G. M. Webb

The Transport of Low-frequency Turbulence in Astrophysical Flows. I. Governing Equations

Numerous problems in space physics and astrophysics require a detailed understanding of the transport and dissipation of low-frequency turbulence in an expanding magnetized flow. We employ a scale-separated decomposition of the incompressible MHD equations (based on an Elssasser description) and develop a moment hierarchy to describe the transport of the total energy density in fluctuations, the cross-helicity, the energy difference, and correlation lengths corresponding to forward- and backward-propagating modes and to the energy difference. The dissipation terms for the various transport equations are derived. One-point closure schemes are utilized. The technical elements of this work that distinguish it from previous studies are (1) the inclusion of the large-scale background inhomogeneous Alfvenic velocity V A at a level of detail greater than before, (2) the introduction of a tractable slow timescale closure to eliminate high-frequency interference terms that is likely to prove a useful approximation for practical problems related to the transport of turbulence in an inhomogeneous flow such as the solar wind or solar corona, and finally, (3) we develop a simplified phenomenology for the energy difference or equivalently residual energy that may be useful for practical applications. This yields a coupled system of six equations that describes the transport of turbulence in inhomogeneous sub-Alfvenic and super-Alfvenic flows. The turbulence transport equations are quasi-linear in their spatial evolution operators and nonlinear in the dissipation terms, making the model equations relatively tractable to analysis.

Structures of Interplanetary Magnetic Flux Ropes and Comparison with Their Solar Sources

Magnetic reconnection is essential to release the flux rope during its ejection. The question remains: how does the magnetic reconnection change the flux rope structure? Following the original study of \citet{Qiu2007}, we compare properties of ICME/MC flux ropes measured at 1 AU and properties of associated solar progenitors including flares, filaments, and CMEs. In particular, the magnetic field-line twist distribution within interplanetary magnetic flux ropes is systematically derived and examined. Our analysis shows that for most of these events, the amount of twisted flux per AU in MCs is comparable with the total reconnection flux on the Sun, and the sign of the MC helicity is consistent with the sign of helicity of the solar source region judged from the geometry of post-flare loops. Remarkably, we find that about one half of the 18 magnetic flux ropes, most of them being associated with erupting filaments, have a nearly uniform and relatively low twist distribution from the axis to the edge, and the majority of the other flux ropes exhibit very high twist near the axis, of up to

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

\gtrsim 5

Diffusive Shock Acceleration and Reconnection Acceleration Processes

turns per AU, which decreases toward the edge. The flux ropes are therefore not linear force free. We also conduct detailed case studies showing the contrast of two events with distinct twist distribution in MCs as well as different flare and dimming characteristics in solar source regions, and discuss how reconnection geometry reflected in flare morphology may be related to the structure of the flux rope formed on the Sun.

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

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.

PICKUP ION DYNAMICS AT THE HELIOSPHERIC TERMINATION SHOCK OBSERVED BY VOYAGER 2

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

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.

Local and nonlocal advected invariants and helicities in magnetohydrodynamics and gas dynamics I: Lie dragging approach

The recent Voyager 2 (V2) observations of the termination shock (TS) indicate that it is a plasma shock unlike any other in the heliosphere with the dynamics and structure heavily influenced by the presence of an energized population of pickup ions (PUIs). The unexpected finding of cold plasma downstream of the TS in the heliosheath, corresponding to very little heating of the thermal solar wind (SW), suggests that the energy dissipated by the shock is dominated by the energization of PUIs at the TS. We examine the shock surfing mechanism at the test particle level, where multiply reflected ions (MRIs) gain energy from the motional electric field as a consequence of reflection from the cross-shock potential (CSP), for a model of the TS3 (the third TS crossing measured by V2). The energization of PUI filled-shell distributions at a stationary, perpendicular model of the TS3 indicates that shock surfing can provide both substantial PUI acceleration and a dissipation mechanism at the TS. For a sufficiently strong CSP and sufficiently narrow shock ramp MRI acceleration can account for the missing energy of the downstream SW plasma.

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

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.

Nonlinear field line random walk for non-Gaussian statistics

In this paper advected invariants and conservation laws in ideal magnetohydrodynamics (MHD) and gas dynamics are obtained using Lie dragging techniques. There are different classes of invariants that are advected or Lie dragged with the flow. Simple examples are the advection of the entropy S (a 0-form), and the conservation of magnetic flux (an invariant 2-form advected with the flow). The magnetic flux conservation law is equivalent to Faradays equation. The gauge condition for the magnetic helicity to be advected with the flow is determined. Different variants of the helicity in ideal fluid dynamics and MHD including: fluid helicity, cross helicity and magnetic helicity are investigated. The fluid helicity conservation law and the cross-helicity conservation law in MHD are derived for the case of a barotropic gas. If the magnetic field lies in the constant entropy surface, then the gas pressure can depend on both the entropy and the density. In these cases the conservation laws are local conservation laws. For non-barotropic gases, we obtain nonlocal conservation laws for fluid helicity and cross helicity by using Clebsch variables. These nonlocal conservation laws are the main new results of the paper. Ertels theorem and potential vorticity, the Hollman invariant, and the Godbillon?Vey invariant for special flows for which the magnetic helicity is zero are also discussed.