J. A. le Roux

The simulated features of heliospheric cosmic-ray modulation with a time-dependent drift model. I: General effects of the changing neutral sheet over the period 1985-1990

A time-dependent drift model is used to simulate the heliospheric modulation of galactic cosmic rays, with emphasis on the effects of the wavy heliospheric neutral sheet (HNS) as a function of time during the period 1985-1990. The model predicted a clearly defined time difference between the time when minimum modulation occurred at earth and at radial distances farther away from the sun, indicating that the HNS plays an important part in establishing this observed time delay. The movements of the Voyager 1 and 2 and Pioneer 10 spacecraft were simulated in order to calculate instantaneous radial and latitudinal gradients. The time dependencies of these gradients were found to follow the observations well. The model study indicates that the HNS cannot be ignored, especially during periods of low to moderate solar activity, when the modulation of cosmic rays is described.

The role of merged interaction regions and drafts in the heliospheric modulation of cosmic rays beyond 20 AU - A computer simulation

The time-dependent, axially symmetric transport equation of cosmic rays in the heliosphere was solved numerically using the Voyager 2 magnetic field measurements to simulate merged interaction regions (MIRs) and rarefaction regions (RRs) for the period 1985-1989. Concurrently, the wavy neutral sheet was used as a time-dependent, drift parameter. The two approaches were first modeled separately and then in combination in this exploratory study. The drift approach proved to be the more successful for 1985 to early 1988 because it could reproduce the intensity levels, the factor modulation, and latitudinal gradients for 1 GeV protons at 23 AU without difficulties

TIME-DEPENDENT ACCELERATION OF INTERSTELLAR PICKUP IONS AT THE HELIOSPHERIC TERMINATION SHOCK USING A FOCUSED TRANSPORT APPROACH

A time-dependent focused transport approach to modeling diffusive shock acceleration of interstellar pickup ions at the termination shock is discussed. By taking into account time variations in the magnetic field angle, and thus by implication in the shock obliquity and injection speed at the termination shock using Voyager 1 observations as guide, we show that unaccelerated core interstellar pickup protons can be accelerated by the nearly perpendicular termination shock at the heliospheric nose. Many features of anomalous cosmic rays, observed by the Voyager spacecraft at energies below the big spectral dip starting at ~3 MeV, can be successfully reproduced with this approach. This includes multiple power-law spectral slopes with stable breaking points, power-law spectral slopes that are harder than predicted by standard diffusive shock acceleration, large pitch-angle anisotropies with strong time variations upstream that converge to steady-state isotropy in the heliosheath, upstream pitch-angle anisotropies that peak at ~1 MeV, episodic strongly anisotropic intensity spikes at the termination shock, and strong spectral volatility upstream that is reduced downstream and almost completely disappears farther downstream.

Diffusive Shock Acceleration and Reconnection Acceleration Processes

Shock waves, as shown by simulations and observations, can generate high levels of downstream vortical turbulence, including magnetic islands. We consider a combination of diffusive shock acceleration (DSA) and downstream magnetic-island-reconnection-related processes as an energization mechanism for charged particles. Observations of electron and ion distributions downstream of interplanetary shocks and the heliospheric termination shock (HTS) are frequently inconsistent with the predictions of classical DSA. We utilize a recently developed transport theory for charged particles propagating diffusively in a turbulent region filled with contracting and reconnecting plasmoids and small-scale current sheets. Particle energization associated with the anti-reconnection electric field, a consequence of magnetic island merging, and magnetic island contraction, are considered. For the former only, we find that (i) the spectrum is a hard power law in particle speed, and (ii) the downstream solution is constant. For downstream plasmoid contraction only, (i) the accelerated spectrum is a hard power law in particle speed; (ii) the particle intensity for a given energy peaks downstream of the shock, and the distance to the peak location increases with increasing particle energy, and (iii) the particle intensity amplification for a particular particle energy, f(x,c/c_0)/f(0,c/c_0), is not 1, as predicted by DSA, but increases with increasing particle energy. The general solution combines both the reconnection-induced electric field and plasmoid contraction. The observed energetic particle intensity profile observed by Voyager 2 downstream of the HTS appears to support a particle acceleration mechanism that combines both DSA and magnetic-island-reconnection-related processes.

A KINETIC TRANSPORT THEORY FOR PARTICLE ACCELERATION AND TRANSPORT IN REGIONS OF MULTIPLE CONTRACTING AND RECONNECTING INERTIAL-SCALE FLUX ROPES

Simulations of particle acceleration in turbulent plasma regions with multiple contracting and merging (reconnecting) magnetic islands emphasize the key role of temporary particle trapping in island structures for the efficient acceleration of particles to form hard power-law spectra. Statistical kinetic transport theories have been developed that capture the essential physics of particle acceleration in multi-island regions. The transport theory of Zank et al. is further developed by considering the acceleration effects of both the mean and the variance of the electric fields induced by the dynamics of multiple inertial-scale flux ropes. A focused transport equation is derived that includes new Fokker-Planck terms for particle scattering and stochastic acceleration due to the variance in multiple flux-rope magnetic fields, plasma flows, and reconnection electric fields. A Parker transport equation is also derived in which a new expression for momentum diffusion appears, combining stochastic acceleration by particle scattering in the mean multi-flux-rope electric fields with acceleration by the variance in these electric fields. Test particle acceleration is modeled analytically considering drift acceleration by the variance in the induced electric fields of flux ropes in the slow supersonic, radially expanding solar wind. Hard power-law spectra occur for sufficiently strong inertial-scale flux ropes with an index modified by adiabatic cooling, solar wind advection, and diffusive escape from flux ropes. Flux ropes might be sufficiently strong behind interplanetary shocks where the index of suprathermal ion power-law spectra observed in the supersonic solar wind can be reproduced.

A Generalized Nonlinear Guiding Center Theory for the Collisionless Anomalous Perpendicular Diffusion of Cosmic Rays

The original nonlinear guiding center (NLGC) theory by Matthaeus et al. was a breakthrough in establishing a theory that promised to reproduce for the first time the anomalous perpendicular diffusion results from test particle trajectory calculations in prescribed static magnetic field turbulence dominated by a two-dimensional component. The assumptions used in this approach guaranteed anomalous diffusion to be a classical process (the variance � Δx 2 �∝ (Δt) α with α = 1). However, Shalchi & Kourakis showed that similar calculations can be even better reproduced within the context of a generalized compound diffusion model for anomalous perpendicular diffusion whereby anomalous diffusion is nonclassical (0 <α< 1 (subdiffusion) or 1 <α< 2 (superdiffusion)). In this paper, it is shown how NLGC theory can be generalized to model compound diffusion conditions consistent with the generalized compound diffusion model in terms of a nonuniform plasma medium. Such a medium is assumed to generate a distribution of cosmic-ray pitch-angle scattering times for cosmic rays interacting resonantly with a minor small-scale slab turbulence component. This suggests that the anomalous perpendicular diffusion from the test particle simulations might be explained best within the framework of anomalous diffusion in a nonuniform plasma medium. It is argued that, during intermediate times when the magnetic field appears to be static in the quiet solar wind near Earth, generalized NLGC theory predicts possibly subdiffusive anomalous perpendicular transport for intermediate-energy (E � 400–500 MeV) cosmic rays because they experience nonuniform scattering conditions. These conditions are suggested to be a product of stochastic wave growth of small-scale slab turbulence resulting intermittently in large patches of intense slab turbulence where scattering times are reduced and cosmic rays are trapped. Classical anomalous diffusion is expected to be restored at high cosmic-ray energies (E � 400–500 MeV) in accordance with original NLGC theory because they encounter more uniform scattering conditions due to weaker stochastic wave growth. Magnetic turbulence observations near Earth during 2003 were used to show that the solar wind can be characterized as a nonuniform medium sustaining a power-law distribution of cosmic-ray pitch-angle scattering times. However, application of generalized NLGC theory to the 2003 observations produced unexpectedly a negative exponent for the variance of the anomalous diffusion ( α< 0). This indicates that particle bunching occurs as cosmic rays are trapped in plasma regions associated with strong particle scattering. Such effects can plausibly be explained by the intermittent presence of strong compressive turbulence downstream of shocks, stream, and interaction regions.

The simulated features of heliospheric cosmic-ray modulation with a time-dependent drift model. IV: The role of heliospheric neutral sheet deformation

Previous calculations with a time-dependent drift model revealed the model to be less successful in describing time-dependent modulation during periods of moderate to large solar activity. In this paper, it is argued that a major reason for this is that the previously used wavy heliospheric neutral sheet (HNS) description was based on an idealized HNS not subject to any spatial evolution while propagating radially outward. It is suggested that the deformation and compression of HNS wave peaks will lead to significant increases in the crossfield diffusion across these peaks (short-circuiting). The cosmic rays will effectively experience reduced tilt angles and therefore a reduction in the integrated HNS modulation effect between an observer and the heliospheric boundary. During periods of moderate to large solar activity these HNS deformation processes are progressively more frequent and should lead to a significant reduction in time-dependent modulation as predicted by drift models. Calculations done with radially propagating tilt angles that effectively decrease with radial distance give the expected reduction which improves the general description of modulation from 1987-1988.

The simulated features of heliospheric cosmic-ray modulation with a time-dependent drift model. III - General energy dependence

The time-dependent cosmic-ray transport equation is solved numerically in an axially symmetric heliosphere. Gradient and curvature drifts are incorporated, together with an emulated wavy neutral sheet. This model is used to simulate heliospheric cosmic-ray modulation for the period 1985-1989 during which drifts are considered to be important. The general energy dependence of the modulation of Galactic protons is studied as predicted by the model for the energy range 1 MeV to 10 GeV. The corresponding instantaneous radial and latitudinal gradients are calculated, and it is found that, whereas the latitudinal gradients follow the trends in the waviness of the neutral sheet to a large extent for all energies, the radial gradients below about 200 MeV deviate from this general pattern. In particular, these gradients increase when the waviness decreases for the simulated period 1985-1987.3, after which they again follow the neutral sheet by increasing rapidly.

Compound and perpendicular diffusion of cosmic rays and random walk of the field lines: II. Non-parallel particle transport and drifts

Compound transport of energetic charged particles across the mean magnetic field due to field line random walk is investigated by means of a Chapman–Kolmogorov equation. The probability distribution function (pdf) for the particle transport across the field P⊥ is given as a convolution of the pdf for random walk of the magnetic field, PFRW, with the pdf Pp, for particle transport relative to the random walking field. The particle propagator Pp includes the effects of advection, drift, parallel diffusion and local perpendicular diffusion of particles relative to the random walking field. At early times, the particles sub-diffuse across the field due to field line random walk. At late times, the effective cross-field diffusion coefficient has the form κ⊥e = κ⊥ + κF. The diffusion coefficient κ⊥ is the local cross-field diffusion coefficient due to particle scattering in the random magnetic field. The diffusion coefficient κF is due to coherent particle advection parallel to the mean magnetic field B0 coupled with transverse random walk of the magnetic field. Estimates of cross-field diffusion due to field line random walk, advection and drift are obtained both near to the heliospheric current sheet at Earth and at higher helio-latitudes. Cross-field diffusion due to field line random walk and advection is shown to be an important transport mechanism for low-energy particles near the current sheet, where the effects of drifts are negligible. Drift effects and field line random walk are also assessed at higher helio-latitudes off the current sheet, for a model interplanetary magnetic field, with a flat current sheet in the helio-equatorial plane.

Nonlinear field line random walk for non-Gaussian statistics

We investigate analytically the random walk of magnetic field lines. In previous articles about this subject, a Gaussian model has been used for replacing the field line distribution function. Here we employ a Kappa distribution to investigate the influence of a non-Gaussian statistics. As shown, only the amplitude of the field line mean square deviation and the field line diffusion coefficient are different from the Gaussian model if we assume κ > 2. It seems that the exact form of the field line distribution is less important for computing field line diffusion coefficients in this case. This conclusion confirms previous investigations performed within the framework of a Gaussian statistics.