Slab Turbulence Research Articles

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.

Perpendicular diffusion in magnetostatic slab turbulence: The theorem on reduced dimensionality and microscopic diffusion

In the present paper we revisit the problem of perpendicular diffusion in magnetostatic slab turbulence. It is shown that the particles cannot be scattered away from the magnetic field line by more than an unperturbed Larmor radius. This result confirms the theorem on reduced dimensionality proposed earlier for this special case. Furthermore, we estimate the motion of the particles away from the field line often called microscopic diffusion. The relation between the microscopic diffusion coefficient derived in the present paper and well-known results such as classical scattering theory and Bohm diffusion is also discussed.

On numerical turbulence generation for test-particle simulations

A modified method is presented to generate artificial magnetic turbulence that is used for test-particle simulations. Such turbulent fields are obtained from the superposition of a set of wave modes with random polarizations and random directions of propagation. First, it is shown that the new method simultaneously fulfils requirements of isotropy, equal mean amplitude and variance for all field components, and vanishing divergence. Second, the number of wave modes required for a stochastic particle behavior is investigated by using a Lyapunov approach. For the special case of slab turbulence, it is shown that already for 16 wave modes the particle behavior agrees with that shown for considerably larger numbers of wave modes.

Analytical description of nonlinear particle transport in slab turbulence: High particle energies and stochastic acceleration

Pitch-angle scattering, parallel spatial diffusion, and stochastic acceleration of cosmic rays are investigated analytically. Based on a second-order quasilinear theory, we derive analytical expressions for the aforementioned transport parameters for all possible magnetic field strengths and particle energies. This work complements previous work where only parallel diffusion for low energetic particles was considered. Furthermore, we compute the first time the momentum diffusion coefficient. It is also shown that the relation between the momentum diffusion coefficient and the parallel spatial diffusion coefficient is more complicated than assumed in previous work.

DRIFT COEFFICIENTS OF CHARGED PARTICLES IN TURBULENT MAGNETIC FIELDS

Using test-particle simulations, the off-diagonal elements of the diffusion tensor are evaluated numerically. The comparison of the so-obtained time-dependent drift coefficients with analytical approximations shows that, for weak turbulence strengths or for slab turbulence geometry, the weak scattering result provides an excellent agreement with the numerical results. For two- or three-dimensional turbulence geometry, however, neither the classical scattering result nor alternative analytical approaches provide an accurate description of the numerically obtained values. Furthermore, the influence is discussed of a non-constant energy range in the turbulence spectrum and of non-static turbulence, for which the time dependence is modeled using magnetohydrodynamic plasma waves.

THE EFFECT OF INTERMITTENT GYRO-SCALE SLAB TURBULENCE ON PARALLEL AND PERPENDICULAR COSMIC-RAY TRANSPORT

Earlier work based on nonlinear guiding center (NLGC) theory suggested that perpendicular cosmic-ray transport is diffusive when cosmic rays encounter random three-dimensional magnetohydrodynamic turbulence dominated by uniform two-dimensional (2D) turbulence with a minor uniform slab turbulence component. In this approach large-scale perpendicular cosmic-ray transport is due to cosmic rays microscopically diffusing along the meandering magnetic field dominated by 2D turbulence because of gyroresonant interactions with slab turbulence. However, turbulence in the solar wind is intermittent and it has been suggested that intermittent turbulence might be responsible for the observation of 'dropout' events in solar energetic particle fluxes on small scales. In a previous paper le Roux et al. suggested, using NLGC theory as a basis, that if gyro-scale slab turbulence is intermittent, large-scale perpendicular cosmic-ray transport in weak uniform 2D turbulence will be superdiffusive or subdiffusive depending on the statistical characteristics of the intermittent slab turbulence. In this paper we expand and refine our previous work further by investigating how both parallel and perpendicular transport are affected by intermittent slab turbulence for weak as well as strong uniform 2D turbulence. The main new finding is that both parallel and perpendicular transport are the net effect of an interplay between diffusivemore » and nondiffusive (superdiffusive or subdiffusive) transport effects as a consequence of this intermittency.« less

TIME-DEPENDENT PERPENDICULAR TRANSPORT OF FAST CHARGED PARTICLES IN A TURBULENT MAGNETIC FIELD

We present an analytic derivation of the temporal dependence of the perpendicular transport coefficient of charged particles in magnetostatic turbulence, for times smaller than the time needed for charged particles to travel the turbulence correlation length. This time window is left unexplored in most transport models. In our analysis all magnetic scales are taken to be much larger than the particle gyroradius, so that perpendicular transport is assumed to be dominated by the guiding center motion. Particle drift from the local magnetic field lines (MFLs) and magnetic field line random walk are evaluated separately for slab and three-dimensional (3D) isotropic turbulence. Contributions of wavelength scales shorter and longer than the turbulence coherence length are compared. In contrast to the slab case, particles in 3D isotropic turbulence unexpectedly diffuse from local MFLs; this result questions the common assumption that particle magnetization is independent of turbulence geometry. Extensions of this model will allow for a study of solar wind anisotropies.

FOCUSED ACCELERATION OF COSMIC-RAY PARTICLES IN NON-UNIFORM MAGNETIC FIELDS

The Fokker-Planck equation for cosmic-ray particles in a spatially varying guide magnetic field in a turbulent plasma is analyzed. An expression is derived for the mean rate of change of particle momentum, caused by the effect of adiabatic focusing in a non-uniform guide field. Results of an earlier diffusion-limit analysis are confirmed, and the physical picture is clarified by working directly with the Fokker-Planck equation. A distributed first-order Fermi acceleration mechanism is identified, which can be termed focused acceleration. If the forward- and backward-propagating waves have equal polarizations, focused acceleration operates when the net cross helicity of an Alfvenic slab turbulence is either negative in a diverging guide field or positive in a converging guide field. It is suggested that focused acceleration can contribute to the formation of the anomalous cosmic-ray spectrum at the heliospheric termination shock.

Ultra-high-energy galactic cosmic ray hadrons from distributed focused acceleration

Large-scale spatial variations of the galactic guide magnetic field, characterized by the focusing length L = − [ d ( ln B ( z ) ) / d z ] − 1 , together with a nonzero value of the cross helicity state of Alfvenic slab turbulence H give rise to first-order Fermi, betatron-like acceleration of energetic cosmic ray hadrons, if the product H L 0 is negative. The equilibrium momentum spectrum of accelerated particles by this focus ed acceleration mechanism is determined. Assuming a simple chemical composition of injected cosmic rays (99 percent protons and 1 percent Fe), it is shown that the ultra-high-energy Fe/p-ratio is a rapidly changing function of the total energy. The derived chemical composition of cosmic rays is first proton-dominated at low energies, then becomes Fe-dominated between 0.1 and 2.4 PeV, before becoming strongly proton-dominated between 2.4 PeV and 1100 PeV, and strongly Fe-dominated at energies beyond 1100 PeV.

A UNIFIED PARTICLE DIFFUSION THEORY FOR CROSS-FIELD SCATTERING: SUBDIFFUSION, RECOVERY OF DIFFUSION, AND DIFFUSION IN THREE-DIMENSIONAL TURBULENCE

A new nonlinear theory for cosmic-ray scattering across the mean magnetic field is derived. This theory can be applied for arbitrary turbulence geometry. Previous theories such as the extended nonlinear guiding center theory are deduced as special limits. Furthermore, the new theory can explain subdiffusive transport for slab turbulence and the recovery of diffusion for slab/two-dimensional and three-dimensional turbulence. The nonlinear standard theory for field line wandering can be obtained as a special limit.

Random walk of magnetic field lines in dynamical turbulence: A field line tracing method. I. Slab turbulence

To study the wandering of magnetic field lines is an important subject in theoretical physics. Results of field line random walk theories can be applied in plasma physics as well as astrophysics. Previous investigations are based on magnetostatic models. These models have been used in analytical work as well as in computer simulations to warrant mathematical and numerical tractability. To replace the magnetostatic model by a dynamical turbulence model is a difficult task. In the present article, a field line tracing method is used to describe field line wandering in dynamical magnetic turbulence. As examples different models are employed, namely, the plasma wave model, the damping model of dynamical turbulence, and the random sweeping model. It is demonstrated that the choice of the turbulence model has a very strong influence on the field line structure. It seems that if dynamical turbulence effects are included, Markovian diffusion can be found for other forms of the wave spectrum as in the magnetostatic model. Therefore, the results of the present paper are useful to specify turbulence models. As a further application we consider charged particle transport at early times.

INHOMOGENEOUS NEARLY INCOMPRESSIBLE DESCRIPTION OF MAGNETOHYDRODYNAMIC TURBULENCE

The nearly incompressible theory of magnetohydrodynamics (MHD) is formulated in the presence of a static large-scale inhomogeneous background. The theory is an inhomogeneous generalization of the homogeneous nearly incompressible MHD description of Zank & Matthaeus and a polytropic equation of state is assumed. The theory is primarily developed to describe solar wind turbulence where the assumption of a composition of two-dimensional (2D) and slab turbulence with the dominance of the 2D component has been used for some time. It was however unclear, if in the presence of a large-scale inhomogeneous background, the dominant component will also be mainly 2D and we consider three distinct MHD regimes for the plasma beta ? 1, ? ~ 1, and? 1. For regimes appropriate to the solar wind (? 1, ? ~ 1), compared to the homogeneous description of Zank & Matthaeus, the reduction of dimensionality for the leading-order description from three dimensional (3D) to 2D is only weak, with the parallel component of the velocity field proportional to the large-scale gradients in density and the magnetic field. Close to the Sun, however, where the large-scale magnetic field can be considered as purely radial, the collapse of dimensionality to 2D is complete. Leading-order density fluctuations are shown to be of the order of the sonic Mach number and evolve as a passive scalar mixed by the turbulent velocity field. It is emphasized that the usual pseudosound relation used to relate density and pressure fluctuations through the sound speed as ?? = c 2 s ?p is not valid for the leading-order density fluctuations, and therefore in observational studies, the density fluctuations should not be analyzed through the pressure fluctuations. The pseudosound relation is valid only for higher order density fluctuations, and then only for short-length scales and fast timescales. The spectrum of the leading-order density fluctuations should be modeled as k ?5/3 in the inertial range, followed by a Bessel function solution K ?(k), where for stationary turbulence ? = 1, in the viscous-convective and diffusion range. Other implications for solar wind turbulence with an emphasis on the evolution of density fluctuations are also discussed.

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.

Simulation results on the influence of magneto-hydrodynamic waves on cosmic ray particles

The problem of cosmic ray scattering in magnetohydrodynamic turbulence is revisited using a recently developed Monte Carlo simulation code. Diffusion coefficients are obtained using magnetostatic turbulence, slab Alfvénic turbulence with and without electric fields and isotropic magnetosonic turbulence with and without electric fields. All non-magnetostatic simulations are carried out for different values of the Alfvén velocity so that the effects of different particle to Alfvén velocity ratios can be studied. For slab turbulence, it is shown that Alfvén waves guarantee the recovery of diffusion, as predicted by analytical studies. The inclusion of electric fields, however, immediately leads to superdiffusion, if the Alfvén velocity is not too small. Furthermore, the energy gain through stochastic acceleration processes is investigated both qualitatively and quantitatively. The comparison with previous simulation reveals both similarities and interesting differences.

On the diffusivity of cosmic ray transport

A common assumption in particle transport theory is the assumption of diffusive propagation, corresponding to a linear increase of the mean square displacement of particle orbits with time. In recent years, however, several authors have discovered nondiffusive transport in observations and in test particle simulations. Some analytical theories also predict subdiffusion or superdiffusion of charged particles. In this article, a recently developed test particle code is employed to explore diffusive and nondiffusive transport regimes. For slab, two‐component, and isotropic turbulence the simulations are performed for different particle energies. It is demonstrated that, in most cases, cross‐field transport is indeed subdiffusive. The level of subdiffusivity is sensitively controlled by the turbulence geometry.

A NOTE ON PERPENDICULAR SCATTERING LENGTHS

The problem of cosmic ray diffusion in magnetostatic slab turbulence is revisited. It is known that, for large timescales, the perpendicular diffusion coefficient is subdiffusive. Although, for small timescales, the field line random walk limit should apply, it is shown that the perpendicular motion is dominated by the Larmor orbit, and that no constant scattering length can be seen. It is therefore concluded that, in magnetostatic slab turbulence, perpendicular transport is completely suppressed.

Confinement and anisotropy of ultrahigh-energy cosmic rays in isotropic plasma wave turbulence

The mean free path and anisotropy of galactic cosmic rays is calculated in weak plasma wave turbulence that is isotropically distributed with respect to the ordered uniform magnetic field. The modifications on the value of the Hillas energy, above which cosmic rays are not confined to the Galaxy, are calculated too. The original determination of the Hillas limit has been based on the case of slab turbulence where only parallel propagating plasma waves are allowed. We use quasilinear cosmic ray Fokker-Planck coefficients to calculate the mean free path and the anisotropy in isotropic plasma wave turbulence. In isotropic plasma wave turbulence the Hillas limit is enhanced by about four orders of magnitude to Ec = 2.03- 105 Ane 1/2 (Lmax/10 pc) PeV resulting from the dominating influence of transit-time damping interactions of cosmic rays with obliquely propagating magnetosonic waves. Below the energy Ec the cosmic ray mean free path and the anisotropy exhibit the well known E1/3 energy dependence. At energies higher than Ec both transport parameters steepen to a E3-dependence. This implies that cosmic rays even with ultrahigh energies of several hundreds of EeV can be rapidly pitch-angle scattered by interstellar plasma turbulence, and are thus confined to the Galaxy.

Analytical forms of correlation functions and length scales of astrophysical turbulence

In this article correlation-functions, correlation-scales, and field line diffusion coefficients of magnetic turbulence are discussed. For the slab turbulence model these parameters are calculated analytically for different forms of the wave-spectrum of the magnetic fluctuations. It is also demonstrated in this article that the definition of the correlation-length used previously yields infinity in some cases. Furthermore, correlation-functions and correlation-scales are calculated analytically for the well-established slab/2D composite model. It is demonstrated that the 2D correlation function decays much slower than the slab correlation function. These results are essential for understanding previous theoretical results of superdiffusive field line random walk and very recent spacecraft measurements of magnetic correlations in the solar wind.

Perpendicular transport of charged particles in slab turbulence: recovery of diffusion for realistic wavespectra?

It is a standard assumption that parallel scattering suppresses perpendicular diffusion to a subdiffusive level when the turbulence lacks transverse structure. In several previous papers, it has been demonstrated that a running or time-dependent diffusion coefficient decreases with increasing time. In most studies, it was assumed that this subdiffusive behaviour is directly related to the slab structure of turbulence. By applying reliable analytical calculations in combination with realistic forms of the wavespectrum of interplanetary turbulence, it is demonstrated in this paper that the long-time behaviour of running diffusion coefficients depends on the wavespectrum in the energy range. If the energy range spectral index exceeds unity, diffusion is recovered. This recovery of diffusion obtained within the magnetostatic slab model is an unexpected result.

Field line wandering and perpendicular scattering of charged particles in Alfvénic slab turbulence

We describe analytically the magnetic field line random walk in Alfvenic slab turbulence. This result can easily be combined with a compound transport model to describe analytically the perpendicular scattering of charged cosmic rays. We demonstrate that the well known subdiffusive solution for magnetostatic turbulence is only valid for shorter time scales. For longer times, perpendicular diffusion of charged test-particles is recovered due to wave propagation effects. For low energy cosmic rays, this new result could be important for understanding heliospheric observations.