TOWARD AN AB INITIO THEORY OF THE SOLAR MODULATION OF ...

The Astrophysical Journal, 585:502–515, 2003 March 1 # 2003. The American Astronomical Society. All rights reserved. Printed in U.S.A.

TOWARD AN AB INITIO THEORY OF THE SOLAR MODULATION OF COSMIC RAYS S. Parhi, J. W. Bieber, and W. H. Matthaeus Bartol Research Institute, University of Delaware, Newark, DE 19716

and R. A. Burger School of Physics, Potchefstroom University for CHE, Potchefstroom, South Africa Received 2002 August 26; accepted 2002 November 6

ABSTRACT Recent efforts to improve and complete modulation theory are described, emphasizing factors that arise in attempts to understand the diffusion tensor based on turbulence theory and the theory of charged particle scattering. Direct numerical solutions of the transport equations are used to illustrate several sensitive factors affecting modulation that are difficult to understand either observationally or physically. First, the nature of perpendicular diffusion is still poorly understood. We introduce a new perpendicular diffusion formalism that can smoothly vary between the field line random walk limit and a recent treatment based on the TaylorGreen-Kubo equation. Second, this new perpendicular diffusion coefficient is shown to be sensitive to an outer scale of turbulence, the ‘‘ ultrascale,’’ which we know almost nothing about. Third, the radial variation of the ordinary correlation length is still poorly characterized, in part owing to the uncertain impact of pickup ion–driven turbulence in the outer heliosphere. These three factors encompass a very diverse set of possible solutions of the modulation equation, only a small part of which even remotely resembles available observations. Subject headings: cosmic rays — diffusion — scattering — solar-terrestrial relations — Sun: particle emission — turbulence

tions (Giacalone & Jokipii 1999). Further, dynamical effects (Bieber et al. 1994), wave damping (Schlickeiser & Miller 1998), and resonance broadening (Dro¨ge 2000) at lower energies are still active areas of study. For reasons such as these, it is obvious why there is at present no accepted ab initio modulation theory, that is, one in which the diffusion coefficients are determined on the basis of scattering theory and the underlying magnetic fluctuation parameters are computed from plasma theory and known features of heliospheric structure. Recently, we have been attempting to develop an ab initio approach to modulation, building on recent efforts along these lines (Pauls, Zank, & Williams 1995; Zank et al. 1998; Burger & Hattingh 1998; Le Roux, Zank, & Ptuskin 1999; Burger, Potgieter, & Heber 2000; Parhi et al. 2001, 2002). The development of an ab initio theory faces at least three major challenges: First, we do not yet have a satisfactory theory of diffusion perpendicular to the large-scale magnetic field. Available theoretical formulations (Jokipii & Parker 1969; Forman, Jokipii, & Owens 1974; Bieber & Matthaeus 1997, hereafter BAM) and numerical simulations (Giacalone & Jokipii 1999; Mace et al. 2000) of perpendicular diffusion are rather widely divergent. Second, perpendicular diffusion in two-component slab/two-dimensional turbulence depends critically on an outer scale termed the ‘‘ ultrascale ’’ (Matthaeus et al. 1995, 1999a, 1999b; Gray Pontius, & Matthaeus 1996), about which we have little observational information. Third, the radial variation of both the parallel and perpendicular diffusion coefficients is strongly dependent on radial variation of the ordinary correlation length. However, variation of the correlation length is very poorly understood, in part owing to the uncertain impact of pickup ion–driven turbulence in the outer heliosphere (Zank, Matthaeus, & Smith 1996; Zank et al. 1998; Smith

1. INTRODUCTION

Cosmic-ray modulation in the heliosphere is a complex process by which the Sun alters the intensity and energy spectrum of Galactic cosmic rays that enter into our heliosphere. This involves at a detailed level important and incompletely understood theoretical features of charged particle scattering and plasma turbulence. Scattering theory involves turbulence parameters, and thus one needs to understand how plasma turbulence evolves throughout the heliosphere. Even in the simplest formulations of scattering and turbulence theories, this would involve specification of a turbulence energy density and a correlation length scale everywhere in the three-dimensional heliosphere. Boundary data in the inner heliosphere can stand in place of a complete understanding of the origins of solar wind turbulence, but it is also necessary to understand how turbulence is driven within the heliosphere, for example, by shear and by excitation of fluctuations by scattering of interstellar pickup ions. In more elaborate models of interplanetary turbulence, the fluctuations may comprise two or more components, each with a known symmetry (Marsh & Tu 1989; Matthaeus, Goldstein, & Roberts 1990; Matthaeus, Smith, & Bieber 1999a; Bieber et al. 1994; Bieber, Wanner, & Matthaeus 1996). A useful example is the two-component model, which includes a one-dimensional ‘‘ slab ’’ and twodimensional ingredients. It is also important to note the effect of the charged test particle diffusion tensor on modulation. Perpendicular diffusion, especially at lower energies (Giacalone & Jokipii 1999; Mace, Matthaeus, & Bieber 2000), is not well accounted for theoretically. While the situation appears to be better for parallel diffusion (Bieber et al. 1994), there are still reported discrepancies relative to test particle simula502

AB INITIO THEORY OF SOLAR MODULATION

where the magnetic fluctuation at position x is bðxÞ and the new position x0 is displaced from the old position x by a spatial lag z parallel to the mean magnetic field; b2 ¼ b2x þ b2y is the rms turbulent magnetic field, and h i denotes an ensemble average over the relevant distribution of field lines. Correlation lengths measured in the solar wind by single spacecraft differ from the actual parallel correlation length in that the lags are generally radial and not aligned with the mean magnetic field; such observed correlation scales are typically 1011–1012 cm at 1 AU. Despite recent progress (Zank et al. 1998; Le Roux et al. 1999; Burger et al. 2000), we still face major challenges, which we illustrate here by the direct numerical solution of transport equations. This is accomplished by introducing a new perpendicular diffusion formalism that smoothly varies between the field line random walk (FLRW) limit and a recent model (BAM) that emerged from the use of the Taylor-Green-Kubo equation. We then study the sensitivity of modulation to the form of the perpendicular diffusion coefficient (BAM vs. FLRW, as well as intermediate forms), to the ultrascale, and to the radial variation of the ordinary correlation length.

2. DISCUSSION OF SIGNIFICANT FACTORS

2.1. Nature of Fluctuations Fluctuations in the wind and interplanetary magnetic field may be waves that are remnants propagating out of the solar corona (Smith et al. 2001). A traditional view (Belcher & Davis 1971) is that the wave can propagate independently, but interaction with inhomogeneity can lead to reflection and the possibility of wave-driven turbulence (Marsh & Tu 1989; Matthaeus et al. 1999b). Fluctuations may also arise as a result of large-scale wind shear and evolve nonlinearly in a manner analogous to classical hydrodynamic turbulence. Single-point observations cannot distinguish among these possibilities. Theoretical treatments from both MHD wave and MHD turbulence perspectives coupled with observations have brought many insights into the study of these fluctuations and their local properties. There has been continuing interest in describing and explaining the spatial and temporal transport of fluctuations that interact with inhomogeneous large-scale solar wind features and in addressing the transport of noninteracting waves (Hollweg 1974; Whang 1980; Marsh & Tu 1989; Zhou & Matthaeus 1990). More recent theories attempt to describe both transport and turbulence effects (Matthaeus et al. 1999b; Smith et al. 2001). Spacecraft instruments on board Voyager and Pioneer have supplied magnetic field data and plasma data that suggest organized large-scale fluctuations in heliospheric plasma. These largescale solar wind fluctuations are possibly generated by shocks, merged interaction regions, coronal mass ejections, or some nonuniform admixture of fluctuations.

Because of the complexity of a general model of the solar wind, various simplifying assumptions have been considered in recent years to derive a model that explains both fast and slow fluctuations. Zhou & Matthaeus (1990) describe a one-fluid model approach regarding the separation of the time and space scales that are associated with large-scale solar wind fields evolving over slow timescales and the small-scale, faster evolving MHD fluctuations that are considered to be incompressible. This requires that the correlation length of the fluctuations be much less than the scale of the large inhomogeneities. This allows the full compressible MHD equations to be separated into dynamical equations for large- and small-scale fields and provides a first approximation to the spectral evolution for the radial dependence of interplanetary fluctuation spectra. 2.2. Diffusion of Magnetic Field Lines Recently, a nonperturbative approach has been developed for a two-component model (slab plus two-dimensional turbulence) to better understand the diffusion of magnetic field lines (Matthaeus et al. 1995; Gray et al. 1996). In the two dimensions plus slab model of magnetic turbulence, we assume B ¼ B0 þ bðx; y; zÞ, where B 0 ¼ B0^z; b ? ^z; and b ¼ b2D ðx; yÞ þ bslab ðzÞ. In general, we can write b2D ðx; yÞ ¼ µ ½aðx; yÞ^z, where a^z is the vector potential for the two-dimensional component of magnetic turbulence. The magnetic field line diffusion coefficients are found to be a nonlinear combination of magnetic field line wandering for slab turbulence with lslab as the parallel correlation length and the same associated with two-dimensional fluctuations with ultrascale ~l as an outer scale weighted by the two-dimensional magnetic fluctuations. The field line diffusion coefficient in composite turbulence, D? , is related to the separate slab and two-dimensional diffusion coeffiand D2D cients Dslab ? ? as follows (Matthaeus et al. 1995; Ruffolo & Matthaeus 2001): D

et al. 2001) and in part owing to difficulties in measuring (especially in the outer heliosphere) the correlation length in the relevant direction, namely, parallel to the magnetic field. The parallel correlation scale can be defined as R1 hbðxÞ x bðx0 Þidz ; ð1Þ c ¼ 0 hb2 i

503

D? ¼ Dslab ? þ

2 ðD2D ? Þ ; D?

where Dslab ? ¼ D2D ?

Z 1 b2slab lslab 1 Pslab xx ð0Þ ¼ ðkz ÞPslab ; xx ðkz Þdkz ¼ 2 2 2 B20 2B0 2B0 1 1=2 Z 1 Z 1 2D Pxx ðkx ; ky Þ b2D~l 1 ¼ ¼ dkx dky : 2 B0 2B20 1 1 k?

2D The above equations for Dslab ? and D? are slightly different from the corresponding equations in Ruffolo & Matthaeus (2001) because Ruffolo & Matthaeus employ symmetric normalization of Fourier transforms, whereas we use a normalization such that the turbulence power spectrum Pxx can be written in terms of a Fourier transform of the magnetic correlation function Rxx with respect to its spatial coordinate as follows (Bieber et al. 1994): Z 1 Rxx ðxÞ ¼ Pxx ðkÞeik x x d 3 k ; 1

Pxx ðkÞ ¼ Here,

variance

Z

1 3

ð2Þ

of

the

1

Rxx ðxÞeik x x d 3 x :

1

slab

magnetic

fluctuation

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The modulation of Galactic cosmic rays is described by Parker’s transport equation (Parker 1965) for the omnidirectional distribution function f ðr; p; tÞ for particles with momentum p at position r as diffusion

@f ¼ @t

drift convection zfflfflfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflfflfflffl{ zfflfflfflfflffl}|fflfflfflfflffl{ zfflfflffl ffl}|fflfflfflffl{ ðsÞ x K x f hvd i x f V w x f adiabatic energy loss

zfflfflfflfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflfflfflfflffl{ source zfflfflfflfflffl}|fflfflfflfflffl{ 1 @f þ ð x V wÞ þ Qðr; p; tÞ ; 3 @ ln p D

Pickup ions are connected to a number of ambiguities. Although wave-particle theory (Lee & Ip 1987) suggests the presence of power near resonant wavenumbers due to pickup-induced waves in the outer heliosphere, direct evidence of these waves is lacking. However, there is some evidence that the observed overall fluctuation level at greater than 10 AU requires injection of additional power, e.g., by the pickup process (Matthaeus et al. 1999b; Smith et al. 2001). Turbulence evolution also affects the correlation scale (Zank et al. 1998). To the extent that lslab can be determined in the outer heliosphere, there is no evidence that it decreases in observations. In fact, correlation-scale varying proportional to r is suggested by certain observations (Horbury et al. 1996; Smith et al. 2001). However, the correlation

3.1. Models and Strategy

D

2.3. Observation/Modulation Discussion

3. MODULATION CALCULATIONS

D

where fs is the fraction of the turbulent energy in the slab component (the remainder being in the two-dimensional component). Physically, the ultrascale can be considered as the length obtained from the ratio of mean square flux to mean fluctuation energy. It is therefore a measure of the mean size of poloidal (two-dimensional) flux structures, which could be of the order of the system size. In other words, a finite ultrascale could relate to the question of whether all poloidal structures are contained within a given system size. If so, then for still larger system size, the value of the ultrascale will remain finite. However, there are physical reasons indicating that it could be divergent (Matthaeus et al. 1999a). Another issue concerning the ultrascale is whether there are signatures in the correlation functions that are connected with the ultrascale. Physically, the ultrascale is governed by large-scale solar wind fluctuations. It is not exactly known how much fluctuation energy resides in ultrascale fluctuations, although it is known that a substantial amount of energy resides in spatial scales between the ion inertial scale (50 km) and the observed correlation scale (1:5 1011 cm). Still, very little appears to have been done to describe these fluctuations intuitively or in terms of the mathematics of the FLRW. Nevertheless, it is to be noted from the definitions of D2D ? slab 2D 2 and Dslab ? that D? / ðb2D =B0 Þ, whereas D? / ðbslab =B0 Þ , and hence for small-amplitude magnetic field fluctuations, which means b=B0 0, the correct model should include the ultrascale for the two-dimensional diffusion coefficient and should not simply consider the slab or quasi-linear approach. Even though there remain observational questions regarding the ultrascale, it is in fact very easy to control in numerical models, as described below. Thus, one of our goals in this paper is to explore the sensitivity of modulation to the ultrascale.

scale that is measured describes correlations along a radial profile. In the inner heliosphere (where the large-scale field is nearly radial), this corresponds to lslab , but in the outer heliosphere (where the field becomes azimuthal), it describes a physically distinct quantity related to the two-dimensional component of the turbulence. For simple models of the slab spectrum, the correlation length delimits two regimes with different rigidity (P) scalings of the parallel mean free path k . By definition, P ¼ pc=Ze, where c is the speed of light, Ze is the particle charge, and p is the momentum of the particle. The scalings are as k / P1=3 or k / P2 (for a power spectrum that has a k5=3 inertial range and a flat k0 energy range), depending on whether the particle Larmor radius is respectively smaller than or larger than the correlation length, as described in detail in x 3.1.3 and Parhi et al. (2002). For a magnetic field magnitude scaling as r1 , the Larmor radius scales proportionally to r. If lslab / r0:3 , then there is a rapid migration of particles from the k / P1=3 regime to the k / P2 regime as we trace them backwards into the outer heliosphere. With larger mean free paths, diffusion becomes relatively more important, leading to diffusiondominated modulation. In contrast, if lslab / rþ1 , then greater than 10 GV particles remain in the k / P1=3 regime throughout the modulation process. With small mean free paths, modulation becomes drift dominated.

D

b2slab ¼ hj bslab ðzÞ j 2 i, variance of the two-dimensional mag2 ¼ k2 þ k2 , and netic fluctuation b22D ¼ hj b2D ðx; yÞ j 2 i, k? x y k ¼ ðkx ; ky ; kz Þ denotes the wavevector. In particular, 2D 2D R2D xy ðx; yÞ hbx ð0; 0Þby ðx; yÞi (assuming homogeneity of 2 the turbulence). Thus, we get P2D xx ðkx ; ky Þ ¼ ky Aðkx ; ky Þ and 2 Aðk ; k Þ, where Aðk ; k Þ is the Fourier ðk ; k Þ ¼ k P2D x y x y x y yy x transform of the autocorrelation hað0; 0Þaðx; yÞi (Ruffolo, Chuychai, & Matthaeus 2001). The ultrascale ~l for nonaxisymmetric magnetic fluctuations can also be defined mathematically (Ruffolo et al. 2001) as R1 R1 ha2 i 1 1 A dkx dky ~l ¼ R 1 R 1 ¼ ; 2D 2D ð1 fs Þb2 1 1 ðPxx þ Pyy Þdkx dky

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D

504

ð2Þ

where V w is the solar wind velocity, Q is a source term, and hvd i is the particle drift velocity. The term KðsÞ is the symmetric part of the diffusion tensor with the elements k , ?, , respectively the diffusion coefficients parallel to and and r ? in the two directions perpendicular to the mean background magnetic field. Here r and h are the heliocentric radial distance and colatitude (polar angle), respectively. The distribution function f is related to the differential intensity with respect to the kinetic energy, jT , by jT ¼ p2 f . For our steady state two-dimensional model (@f =@t ¼ 0) with no source term, we describe the anisotropic diffusion by the radial difr sin2 , with the spifusion coefficient rr ¼ k cos2 þ k? ral angle (the angle between the mean field and radial direction), and by ? , the diffusion coefficient perpendicular to the mean magnetic field in the polar direction. In our r acts only in the radial direction two-dimensional model, k? (because @f =@ ¼ 0), and the model simulates the effect of a wavy current sheet (Burger & Hattingh 1995; Hattingh & Burger 1995) by using an averaged drift field with only an r and a h component. The tilt angle of the wavy current sheet is set at 20 , a value appropriate for moderate solar modulation conditions. The solar wind velocity is radial.

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The solar wind speed is 400 km s1 in the equatorial plane and increases to 800 km s1 in the polar regions. 3.1.1. Boundary Conditions

The interstellar proton spectrum as a function of rigidity is taken to be (Bieber et al. 1999; Burger et al. 2000) fIS ðPÞ ¼ 8 expð9:472 1:999 0:69382 þ 0:29883 0:047144 Þ ; > > > < P < 7 GV ; > 1:9 104 P2:78 ; > > : P 7 GV ; with P in units of GV and fIS in units of particles (GeV sr m2 s)1, where ¼ log10 P. Please note that this equation is slightly different from Burger et al. (2000) because of a typographical error in their paper. The heliosphere is taken to be spherical with an outer boundary at rB ¼ 100 AU. The following boundary conditions are used: 1. At the outer boundary of the heliosphere, r ¼ rB , the local interstellar spectrum is taken as the input spec-

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trum; i.e., f ðrB ; ; pÞ ¼ fIS ðpÞ for all values of h. This condition is valid since the termination shock is not considered in this time-independent model. We are aware that the region beyond the termination shock may modify the modulation (Webber & Lockwood 2001), but it is beyond the scope of this paper to include this region in the simulation. 2. The inner boundary is taken at r1 > r , where r is the radius of the Sun. Here we use a vanishing derivative condition, namely, df =dr ¼ 0. This is also called the Neumann boundary condition. 3. On the polar lines ( ¼ 0; ), there cannot be any streaming in the meridional direction, and for the Parker field it follows that @f ¼0: @ ¼0; 3.1.2. Perpendicular Diffusion Coefficients r We consider k? as some parametric combination of one limit established by the Green-Kubo-Taylor equation (BAM) and the other limit established by the FLRW. Thus,

Fig. 1.—Results from an ab initio modulation model for ¼ 0 (BAM), lslab / r0:3 , and ~l ¼ 200lslab . The left-hand panels display model predictions for (a) the Galactic and 1 AU spectrum, equatorial plane, (c) the radial profile for 200 MeV particles, equatorial plane, and (e) the average latitudinal gradient from the equator to 80 at 2.1 AU. The red diamonds and blue stars used in the left panels for both model and observational results are for negative and positive solar polarities, respectively. The Galactic spectrum in (a) is indicated by green. The right-hand panels display key elements of the diffusion tensor. Specifically, the radial mean free path rr , latitudinal mean free path , and drift scale A are plotted vs. (b) rigidity, (d ) radius at high latitude (80 ) for 200 MeV protons, and ( f ) radius in the equatorial plane for 200 MeV protons. The radial profile data (Webber & Lockwood 2001) are from Voyager and IMP, the latitudinal gradient data (Heber et al. 1996) are from Ulysses, and the 1 AU spectrum is from IMP data (McDonald et al. 1992). The error bars on the observations are comparable to the size of the data points and hence omitted.

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r we define k? as r k? ¼

VRL 1 2 2 3 1 þ 2 2

1 ;

ð3Þ

where ¼

2 RL : 3 D?

Here RL is the particle Larmor radius, V is the particle speed, and ¼ V =RL is the angular gyrofrequency. We also have introduced the timescale , which is the effective timescale for the decorrelation of the particle trajectories. Here is the parameter such that when ¼ 0 it corresponds to the BAM limit and when ¼ 1 it corresponds to FLRW limit. Note that when RL 4D? both the FLRW and BAM limits r =V is rigidity independent. On the other are the same and k? r =V / P22 . hand, if RL 5 D? , then 5 1 and hence k? Thus, ? 3? =V is independent of rigidity for the FLRW limit, and ? / P2 for the BAM limit (Cummings, Stone, & Webber 1994) and for simulations involving some intermediate values of (Giacalone & Jokipii 1999; Qin, Matthaeus, & Bieber 2002), such as ¼ 0:5, ? / P. r Anisotropic perpendicular diffusion with ? > ? follows naturally from the theoretical work of Jokipii (1973).

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Ulysses observed an increase in the variance of the components of the magnetic field as it moved from the equatorial plane to a latitude of 80 . The variance in the transverse and normal directions increases by a factor of 2.5 more than that in the radial direction (Balogh et al. 1996). In this r paper we set ? as 5 times ? at high latitudes. For drift we use the form A ¼ VRL =3. The motivation for this weak scattering form is given in Burger et al. (2000). 3.1.3. Parallel Diffusion Coefficients

We consider k 3k =V as defined in Zank et al. (1998): 5=3 1=3 B0 P 7=9A 2=3 k ¼ 3:1371 2 lslab 1 þ ; ðq þ 1=3Þðq þ 7=3Þ bx;slab c ð4Þ where A ¼ ð1 þ s2 Þ5=6 1 ; s¼ q¼

0:746834RL ; lslab 5s2 =3

1 þ s2 ð1 þ s2 Þ1=6

;

Fig. 2.—Same as in Fig. 1, but for ¼ 0:5. This corresponds to a model between the BAM and FLRW limits. At 1 AU, the modulation is weaker and the drift is stronger compared to those quantities in Fig. 1. There is a significant increase in , resulting in a steep drop in the latitudinal gradient, which almost vanishes for both polarities.

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Fig. 3.—Same as in Fig. 2, but for lslab / r. The modulation is much weaker than indicated by observations.

and b2x;slab is the variance of the x-component of the slab geometry fluctuations. Equation (4) is in very close accord with the exact Fokker-Planck result (Zank et al. 1998). One should note that the expression for k may not be valid for small rigidities when dynamical MHD turbulence effects are important (Bieber et al. 1994). The term in front of the square brackets on the right-hand 1=3 2=3 side of equation (4) can be written as ðB0 =bx Þ2 RL lslab . The fractional term inside the brackets is important for certain energies when RL is larger than or comparable to lslab . When this happens there is no resonant scattering with turbulent fluctuations in the inertial range 1=3 2=3 (k / RL lslab when s ! 0); rather, scattering occurs with fluctuations residing in a much flatter energy-containing range (k / R2L =lslab when s ! 1; Zank et al. 1998). Thus, it is apparent that the scaling of k with respect to both rigidity and correlation length can change from the inner to the outer heliosphere depending on how the correlation length evolves with heliocentric distance. Note that the radial dependence of b2 follows from the models of Zank et al. (1996). In the equatorial plane its radial dependence changes from r3:6 to r2:2 at about 7 AU (Burger et al. 2000). Since the radial dependence of the magnitude of the average magnetic field changes from r2 close to the Sun to r1 in the outer heliosphere that we consider, the ratio b2 =B20 has only a weak radial dependence in most of the heliosphere. This is also true in polar regions.

3.2. Numerical Results We employ the direct numerical simulation (Burger & Hattingh 1995; Hattingh & Burger 1995) of the steady state version of equation (2) with no source term by varying , ~l =lslab , and the correlation length and study its impact on particle intensity, mean free paths and drift scale, and radial and latitude gradients in our attempt to build an ab initio model of modulation. Our base line quantities are ¼ 0 (BAM limit) and ~l =lslab ¼ 200, and the correlation length of the magnetic field varies as r0:3 . Consider first the left-hand panels in Figures 1–9. The panels show (a) the predicted energy spectrum for positive and negative polarity at 1 AU in the equatorial plane, (c) the radial profile in the equatorial plane for 200 MeV protons, and (e) the energy dependence of the latitude gradient at 2.1 AU. The latitude gradient G is defined as Iðr; 2 Þ 100 G ðrÞ ¼ ln ; Iðr; 1 Þ 2 1 with Iðr; i Þ the intensity at position (r; i ), where i denotes latitude. Ulysses observations of this quantity are currently available (Heber et al. 1996) only for positive solar polarity. The radial profile is compared with data from Voyager and the Interplanetary Monitoring Platform (IMP; Webber & Lockwood 2001). The 1 AU spectrum is compared with IMP data (McDonald et al. 1992). The right-hand panels in

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Fig. 4.—Same as in Fig. 1, but for ¼ 1:0. This corresponds to the FLRW limit. The modulation is extremely weak. Positive and negative polarity curves are indistinguishable.

Figures 1–9 display the radial mean free path rr ¼ 3rr =V , the latitudinal mean free path ¼ 3 ? =V , and the drift scale A ¼ 3A =V versus (b) rigidity and versus heliocentric distance at (d ) high latitudes and ( f ) in the equatorial plane for 200 MeV protons. The strength of the drift effects is characterized by a ‘‘ drift factor ’’ S defined as ½iðþÞ iðÞ=½iðþÞ þ iðÞ, where iðþÞ and iðÞ denote the intensities of 200 MeV protons for positive and negative solar polarities, respectively. The terms S1 and S56 denote drift factors at 1 and 56 AU, respectively. We choose 1 and 56 AU to characterize the influence of drifts in both the inner and outer heliosphere. Note that the drift factor ranges from 1 to +1. A negative number for S at a given radial distance simply indicates that the intensity for negative polarity is higher than that of positive polarity. Similarly, we characterize the overall strength of modulation by a ‘‘ modulation factor ’’ M defined as ½iðþÞ þ iðÞ=2, where iðþÞ and iðÞ are normalized to the interstellar intensity at 200 MeV. The drift and modulation factors for our chosen parameters are provided in Table 1. For all our calculations we assume the scaling of the ultrascale as ~l / lslab . In addition, is generally less than unity for the parameters we employ. Thus, it is easy to find that ? 2R2L =ð3D? Þ / 1=lslab for ¼ 0 (BAM model) and that ? RL for ¼ 0:5 and ? 3D? =2 / lslab for

¼ 1:0 (FLRW model). An easy comparison involving our two types of correlation lengths suggests that a smaller correlation length lslab leads to a larger perpendicular diffusion coefficient for the BAM model, resulting in reduced drift effects (evident clearly in the outer heliosphere, as seen in Table 1), and to a smaller perpendicular diffusion coefficient for the FLRW model, resulting in increased drift effects (evident clearly in both the inner and the outer heliosphere, as also seen in Table 1), which we see below in detail. However, the correlation length has no impact on the perpendicular diffusion coefficient in the case of ¼ 0:5. The table essentially generalizes the conclusion of Parhi et al. (2002), namely, that a larger correlation length favors drift domination in the outer heliosphere for the BAM model and for a wide range of ultrascale selections. In Figure 1c, showing results for ¼ 0 and ~l ¼ 200lslab , with lslab / r0:3 , the radial profiles are similar in the two polarity states, crossing each other at around 20 AU. The drift is also small, as seen from the drift factors (Table 1) at both 1 and 56 AU. A calculation for the drift factor for the same parameters but with lslab / rþ1 (Parhi et al. 2002) turns out to be S1 ¼ 0:18, S56 ¼ 0:65, and M1 ¼ 0:09. The latter has been described as drift-dominated modulation. The transition from diffusion-dominated modulation to driftdominated modulation is displayed by the amount of

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Fig. 5.—Same as in Fig. 4, but for lslab / r. The modulation is practically nonexistent. The curves are indistinguishable.

separation in the radial profiles in the two polarities available quantitatively from the drift factor (Table 1). The latitude gradients (Fig. 1e) for both polarities have the same sign in most of the energy ranges, but they do not agree well with the data. Keeping open baseline values as in Figure 1, is changed from 0 (BAM limit) to 0.5 in Figure 2. This value of is

exactly halfway between the BAM and FLRW limits and is reminiscent of the intermediate behavior found in numerical simulations of perpendicular diffusion (Giacalone & Jokipii 1999; Qin 2002). We have already seen that for this case ? RL , and thus perpendicular diffusion does not depend on lslab . The changes in the intensity between Figures 1a and 2a appear to be rather small, but the modulation factor M1

TABLE 1 Drift Factor and Modulation Factor for Each Figure Scan

¼ 0, ~l =lslab ¼ 200 ........... ¼ 0:5, ~l =lslab ¼ 200 ........ ¼ 1:0, ~l =lslab ¼ 200 ........ ¼ 0:0, ~l =lslab ¼ 5 ............ ¼ 0:0, ~l =lslab ¼ 200 ........ ¼ 0:0, ~l =lslab ¼ 5000 ......

lslab / r0:3

slslab / r

S1 = 0.04, S56 = 0.18, M1 = 0.08 (Fig. 1) S1 = 0.15, S56 = 0.06, M1 = 0.18 (Fig. 2) S1 = 0.01, S56 = 0.01, M1 = 0.57 (Fig. 4) S1 = 0.54, S56 = 0.18, M1 = 0.13 (Fig. 6) S1 = 0.04, S56 = 0.18, M1 = 0.08 (Fig. 1) S1 = 0.29, S56 = 0.52, M1 = 0.05 (Fig. 8)

S1 = 0.18, S56 = 0.65, M1 = 0.09 (Parhi et al. 2002, Fig. 2) S1 = 0.08, S56 = 0.07, M1 = 0.19 (Fig. 3) S1 = 0.00, S56 = 0.00, M1 = 0.99 (Fig. 5) S1 = 0.34, S56 = 0.25, M1 = 0.12 (Fig. 7) S1 = 0.18, S56 = 0.65, M1 = 0.09 (Parhi et al. 2002, Fig. 2) S1 = 0.66, S56 = 0.88, M1 = 0.06 (Fig. 9)

Notes.—The drift factor S ¼ ½iðþÞ iðÞ=½iðþÞ þ iðÞ and modulation factor M ¼ ½iðþÞ þ iðÞ=2 are displayed for each figure. Here iðþÞ and iðÞ denote the normalized intensities (relative to interstellar value) for positive and negative polarities, respectively. The terms S1 and S56 denote the quantities at 1 and 56 AU, respectively, and M1 denotes the modulation factor at 1 AU. The data for negative polarity at 56 AU are estimated from an exponential interpolation of the Webber & Lockwood 2001 data. The observed values are S1obs ¼ 0:13, obs ¼ 0:24, and M1obs ¼ 0:07. The first three rows present a scan of the parameter from the BAM ( ¼ 0) to FLRW ( ¼ 1) limits. The last S56 three rows present a scan in ultrascale from ~l ¼ 5lslab to ~l ¼ 5000lslab . Note that the first row is repeated as the fifth row.

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Fig. 6.—Same as in Fig. 1, but for ~l =lslab ¼ 5. The smaller ultrascale increases the drift factor at 1 AU but reduces the modulation.

at 1 AU is actually more than double that obtained in the case of ¼ 0 (Fig. 1). This means modulation is weaker for the intermediate case between the BAM and FLRW models. In other words, modulation is sensitive to . The magnitude of the drift factor is now increased at 1 AU but reduced in the outer heliosphere and is very small at 56 AU. Given the similarity of rr in Figures 1b and 2b, it is not surprising that the shapes of the radial profiles are similar, but there is a significant increase in , resulting in a steep drop in the latitudinal gradient, which almost vanishes for both polarities. In Figure 3, the correlation length varies with heliospheric radial distance r. This changes the modulation factor or latitudinal gradient very little relative to Figure 2, but the drift becomes somewhat weaker (Table 1) in the inner heliosphere. Returning to the baseline quantities ~l ¼ 200lslab and lslab / r0:3 but changing to 1, the FLRW limit, as seen in Figure 4b shows large increases in both rr and in the inner heliosphere. Particles therefore have easier access to these regions, leading to dramatically higher intensities, as seen in the 1 AU spectra (Fig. 4a; also from M1 ¼ 0:57 in Table 1) and the flattening of the radial profiles (Fig. 4c). Drift effects are reduced because of the larger . In fact, is now large enough to eliminate latitudinal gradients completely in the inner heliosphere during both polarity cycles, as shown in Figure 4e.

Figure 5 shows the results obtained by retaining the FLRW limit as in Figure 4 but changing the correlation length to be proportional to radial distance. Now rr and further increase in the inner heliosphere so that particles have better access to these regions, leading to further increase in intensity (Fig. 5a). This is also seen from the weaker modulation factor M1 ¼ 0:99. The drift effect is almost zero, and the modulation factor is close to 1, indicating that very little modulation occurs in this case. Hence, one should conclude from Figures 1–5 that the modulation factor at 1 AU increases and that the magnitude of the drift factor at 56 AU decreases as increases. In other words, as one moves from the BAM to the FLRW limit, modulation at 1 AU decreases, and drift effects in the outer heliosphere become weaker, but the effect of drift in the inner heliosphere is not well ordered by . Now we examine the effect of returning to the baseline values ( ¼ 0 and ~l ¼ 200lslab , with lslab / r0:3 ) but decreasing ~l =lslab to 5 from 200. Figure 6 shows that this has only a moderate effect on the solutions. The smaller in the inner heliosphere yields larger drift effects at 1 AU, as can be seen in Figures 6a and 6c. The drift effect here is in fact several times larger than some of the drift factors at 1 AU in Figures 1–5; note that the drift factors are S1 ¼ 0:54 and S56 ¼ 0:18. The modulation factor at 1 AU does not change so much compared to Figure 1. The radial profile for

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Fig. 7.—Same as in Fig. 6, but for lslab / r. The latitudinal gradient for positive polarity is much higher than that in Fig. 6, and the gradient for negative polarity has the opposite sign. This latter pattern is generally more prominent for the case in which lslab / r.

the positive polarity cycle (Fig. 6c) flattens (and therefore stays too high compared to the data), and the latitudinal gradient for the same cycle decreases. When the correlation length is changed and is considered to be proportional to r, as in Figure 7, the intensity pattern does not change much relative to Figure 6, as seen from the two almost identical modulation factors, but the latitudinal gradient for positive polarity is much higher, and that for negative polarity changes sign. This latter pattern seems to be more prominent for the cases in which lslab / r. In Figure 8, ~l =lslab is increased from 200 (Fig. 1) to 5000 (this is a huge value, comparable to the size of the heliosphere). The behavior of the profiles with respect to observational data is somewhat similar to that for the case in which the correlation length increases with radial distance (Fig. 2 of Parhi et al. 2002). Although the correlation length now decreases with radial distance, the effects due to this decrease are countered to some extent by those arising because of the large increase in the value of ~l =lslab . Varying ~l =lslab from 200 to 5000, one notices that the modulation factor remains almost the same, indicating that there is very little change in modulation level, but drift effects become more prominent. Keeping the large value of ~l but taking lslab to be proportional to radial distance as in Figure 9, the solutions dramatically change. The drift effects are much stronger

now, although the modulation factor is almost the same in both cases (Figs. 8 and 9). The radial profile for positive polarity does not compare well with observations. The observational data for the latitudinal gradient are much lower than the simulation results for the positive cycle. Generally, drift effects are much too strong in the BAM model when the ultrascale ~l is very large. 4. DISCUSSION AND CONCLUSIONS

This work has taken an ab initio approach to modulation modeling, in which elements of the diffusion tensor are computed from known properties of solar wind turbulence based on particle transport theory and the variation of turbulence properties through the heliosphere is determined by turbulence transport models. We have focused on three important but poorly understood aspects of ab initio modeling: (1) the radial variation of the magnetic correlation length for parallel (slab) turbulence modes, (2) the size of the turbulence ultrascale, which determines the rate of FLRW for two-dimensional turbulence modes, and (3) the rigidity dependence of the perpendicular diffusion coefficient. For the latter topic, we introduced a parameter , which can be used to smoothly vary ? from the BAM limit to the FLRW limit.

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Fig. 8.—Same as in Fig. 1, but for ~l =lslab ¼ 5000. The modulation is stronger than in Fig. 1.

4.1. Remarks on Radial Profile Results for the radial profile are summarized in Table 1 and are presented graphically in Figure 10. The axes represent the difference between the predicted and observed (Webber & Lockwood 2001) drift factors at 1 AU (abscissa) and 56 AU (ordinate). The overall strength of modulation is indicated by the vertical bar attached to each data point. The length of each bar scales logarithmically with the ratio of the predicted to the observed modulation factor at 1 AU, with upward (blue) bars corresponding to too little modulation predicted and downward (red) bars corresponding to too much modulation predicted. A good fit to the data would be indicated by a point near the center of the plot with a very small bar. From Figure 10 we see that the best overall fit to the radial profile is given by Figures 1 and 7. These have in common that they both use the BAM model ( ¼ 0) for ? and that they both have a comparatively small or intermediate ultrascale ~l . The two best fits show no preference for the radial variation of the correlation length, however. One (Fig. 1) has the correlation length scaling with r0:3 , while the other has the correlation length scaling with r. (We note that Fig. 1 was previously presented in Parhi et al. 2002 but was not considered the best fit in that work. The reason for the change is that the latest Webber & Lockwood

[2001] observations show less differentiation between positive and negative magnetic polarity than their prior results did. Thus, our conclusions have shifted toward a less driftdominated picture of solar modulation.) There is a general tendency for the theoretical models to predict too little modulation (blue bars). Only Figures 8 and 9 overpredict modulation (red bars), and only by a small amount. These models both assume very large ultrascale, but their drift factors do not agree well with observations. It is curious that the results of this work clearly favor the BAM model over the FLRW model of ? . The FLRW model ( ¼ 1) is represented by Figures 4 and 5, and both predict far too little modulation. They also predict almost no differentiation between positive and negative solar magnetic polarity, as shown by the near-zero drift factors in Table 1. Recent numerical studies, in contrast, have generally concluded that neither BAM nor FLRW correctly describes perpendicular diffusion and that the correct value lies somewhere between these limits (Giacalone & Jokipii 1999; Qin 2002). Further complicating matters are the outer heliosphere observations suggesting that rr =V has a P2 rigidity dependence at lower rigidities and flattens at higher rigidities (Cummings & Stone 2001), exactly as predicted by the BAM model. Considering all this, we might conclude that ‘‘ the jury is still out ’’ on the question of whether BAM is an appropriate model for ? in the solar wind.

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Fig. 9.—Same as in Fig. 8, but for lslab / r. This combination of parameters produces the strongest drift effects.

4.2. Remarks on Latitude Gradient For completeness we have also shown predictions of the energy dependence of the latitude gradient for protons at 2.1 AU. As noted by Burger et al. (2000), this energy dependence provides a sensitive, demanding test of modulation models. Comparing our model predictions (panel e in Figs. 1–9) with Ulysses observations during positive solar polarity (Heber et al. 1996), it is clear that the present version of our ab initio modulation model is not ready for such a rigorous test. This indicates that the model is seriously deficient in its treatment of the high-latitude heliosphere. We are aware of three areas where improvements must be made: 1. We must consider models of the large-scale magnetic field that permit excursions of the large-scale field in heliolatitude (Fisk 1996; Burger & Hattingh 2001). This is crucial for ab initio models, because these models generally have the property that parallel diffusion is much more efficient than perpendicular diffusion (k 4? ). It is the large-scale field that determines how the latitudinal diffusion coefficient partitions into ? and k . Our current model assumes the Parker field, for which k contributes nothing to latitudinal diffusion, no matter how large it is. However, if the large-scale field has a latitudinal component, then k will contribute to the latitudinal diffusion coefficient, possibly increasing it dramatically.

2. We need improved models of turbulence transport at high latitude. Our turbulence transport model is strongly influenced by the degree of ‘‘ mixing ’’ of inward- and outward-propagating modes. In this work we assume zero cross-helicity at all latitudes, but there are clear indications that modification of this assumption in the high-latitude region is required (Goldstein, Roberts, & Matthaeus 1995b), since Ulysses observations show dominance of outward-propagating waves, out to at least 4 AU (Goldstein et al. 1995a; Bavassano, Pietropaolo, & Bruno 2001). 3. We need improved models of anisotropic perpendicular diffusion at high latitude. Anisotropic perpendicular diffusion is motivated both by theoretical considerations (Jokipii & Ko´ta 1989) and by Ulysses observations (Balogh et al. 1996) of anisotropic magnetic field fluctuations in the high-latitude heliosphere. We have taken this into account in a somewhat ad hoc way by setting ? to be a fixed (factor of 5) multiple of r ? at high latitudes. This procedure is motivated by the FLRW model in pure slab turbulence, in which case the anisotropic perpendicular diffusion coefficients scale proportionally with the anisotropic field variances. However, the FLRW model has been called into question here and by prior theoretical (Urch 1977; Ko´ta & Jokipii 2000) and numerical (Giacalone & Jokipii 1999; Qin et al. 2002) studies. Moreover, the field line diffusion coefficient requires modification in two-component turbulence (Ruffolo et al. 2001). For such turbulence the two

obs against S S obs . The vertical bars denote the value M =M Fig. 10.—Plot of S56 S56 1 1 obs on a logarithmic scale, as shown on the lower right. Blue or red 1 indicates that the predicted modulation factor is larger or smaller, respectively, than the observed one.

Fig. 11.—Top panels: Ratio r ? =k vs. heliocentric distance. Bottom panels: Relative contributions of perpendicular diffusion (red curves) and parallel diffusion (green curves) to the radial diffusion coefficient rr . For the left-hand panels, the parameters are ¼ 0, ~l ¼ 200lslab , and lslab / r0:3 , which corresponds to Fig. 1, and for the right-hand panels, they are ¼ 0, ~l ¼ 5lslab , and lslab / r, corresponding to Fig. 7. Note that Figs. 1 and 7 are our best scans (see Fig. 10) because they match the observations well.

AB INITIO THEORY OF SOLAR MODULATION perpendicular variances contribute nonlinearly to field line diffusion; the directions are not decoupled as in the slab model. 4.3. Implications for Perpendicular Diffusion It is clear that perpendicular diffusion is a key factor governing solar modulation. Note that the only thing changing in the different simulations with lslab / r0:3 (Table 1, second column) is ? , yet these simulations produce a very broad range of modulation conditions. The same holds for the lslab / r case (Table 1, third column). (Note, however, that k does change from the second to the third column.) In Figure 11 we present further information on ? for the two cases that produced the best fit to the radial profile. The top panels display the ratio r ? =k for these two cases. They show that r ? =k is extremely small, generally less than 103 . (By virtue of our assumption of anisotropic perpendicular diffusion, the corresponding ratio for ? is 5 times larger.) Although recent numerical simulations (Giacalone & Jokipii 1999; Qin 2002) have favored a larger ratio ? =k 0:02 0:04, our experience with larger ratios is that they do not produce enough modulation.

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It is commonly assumed that ? dominates radial diffusion in the outer heliosphere, because the field angle approaches 90 in this region. (Recall that rr ¼ k cos2 þ ? sin2 , with the field angle.) The bottom panels of Figure 11 display the separate contributions of ? and k to the radial diffusion coefficient. Figure 11c (corresponding to Fig. 1) is in accord with the conventional wisdom that perpendicular diffusion is dominant in the outer heliosphere, but Figure 11d (corresponding to Fig. 7) is not. In the latter case, parallel diffusion dominates rr through most of the heliosphere and remains competitive with perpendicular diffusion out to the assumed modulation boundary. Under some circumstances, then, cosmic-ray particles may find it easier to enter the heliosphere by spiraling in along the large-scale field rather than by crossing field lines via perpendicular diffusion. The authors wish to thank the referee for useful suggestion. The work is supported by NASA grant NAG5-8134 (SECTP) and by NSF grants ATM 00-00315 and ATM 01-05254.

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