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 We present a new axisymmetric model of the heliosphere that includes three principal particle species (plasma with magnetic field, interstellar neutral atoms, and galactic cosmic rays). Unlike previous modulation models, which necessarily included a modulation boundary, we study the entry of the galactic cosmic rays into the heliosphere directly from the local interstellar medium population. An important improvement over previous global heliospheric models is that a kinetic description for the cosmic rays is used instead of the usual fluid approach. We compute the cosmic-ray diffusion coefficients from quasi-linear theory, assuming a constant (but different) power spectrum of the fluctuations in both the solar wind and the interstellar medium. Our results show that low-energy particles are strongly attenuated by the magnetic wall in the inner heliosheath and do not reach the termination shock. The model predicts small cosmic rays gradients in most of the heliosphere, implying little modification to the termination and bow shocks. The exception is the inner heliosheath region, where cosmic-ray gradients are large and the energetic particles can, consequently, affect the plasma flow. Particle spectra in the inner heliosphere are found to be in agreement with the observations. We also find that the inner heliospheric galactic cosmic-ray population is not sensitive to the upwind-downwind asymmetry of the termination shock.
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 The solar system moves through interstellar space populated by three principal particle species: the thermal electron-ion plasma, neutral atoms, and the galactic cosmic rays (GCR). The latter is a highly nonthermal ionic population. One measure of the relative importance of the species is total energy density of the corresponding “gas.” In the local interstellar medium (LISM), also called the local cloud, these three species have energy densities of approximately 0.38, 0.61, and 0.55 eV cm−3, respectively [Ip and Axford, 1985; Frisch, 1997]. In addition, the plasma component carries with it an embedded magnetic field with an energy density of the order of 0.1 eV cm−3 [Frisch, 2000], although the last value is highly uncertain.
 From the energy density estimates, GCR appear to be the dominant species in the LISM and are expected to have a profound effect on the plasma flow in and around the heliosphere. The issue is, however, complicated by the fact that these highly energetic particles may be only weakly coupled to the background plasma. Indeed, the lengthscale of the plasma-cosmic ray interaction is the diffusion length κ/u, where κ is the cosmic-ray diffusion coefficient and u is the average velocity of the scattering centers. The diffusion lengthscale can become very large at particle energies above several GeV so that cosmic rays may not interact with the heliosphere at all. On the other hand, MeV particles are expected to be strongly coupled to the plasma and as a result may change the flow of plasma inside the heliosphere.
 Galactic cosmic ray propagation in the heliosphere has traditionally been the domain of the modulation studies for the past three decades. It is beyond the scope of this paper to give even a brief overview of the modulation models in existence; recent reviews are given by Potgieter  and Jokipii and Kota . However, despite the detailed treatment of the GCR modulation by the the solar wind, virtually all models lack a self-consistent treatment of the three particle species. In particular, the plasma flow geometry and the magnetic field are typically described by a spherically expanding solar wind and the resulting Parker spiral field, with possible modifications. Many models now correctly include a subsonic post-shock solar wind region, an approach first used by Kota and Jokipii . This approach is especially relevant in view of observations showing that galactic cosmic rays experience modulation in the inner heliosheath [Webber and Lockwood, 1997; McDonald et al., 2000]. However, both the external “modulation boundary” and the termination shock (TS) are typically taken to be spherically symmetric. Notable exceptions are works by Fichtner et al. , Haasbroek and Potgieter , and Sreenivasan and Fichtner . In the first paper, the authors studied GCR modulation inside an ellipsoidal domain, assumed to be produced by the TS elongation along the solar polar axis, with the help of a fluid model. By contrast, Haasbroek and Potgieter  solved the momentum-dependent transport equation inside an asymmetric domain designed to simulate the shape of the TS produced by the solar wind interaction with the interstellar flow. Sreenivasan and Fichtner  solved a similar problem for anomalous cosmic rays (ACR) using a fully three-dimensional (3-D) model. One of the main results of these two papers is that the cosmic-ray asymmetry with respect to the interstellar wind direction becomes smaller at small heliocentric distances; this result will be discussed below in some detail.
 It has been realized for some time that the inner heliosphere is not the only region where cosmic rays and solar wind interact. In particular, gas dynamic and MHD models show that the heliospheric magnetic field is strongly amplified near the nose of the heliopause because of flow deceleration [Nerney et al., 1993; Washimi and Tanaka, 1996; Linde et al., 1998]. The amplified magnetic field will inhibit particle diffusion and is likely to stop the lower energy galactic cosmic rays from propagating past the inner heliosheath. Dissatisfaction with the “modulation boundary” approach led to development of global GCR models including not only the solar wind but the LISM interaction region as well [Izmodenov, 1997; Fahr et al., 2000; Myasnikov et al., 2000a, 2000b]. However, every global heliospheric model has been based on the fluid GCR description, where a momentum averaged quantity (pressure) is used instead of the phase space density. The deficiency of this approach stems from the necessity of introducing momentum-averaged diffusion coefficients and a cosmic-ray “adiabatic index” as input parameters. This makes the model less consistent because the average diffusion coefficients depend on the cosmic ray gradients themselves.
 Energy-dependent cosmic-ray modulation in the heliosheath was first studied by Grzedzielski et al.  with a non-self-consistent model. The authors used an incompressible flow assumption and constant magnetic field in the heliosheath. The model of Florinski and Jokipii  represented the first attempt to study the GCR-solar wind interaction at a self-consistent two-dimensional level without the assumption of spherical symmetry. While the solar wind flow was computed self-consistently, the model did not extend into the interstellar medium and only contained a limited region beyond the TS. As a result, the model was only applicable in the upwind direction with respect to the heliopause. This paper overcomes some of the difficulties encountered in the earlier study by introducing a two-dimensional, axisymmetric global heliospheric model of galactic cosmic ray propagation in the heliosphere and the surrounding LISM. Since neutral atoms have a profound effect on the shape of the heliospheric boundaries and the solar wind flow [Baranov and Malama, 1993; Pauls et al., 1995], we include them in the simulation using a fluid approximation. The model treats cosmic rays as a nonthermal distribution in phase space, rather than a fluid, and uses realistic diffusion coefficients calculated from our understanding of the magnetic field and the turbulence levels in both the heliosphere and the LISM. To relate out approach to the fluid models, we compute the momentum-averaged diffusion parameters from our model and compare them to those used by models based on the hydrodynamic approximation.
2. Model Description
 The model presented here is essentially two-dimensional, i.e., axisymmetric with respect to the interstellar wind direction. What we mean by “essentially” is that even though the plasma and neutral components and the interstellar magnetic field are axisymmetric, no such assumption is made about the heliospheric magnetic field. Since the symmetry axis of the solar wind (the direction perpendicular to the ecliptic plane) and the direction of the LISM flow do not coincide and may be almost orthogonal to each other [Frisch, 1997], the correct treatment of the problem is necessarily three-dimensional. While a 3-D kinetic self-consistent heliospheric model presents serious computational challenges, we believe that a two-dimensional model can provide insight into the process of modulation in the outer heliosphere.
2.1. Plasma, Neutrals, and Magnetic Field
 MHD equations describe the plasma flow which is axially symmetric about the direction of the LISM flow (z-axis). The MHD system of conservation laws for the thermal plasma is
where ρp and up are the density and velocity of the electron-proton plasma, respectively, and pp = pg + B2/(8π) and ep = ρpup2/2 + B2/(8π) + pg/(γ − 1) are the total pressure and energy density of the plasma, including the magnetic field (pg is the thermal or “gas” pressure of the plasma only). The source terms on the right-hand side describe the transfer of density, momentum, and energy density in charge exchange collisions between the ionic and neutral components. The expressions we use are based on the formalism developed by Pauls et al. . Equations (1)–(3) are supplemented by Faraday's law for the magnetic field B
Note that the magnetic force involved in equations (2) and (3) is only included in the interstellar flow, i.e., in the region beyond the heliopause. As we show below, the heliospheric magnetic field is included in the kinematic approximation, i.e., only equation (4) is used. This approach is necessary to maintain the axial symmetry of the problem as the j × B force will necessarily destroy this symmetry. A stationary kinematic model has been used by Barsky  to study the effects of the neutral component, computed using the Baranov and Malama  model, on the magnetic field in the inner and outer heliosheath regions.
 Neutral atoms are described by a set of equations similar to the MHD system above but without the magnetic field. At this time we only include the local interstellar hydrogen component (the so-called two-fluid approximation). This approach, used by Pauls et al. , retains only those hydrogen atoms that are primordial (LISM) or have experienced charge exchange in the outer heliosheath. The remaining two neutral populations, originating in the inner heliosheath and the supersonic solar wind, can be included in a multi-fluid model [Zank et al., 1996b]. The simpler approach by Pauls et al.  captures the basic features of the solar wind–LISM interaction, such as the deceleration of the supersonic wind, the decrease in the TS and heliopause stand-off distances, and the formation of the hydrogen wall. Since this approach is also computationally less demanding than the multi-fluid approach of Zank et al. [1996b], we have accordingly adopted this approach for our initial study and concentrated on the entrance of cosmic rays into the heliosphere and their subsequent modification. A subsequent paper will address the full complexity of the multifluid model.
 In the two-fluid (plasma and H atoms) approximation, the conservation laws for the neutral component are
where the notation is similar to that for the plasma with the subscript “H” designating the neutral atoms.
 Both sets of equations (1)–(4) and (5)–(7) are solved numerically for Cartesian vector components on a polar (r–θ) grid lying in a half-plane ϕ = const, passing through the z-axis. The angle ϕ is measured from the solar polar axis, which is assumed to be orthogonal to the LISM flow direction. Note that this axis coincides with the x-axis of the problem when ϕ = 0. Our geometry is therefore similar to that of Nerney et al. , except that they measure the ϕ-angle from the ecliptic plane. The equations are solved using the total variation diminishing (TVD) Lax–Friedrichs numerical scheme [Kulikovskii et al., 2001]. This method was used successfully by Pogorelov and Semenov  and Pogorelov and Matsuda [1998, 2000] to study the LISM–solar wind interaction. LISM parameters (see Table 1) are chosen such that the interstellar flow is both supersonic and super-Alfvénic.
Mp, MH and MA are the plasma and neutral hydrogen sonic Mach numbers and the plasma Alfvén number, respectively.
5 × 107 m s−1
2.5 × 106 m s−1
1.5 × 105 K
8 × 103 K
1.88 × 10−6 G
0.23 eV cm−3
A2 (slab only)
4.5 × 1011 cm
 To derive equations describing the three-dimensional heliospheric field in a 2-D model, we begin with Faraday's law under the assumption of axial symmetry of the plasma flow, i.e., ∂u/∂ϕ = ∂w/∂ϕ and v = 0, where u, v and w are the x-, ϕ-, and z-components of the plasma velocity up, respectively. We can therefore write
Equation (10) is already in the required form, but equations (8) and (9) should be transformed using the divergence-free condition ∇ · B = 0, which yields
 Note that the system of equations (10)–(12) does not involve the angle ϕ at all and is therefore suitable for the axisymmetric geometry. The inner boundary conditions do depend on ϕ, and as a result, the complete solution depends on ϕ parametrically. We use the Parker spiral field as a condition at the inner boundary, which is located at 10 AU in our simulation. The Parker field can be written out using the polar coordinates r and θ as
where Ω0 is the angular velocity of the solar rotation and U0 is the radial solar wind speed at 1 AU (see Table 1). The radial field is Br = B0(r0/r)2, where r0 = 1 AU. The system of equations (10)–(12) is solved using a numerical method similar to that used for the plasma and neutral components.
2.2. Cosmic-Ray Transport
 Galactic cosmic rays can be described by Parker's [Parker, 1965; Skilling, 1975] transport equation for the phase space density f,
where κ is the diffusion tensor. Drift velocities are not included in the equation because they contain a component perpendicular to the half-plane ϕ = const. We also note that diffusion in the direction perpendicular to the half-plane is necessarily ignored. This is a serious limitation of the axial symmetry. To better determine the amount of GCR modulation, one needs to consider the angle-averaged cosmic-ray distribution computed for a range of ϕ. We plan to include such an analysis in a future publication.
 The diffusion tensor is determined by the formula
To find κ∥ we use an expression from quasi-linear theory describing diffusion as a resonant scattering process on MHD turbulence embedded in the plasma. This expression, derived by Zank et al.  and Giacalone and Jokipii , assumes the combined energy-inertial power spectrum of turbulence with transition occurring near the correlation length
where lc is the correlation length (turbulence outer scale), rg is the cyclotron radius, V is the particle velocity, and Asl2 = 〈δB⟂2〉/B2 is the amplitude of slab turbulence (the only kind of fluctuations that affect parallel transport). Equation (18) was used in GCR modulation studies by Burger and Hattingh .
 Perpendicular diffusion is considerably less well understood. Here we rely on the study of le Roux et al.  who compared three different models of perpendicular diffusion, all derived theoretically. We use the “modified anomalous” model that assumes that perpendicular diffusion is caused by parallel diffusion along divergent or “wandering” field lines. The expression is
where Atot is the total turbulence amplitude. From theoretical models [Zank and Matthaeus, 1992] and in situ measurements [Bieber et al., 1996], slab turbulence contributes 20% and 2-D turbulence contributes the remaining 80% to the total turbulence level in the solar wind. The coefficient α (see Table 1) expresses the fact that the model tends to predict a much larger κ⟂ than computed in hybrid simulations [Giacalone and Jokipii, 1999].
 Evolution of the turbulent spectra in the solar wind and associated diffusion coefficients were studied by Zank et al. [1996a, 1998]. Those authors derived equations for the evolution of A and lc under the assumption of incompressibility of fluctuations. Turbulence driving by shear in the solar wind and pickup ion isotropization as well as dissipation to the turbulent cascade were all included. In principle this theory could be used to compute the diffusion coefficients for cosmic-ray transport. However, the theory has several limitations and needs additional refinement before it can be applied to multidimensional models such as ours. Specifically, the model of Zank et al. [1996a, 1998] assumes that the solar wind plasma flow is spherically symmetric and highly super-Alfvénic and is therefore not applicable to the inner heliosheath region.
 Since the theory of Zank et al. [1996a] cannot be easily incorporated into our model, we adopted a simpler alternative. We note, in particular, that the correlation length lc is generally weakly dependent on the heliocentric distance (although it can, in theory, vary with latitude; see, e.g., Burger and Hattingh ), while 〈δB2〉 decreases more slowly than the r−3 law predicted by the WKB theory [Zank et al., 1996a]. In our approach we “freeze” the amplitude and the correlation length at the distance of 10 AU (our inner boundary) for the solar wind and at 1000 AU (the outer boundary) for the LISM.
 We expect the turbulence parameters to be very different in the LISM and solar wind. A large number of measurements of radio wave scattering in the interstellar gas indicate that the medium may be fully turbulent [Rickett, 1990; Armstrong et al., 1995]. Thus there are compelling reasons to believe that the interstellar medium is turbulent with the same partition into the energy and inertial range [Spangler, 1991; Norman and Ferrara, 1996]. The amplitude of magnetic field fluctuations (A2) is not well determined due to the lack of our knowledge of the type of turbulence, which makes it difficult to infer the magnetic field fluctuation spectrum from the observed plasma density fluctuations [Armstrong et al., 1995]. Here we have adopted a factor of 10 larger turbulence level in the LISM than in the solar wind, partly to compensate for the larger κ⟂/κ∥. Armstrong et al.  and Ferrara  estimate the interstellar turbulence outer scale at 1018 cm. Since the direction of the interstellar field is poorly known, we use isotropic diffusion in this region (see Table 1).
 We also note that Parker's spiral field and its extension beyond the termination shock in the polar region produces an extraordinary large diffusion coefficient. To remedy the situation, we add the Jokipii-Kota modified field [Jokipii and Kota, 1989; Jokipii et al., 1995] to the system of magnetic field equations (10)–(12). On assuming that the fluctuating field only has a ϕ component, it is easy to show (Appendix A) that its amplitude is described by the same equation as Bϕ [Florinski, 2001], i.e.,
We assume that Bm = 0.2 Br and is independent of θ and ϕ at 1 AU. The total field B2 in equation (17) is computed according to
It should be understood that the modified field is still part of the “regular” magnetic field B and not part of the turbulent field which is described by A and lc.
 We begin with the interstellar spectrum at the external boundary by approximating the spectrum obtained by Ip and Axford , which was based on primordial (supernova shock-accelerated) particle propagation in the galaxy subject to ionization losses. The spectra asymptotically decreases as p−4.7 at high energies and approaches a constant at low energy. The normalization was chosen such that the spectrum fits the BESS balloon measurements [Sanuki et al., 2000] at the highest energy that we consider (10 GeV), for which no heliospheric modulation is expected.
 The transport equation (16) is solved on the polar grid using the fully implicit alternating direction (ADI) method [Florinski, 2001]. The method was also used by Florinski and Jokipii  in the context of a local self-consistent GCR heliospheric model. We use a zero gradient inner boundary condition for the cosmic rays, which can be shown to give similar results to a reflecting (i.e., zero streaming flux) condition. We assume that no modulation occurs at the highest energies (10 GeV) and therefore keep the intensity fixed at the upper momentum boundary. At low energies, we assume particles below (10 MeV) are unable to enter the heliosphere, given their short diffusion length. Accordingly, we set f = 0 at this momentum boundary.
3. Model Results
 In this paper we present results obtained for the meridional half-plane ϕ = 0. Preliminary results have been published by Florinski et al. . We plan to report the results for other cross-sections in a future publication. The simulation grid extended 400 cells from 10 to 1000 AU, using a variable radial spacing in the range between 0.8 and 5.7 AU, while the angular direction θ contained 100 cells between 0 and 180°. The momentum interval contained 170 points, logarithmically spaced between 10 MeV and 10 GeV.
3.1. Heliospheric Structure, Magnetic Field, and Diffusion
Figures 1 and 2 show the general heliospheric plasma interface structure for the case without interstellar neutral atoms and with neutral atoms included, respectively. The general structure and the effects of neutrals are very well known [see, e.g., Baranov and Malama, 1993; Pauls et al., 1995; Zank et al., 1996b; Müller et al., 2000] and will not be discussed here. However, since fundamental structural differences exist between heliospheric models with and without neutrals included, we point out below how these differences can affect cosmic-ray propagation.
 The inclusion of neutrals self-consistently in heliospheric models reduces the size of the solar cavity compared with plasma-only models. This effect is almost entirely caused by the removal of momentum from the supersonic solar wind by charge exchange. For the parameters chosen, comparison of Figures 1 and 2 shows that the termination shock moves inward from 165 to 100 AU in the upwind direction; the stagnation point on the heliopause moves in from 255 to 165 AU, and the bow shock is accordingly located closer to the Sun (220 AU versus 360 AU without neutrals). As shown below, cosmic-ray modulation occurs only in the heliosphere (the region bounded by the heliopause). This implies a significant reduction in the modulation cavity size. Since the supersonic solar wind is decelerated by charge exchange, the cooling rate, which is proportional to ∇ · u, may be expected to be smaller when neutrals are included. Finally, the heliosphere contracts in the tail region also, due to cooling of the shocked solar wind by charge exchange, thus reducing the thermal pressure supporting the heliopause. This cooling is apparent in the temperature plot of Figure 2. Combined with the deceleration by charge exchange, the tail flow is more convergent with neutrals, which leads to acceleration of the GCR to higher energies. We briefly discuss this effect in section 5.
Figure 3 shows the three components, Bx, Bz, and Bϕ, of both the interstellar and heliospheric magnetic fields for the neutrals case. Since the interstellar field is aligned with the flow direction, we have Bϕ = 0 in the LISM. Conversely, the heliospheric field in the meridional plane is almost entirely in the ϕ direction. The transition between the fields is apparent along the heliopause. The sawtooth shape is produced by the algorithm that separates the two fields (heliopause detection) and the granularity of the polar grid. The interstellar field is seen to be amplified in the outer heliosheath past the bow shock transition and approaching the stagnation point.
 One important feature of the interaction shown in Figure 3 is the existence of a region with a highly amplified heliospheric magnetic field (“magnetic wall”) in the inner heliosheath. The Bϕ component is amplified near the stagnation point due to the flow deceleration (“Axford-Cranfill” effect) and is then convected to higher latitudes [see Nerney et al., 1993; Washimi and Tanaka, 1996; Linde et al., 1998; Zank, 1999]. While the increased magnetic pressure has no effect on the flow due to the kinematic approximation used here, the increased magnetic field will affect the diffusion coefficients and thus the cosmic-ray transport.
 The dependence of the heliospheric magnetic field on the radial distance is shown in Figure 4. Again, only the case with the charge exchange is shown. We transform the x- and z- magnetic field components to the spherical components Br and Bθ in the frame with the solar axis aligned with the z-direction, which have more explicit physical meaning in the solar wind. Note that for the polar plane, Bϕ corresponds to the ϕ- component of Parker spiral. The third spherical component, Bθ, is not shown because it is generally very small. Figure 4 shows that Br decreases exactly as r−2, as expected, since the charge exchange in the supersonic wind has no effect on it in the meridional plane (this is not the case for other cross-sections). Br does not change across the shock at 0 and 180° but undergoes a small jump at 90°, where the shock normal is not aligned with the radial direction. However, the Bϕ component does not follow r−1 law upstream of the TS exactly, due to the charge-exchange induced deceleration of the solar wind. This field then undergoes a jump at the shock by a factor of 4 and continues to increase with radial distance eventually forming a magnetic wall in the upwind and the crosswind directions. The “height” of the wall is only a factor of 2 different at 0 and 90°. We emphasize here that the magnetic field at 90° on the heliopause is an extension of the equatorial field that is convected to higher latitudes and is unrelated to the polar field, which is generally very small.
 We show 1 GeV proton diffusion coefficients in Figure 5. Note that κ is very large in the LISM (∼1027 cm2 s−1) and causes no modulation even at the lowest energy on the scales of the problem. This large diffusion coefficient is expected from the turbulence parameters, specifically, the extremely large outer scale, in the LISM. It is, however, still 100 times smaller than the “canonical” value of 1029 cm2 s−1 for the 1 GeV GCR diffusion coefficient in the galaxy [Hartquist and Morfill, 1994]. Most notably, the diffusion barrier is clearly seen in the inner heliosheath, where κ is 100–1000 times smaller than in the solar wind upstream of the termination shock.
 Diffusion coefficients shown in Figure 5 are in sharp contrast with the “standard” (Arizona-Potchefstroom) modulation models [Jokipii and Kota, 2000; Potgieter et al., 2001]. The latter typically either ignore the heliosheath modulation or use a uniform in width region filled with shocked solar wind, where the diffusion coefficients are nearly independent of the radial distance. In contrast, our results show that the magnitude of the diffusion coefficient in the heliosheath decreases sharply toward the heliopause. The thickness of the barrier (i.e., the heliosheath) is different in the upstream and the sidewind directions and its size (65 and 100 AU, respectively) is significantly larger than that commonly employed by the modulation models. Note also that as the barrier is of roughly uniform strength at all latitudes in the upwind part of the heliosheath, galactic cosmic rays no longer have an easy access to the inner heliosphere along the polar solar axis. Both κrr and κθθ are relatively small in the tail region but are a factor of 10 larger than in the barrier. The barrier persists at least as far downstream into the tail as our simulation domain extends.
 Finally, Figure 6 shows the effect of the field modification according to equation (21) on the diffusion coefficient in the heliotail region. It is well known that κ can become very large in the polar region where the magnetic field is directed radially and very small. This effect is exacerbated beyond the termination shock making it difficult for the numerical code to deal with the situation where κ changes by a factor of 104 within a very short distance, as shown by a dashed line in Figure 6. The modified field is seen to resolve the problem without appreciably affecting the diffusion coefficient in the rest of the heliosphere.
3.2. Cosmic-Ray Intensity and Spectra
Figure 7 shows the cosmic-ray distribution throughout the simulation domain at 30 and 500 MeV energies for the simulation run without neutral atoms. Figure 8 shows the same quantities for the simulation with hydrogen atoms included. Generally, modulation patterns are similar in these two cases. Modulation slightly increases in the presence of neutral hydrogen and GCR intensities in the inner heliosphere are smaller. It can be seen that low-energy particles are mostly filtered by the small diffusion region (“magnetic wall”) in the inner heliosheath. The particle distribution inside the termination shock shows a typical lobe structure characteristic of the modulation models [Potgieter, 1998], which is produced by a rapid change of diffusion with heliolatitude. A visible difference between the two cases is the increased intensity in the heliotail region in the second case. This could indicate reacceleration of cosmic rays by the convergent flow.
Figure 9 shows the GCR differential intensity J = fp2 at several different points in the modulation region for the no-neutrals run, while Figure 10 plots the spectra for the case with H atoms. In these figures, rs is the heliocentric distance to the TS for a given latitude. For the upwind and crosswind directions we plot spectra at 1.1 rs, while the tail spectrum is shown for a fixed heliocentric distance of 800 AU. While intensities are slightly smaller in the heliosheath in the neutral-modified case, the difference is generally very small. Note that particles with energies 30 MeV and below are incapable of penetrating into the heliosphere and are absent from the heliosheath spectrum. While low-energy GCR can be seen in the inner heliosphere at 10 AU, they are merely the product of cooling at higher energies and do not reflect the interstellar population at this energy. The 10 AU spectra are almost indistinguishable from each other, a result that we will discuss further in the next section. In both cases the inner heliospheric spectra are seen to be in a fair agreement with observations for the past solar minimum at both high [Sanuki et al., 2000] and low [McDonald, 1998] energies.
 From studying the particle spectra at various locations in the heliosphere, we can draw an important conclusion about the nature of the modulation process. In particular, GCR attenuation in the outer heliosheath is comparable to or larger than modulation by the solar wind upstream of the TS. Thus our results indicate that it is more appropriate to use premodulated spectra as a boundary condition in local heliospheric (modulation) GCR models. Note that Kota and Jokipii  arrived at the same conclusion using a simpler GCR model with a spherically symmetric solar wind flow.
 We should mention here that the kinematic model, due to its neglect of the Lorentz force on the plasma, tends to overestimate the magnetic field in the heliosheath. In particular, the 3-D model of Linde et al.  predicted 5 μG magnetic field in the ridge, which is smaller than calculated in our model in the immediate vicinity of the heliopause. This, in turn, would imply that the decrease in diffusion coefficient in this region would not be as dramatic. While GCR modulation in the magnetic wall would still be important, the effect may be smaller than predicted here. We plan to verify this conclusion in the future by using a three-dimensional model for the background (plasma + neutrals) component.
 Another difference between the plots is the greater intensity of 1 GeV cosmic rays in the heliotail at 800 AU than in the LISM. This may be a result of an additional acceleration in the heliotail region. A similar effect has been seen in ACR simulations of Czechowski et al. . However, our simulation did not extend far enough in the radial direction to achieve measurable deceleration of the solar wind flow in the heliotail, and it is possible that the results are affected by an external boundary that is not sufficiently far downstream.
 The extension of the current 2-D model to three dimensions is not trivial. Since the magnetic ridge is expected to persist at other ϕ-angles [Linde et al., 1988], filtration in the modulation wall will remain a dominant effect. Introduction of drifts would cause the modulation pattern to vary with the 22-year solar cycle. While drifts are more important inside the termination shocks, radial drifts downstream of the shock may lead to asymmetry in the GCR distribution, which, in turn, may affect the solar wind plasma flow [Florinski and Jokipii, 1999].
4. Cosmic-Ray Asymmetry in the Inner Heliosphere
 In this section we compare the results of our global heliospheric model with traditional modulation models. In particular, we are interested in the possibility of detecting the asymmetry of the TS from a corresponding asymmetry of the cosmic-ray distribution in the inner heliosphere with respect to the direction of the interstellar flow. The cosmic-ray transport equation has been solved by Nosov [1977, 1981] for an ellipsoidal heliospheric boundary under the assumption of weak modulation and small deviation from spherical geometry, respectively. However, because the author assumed κ = const, the important effect of the latitudinal variability of the diffusion coefficient has been ignored. Since high energy (several GeV) cosmic rays are not modulated and low energy particles are products of adiabatic cooling, we focus on intermediate energies (hundreds of MeV to 1 GeV).
 We show a more detailed view of the GCR phase space density for 1 GeV particles in Figure 11. A “lobe” structure is exhibited that is very characteristic of local modulation models. A clear feature of Figure 11 is the disappearance of the asymmetry in particle intensity with decreasing heliocentric distance. This effect is evident from looking at the contour crossing points with the horizontal axis. It is interesting to note that this result agrees with that of Haasbroek and Potgieter  and Sreenivasan and Fichtner , who calculated GCR and ACR modulation, respectively, in a domain bounded by a simulated asymmetric shock. The difference in the degree of modulation (our model predicts smaller radial gradients than Haasbroek and Potgieter ) is due to our using relatively large diffusion coefficients and neglecting drifts. Furthermore, the inner boundary at 10 AU is still quite far from the Sun and the condition ∂f/∂r = 0 at the boundary can suppress radial gradients here.
 A simple analytic model of cosmic ray transport can be constructed using a latitudinally varying diffusion coefficient in the inner heliosphere. We wish to confirm analytically that the cosmic-ray intensity at small r is insensitive to changes in the shock stand-off distance. A strict analytic verification is very complicated and we are obliged to make a number of simplifying assumptions. First, we introduce a spherical coordinate system with the polar axis (z-axis) aligned with the solar rotational axis (this system is different from the polar coordinate system described in section 2, which has the z-axis pointing in the interstellar flow direction). We further assume that the magnetic field has only the ϕ component and that the system in axially symmetric. As a result diffusion is isotropic, i.e.,
This assumption is generally good beyond several AU heliocentric distance. We further assume that the diffusion rate scales linearly with the gyroradius but is independent of momentum. While the last assumption is generally incorrect, it provides a particularly simple solution of the two-dimensional transport equation. Then
where B0 is the magnetic field strength at some reference distance r0. Since we assume that Br = 0, the last equation may be written as
where U0 is the solar wind velocity and δ is the Jokipii-Kota magnetic field modification due to the large-scale movement of the footpoints. This field prevents diffusion from becoming infinite at θ = 0. Equation (16) is still difficult to solve. We therefore set δ = 0 and redefine κ0 so that
 The cosmic-ray transport equation in the supersonic solar wind is
This equation was solved by Owens and Jokipii  for the case when the diffusion coefficients and the wind velocity were slowly varying functions of θ (represented by low-order Legendre polynomials). Here we consider the opposite case when κ becomes infinite at θ = 0.
 As κ is independent of p, there is no distinction between particles at different energies and the last term may be written as
where σ is the slope of the power-law spectrum, which is in our case preserved in the modulation region. Introducing η = r0U0/κ0 (“modulation parameter”) and using equation (25), equation (26) can be rewritten as
We impose the condition f(rs, θ) = fs at the spherical outer boundary. The inner boundary condition at r = 0 must be chosen such that f is limited. From the form of equation (28) we see that f is a power-law function of r, which means the power-law index cannot become negative to avoid singularity at r = 0. For the latitudinal boundary conditions, it is clear that the GCR distribution must be symmetric about the equatorial plane which means ∂f/∂θ(r, π/2) = 0. The boundary condition at θ = 0 is left undefined.
 We now observe that for the energies of our interest the condition η ≪ 1 is usually satisfied. For example, from Figure 5 we see that κ = 3.5 × 1023 cm2 s−1 at the reference distance r0 = 100 AU for 1 GeV protons, which gives η = 0.21 for U0 = 500 km s−1. Then treating η as a small parameter, we obtain the first-order solution as
Note that equation (30) balances diffusion with cooling, while the convective term is of the second order and therefore omitted. We seek a solution of the form
with the condition that Φ(rs) = 0. One can easily see that
 We calculate f according to equation (32) with rs = 100 AU and 150 AU, σ = 3.5 and show the results in Figure 12. The general structure of the solution is well reproduced by equation (32) (compare with Figure 11). Further, from the above equation one can see that the dependence of f on rs weakens as r → 0. In other words the amount of modulation experienced by a cosmic ray particle as it propagates to small heliocentric distances is essentially independent of the location of the modulation boundary. This effect is clearly seen in Figure 12, which shows that the difference between the two cases is the largest in the outer heliosphere and disappears with decreasing heliocentric distance.
 The above result is a combination of two factors. First, most attenuation in the ecliptic plane takes place at small r, and second, large diffusion in the polar region provides an easier path for the particles to get inside and further diffuse to lower latitudes. We should note that an inclusion of drifts could change this conclusion for the negative solar cycle, when they tend to counterbalance the polar diffusion.
 The result implies that it is very difficult to detect an asymmetry of the cosmic-ray distribution, which has been imposed by the outer heliospheric structures, by measuring the cosmic-ray intensity at 1 AU. Galactic cosmic rays do not therefore provide a very useful source of information about the outer heliosphere. It is interesting to note that Sreenivasan and Fichtner  arrived at the same conclusion but for anomalous cosmic rays. Still, it may be possible to measure the energetic neutral atoms, produced by the charge exchange of the neutral hydrogen atoms on cosmic rays. These secondary particles are not affected by the magnetic fields and can potentially be used to detect the asymmetry of the heliosphere [Hsieh and Gruntman, 1993; Czechowski et al., 2001].
5. Comparison With Global Fluid Models
 In this section we show results for the momentum averaged cosmic-ray properties, such as their pressure pc, average diffusion coefficients , and the adiabatic index γc. These may be found by comparing the transport equation for the cosmic-ray pressure with the original Parker equation to obtain
(no summation over indices) and
where p is momentum and c is the speed of light. Equation (34) shows that the average diffusion coefficient depends on the phase space diffusion coefficient κ as well as on the cosmic-ray gradients.
Figure 13 shows the dynamic pressure of the plasma, ρpup2/2, and the three “thermal” pressures pg, pH, and pc for three different angles relative to the interstellar flow direction. The hydrogen wall is visible in the top panel (outer heliosheath region). Note that the GCR pressure gradients are generally very small throughout the heliosphere. The difference in pressure between the external boundary at 1000 AU and the inner boundary at 10 AU is only 0.07 eV cm−3. The only place where the GCR pressure gradient is large is the modulation wall in the inner heliosheath. The wall is quite effective at diverting even the higher-energy particles, which carry most of the pressure in the interstellar spectrum. Since pc gradients in the inner heliosphere and at the TS are small, very little modification to the shock structure is expected. In view of the large diffusion coefficient, galactic cosmic rays may be capable of at most producing a wide and shallow precursor to the termination shock without appreciably affecting the subshock compression ratio. This conclusion is consistent with the results of the spherically symmetric self-consistent model of le Roux and Fichtner .
Figure 14 shows the average diffusion coefficient for the simulation run with the LISM neutrals atoms. Oscillations in the LISM are due to very small GCR intensity and pressure gradients in that region resulting in a decrease in accuracy of equation (34). Inner heliospheric diffusion coefficients are considerably larger than those used in the self-consistent simulations of Myasnikov et al. [2000a, 2000b], even in their case 4, which used a comparable κ (assumed isotropic in their paper) at small heliocentric distances only. Fahr et al.  used a more realistic diffusion coefficient inside the heliopause, but the value of κ in the LISM was also at least 3 orders of magnitude smaller than in this paper. Additionally, neither paper included the possibility of κ being small in the inner heliosheath, inside the magnetic wall.
 The results for pc are generally in agreement with those obtained by the fluid models for the case when the largest values of κ for each particular model were used. We do not observe any GCR acceleration by the TS or by the compressive flow in the outer heliosheath, as reported by Myasnikov et al. [2000a]. However, we see a large gradient in pc in the inner heliosheath, which is absent from the above-mentioned models. Note also that the inner structure and the strong latitudinal dependence of GCR intensity inside the TS is absent from the fluid models.
 Since the diffusion coefficient in the LISM is considerably larger due to the larger size of the dominant turbulent eddies, there is no change in GCR intensity in this region and consequently, no modification to the bow shock. A possibility remains, however, for flow modification in the heliosheath (magnetic wall) region, where the diffusion coefficient is very small. We note, in particular, that the cosmic-ray pressure gradient is comparable to the charge exchange source term in the plasma momentum equation in the region close to the heliopause. A positive radial GCR pressure gradient may be expected to slow down the solar wind, which could result in a reduction of the distance to the heliopause. We expect cosmic rays to have a more profound influence on the dynamics of the inner heliosheath than in any other heliospheric region. It is interesting to note that Fahr et al.  found no change in the location of the heliopause when GCR were included self-consistently for their largest interstellar κ. We plan to verify their conclusion by including the GCR pressure term, ∂pc/∂xi, in the plasma flow equations (1)–(3) for a fully self-consistent treatment of the problem.
 The cosmic-ray “adiabatic index,” computed according to equation (35), is shown in Figure 15. The value of γc is 1.42 in the LISM, dropping to about 1.38 in the inner heliosphere, an effect caused by the filtering of predominantly lower energy particles at the “magnetic wall.” The lower values are closer to the ultra-relativistic value of 4/3 used in the work of Fahr et al.  than to nonrelativistic 5/3 or 1.5 used by Myasnikov et al. [2000a, 2000b]. Values of γc calculated with our model are very similar to those obtained by le Roux and Fichtner .
 The possibility of GCR reacceleration in the heliotail remains an open problem at this time. Our results (see Figures 8 and 10) show that acceleration takes place only when the neutral atoms are included; otherwise, plasma deceleration in the tail does not occur. If the diffusion lengthscale, κ/u, is much less than the lengthscale of velocity deceleration, it may be shown that the cosmic-ray intensity will change appreciably on distances of order u∣∂u/∂z∣−1. A recent paper of Izmodenov and Alexashov  shows that the shocked solar wind plasma will be decelerated to the LISM speed by 3000 AU along the heliopause, but it requires tens of thousands of AU for the neutrals-modified solar wind to fully equilibrate with the interstellar medium along the symmetry axis. A major difficulty with numerical modeling of the heliotail is a loss of the transverse (x-direction) resolution on a spherical grid with increasing heliocentric distances. A more detailed numerical model, possibly based on a rectangular grid, may be needed in this case.
 This paper presents an effort to develop a global self-consistent model of the interaction of the galactic cosmic rays with plasma flows in the heliosphere and the surrounding LISM. Our model includes the charge exchange between the interstellar neutral hydrogen atoms and the plasma protons as well as the convection, diffusion, and energy change of cosmic rays caused by the compressible plasma flow and its frozen-in magnetic field. The main advantage over earlier fluid models is the self-consistent determination of the energy-dependent diffusion coefficients and the availability of the GCR spectra. Additionally, an important advance over traditional modulation models lies in the inclusion of cosmic-ray modulation in the heliosheath region. The major results of our axisymmetric models may be summarized as follows.
 1. A small diffusion region exists in the inner heliosheath that prevents the entry of the lower-energy (<100 MeV) cosmic rays into the heliosphere. Small diffusion is caused by magnetic field (Bϕ) amplification near the stagnation point in the upstream direction and its convection along the heliopause and into the tail. All low-energy cosmic rays in the inner heliosphere are produced by cooling at high energies. The barrier also significantly attenuates higher-energy (GeV) particles so that the amount of modulation inside this barrier in comparable to or exceeds that in the unshocked solar wind. We note that the kinematic magnetic field model tends to overestimate this effect.
 2. Galactic cosmic rays are generally weakly coupled to the plasma in the heliosphere because diffusion coefficients are large. GCR are not expected to modify the structure of bow shock or the termination shock. An exception is the inner heliosheath region that has relatively large radial gradients in the GCR pressure. We expect this region to be more strongly influenced by cosmic-ray dynamics.
 3. The GCR distribution in the heliosphere is highly asymmetric at large heliocentric distances. However, the asymmetry disappears at smaller distances due to a specific diffusion pattern in a Parker field. This feature makes it difficult to use GCR to directly probe the structure of the outer heliosphere.
 4. A possibility for the reacceleration of GCR in the decelerated flow in the heliotail was detected. This effect is noticeably not present when charge exchange processes are ignored. However, our simulation domain did not extend to sufficiently large distances in the downstream direction and a more detailed model may be needed to properly study this effect.
Appendix A:: Modified Magnetic Field in the Axisymmetric Model
 In this section we derive equation (20) describing the evolution of the modified magnetic field. This field is produced by random motion of the footpoints associated with supergranulation [Jokipii and Parker, 1968] and is assumed to lie in the plane perpendicular to the radial direction at the origin (footpoint). For the sake of simplicity we take the modified field to be in the ϕ direction.
 We now assume that Bϕ has a regular component Bϕ0, described by the standard Archimedian spiral, and a random component Bmg, where g(l, t) is a random function of distance l along the streamline and time t (actually, a map of the transverse velocity of supergranulation uϕ(t)). Note that Bm need not be small compared to Bϕ0. Both Bϕ0 and Bm are assumed to vary on time- and lengthscales much larger than the rapidly oscillating function g. Indeed, for a typical supergranulation period of τ = 1 day and taking the solar wind velocity of 700 km s−1, we obtain the wavelength of λ ≃ 6 × 1012 cm = 0.4 AU, which is considerably smaller than the typical lengthscales of the problem (several AU to tens of AU).
 The derivation of equation (20) is straightforward. Equation (10) is multiplied by g and is averaged over a time Δt that is much larger than τ and over a distance Δl in the direction of the local wind velocity u = (u, 0, w) much larger than λ to obtain
Then, using the fact that any random function satisfies the conditions
 This work was supported, in part, by NASA grant NAG5-11621 and by NSF grants ATM-0296113 and ATM-0296114. V. F. acknowledges additional support from the International Space Science Institute (Bern) under the “Physics of the Heliotail” program. N. V. P. thanks Professor G. P. Zank for his hospitality and support while a visiting scholar at IGPP, where this work was accomplished. N. V. P. was also supported by the Russian Foundation for Basic Research grant 02-01-0948.
 Shadia Rifai Habbal thanks Horst Fichtner and Vladislav V. Izmodenov for their assistance in evaluating this paper.