Transport Theory of Cosmic Rays with application to the acceleration of charged particles at interplanetary shock waves
Final Report Abstract
Solar energetic particles (SEPs) are charged particles (mostly protons, electrons and heavier ions) coming from the sun, which are accelerated to energies reaching from a few keV to several GeV. The acceleration sites are solar flares and shock waves in the solar corona and interplanetary space. As pointed out by Schwenn, "these particles are of particular concern in the space weather context since they can penetrate even the skins of spaceprobes traveling outside the Earth’s magnetosphere and blind or even damage sensitive technical systems. The strongest events like the ones in August 1972 or in October/November 2003 produce radiation doses that might be lethal to unprotected astronauts while traveling in space outside our protective magnetosphere." Apparently, the study of the acceleration of charged particles in magnetized plasmas is not only of academic interest but has also significant implications on space travel and space probes. By the time of writing it is widely and commonly accepted that first order Fermi acceleration (diffusive shock acceleration, DSA) is the driving mechanism for particle acceleration to such high energies. Here we considered an interplanetary and mostly oblique CME-driven shock. A (preliminary) numerical analysis indicates that (at an early stage of the shock) over a large range along the shock front protons are accelerated to energies of several tens of MeVs; occasionly even to several hundreds of MeVs. The Padsa-code (Particle Acceleration through Diffusive Shock Acceleration, written in C++) calculates the maximum momentum (resp. energy) of the accelerated particles at each position along the shock front and uses currently results from a numerical simulation of an interplanetary shock. For qualitative investigations the code allows also to investigate the influence of several different parameters on the result (e.g., particle species, shock compression ratio, energy spectrum of the magnetic turbulence, etc.) and it was found that the maximum energy depends rather sensitively on the shock compression ratio and the shock speed. The next steps include the application of the code to observational data and a possible application to real time shock analysis. One important input parameter that governs the result for diffusive shock acceleration is the total diffusion coefficient, which describes the stochastic motion of charged particles in turbulent magnetized plasmas. Usually on distinguishes between a component perpendicular and parallel to the ambient magnetic background field. Here, however, focus was set on the perpendicular component, which is still not fully understood. A recent approach to develop an improved theory of perpendicular diffusion uses the gyrophase-averaged Fokker-Planck equation to describe fourth order correlation functions (which are a fundamental input parameter and have usually been approximated by two second order correlations). Here, this Unified Nonlinear Transport (UNLT) theory was combined with the Newton-Lorentz equations of motion, which describe the exact particle motion in turbulent magnetized plasmas (hitherto, most investigations assume that the particles’ gyrocenters follow magnetic field lines). It was found that, in contrast to some previous results, the diffusion coefficient can be described as a superposition of two contributions; the first contribution describes the motion of the gyrocenter, while the second contribution seems to become important only for high particle energies and could be interpreted as particles scattering away from field lines. Another important input parameter is the strength (energy) of the magnetic turbulence. A detailed understanding of how magnetic turbulence evolves in the solar wind and across shocks is crucial for a proper description of diffusion coefficients and, consequently, diffusive shock acceleration. Therefore, a two-scale separated decomposition of the incompressible MHD equations (based on the Elsässer description) has been used to derive a set of transport equations for the kinetic and magnetic energy residing in the turbulence, and for the energy difference (residual energy). Hereby, the nonlinear terms for the energy difference have been modeled separately by assuming a counteraction between Alfvén effect (equipartition between kinetic and magnetic energy) and small-scale dynamo (MG-model). To close the system we also introduced transport equations for the correlation lengths corresponding to the forward and backward propagating Elsässer variables and to the energy difference. Correlation lengths are also an important input parameter in particle transport theory, since they influence the shape of the power spectrum of the magnetic turbulence. This new set of transport equations includes (i) the large-scale background inhomogeneous Alfvénic velocity at a level of detail greater than in previous studies and (ii) a tractable slow timescale closure to eliminate highfrequency influence terms. To simplify the set of transport equations one can assume that the correlation lengths of the forward and backward propagating modes of the Elsässer variables are equal. We have shown that in this case certain implications and restrictions are imposed on the set of transport equations, so that some parameter cannot be chosen arbitrarily anymore, e.g., the cross helicity, both Elsässer variables, and the correlation length for the energy difference. As a first application we investigated the evolution of turbulence across a shock, using the simplified set of transport equations with arbitrary cross helicity. We found that the amplification of magnetic energy was not as high as has been seen in observations. We also found that the MG-model leads to unphysical results.
Publications
- The Transport of Low-frequency Turbulence in Astrophysical Flows. I. Governing Equations, Astrophys. J., 745, 35, 2012
Zank, G.P., Dosch, A., Hunana, P., Florinski, V., Matthaeus, W.H., & Webb, G.M.
(See online at https://doi.org/10.1088/0004-637X/745/1/35) - The transport of low-frequency turbulence in astrophysical flows: Correlation lengths, AIP Conference Proceedings: Solar Wind 13, 2012
Dosch, A., Adhikari, L. & Zank, G.P.
