KET (Heber et al. 1996, 1999, 2002). (b) Two computed p=e values as a function of Ulysses latitude for the fast latitude scan period around 1995 in comparison with the observed p=e from KET. (c) Cases A and B for two different assumptions for K? in eq. (3), showing FðÞ as a function of polar angle. Case B (dotted line) corresponds to d ¼ 8 and F ¼ 35 , and case A (solid line) to d ¼ 6 and F ¼ 15 in eq. (3).

reduced to less than 10% for most of the period, while during the 1989–1992 solar maximum they reduced to essentially 0% for two relatively short periods and remained below 10% for 3 out of the 4 years. In x 3 the model solutions are shown to illustrate that this approach and these drift levels are consistent with the data for the two maximum epochs. 3. LONG-TERM COSMIC-RAY MODELING

ð4Þ

where Bm is the Parker HMF, modified only in the heliospheric polar regions and similar to what Jokipii & Ko´ta (1989) used; Dfak ¼ 10:0 in units of (rigidity)2, which causes drifts to be somewhat reduced at lower rigidities, as explained by Burger et al. (2000); ðKA Þ0 ¼ 1:0, and is the ratio between the particle speed and the speed of light. This means that f2 ðtÞ as shown in Figure 2, together with ðKA Þ0 , is also indicative of the drifts required over a full 22 year cycle. During solarminimum periods drifts are obviously large, varying between 80% and 100% for at least 3 years around every minimum. For A < 0 cycles, around, e.g., 1985, drifts drop sharply with increased modulation, but not for the A > 0 cycle, around, e.g., 1997, which is the cause of the well-known flat versus sharp intensity-time profiles in cosmic-ray observations of all energies during A > 0 and A < 0 polarity cycles, respectively. During the solar-maximum conditions in 1979–1982, drifts

Figure 4 shows the results of the compound modeling approach, in which the computed intensities are compared with the 1.2 GV electron observations and 1.2 GV helium observations at Earth. As illustrated, this approach produces the correct modulation amplitude and most of the modulation step decreases in the observations. Some of these simulated steps do not have the correct magnitude and phase, indicating that some refinement of this approach is still needed, allowing, for example, for some global merging of the PDBs. However, the gratifying aspect of these results is that solar-maximum modulation could indeed be largely reproduced for different cosmic-ray species using a relatively simple concept, while maintaining the major modulation features during solar minimum, such as the flatter modulation profile for electrons (helium) in 1987 (1997), but a sharper profile in 1997 (1987). A major challenge for modeling is replicating the observed latitudinal gradients for cosmic-ray protons, which are significantly smaller (as observed by Ulysses for the A > 0 polarity cycle; see the review by Heber & Potgieter 2000) than

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Fig. 4.—(a) Computed 1.2 GV electron intensity at Earth compared with observations from ICE (squares; e.g., Clem et al. 1996; Evenson 1998) and Ulysses (triangles; e.g., Heber et al. 2002; Clem et al. 2002). (b) Computed 1.2 GV helium intensity at Earth compared with observations from IMP (McDonald 1998; McDonald et al. 2001).

predicted by classical drift models (e.g., Haasbroek & Potgieter 1995). For A > 0 polarity cycles, positive charged particles, such as cosmic-ray protons, drift in from the poles of the heliosphere to the Sun and outward along the HCS. For these periods, large latitudinal gradients were expected in drift-dominated models. However, by comparing the model results with the Ulysses observations, which observed much smaller gradients, it quickly became evident that this was due to the overestimation of drifts in the polar regions of the heliosphere in these models (Potgieter et al. 1997). In addition, when the clear effects of corotating interaction regions were observed (McKibben et al. 1995; Paizis et al. 1999) in cosmic-ray intensities at high heliolatitudes, without accompanying structures in the solar wind and the HMF, it became obvious that efficient latitudinal transport occurs in the heliosphere. It could not result from drifts, because then the latitudinal gradients would be large. One of the possibilities to account for these effects is large perpendicular diffusion in the polar region, K? (e.g., Ko´ta & Jokipii 1998). Drift effects can be reduced by increasing K? (Potgieter 2000; Ferreira et al. 2000). The exact enhancement of this latitudinal diffusion coefficient in the polar direction is still somewhat controversial and may change if a non-Parker HMF is used. This may also be another possibility for reducing latitudinal gradients. Encouraged by the results shown in Figure 4, the compound concept was applied in Figure 5 to illustrate the success of reducing drifts effectively through a large K?. In Figure 5a the computed proton intensity is shown, and in Figure 5b, the 2.5 GV electron intensity along the Ulysses trajectory. Shown in comparison are the Ulysses Kiel Electron Telescope (KET) data (Heber et al. 2002). For both species, three computed scenarios are shown, corresponding to three assumed values of

b in equation (3), which gives the magnitude of K? in the equatorial plane. In accordance with observations, there are noticeable computed latitudinal gradients for the proton intensity, as is especially evident during the fast latitude scan in 1995, but almost none for the electron intensities. This feature has recently been studied in great detail with the availability of the Ulysses KET simultaneous observations of electrons and protons (e.g., Heber et al. 1999). Comparing the model with observations shows that b ¼ 0:03 in equation (4) does produce realistic latitudinal gradients for both species and that a large K? is indeed necessary to fit the observed charge-sign– dependent modulation over the full period shown. Decreasing K? by assuming b ¼ 0:01 in equation (4) obviously results in latitudinal gradients that are too large, because drift effects then become too large. It is also evident in Figure 5 that decreasing b results in a larger computed modulation amplitude, especially for the 2.5 GV electrons, indicating that the compound approach is sensitive to this diffusion coefficient. The compound model can simulate the 11 year modulation cycle in the inner heliosphere without GMIRs. In Figure 3a the electron-to-proton ratio (e=p) is calculated along the Ulysses trajectory and at Earth. Comparing the model computations with observations shows that the chargesign dependence, as observed by Ulysses, could also easily be reproduced by the compound model. The presence of chargesign dependence caused by controlled drifts is, therefore, well understood for solar-minimum and intermediate-activity periods. However, for solar-maximum periods the modeling is below the observations (see also Ferreira et al. 2003b). This aspect is discussed in x 4. The difference between the e=p along the Ulysses trajectory and at Earth during the fast latitude is indicative of the drift-related latitude dependence of the protons but not of the electrons during this period.

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Fig. 5.—(a) Computed 2.5 GV proton intensity and (b) 2.5 GV electron intensity along the Ulysses trajectory in comparison with 2.5 GV data from the Ulysses KET instrument (Heber et al. 1996, 1999, 2002).

Looking into greater detail on a shorter timescale, Figure 3b highlights the latitude dependence of 2.5 GV protons during the Ulysses fast latitude scan period. Shown here are the observed and the model p=e as a function of Ulysses’s latitude (Heber et al. 1996, 1999, 2002). All the p=e’s are normalized. Two computed scenarios, corresponding to two assumptions of K? in equation (3) done through different FðÞ as shown in Figure 3c, are shown for comparison with the observed values. Case B is computed with d ¼ 8 and F ¼ 35 , which was previously used by Burger et al. (2000) and Ferreira et al. (2001), while for case A, d ¼ 6 and F ¼ 15 . For case B the model yields a strong latitudinal dependence close to the equatorial plane but much less over the poles, inconsistent with the observations. To rectify the situation, FðÞ is changed to that for case A, which illustrates that the enhancement of K? should be somewhat less and that it must occur closer to the equatorial plane for this period than previously assumed. Evidently, case A produces better agreement with the observed p=e at solar minimum as an illustration of the important role the latitudinal perpendicular diffusion plays: in particular, the fact that it must be anisotropic in the sense of having a significant enhancement toward the polar regions of the heliosphere.

simulated reasonably well for these periods. The exception is the periods of large solar activity, e.g., 1979–1980, 1981– 1983, and 1990–1991. During periods of very high solar activity, the computed e=He increased faster than the observations, which is particularly evident for the period of 1981– 1982, while the computed ratio decreased faster than the observations for the period of 1990–1992, although not to the extent that a steady state approach would predict. The two periods, indicated by ‘‘A’’ and ‘‘B’’ in the figure, are selected for further study.

4. MODULATION AT SOLAR MAXIMUM As shown in Figure 4, the compound approach simulates the 1.2 GV electron and helium intensity-time profiles at Earth for the period of 1976–2000. In Figure 6 the corresponding computed 1.2 GV e=He is shown at Earth for the same period in comparison with the observed e=He (Clem et al. 1996; Clem, Evenson, & Heber 2002; Evenson 1998; McDonald et al. 2001). The overall trends in the computed ratios are markedly compatible with the observed values, especially during low to moderate solar activity. The basic characteristics of this time profile for the A > 0 and A < 0 polarity cycles, e.g., the flattish ‘‘M’’ and ‘‘W’’ shapes, are also evident for this approach, although not nearly as pronounced as in steady state models (Burger & Potgieter 1999). Obviously, drifts and the response to changes in are

Fig. 6.—Computed 1.2 GV e=He at Earth for 1976–2000 in comparison with the observed e=He obtained from electron measurements from ICE (Clem et al. 1996; Evenson 1998), helium measurements from IMP (e.g., McDonald 1998; McDonald et al. 2001), and electron measurements from KET (Clem et al. 2002). Two periods with relatively large differences between the computed ratios and the observations are selected (A and B). The shaded areas correspond to the period when there was not a well-defined HMF polarity.

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In Figures 7a and 7c the computed and observed 1.2 GV electron and helium intensities are displayed for the period of 1980–1983, period A in Figure 6. Shown here is that the computed helium intensity trends are compatible with observations, especially for the period of 1981–1982. However, for the period of 1980–1982, the computed electron intensities are not compatible with what is observed. Figures 7b and 7d show the situation for the 1989–1992 period, period B in Figure 6. Shown here is that the computed trends in the electron intensities are largely compatible with the observations, while the computed helium intensities are too large when compared to the Interplanetary Monitoring Platform (IMP) data for the period of 1990–1991. These differences at solar maximum can be interpreted to mean that drifts need to be further reduced to compute charge-sign–dependent modulation compatible with observations for very high solar activity periods, an aspect studied in greater detail below. Another possibility could be to use different diffusion coefficients, not only varying in magnitude but in spatial dependence. Currently, this aspect is also under investigation. Recently, Ulysses finished its second (solar-maximum) fast latitude scan and continued its measurement toward the heliographic equator. Since then, the 1.2 GV helium measurements from IMP-8 have not been available, and to continue our investigation, we need to switch to 2.5 GV particles observed by the KET instrument. In Figure 8 the computed e=p along the Ulysses trajectory is shown for solar-maximum conditions, 1999–2002, in comparison with the observed 2.5 GV e=p from KET. Four scenarios are shown, corresponding to different values of ðKA Þ0 in equation (4), namely, 1.0, 0.6, 0.1, and 0, where ðKA Þ0 is a constant through which the amount of drifts is changed explicitly. Note that for the

Fig. 7.—Computed 1.2 GV electron and helium intensity-time profiles compared with observations for the two selected periods in Fig. 6. HMF polarity changes from A > 0 to A < 0 in 1980–1981 and vice versa in 1990– 1991.

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Fig. 8.—Top: Computed e=p along the Ulysses trajectory for solarmaximum conditions (1999–2002) in comparison with the observed e=p from KET (Heber et al. 2001). Four computed scenarios are shown, corresponding to four different values of ðKA Þ0 , namely, 1.0, 0.6, 0.1, and 0 in eq. (4). The shaded area shows the period with no well-defined HMF polarity. Bottom: ðKA Þ0 f2 ðtÞ over the same period as a representation of the total amount of drifts that occurred according to the model.

compound approach presented earlier, drifts (eq. [4]) are also implicitly scaled with time using the function f2 ðtÞ given by equation (2). Therefore, the total fraction of time-dependent drifts is given by jðKA Þ0 f2 ðtÞj, shown in Figure 8 (bottom). The shaded area shows the period in which there was not a welldefined HMF polarity. For 1999.0–1999.8, the scenarios in which ðKA Þ0 ¼ 0:6 1:0 represent the data well along the Ulysses trajectory, with the total drifts varying from 50% to 15% to 60% to 15% over this period. For 1999.8–2000.3, the scenario ðKA Þ0 ¼ 0:1 is clearly better, indicating that the total drifts had dropped below 10%, with values as low as 1%–3%. For the period of 2000.3–2000.8, the ðKA Þ0 ¼ 0 scenario fits the observations better, indicating that global drifts had vanished for the duration of the HMF polarity reversal. After the end of 2000, the nondrift case became poorly compatible with the data, indicating that drifts recovered very quickly after the polarity reversal. Thus, for 2001.3–2002.0 the scenario ðKA Þ0 ¼ 0:6 is already good, although the ðKA Þ0 ¼ 1:0 scenario gives a totally unrealistic increase in intensities owing to the reversal in polarity. The sharp increase in the computed intensities for the scenarios in which ðKA Þ0 ¼ 0:6 1:0 is clearly not evident in the data. The increase in the observed e=p indicates that drifts, although small, must have been present close to the period of polarity reversal. The compound approach without modification of drifts through ðKA Þ0 evidently results in too large a charge-sign–dependent modulation around solar maximum, so that KA also needs to be decreased for there to be no drifts for at least 6 months, as

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illustrated. To produce the observed charge-sign–dependent modulation during extreme solar-maximum conditions, the heliosphere must become ‘‘diffusion-dominated’’ (Haasbroek & Potgieter 1995; Heber et al. 2001). Additional mechanisms that may naturally reduce global drifts toward solar maximum but are not considered here are multiple HCSs at solarmaximum conditions (Ko´ta & Jokipii 2001), an increase in the perpendicular diffusion coefficient in the polar direction toward solar maximum, and/or a different type of HMF geometry for solar-maximum conditions and possible effects in the heliosheath. 5. MODULATION IN THE OUTER HELIOSPHERE Finally, the compound approach was tested for the outer heliospheric regions by comparing in Figure 9 the computed results with Voyager 2 135–225 MeV proton observations. It follows that this approach produces results that are largely compatible with the observations when Voyager 2 was moving from Earth to 50 AU. However, some of the fine structure in the model computations at Earth, as shown in Figure 4, is not present at these distances, where the computations are much ‘‘smoother.’’ Some merging between neighboring PDBs therefore seems necessary for the model to realistically simulate, e.g., the two large steps in 1981 and 1983. The outstanding question is how much merging occurs and what happens with diffusion barriers in the heliosheath. These aspects are currently being investigated. 6. SUMMARY AND CONCLUSIONS The long-term and charge-sign–dependent modulation of cosmic rays over 11 and 22 year modulation cycles has been studied. A new compound approach to modeling cosmic-ray intensities over these periods is applied. It combines the effects of the global changes in the HMF magnitude with drifts, and therefore also time-dependent current-sheet tilt angles, to establish a time dependence for the diffusion coefficients. It is shown that this approach applied to modulation at Earth is remarkably successful when compared with data over a period of 22 years; for example, when compared with 1.2 GV electron and helium observations at Earth, this approach produces the correct modulation amplitude and most of the modulation steps in the observations. Some of these simulated steps do not have the correct magnitude and phase, indicating that some refinement of this approach is still needed, allowing, for example, for some global merging of the PDBs. However, the gratifying aspect of these results was that solar-maximum modulation could indeed be largely reproduced for different cosmic-ray species using a relatively simple concept and without any merging in propagating barriers, while maintaining the major modulation features during solar minimum, such as the flatter modulation profile for electrons (helium) in 1987 (1997), but a sharper profile in 1997 (1987). Apart from describing the modulation of cosmic rays over long periods, the model also produces the observed chargesign–dependent modulation from minimum to maximum solar activity. Charge-sign–dependent modulation is one of the important features of cosmic-ray modulation because it is the most direct indication of gradient, curvature, and current-sheet drifts in the heliosphere. Studying it along the Ulysses trajectory emphasizes the difference in the latitude dependence caused by drifts of protons and electrons at 2.5 GV. It is shown that this model could straightforwardly account for the

Fig. 9.—(a) Computed cosmic-ray proton intensities along the Voyager 2 trajectory using the compound approach. In comparison, the observed 135– 225 MeV protons from Voyager 2 are shown (e.g., Fuji & McDonald 2001). (b) Polar angle–dependent and (c) radially dependent trajectory of Voyager 2.

latitude dependence of 2.5 GV cosmic-ray protons and the lack thereof for electrons along the Ulysses trajectory. In addition to the time dependence of the diffusion coefficients, a significant larger perpendicular diffusion toward the polar region is required to reduce the large latitudinal effects caused by unmodified drifts. It was also shown that during periods of large solar activity, the compound approach needs modification in the sense that drifts must be reduced further to better describe the observed e=He (at Earth) and e=p (along the Ulysses trajectory) during a year in which the HMF polarity reverses. We found that drifts reduced from a 50% level at the beginning of 1999 to a 10% level by the end of 1999, vanished during 2000, but quickly recovered after the polarity reversal during 2001 to levels above 10%. After 2001 our model predicts a steady increase in drifts from 10% up to 20% at the end of 2002. This indicates that to produce realistic charge-sign–dependent modulation during extreme solar-maximum conditions, the heliosphere must become diffusion-dominated (Haasbroek & Potgieter 1995; Heber et al. 2001). Additional mechanisms that may naturally reduce global drifts toward solar maximum but not considered here are multiple heliospheric current sheets at solar-maximum conditions (Ko´ta & Jokipii 2001), an increase in the perpendicular diffusion coefficient in the polar direction toward solar maximum, and/or a different type of HMF

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geometry for solar-maximum conditions and possible effects in the heliosheath By comparing results with Voyager 2 observations, when it was moving from Earth to 50 AU, it was found that the model could also account for cosmic-ray modulation in the outer heliosphere without merged interaction regions. However, some of the fine structure in the model computations is not present at these distances, where the computations are much smoother. This indicates that some merging between neighboring propagating diffusion barriers seems necessary for the model to realistically simulate, e.g., the two large steps

in 1981 and 1983. The outstanding question is how much merging occurs and what happens with these diffusion barriers in the heliosheath. These aspects are currently being investigated.

We thank Gerd Wibberenz, Bernd Heber, and Horst Fichtner for interesting discussions on long-term/charge-sign– dependent modulation. We also thank the South African National Research Foundation for partial financial support.

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