1995). The other Ðnding is the surprisingly small positive latitude gradients in the Ñuxes of most cosmic-ray and

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1 THE ASTROPHYSICAL JOURNAL, 488:841È853, 1997 October 20 ( The American Astronomical Society. All rights reserved. Printed in U.S.A. A LINEAR RELATIONSHIP BETWEEN THE LATITUDE GRADIENT AND 26 DAY RECURRENT VARIATION IN THE FLUXES OF GALACTIC COSMIC RAYS AND ANOMALOUS NUCLEAR COMPONENTS. I. OBSERVATIONS MING ZHANG Enrico Fermi Institute, University of Chicago, 933 E. 56th Street, Chicago, IL Received 1997 February 6; accepted 1997 May 30 ABSTRACT We Ðnd that there exists a linear relationship between the magnitude of latitude gradient and the amplitude of 26 day recurrent variations in the Ñuxes of galactic cosmic rays and anomalous nuclear components. This result is based on energetic charged particle measurements from the University of Chicago High Energy Telescope (HET) in the Cosmic Ray and Solar Particle Investigation (COSPIN) consortium on the Ulysses spacecraft during its mission to the solar poles and on similar measurements from IMP-8 spacecraft orbiting around Earth at 1 AU in the solar ecliptic. The linear relationship holds for recurrent cosmic-ray Ñux variations observed in the inner heliosphere at all latitudes including the high-latitude regions covered by Ulysses and in the equatorial by IMP-8, and it is roughly independent of particle energy and species. The relationship means that particles with large-latitude Ñux gradients are also strongly modulated by recurrent solar wind structures such as corotating interaction regions that appear in low- and middle-latitude regions. In the previous solar cycle, when the solar magnetic Ðeld had an opposite polarity, cosmic-ray measurements from the Voyager spacecraft and IMP-8 also show a linear relationship between the latitude gradient and the amplitude of recurrent variations even though the latitude gradient had a negative sign. With these observational facts, the three-dimensional Ñux distributions of cosmic rays in the inner heliosphere can be more easily measured and better understood because the linear relationship implies that there are similarities among di erent kinds of charged particles in their modulated Ñux distributions. These observations suggest that there is a common dominant modulation mechanism controlling both the global latitudinal distribution and the short-term temporal variation of cosmic-ray Ñuxes. A theoretic model to account for these observations and to understand cosmic-ray modulation in the three-dimensional heliosphere will be presented in Paper II. Subject headings: cosmic rays È interplanetary medium È solar-terrestrial relations 1. INTRODUCTION Studies of data returned from the Cosmic Ray and Solar Particle Investigation (COSPIN) experiment onboard the Ulysses spacecraft, during its out-of-ecliptic journey to the south polar region of the Sun (maximum latitude 80.2 S in 1994 September), back to the solar equator (1995 March), and continuing on to the north polar region (80.2 north in 1995 July), have led to two signiðcant and surprising Ðndings about solar modulation of galactic cosmic rays and anomalous nuclear components in the three-dimensional inner heliosphere (Simpson et al. 1995a, references therein; Simpson et al. 1995b). One is the 26 day recurrent variations in the cosmic-ray Ñuxes which were observed continuously all the way up to highest heliographic latitude of 80.2 S, where there are no recurrent solar windèmagnetic Ðeld variations because it is far beyond the maximum latitudinal extension (D40 ) of both the heliospheric current sheet and corotating interaction regions (CIRs) (Kunow et al. 1995; McKibben et al. 1995; Simpson et al. 1995a; Zhang et al. 1995). The other Ðnding is the surprisingly small positive latitude gradients in the Ñuxes of most cosmic-ray and anomalous nuclear particles (Simpson et al. 1995a), which have been reconðrmed by Simpson et al. (1995b) and McKibben et al. (1996) with greater accuracy using more recent Ulysses measurements during its fast latitude scan from the south pole to the north pole pass. Latitude gradients in both the northern and southern hemispheres are nearly identical (Simpson, Zhang, & Bame 1996; McKibben et al. 1996). 841 Although it is not yet clear how the recurrent cosmic-ray Ñux variations reach high-latitude regions in the inner heliosphere, where there are no recurrent structures in the solar wind and interplanetary magnetic Ðelds, the mechanism for the recurrent modulation of cosmic-ray Ñuxes in middle and low latitudes is fairly well known. In general, short-term cosmic-ray intensity variations are driven by solar wind structures in low- and middle-latitude regions, such as, for example, corotating interaction regions (CIRs). The change of particle di usion properties associated with passage of CIRs in the local interplanetary medium is the main mechanism causing cosmic-ray intensity variations observed in low and middle latitudes (Chih & Lee 1986; Ko ta & Jokipii 1991, and references therein). The latitude gradient of the cosmic-ray intensity, however, should be determined by large-scale or perhaps even the global magnetic Ðeld structure in the heliosphere. While the role of particle di usion cannot be neglected in all the considerations of the three dimensional modulations of cosmic rays, particle drifts in the average heliospheric magnetic Ðeld are thought to be an important process in controlling the latitudinal Ñux distribution, for opposite signs of the latitude gradient have been observed in the opposite polarity phase of the 22 yr solar magnetic Ðeld cycle (see the review by McKibben 1989, and references therein). Since the recurrent variation and latitude gradient of cosmic-ray Ñuxes are controlled by two independent particle transport mechanisms and by magnetic Ðelds of vastly di erent scales and Ñuctuation characteristics, these two

2 842 ZHANG Vol. 488 phenomena are then usually thought to be unrelated. Even in the models of Ko ta & Jokipii (1991, 1995) and Jokipii & Ko ta (1995), who numerically simulated these two phenomena simultaneously, the treatments for them are separate mechanisms, i.e., the large-scale modulation is controlled by drift e ects in conjunction with di usion, while small-scale structure is caused by di usion e ects in transient and corotating structures. There is no correlation found between them in the results of their numerical calculations. However, as we will demonstrate in this paper, there exists a close relationship between the magnitude of latitude gradients and the amplitude of the recurrent variations in the cosmic-ray and anomalous component Ñuxes. It is found to be true everywhere in the inner heliosphere for the present solar cycle with a positive magnetic polarity and the previous solar cycle with the opposite polarity. We will discuss the importance of this discovery to our understanding and measurement of cosmic-ray distributions in the three-dimensional inner heliosphere. A preliminary account of this linear relationship was presented recently (Zhang 1996). 2. OBSERVATION AND ANALYSIS 2.1. Overview of Observation The present investigation is mainly based on the measurements of galactic cosmic-ray and anomalous nuclear components made by the University of Chicago High Energy Telescope (HET) in the COSPIN consortium. For more information about the instrument and descriptions of the data channels, see previous publications (e.g. Simpson et al. 1992). The measurements we use in this paper cover the period from Ulysses encounter with Jupiter at 5.3 AU in 1992 February to the south solar polar encounter in 1994 September, and the fast latitude scan which extended from the south polar region in 1994 September to the north polar region in 1995 August (see Smith, Marsden, & Page 1995 for Ulysses trajectory information). From the time of Ulysses-Jupiter encounter in 1992 February to the arrival at maximum latitude, the level of cosmic radiation increased globally as a result of decreasing solar activities on the Sun. To separate this long-term temporal variation from changes due to the motion of the spacecraft in the heliosphere, we use as a baseline reference measurements from the University of Chicago charged particle telescope on IMP-8 spacecraft in Earth orbit (see Garcia-Munoz, Mason, & Simpson 1977a for a description of the instrument). We exclude the time periods when our particle channels are contaminated with particles of solar origin which are accelerated either on the Sun or by interplanetary shocks. Because of the near minimum condition of solar activity during the UlyssesÏ high-latitude observations, the omission of these periods does not signiðcantly a ect our analysis. The e ect of latitudinal cosmic-ray intensity gradients is demonstrated by graph D in Figure 1, where measurements in similar energy ranges from both Ulysses and IMP-8 are displayed. The H9 counting rate (solid line) obtained on Ulysses, which mainly measures the integral Ñux of E [ 92 MeV cosmic-ray protons (average energy D1 GeV), increased relative to the RCK integral rate of E [ 106 MeV (average energy also D1 GeV) cosmic rays (dashed line) measured by IMP-8 as Ulysses was climbing to high south and north latitudes, indicative of a deðnitive positive latitude gradient in cosmic-ray intensity. But the intensity di erence from the equator to the poles is small compared to the long-term (11 yr) intensity increase from the near solar maximum condition in early 1992 to the near solar minimum condition in late 1995, consistent with the small latitude gradients as reported by Simpson et al. (1995a). Anomalous cosmic rays generally have larger latitude gradients than galactic protons, but still the overall pole-to-equator Ñux di erence is much smaller than their solar cycle variations, meaning that the cosmic rays in the polar regions are also heavily modulated (Simpson et al. 1995a). In order to show the recurrent variations more clearly, we have detrended the Ñux by subtracting the daily intensity measurement and then normalized by its 27 day running average i.e., *H9 \ (H9 [ SH9T)/SH9T, where SH9T denotes 27 day running average of the H9 rate. In graph C in Figure 1, the recurrent variations in the H9 rate with a period of 26 days are clearly present up to the highest solar latitudes, particularly in the southern hemisphere, while the recurrent variations in the solar wind speed (graph A, Fig. 1) and magnetic Ðelds (graph B, Fig. 1) are all conðned to middle- and low-latitude regions. Recurrent variations are simultaneously observed with a synodic period of D27 days or sometimes D13.5 days near the solar equator by the Earth-orbiting IMP-8 spacecraft and ground-based neutron monitors which extend the measurements to the very high energies of greater than 13 GeV (Simpson et al. 1995a). To remove the long-term, global temporal Ñux variations as a result of changing global heliospheric conditions in the 11 yr solar activity cycle, we normalize Ulysses Ñux or counting rate measurements to 27 day averages of IMP-8 measurements with similar particle and energy ranges. Figure 2 displays such a ratio for three selected particle channels with di erent modulation levels. The ratio retains the e ect of Ñux changes due to the latitudinal and radial motion of Ulysses as well as short-term changes, such as the recurrent 26 day variation seen by Ulysses. The 27 day (synodic period) recurrent variations seen by IMP-8 have been removed from the ratio through taking 27 day average of the IMP-8 Ñuxes. The Ulysses to IMP-8 intensity ratios have been further renormalized to their own values at 10 S, in accordance with observations by Simpson et al. (1996) that the minimum or the symmetry plane of cosmic-ray latitudinal distribution is a conical surface at 10 south of the solar equator. In this way the polar Ñux excesses can be easily read from Figure 2. The heavy open rectangular boxes which cover the 70 È80 latitude ranges in both the south and north polar regions represent the magnitudes of polar Ñux excess with the vertical sizes indicating their uncertainties. The higher polar Ñux excess at the north pole than at the south pole in each panel is the result of the 10 southward shift of the symmetry surface. For the D40È70 MeV per nucleon helium channel (graph A, Fig. 2), which contains mostly singly charged anomalous helium, which is originally interstellar neutral particles ionized inside the heliosphere and then accelerated by the solar wind termination shock (Fisk, Kozlovsky, & Ramaty 1974; Cummings, Stone, & Webber 1984; Simpson 1995), the polar Ñux excess over the 10 S level is the largest, around 65% at the south pole and 90% at the north pole.

3 No. 2, 1997 LATITUDE GRADIENT AND RECURRENT VARIATION 843 FIG. 1.ÈDaily average (A) solar wind velocity and (B) magnetic Ðeld strength at Ulysses, (C) detrended percentage variations of the H9 integral counting rate measured by Ulysses, and (D) actual rates of the Ulysses HET H9 channel and IMP-8 RCK channel, both of which measure integral cosmic rays of E [ D100 MeV per nucleon. Meanwhile, the amplitude of 26 day variation in this channel is also the largest. In contrast, the Ñux in the D40È70 MeV proton channel has almost no polar excess, while its 26 day recurrent Ñux variation is dominated by random Ñuctuations perhaps due to counting statistics and thus is not visible in graph C of Figure 2. The E[D100 MeV (average energy D1 GeV) protons have medium levels of both latitude excess and amplitude of 26 day variation (graph B Fig. 2). These observations essentially tell us that the larger the latitude intensity gradient cosmic-ray particles have, the stronger their recurrent variations will be. In addition, it is interesting to note that in graphs A and B of Figure 2 the peak-to-bottom di erences of Ulysses/ IMP-8 intensity ratios in the recurrent variations are almost comparable to their pole-to-equator di erences when Ulysses was at middle latitudes in Relationship between the L atitude Gradient and the Amplitude of Recurrent Variations The trend of the above relationship between the magnitude of latitude gradients and the amplitude of 26 day variations exists in many other energy and particle channels. We have developed a systematic way to quantify their relationship L atitude Gradient To calculate the latitude gradient from cosmic-ray Ñux measurements, we only use the data obtained during the south pole to north pole fast latitude scan period. The reason is the following: As shown in Figure 1, UlyssesÏ climb to the maximum south latitude took more than 2.5 yr from the time of Jupiter Ñyby. During this period, the Ulysses spacecraft traveled from 5.4 to 2.2 AU in radial

4 844 ZHANG Vol. 488 FIG. 2.ÈRatios of Ulysses HET Ñuxes of (A) 3 day average D40È70 MeV/n helium, (B) daily average E [ D100 MeV protons, and (C) 3 day average D40È70 MeV protons to IMP-8 27 day averages in the approximately same energy ranges. The ratios have been normalized to their own values at the symmetry plane of 10 S. The boxes near the maximum latitudes indicate their levels of polar Ñux excess. distance, and the global cosmic radiation level increased by more than 1 order of magnitude in some energy ranges. These changes prevent us from making accurate determination of the small-latitude gradients as reported by Simpson et al. (1995a) for we have to separate the radial variation from the latitudinal variation for the Ñuxes, and furthermore, the responses of the instruments on Ulysses and IMP-8 may have changed slightly relative to each other over time and as a result of the large changes in radiation levels and cosmic-ray spectra. On the other hand, the fast latitude scan from the south pole to the equator and then to the north pole occurred within only 11 months, during which period the E [ 100 MeV cosmic-ray Ñux at IMP-8 increased by less than \7% (see Fig. 1) and the radial range of Ulysses changed from 2.3 to 1.34 AU. Because of the small change in radial distance, contribution from the radial gradient of cosmic-ray Ñuxes, which is generally less than a few percent per AU (see McKibben 1975 for review), can be neglected in our analysis. Thus the fast latitude scan after south pole pass permits more accurate determination of the latitude gradient. Simpson et al. (1996) have found that cosmic-ray and anomalous helium Ñuxes are higher at the north pole than at south pole (see also Fig. 2). After the 10 southward o set of the Ñux minimum was taken into account, they found the latitude gradients are nearly identical in both the hemispheres. For the calculation of latitude gradient, we will use these as observational facts. We Ðrst choose the time period when Ulysses was above 70 S (1994 June 26 to November 6) and 70 N (1995 June 20 to September 29) to represent the high-latitude traversals and the time period when Ulysses was traveling from 15 S to 5 S (1995 February 13 to February 26) to represent Ñux minimum centered at 10 S. Particle Ñuxes or counting rates measured by both Ulysses and IMP-8 are averaged over these three time intervals. We deðne the latitude gradients as follows: gs \ 1 h 70 lncf U (80 S)/F (80 I (1) F (10 S)/F (10 U I S)D for the southern hemisphere, and

5 No. 2, 1997 LATITUDE GRADIENT AND RECURRENT VARIATION 845 gn \ 1 h 90 lncf U (80 N)/F (80 N) I (2) F (10 S)/F (10 U I S)D for the northern hemisphere, where F and F are Ulysses and IMP-8 Ñuxes or counting rates averaged U over I the three intervals, respectively. Error bars for the latitude gradients are calculated from the standard deviations of the averaged Ulysses and IMP-8 Ñuxes in the three intervals by using a standard error propagation method (chap. 2 & 4 in Bevington 1969). As a matter of fact, since the cosmic-ray Ñuxes and its spectra observed at IMP-8 did not change much in the 11 month south pole to north pole pass, the latitude gradients are almost solely determined by Ñux increases from the solar equator to the both poles as seen by Ulysses; therefore, uncertainties in the IMP-8 Ñuxes and their slight mismatch of energy ranges with the particle channels of the HET on Ulysses are not important in determining the latitude gradients. For proton and helium Ñux measurements on IMP-8, there are only two energy channels covering the energy range of Ulysses HET Ñux measurements. We thus use a power-law interpolation method to derive the IMP-8 Ñuxes in the energy ranges corresponding to the Ðner energy channels of Ulysses HET instrument. Di erent methods have been used by Simpson et al. (1995b) and McKibben et al. (1996) for the same data set and their numbers are reasonably consistent with this simple calculation. The latitude gradients derived from the HET measurements also agree with those from Kiel Electron Telescope (KET) measurements in the same COSPIN consortium on Ulysses (Heber et al. 1996) Amplitude of Recurrent Variations A simple method to calculate the amplitude for the recurrent Ñux variation can be normally done in the following way. The daily Ñux is subtracted and then normalized by its 27 day running average to become a detrended percentage Ñux deviation as those shown in graph C of Figure 1. The percentage deviation is averaged separately for the times when it is above 0 (denoted as A`) and below 0 (denoted as A ). The amplitude of the recurrent variation is deðned to ~ be the di erence of and A, that is A \ [ A, A` ~ A` ~ which is roughly proportional to the averaged bottom to peak di erence of recurrent deviation waves. However, we cannot use this method for most of our particle channels because the variations in these channels are dominated by random Ñuctuations due to counting statistics, which can be seen in graph C of Figure 2 as an extreme case, and their random statistical Ñuctuations would also contribute to the derived amplitude in the same way as the real recurrent variations do. For the D40È70 MeV protons (graph C, Fig. 2), for example, it is impossible to obtain the amplitude of 26 day variation using such a simple method. Even for the 3 day averages of D40È70 MeV per nucleon helium Ñux (graph A, Fig. 2), the statistical Ñuctuation level is of the order of ^5%. We cannot average the data over longer time periods in order to increase counting statistics signiðcantly because the 26 day variations may eventually be washed out in the averaging process. Fortunately, the H9 [ D100 MeV proton integral rate channel has a high counting rate throughout the entire mission. The statistical uncertainties for its daily averages are better than 0.6%, which is much less than the amplitude of its 26 day variation (Fig. 1, graph C and Fig. 2, graph B). Therefore, the calculation of the amplitude for this channel can be taken straightforwardly with the above simple method. As to the Ñuxes in other channels, we assume that their wave forms for the 26 day recurrent variation are the same as that for the H9 counting rate. This assumption has been tested among the channels which have large enough counting statistics, such as shown in Figure 3, and it has been proven valid. Also, the correlation in the Ñux variations was found to extend to very high energies of several GeV (Fonger 1953). From a theoretic point of view, since 26 day variations and Forbush decreases are both caused by compression of magnetic Ðelds associated fast solar wind streams (Chih & Lee 1986), the time proðle of the Ñux variations in one cycle of recurrent variations should resemble that of a Forbush decrease in the interplanetary space. Thus, like the Forbush decrease, the wave forms of the recurrent variations is approximately energy-independent (Chih & Lee 1986; Lockwood et al. 1986). The daily detrended percentage deviations of the particle Ñuxes are averaged for times when detrended H9 rate deviation (*H9) is above 0 to get A`, and similarly for A. The amplitude is ~ again [ A. In this way, the statistical random Ñuctua- A` ~ FIG. 3.ÈTime history of percentage daily variations of detrended (top) H4He D40È70 MeV per nucleon helium Ñux, (middle) H9E[D100 MeV proton rate, and (bottom) H13 E [ 173 MeV carbon and E [ 204 MeV per nucleon oxygen rate as measured by Ulysses HET. The H4He Ñux is 3 day running average, H9 rate daily average, and H13 rate 5 day running average.

6 846 ZHANG Vol. 488 tion is essentially removed from the amplitude calculation. This method is equivalent to the superposed epoch method or Chree analysis (Forbush et al. 1982), but allowing the recurrent cycle period to vary slightly according to the deviation in the H9 counting rate. We assign the error bars of the amplitude to be the average statistical uncertainties in their Ñuxes or counting rates during the averaging intervals. Another method for amplitude analysis is to use the Fourier transform: the spectral power density as a function of frequency can be derived from a time series data set of percentage Ñux deviation by using a fast Fourier transform scheme (Press et al. 1992, chap. 13.4). The spectra generally have a peak around f \ 1/26 day~1, even for those particle channels that do not obviously show a 26 day variation. But because of the restriction of the fast Fourier transform and limited length of the data set, which gives rise to frequency leakages in the spectrum, the peak always appears to be a broad peak, covering the frequency range approximately from 1/30 to 1/22 day~1. The total spectral power of the 26 day variation is summed within this frequency range and then subtracted by its background spectral power level, which can be determined from adjacent frequencies. The main advantage of the Fourier analysis method is that calculations in di erent particle channels are done independently of each other. But the accuracy for the spectral power determination is not clear to us (see Press et al. 1992). In addition, fast Fourier transform requires some Ðxed numbers of data points, thus making the analysis less Ñexible. In this paper, this Fourier analysis method is only used for the purpose of checking the validity of our Ðrst amplitude analysis method Results Figure 4 displays the amplitude of 26 day variations as a function of the latitude gradient in both the southern hemisphere (open symbols) and the northern hemisphere ( Ðlled symbols). The amplitudes were calculated from data measured in a sample period from 1993 January 1 to July 1. The reason for choosing this period is that the recurrent variations are the largest in the middle latitudes due to strong compression of the solar wind and magnetic Ðelds (Fig. 1, graphs A and B) and the derived amplitudes are much greater than their estimated errors. As one will see later, the correlation between the amplitude of recurrent variations and the latitude gradient remains the same independent of the time periods we choose to study. Each data point in Figure 4 represents a channel of a particular particle type and energy range. There are 10 proton channels, nine helium channels, and three mostly carbon and oxygen channels. Table 1 is a list of their key parameters. It is clear from Figure 4 that the amplitude and the latitude gradient have a linear relationship to each other roughly independent of particle species and energies. Low energy helium channels, which contain mostly singly ionized anomalous component, have large latitude gradients and as well as large amplitudes of 26 day recurrent variations. Low-energy protons, however, show little latitude gradients and thus have small amplitudes. A few lowenergy proton channels have negative values for the amplitude, meaning that their Ñux variations anticorrelate with the H9 rate for E [ D100 MeV high-energy protons. But the negative amplitude values are comparable to their error bars, perhaps indicating that these channels do not exhibit recurrent variations at all. Similarly, the negative latitude gradients for a few low-energy proton channels are also not signiðcant. FIG. 4.ÈLinear relationship between the latitude gradient and the amplitude of 26 day variations seen by Ulysses at middle latitudes in The data points correspond to the 22 channels listed in Table 1.

7 No. 2, 1997 LATITUDE GRADIENT AND RECURRENT VARIATION 847 TABLE 1 CHANNELS OF COSMIC-RAY MEASUREMENTS ON Ulysses AND IMP-8 LATITUDE GRADIENT AMPLITUDE Ulysses IMP-8 (percent deg~1) (percent) Name Particle Energy Name Particle Energy RIGIDITYa (MV) South North (1993 Jan 1ÈJuly 1) H4p p (39È71) MeV ID4LEp p (30È69) MeV 273È372 [0.03 ^ ^ ^ 1.0 H5p p (71È93) MeV ID4HEp p (63È95) MeV 372È ^ ^ ^ 1.1 D6p p (35È40) MeV * 259È ^ ^ 0.05 [4.9 ^ 3.3 K1p p (39È52) MeV * 273È317 [0.09 ^ ^ 0.03 [2.4 ^ 1.6 K2p p (51È64) MeV * 313È352 [0.07 ^ ^ ^ 1.7 K3p p (62È72) MeV * 347È ^ ^ ^ 1.8 K4p p (71È81) MeV * 372È ^ ^ ^ 1.9 K5p p (78È89) MeV * 390È ^ ^ ^ 2.0 K6p p (87È96) MeV * 413È ^ ^ ^ 2.0 H9 p [92 MeV RCK p [1.06 MeV [ ^ ^ ^ 0.1 H4He He (39È72) MeV/n ID4LEHe He (30È69) MeV/n 547È749 (1093È1498) 0.67 ^ ^ ^ 1.2 H5He He (72È95) MeV/n ID4HEHe He (70È95) MeV/n 749È865 (1498È1730) 0.40 ^ ^ ^ 1.8 D6He He (35È40) MeV/n * 517È554 (1034È1107) 0.79 ^ ^ ^ 3.4 K1He He (40È53) MeV/n * 554È639 (1107È1279) 0.70 ^ ^ ^ 1.8 K2He He (52È65) MeV/n * 633È710 (1267È1421) 0.65 ^ ^ ^ 2.1 K3He He (63È73) MeV/n * 699È754 (1398È1509) 0.67 ^ ^ ^ 2.5 K4He He (72È82) MeV/n * 749È801 (1498È1603) 0.43 ^ ^ ^ 2.9 K5He He (79È91) MeV/n * 786È846 (1572È1692) 0.52 ^ ^ ^ 3.2 K6He He (88È97) MeV/n * 831È875 (1663È1750) 0.24 ^ ^ ^ 3.5 H10 C (26È36) MeV/n HZLE C (18È50) MeV/n 445È ^ ^ ^ 1.0 O (30È42) MeV/n O (21È58) MeV/n 478È568 H11 C (44È127) MeV/n HZME C (50È120) MeV/n 581È ^ ^ ^ 1.0 O (51È149) MeV/n O (58È141) MeV/n 627È1098 H13 C [173 MeV/n HZHE C [50 MeV/n [ ^ ^ ^ 1.0 O [204 MeV/n O [58 MeV/n [1302 NOTE.ÈAsterisk denotes channels in which direct Ñux measurements are not available and Ñuxes are derived from other channels using power-law interpolation. a Rigidities are calculated only for the Ulysses HET channels. Numbers in parenthesis are for singly charged anomalous components that are present in the same energy channel. Latitude gradients in the southern and northern hemispheres derived from the fast latitude scan period are roughly equal (Simpson et al. 1996; and also Fig. 4). There is a little north-south di erence in the latitude gradient for some of the low-energy proton channels, but it does not seem to be signiðcant. Also, the latitude gradients derived from Jupiter to the south pole pass are consistent with those from the fast latitude scan (Simpson et al. 1995b). Therefore, we are conðdent that the latitude gradient of cosmic rays did not change much during the Ulysses mission to both the poles and latitude gradients derived from any time period can be used for correlation studies with recurrent variations in other time periods. In the rest of the paper, only the latitude gradients measured in the southern hemisphere during the fast latitude scan are used for further studies as representative of both hemispheres. Results from the Fourier analysis method are shown in Figure 5, where the total spectral power around f \ 1/26 day~1 is displayed as a function of the latitude gradient in the southern hemisphere. A parabola is drawn for the purpose of comparison. Although error bars for the spectral power are hard to estimate and not shown in the Ðgure, all the data points roughly follow a quadratic law, consistent with the amplitude being proportional to the latitude gradient as shown in Figure 4. Therefore, the amplitude calculation method can be used with conðdence. As shown in Figure 6, the linear relationship remains valid for 26 day variations observed at various solar latitudes and radial distances along the Ulysses trajectory. The six time periods for the calculation of the amplitude shown in Figure 6, each 182 days long, correspond to the six thick bars in Figure 1. These six time periods represent di erent characteristic regions: period A is in the low-latitude region, where the slow solar wind dominates; period B has the strongest variations both in the solar wind and cosmic-ray intensities; period C is in the steady high-speed solar wind, but the cosmic rays still undergo fairly large recurrent varia- FIG. 5.ÈSpectral power centered at frequency f \ 1/26 day~1 as a function of latitude gradient. A quadratic line has been drawn to indicate the consistency with the linear relationship shown in Fig. 4.

8 848 ZHANG Vol. 488 FIG. 6.ÈLinear relationship between the latitude gradient and the amplitude of recurrent variations observed at various locations indicated by the thick bars in Fig. 1. tions; periods D and F are taken from the south and north polar regions, respectively; period E covers the middle- and low-latitude regions during the fast latitude scan, which is the same region where the latitude gradients plotted in Figure 6 were measured. Therefore, we can argue that the linear relationship between the latitude gradient and the amplitude of the recurrent variations may likely exist for cosmic-ray intensity variations occurring everywhere in the inner heliosphere independent of the local conditions of the solar wind. It should be noted that the cosmic-ray intensity variations in the periods A and E seem to be more irregular, probably because the spacecraft was near the solar equator where the stream interaction regions are complicated and less well organized, but the linear relationship still holds. This implies that the linear relationship may be extended to any short-term temporal variations caused by interaction regions of the solar wind streams. In addition, the fact that the recurrent cosmic-ray intensity variations observed at high latitudes, where we Ðnd no recurrent variations in the solar wind and magnetic Ðelds, also hold a linear relationship with the latitude gradient, as those intensity variations in the low- and middle-latitude regions do, further suggests that the recurrent cosmic-ray variations in high-latitude regions probably come from the regions underneath in solar latitude. The linear relationship means that particles whose Ñux undergoes strong short-term time variations will also have a large pole-to-equator Ñux di erence. Cosmic-ray and anomalous nuclei Ñuxes measured at IMP-8 in the ecliptic were also undergoing recurrent modulations with a synodic period of D27 days and sometimes D13.5 days (see Fig. 1, graph D). It was realized 19 years ago by Garcia-Munoz et al. (1977b) that anomalous helium had stronger D27 day variations than galactic helium and protons. Their observation made near the equator is consistent with the Ðnding reported here with Ulysses measurements. We have used the same technique as for the Ulysses Ñuxes to calculate the amplitude for recurrent variations of the IMP-8 Ñuxes. Figure 7 shows a relationship of the amplitude of IMP-8 Ñux variations in a sample time interval with the latitude gradient. The latitude gradient, of course, is based mainly on the measurements from Ulysses. All the data points in Figure 7 form approximately a linear relationship, consistent with the observations by Ulysses. However, more detailed examination of the recurrent variations in other time periods reveals that all the IMP-8 Ñux channels, except occasionally the HZLE channel in Table 1, follow a linear relationship in the amplitude versus the latitude gradient. The reason that the HZLE channel is some-

9 No. 2, 1997 LATITUDE GRADIENT AND RECURRENT VARIATION 849 FIG. 7.ÈLinear relationship between the latitude gradient and the amplitude of recurrent variations seen by IMP-8 near the solar equator for a period in FIG. 8.ÈNegative linear relationship between the amplitude of recurrent variations and the latitude gradient for the previous solar cycle, when the solar magnetic Ðelds had a negative polarity. The latitude gradients were measured by the Voyager spacecraft (Christon, Stone, & Hoeksema 1986; McDonald & Lal 1986; Cummings, Stone, & Webber 1987). times o the linear relationship is that this channel has a lower energy threshold, signiðcantly below that of the H10 rate channel on Ulysses (see Table 1), and thus sometimes may contain substantially more anomalous oxygen, which dominates the spectrum below 30 MeV per nucleon near solar minima (Cummings et al. 1984). The anomalous oxygen of energies 8È20 MeV per nucleon has a latitude gradient of D2% deg~1 as measured by Ulysses Low Energy Telescope (LET; see Trattner et al. 1995), which is even much larger than that for the anomalous helium, but the LET data suitable for 26 day recurrent variation studies are not available to us. If we assume that the linear relationship holds valid for the anomalous oxygen of these energies, the recurrent intensity variations measured in the HZLE particle channel should then be stronger than in the H10 counting rate channel on Ulysses, which the latitude gradient measurement in this particle energy range mainly depends on. In fact, observations of low-energy oxygen from the Energetic Particle Composition (EPAC) experiment on Ulysses (Keppler et al. 1996) indicate that the anomalous oxygen Ñux was indeed undergoing a very strong recurrent modulation for this period. Cummings, Mewaldt, & Webber (1983) also reported very large 26 day recurrent variations in the anomalous oxygen intensity in 1978È1980. Thus it is probably true that the anomalous oxygen also satisðes the linear relationship along with other particle species Solar Magnetic Polarity Dependence It should be noted that the above observations by Ulysses and IMP-8 were made in the epoch when the solar magnetic Ðeld has a positive polarity with magnetic Ðeld lines going out from the north solar pole. During the previous 11 yr solar cycle, when the solar magnetic Ðeld had an opposite polarity, measurements of cosmic-ray Ñux latitude gradients were made by two Voyager spacecraft at radial distances of the order of D20 AU from the Sun (Christon, Stone, & Hoeksema 1986; McDonald & Lal 1986; Cummings et al. 1987). At that time, D27 day recurrent variations were also monitored by IMP-8 spacecraft at Earth. Figure 8 is a graph showing the amplitude of Ñux recurrent variations seen by IMP-8 as a function of the latitude gradient seen by the Voyager spacecraft at D20 AU for a time period in late Only three particle channels on Voyager can be chosen to make the particle energy ranges roughly match on both the spacecraft. The latitude gradients were negative for cosmic-ray and anomalous nucleon components in the 1985 negative solar magnetic polarity, consistent the prediction by the drift model of Jokipii and his collaborators (see, e.g., Jokipii 1989). Because the latitude gradient measurements of the Voyager spacecraft were strongly a ected by the radial gradients of cosmic-ray Ñuxes, the same data set had derived several di erent numbers for the latitude gradient depending upon the assumption of how the radial gradient was treated; thus the three particle channels give rise to six data points in Figure 8. However, even so, the amplitude still shows approximately a linear dependence with the latitude gradient. 3. IMPLICATIONS We have demonstrated a linear relationship between the magnitude of latitude gradients and the amplitude of recurrent 26 day variations in the Ñuxes of cosmic rays and

10 850 ZHANG anomalous components. This relationship exists for observations essentially everywhere in the inner heliosphere for both positive and negative solar magnetic Ðeld polarities. This new Ðnding may enable us to understand further about the cosmic-ray distribution in the three-dimensional heliosphere L atitude Gradient Monitor The linear relationship implies that the sizes of cosmicray recurrent time variations observed in low- and middlelatitude regions can in a certain way represent its polar Ñux excess over its equatorial level. This may enable us to sense the relative strength of the cosmic-ray latitudinal Ñux gradient with a space- or ground-based measurement of time variation of the Ñux near the equator. When it is combined with actual latitude gradient measurements using the diurnal anisotropy method in neutron monitor ranges (e.g., Chen, Bieber, & Pomerantz 1991), we could Ðgure out the latitude gradients for cosmic rays and anomalous components at all other energies. In this way we can monitor cosmic-ray latitude gradients continuously without sending spacecraft to high latitudes. This method is at least valid for the present phase of 11 yr solar cycle, i.e., solar minimum condition. It will be important to test whether this relationship still holds during UlyssesÏ second encounter with solar poles when the Sun is at its maximum activity conditions Similarity of Cosmic-Ray L atitude-l ongitude Distribution Maps Let us assume that the magnetic Ðeld structures including small-scale Ñuctuations and large-scale spiral magnetic Ðelds in the inner heliosphere do not change with time in the reference frame that rotates with the Sun. This can be an approximation to the real condition for a short time period, maybe up to a few solar rotations. In such a reference frame, cosmic-ray distributions can be described to be stationary. The trajectory of any spacecraft that moves very slowly with respect to the solar rotation makes almost a circle around the Sun; therefore, short-term time variations seen by spacecraft such as Ulysses and IMP-8 are actually longitudinal variation of cosmic-ray Ñux at a Ðxed latitude and radial distance. As the Ulysses spacecraft moves slowly in latitude, it makes measurements of an ensemble of longitudinal distributions at various latitudes, which can then be compiled to make a two-dimensional latitude-longitude contour plot of Ñux distribution similar to those typically presented in model calculation by theorists (see, e.g., Jokipii 1989; Ko ta & Jokipii 1991). However, as matter of fact, because the exact rotation period of heliospheric magnetic Ðelds is not known and most probably varies slightly with time, it is almost impossible to assign a simple meaningful longitude for the spacecraft position in the reference frame that rotates with the magnetic Ðelds; therefore, the latitudelongitude Ñux distribution map cannot be made precisely like those from theoretical calculations. In Figure 9 are three gray-scale maps of cosmic-ray latitude-longitude Ñux distributions, but they are shown in a conceptual way. They are actually cycligrams of Ñux time measurements from near Jupiter encounter to the maximum south latitude. The period of each cycle has been chosen to be 26 days. If the heliospheric magnetic Ðeld rotates at that rate, the horizontal axis is then the spacecraft longitude. On the right side of the vertical axis is the Ulysses latitude shown in a nonlinear scale. Therefore, Figure 9 presents distorted pictures of the latitude-longitude Ñux distribution maps, with the only difference being the nonlinearity of longitude and latitude scales. If the rotation period of the heliospheric magnetic Ðelds is not 26 days, then the constant longitude line is slanted instead of vertical. But Figure 9 still more or less exhibits a picture for latitude-longitude Ñux distributions of cosmic rays. Shown in Figure 9 are the deviations of Ulysses to IMP-8 Ñux ratios (same as those plotted in Fig. 2) from their 1 AU equatorial value (which is 1 by deðnition) and then normalized to their own south latitude gradients. Their values are represented by the grayness of each pixel (daily) with the darker being the higher Ñux level. Apparently, there exist similarities among the three Ñux distribution maps. Maximum Ñux in each 26 day cycle occurs near the center of the horizontal axis in all the graphs. Although there are slight di erences in their exact Ñux levels, some of the small structures are common for all the three maps. For example, the dark areas occurring in the center between [22 and [30 and between [50 and [66 appear in all the three maps. In the vertical (latitude) direction, the levels of Ñux change relative to its own latitude gradient are roughly equal, particularly for the top and bottom maps. In the middle map, which is the ratio of Ulysses H9 rate to IMP-8 RCK rate, grayness changes slightly faster in vertical direction. We believe this to be the result of an instrumental e ect, where the lower energy cuto of the IMP-8 RCK integral counting rate shifts upward slowly with time so that it gives rise to an additional false latitude gradient during the 2.5 year slow latitude scan period. In the south pole-to-equator and then equator-to-north pole pass, which occurred in less than 6 months in each of the hemispheres, correction of the RCK rate for the shift in its energy response is not required. Unfortunately a similar way of constructing latitude-longitude Ñux distribution maps is not possible for the south pole to north pole period because the spacecraft moved too fast in latitude (it covered on average 13 latitude every solar rotation). In fact, the similarities in the above distribution maps have a fundamental basis built on the linear relationship between the amplitude of 26 day variations and the latitude gradient. Cosmic-ray di erential density at momentum p, position r, h, / (radius, latitude, and longitude) is written as f (p, r, h, /) \ f (p, r \ 1 AU, h \ 0 )h(p, r, h, /), (3) 0 where h(p, r, h, /) is the Ñux distribution function normalized to its value at 1 AU in the equator. Because of the observed small latitude gradients and 26 day variations, we can often write h(p, r, h, /) \ [1 ] g (p, r)h][1 ] A(p, r, h)u(/)], (4) h where g (p, r) is the latitude gradient, A(p, r, h) is the ampli- tude of h recurrent variation, and u(/) is a periodic function with average over the entire longitude / being 0. The linear relationship found in this paper requires that A(p, r, h) \ a(r, h)g (p, r). (5) h Note that a(r, h) is not a function of p. Neglecting the second-order term of g, we can derive from equation (4) that h h(p, r, h, /) [ 1 \ h ] a(r, h)u(/). (6) g (p, r) h

11 FIG. 9.ÈGray-scale images of daily average Ulysses to IMP-8 Ñux ratios in three particle channels for the period from Jupiter encounter to the highest south latitude. The horizontal axis is the phase in 26 day recurrent cycle or equivalent to a solar longitude coordinate. The vertical axis is the number of rotation cycles, and it corresponds to heliographic latitude in a nonlinear fashion (right axis). Dots are times of actual current sheet crossings as observed by the magnetometer experiment on the Ulysses spacecraft ; open dots are from south to north and Ðlled dots are in the opposite direction. Curves are drawn to indicate an approximate position for the heliospheric current sheet.

12 852 ZHANG Vol. 488 The quantity on the left-hand side of equation (6), [h(p, r, h, /) [ 1]/[g (p, r)], is what we plotted in Figure 9 in gray scales. The right-hand h side of equation (6) does not depend on particle species and momentum, thus giving rise to similar distribution maps for particles of di erent energies as shown by Figure 9. The similarity in the latitude-longitude Ñux maps among di erent particles allows us to construct the threedimensional distribution of one type of particles from that of any other particles, once the latitudinal proðle (which determines g ) and radial proðle are known for both particle Ñuxes. The three h dimensional problem can then be reduced to a two-dimensional problem, which is much easier to solve numerically and to understand analytically T heoretical Interpretation The linear correlation between the amplitude of recurrent variations and the latitudinal gradient in the Ñuxes of cosmic rays and anomalous components indicates that there is a uniðed mechanism that controls both the Ñux variation from poles to the equator and the Ñux variation associated with corotating structures. This mechanism could be either a part of interplanetary magnetic Ðelds or a modulation process. A more detailed analysis on this subject can be found in Paper II of this series. The following is a brief summary. If there is a single interplanetary magnetic Ðeld structure that can control both global latitudinal and longitudinal cosmic-ray Ñux distributions, it is possible that there exists a correlation between the latitude gradient and longitudinal recurrent variations. This magnetic Ðeld then must have both a global inñuence and a recurrent property. In the interplanetary medium, such possible recurrent structures are corotating interaction regions (CIRs) and tilted heliospheric current sheet. CIRs are conðned in low-latitude regions, and they cannot determine the global latitudinal Ñux distribution. This conclusion has been reached by a numerical simulation of Ko ta & Jokipii (1991). In particular, during the present solar cycle with magnetic polarity A [ 0, positively charged cosmic-ray particles come from polar regions of the heliosphere and CIRs in low latitude should have no such e ect on the polar Ñux level. The heliospheric current sheet is also a corotating interplanetary structure, which may satisfy both the global and recurrent requirement. For example, if the cosmic-ray Ñux is organized by helio-magnetic latitude, which corresponds to the distance to the heliospheric current sheet, then the rotation of the tilted current sheet, which gives rise to a periodical changes of the helio-magnetic latitude at the spacecraft, can simulate a recurrent Ñux variation. In this situation, the correlation between the latitude gradient (with respect to the magnetic latitude) and the amplitude of recurrent variations is guaranteed. However, detailed phase analysis (Zhang et al. 1995; and also in Paper II) found that the current sheet does not seem to have a close correlation with the recurrent cosmic-ray variation. Instead, CIRs, with compressed magnetic Ðelds, are more likely to be a dominant source for the recurrent decreases of cosmic-ray Ñuxes, much as those so-called CR-B relationship found by Burlaga et al. (1985). On the other hand, if the latitude gradient and recurrent variation of cosmic rays are determined from modulation of two independent magnetic Ðeld structures, i.e., the recurrent variations are caused by compressed magnetic Ðelds of CIRs and the latitude gradient are determined by the global interplanetary magnetic Ðeld from the poles to the equator, then the correlation requires that there must be a common modulation process controlling the Ñux variation in these phenomena. This is mainly the idea proposed in Paper II. Such a modulation process we found is the adiabatic cooling e ect by expanding solar wind. As cosmic-ray particles transport from polar regions to the equatorial, they must continuously undergo adiabatic energy loss, thus, resulting lower Ñuxes along their path. Similarly, the Ñux decreases caused by compressed magnetic Ðelds of CIRs are results of adiabatic deceleration of particles trapped behind the strong Ðelds (Thomas & Gall 1984). Thus both the latitude gradient and the amplitude of recurrent variations should be determined by the amount of additional energy loss due to the extra modulation in these phenomena and as well as the shape of cosmic-ray spectra at the location of observation (the shape of the spectra determines how the Ñux varies with energy changes). It should be pointed out that both the latitude gradient and the amplitude are a deviation of cosmic-ray Ñuxes caused by a perturbation in the modulation that cosmic-ray particles have experienced because of di erent locations or di erent interplanetary media. A detailed analytical calculation (see Paper II) shows that the variation of cosmic-ray Ñux involved in these phenomena can be written in a form similar to that derived by the force-ðeld approximation (Gleeson & Axford 1968), *j \[3C*', (7) j where C \ 1/3M2 [ [(E ] 2E )/(E ] E )L ln j/l ln E]N is the 0 0 Compton-Getting factor which can be determined from the shape of cosmic-ray energy spectra, and ' is the modulation parameter. In general, ' \ / (V /3i)dl, where V is the sw sw solar wind speed, and i is a kind of di usion coefficient that measures the particle transport either through di usion or gradient/curvature drifts, and the integral is taken along an average trajectory of particles. In the recurrent variation, *' is inversely proportional to the di usion coefficient inside the CIRs (i ), while the *' which determines the cir latitude gradient is related to either perpendicular di usion (i ) or particle drift (i ). If the rigidity dependencies of i, i M,and i are similar, A then the Ñux perturbation (*j)/j cir associated M A the latitude gradient and the recurrent variations should yield a similar form as a function of particle rigidity. Thus there is a linear relationship between the latitude gradient and the amplitude. This conclusion is approximately valid even if the rigidity dependences of i, i, and i are slightly di erent because rigidity dependence cir M of Compton- A Getting factor C is much stronger than that for the ratios of the iïs. In addition, the above derivation suggests that the radial gradient of cosmic-ray Ñuxes should also have a similar linear relationship with the latitude gradient. This theory is tested in Paper II. The linear relationship of the amplitude of recurrent variations versus the latitude gradient is the same for both galactic and anomalous cosmic rays. This indicates that in spite of their di erent origins, anomalous components are modulated in the same way as galactic cosmic rays in the inner heliosphere. The reason that anomalous cosmic rays