R. A. Burger

Rigidity dependence of cosmic ray proton latitudinal gradients measured by the Ulysses spacecraft: Implications for the diffusion tensor

The latitudinal gradient of cosmic ray protons observed by Ulysses during September 1994 to July 1995 is small, and it increases as function of rigidity up to ∼ 2 GV and then decreases. Although previous drift models could reproduce the observed small positive gradient for an A > 0 solar polarity cycle, they produced a maximum at a rigidity well below 1 GV, in contrast to the observations. After exploring various options, it turns out that changing the rigidity dependence of the perpendicular diffusion coefficient (DC) in the polar direction so that it differs from that of the parallel DC is the most effective way to obtain good agreement with data. Specifically, we find that this DC must have a flatter rigidity dependence than the parallel DC in order to reproduce the observed rigidity dependence of the latitudinal gradient of protons during an A > 0 solar polarity cycle. We argue that the present study, combined with studies by other authors, suggests that the perpendicular mean-free path of particles with rigidity R ≳ 0.1 GV has at least two distinct components. One is independent of particle rigidity, and one is proportional to the square of the particle rigidity at low rigidity and flattens to become almost independent of it at higher rigidity. We also show an example of the rigidity dependence of this gradient that Ulysses might observe during an A 0 cycle and could impose stricter constraints on the diffusion tensor.

A Fisk-Parker Hybrid Heliospheric Magnetic Field with a Solar-Cycle Dependence

We present a refinement of the Fisk-Parker hybrid field of Burger and Hitge which now includes a region bordering the solar rotational equator where magnetic field footpoint motion occurs only through diffusive reconnection. The hybrid field, therefore, only occurs above a certain latitude in a given hemisphere, and in the equatorial region the field is a pure Parker field. We also propose a simple qualitative model for the solar cycle dependence of the hybrid field, taking into account changes in the tilt angle of the heliospheric current sheet and the latitudinal extend of the polar coronal hole on the photosphere and on the source surface over the course of a solar activity cycle. We find that the amplitude of magnetic field fluctuations for assumed solar minimum parameters would not be observable above the background noise (see Roberts and coworkers). We also show that for these parameters, periodicities associated with differential footpoint motion would be barely distinguishable from rigid rotation at the solar equatorial rate. We point out that the question of periodicities in magnetic field data is perhaps more complicated than previously thought. We confirm the result of Burger and Hitge that a Fisk-type heliospheric magnetic field provides a natural explanation for the observed linear relationship between the amplitude of the recurrent cosmic-ray variations and the global latitude gradient (see Zhang). We show that this relationship holds for helium, protons, and electrons. Moreover, we show that the constant of proportionality is larger when qA > 0 than when qA < 0, as inferred from observations by Richardson and coworkers.

Suppression of Particle Drifts by Turbulence

We present results from direct numerical simulations showing the suppression of the large-scale drift motion of an ensemble of charged particles in a nonuniform turbulent magnetic field. We find that when scattering is negligible, the ensemble average drift velocity is in the direction predicted by the usual guiding center theory. When scattering is very strong, we find that all large-scale drift motions vanish. For an intermediate amount of scattering we find that the antisymmetric drift velocity is typically suppressed by a larger amount than the antisymmetric drift coefficient. We show that the total drift motion of the ensemble is not necessarily completely contained in the antisymmetric part of the diffusion tensor. Because of the occurrence of scattering, knowledge of the spatial variation of the symmetric part of the diffusion tensor is also needed to fully describe the total drift motion of the ensemble.

Modulation effects of anisotropic perpendicular diffusion on cosmic ray electron intensities in the heliosphere

The modulation of cosmic ray electrons provides a useful tool to study the diffusion tensor applicable to heliospheric modulation. Electron modulation responds directly to the assumed energy dependence of the diffusion coefficients below ∼500 MeV in contrast to protons which experience large adiabatic energy losses below this energy. As a result of this and because drifts become unimportant for electrons at these low energies, conclusions can be made about the appropriate diffusion coefficients. Using a modulation model, we illustrate the role of anisotropic perpendicular diffusion on electron modulation. In general, we find that perpendicular diffusion dominates electron modulation below ∼100 MeV. Enhancing it in the polar direction typically produced an increase in modulation for both the A > O (e.g., ∼1990 to ∼2000) and A < 0 (e.g., ∼1980 to ∼1990) solar magnetic polarity cycles. It also causes the radial dependence of the intensity to become more uniform throughout the heliosphere, and causes a significant reduction in the latitude dependence of the intensities at all radial distances, with the largest effects in the inner heliosphere and at low energies. This agrees with studies of cosmic ray protons, which suggest that perpendicular diffusion enhanced in the polar direction of the heliosphere is required in conventional drift models to explain the small latitudinal gradients observed for protons on board the Ulysses spacecraft. The role of enhanced perpendicular diffusion was further investigated by examining electron modulation as a function of the “tilt angle” α of the wavy current sheet. In general, a reduction occurred between the modulation differences caused by drifts as a function of α for both polarity cycles. This work illustrates that anisotropic perpendicular diffusion has profound effects on the modulation of galactic cosmic ray electrons during both polarity cycles.

An AB initio model for cosmic-ray modulation

A proper understanding of the effects of turbulence on the diffusion and drift of cosmic rays (CRs) is of vital importance for a better understanding of CR modulation in the heliosphere. This study presents an ab initio model for CR modulation, incorporating for the first time the results yielded by a two-component turbulence transport model. This model is solved for solar minimum heliospheric conditions, utilizing boundary values chosen so that model results are in reasonable agreement with spacecraft observations of turbulence quantities in the solar ecliptic plane and along the out-of-ecliptic trajectory of the Ulysses spacecraft. These results are employed as inputs for modeled slab and two-dimensional (2D) turbulence energy spectra. The modeled 2D spectrum is chosen based on physical considerations, with a drop-off at the very lowest wavenumbers. There currently exist no models or observations for the wavenumber where this drop-off occurs, and it is considered to be the only free parameter in this study. The modeled spectra are used as inputs for parallel mean free path expressions based on those derived from quasi-linear theory and perpendicular mean free paths from extended nonlinear guiding center theory. Furthermore, the effects of turbulence on CR drifts are modeled in a self-consistent way, also employing a recently developed model for wavy current sheet drift. The resulting diffusion and drift coefficients are applied to the study of galactic CR protons and antiprotons using a 3D, steady-state CR modulation code, and sample solutions in fair to good agreement with multiple spacecraft observations are presented.

The Effect of a Fisk-Type Heliospheric Magnetic Field on Cosmic-Ray Modulation

The Fisk model of the heliospheric magnetic field (HMF) has proved to be quite difficult to implement in three-dimensional numerical modulation codes. We have developed a divergence-free Fisk-Parker hybrid HMF that is easier to implement and that should be a reasonable approximation for the case when field lines open into the heliosphere both in the polar coronal holes and at low latitudes. To study the effects of this hybrid field on galactic cosmic rays, we solve the three-dimensional steady state Parker transport equation numerically, covering the entire model heliosphere. We find that latitudinal cosmic-ray gradients are reduced as expected, but only during epochs when particles are drifting from the polar regions of the heliosphere toward the ecliptic plane. We investigate 26 day recurrent variations of both protons and electrons and find that the amplitude of these variations are directly proportional to the latitudinal intensity gradient, independent of species, in agreement with observational studies. In the absence of drifts in the code, electrons and protons obey different but still linear relationships.

REDUCTION OF DRIFT EFFECTS DUE TO SOLAR WIND TURBULENCE

Gradient and curvature drift play a key role in the modulation of cosmic rays. Reduction in the drift coefficient due to turbulence has been demonstrated unambiguously through direct numerical simulations, but a theory that can explain these results is still lacking. We introduce a parameterized form of the drift coefficient based on direct numerical simulations and show that good agreement with observed proton energy spectra at Earth can be found when it is used in a numerical modulation model. We show that the turbulence ultrascale, for which no observations currently exist, plays an important role in drift reduction. The magnitude at Earth and spatial dependence of this quantity required to fit cosmic-ray observations at Earth are argued to be plausible based on the required properties of the two-dimensional turbulence spectrum at large scales.

On the Ability of Different Diffusion Theories to Account for Directly Simulated Diffusion Coefficients

We present direct numerical simulations of charged-particle transport in a turbulent magnetic field. The magnetic field model used in the simulations consists of a composite of statistically homogeneous slab and two-dimensional turbulence representative of solar wind conditions at Earth. This turbulent magnetic field is then added to a uniform background magnetic field. We find that the parallel and perpendicular mean free paths are well described by power laws as a function of rigidity at different turbulence levels. At a low level of turbulence we find that quasi-linear theory and the field line random walk theory for the parallel and perpendicular mean free paths, respectively, provide predictions that are in good agreement with the simulated mean free paths. At intermediate turbulence levels the simulated parallel and perpendicular mean free paths are best accounted for by recently proposed nonlinear theories, while quasi-linear theory and the field line random walk theory overestimate the simulated mean free paths. At high turbulence levels neither quasi-linear theory and the field line random walk theory nor the nonlinear theories provide predictions that are in good agreement with the simulated parallel and perpendicular mean free paths.

An AB initio model for the modulation of galactic cosmic-ray electrons

The modulation of galactic cosmic-ray electrons is studied using an ab initio three-dimensional steady state cosmic-ray modulation code in which the effects of turbulence on both the diffusion and drift of these cosmic-rays are treated as self-consistently as possible. A significant refinement is that a recent two-component turbulence transport model is used. This model yields results in reasonable agreement with observations of turbulence quantities throughout the heliosphere. The sensitivity of computed galactic electron intensities to choices of various turbulence parameters pertaining to the dissipation range of the slab turbulence spectrum, and to the choice of model of dynamical turbulence, is demonstrated using diffusion coefficients derived from the quasi-linear and extended nonlinear guiding center theories. Computed electron intensities and latitude gradients are also compared with spacecraft observations.

GLOBAL PROCESSES THAT DETERMINE COSMIC RAY MODULATION

The global processes that determine cosmic ray modulation are reviewed. The essential elements of the theory which describes cosmic ray behavior in the heliosphere are summarized, and a series of discussions is presented which compare the expectations of this theory with observations of the spatial and temporal behavior of both galactic cosmic rays and the anomalous component; the behavior of cosmic ray electrons and ions; and the 26-day variations in cosmic rays as a function of heliographic latitude. The general conclusion is that the current theory is essentially correct. There is clear evidence, in solar minimum conditions, that the cosmic rays and the anomalous component behave as is expected from theory, with strong effects of gradient and curvature drifts. There is strong evidence of considerable latitude transport of the cosmic rays, at all energies, but the mechanism by which this occurs is unclear. Despite the apparent success of the theory, there is no single choice for the parameters which describe cosmic ray behavior, which can account for all of the observed temporal and spatial variations, spectra, and electron vs. ion behavior.