## Numerical modelling of the microphysical foundation of astrophysical particle acceleration

##### Abstract

The acceleration and transport of solar energetic particles have been intensively studied ever since the discovery of relativistic particles originating from the Sun. Both processes are
tightly connected to the dynamics of the solar wind and the turbulent interactions of plasma
waves. While advances in both theoretical modeling and observations have been made over
the years, there are still many details which are not understood yet. Solar wind turbulence
on its own is a complicated matter, and especially the regime of kinetic turbulence poses
many open questions.
Kinetic turbulence involves plasma waves at high frequencies and small wave lengths,
where their interactions with the charged particles in the plasma become important. Compared to the well-understood energy spectrum in the inertial range, a steepening of the
spectral slope is expected in the kinetic regime, which is generally attributed to the effects
of dispersion and energy dissipation by resonant interactions with the particles. However,
no complete model for the composition and behavior of the turbulent waves in this so-called
dissipation range is available, yet. Observations suggest that kinetic Alfvén waves are responsible for turbulence in the dissipation range. However, whistler waves, which are also
detected in various regions of the solar wind, may also contribute. This latter case is especially interesting, because whistler waves allow for the transport of energy to frequencies above the proton cyclotron frequency and may, therefore, interact with both thermal and high energy electrons in the solar wind plasma.
A particle-in-cell code is employed to simulate dispersive waves and their interaction with
charged particles in the plasma. As a preliminary study and a first step towards simulations
of dissipation range turbulence, the cyclotron resonance of thermal protons and dispersive
Alfvén waves and their strongly damped analog at higher frequencies, the proton cyclotron
waves, is modeled. To quantitatively analyze the dissipation of these waves, a method is
developed which allows to extract the waves' damping rates from simulation data. Extensive
tests show that cyclotron damping is recovered correctly in the simulation, which is a crucial
prerequisite for a correct model of dissipation range turbulence.
Similar to the case of turbulence, sophisticated models for the transport of solar energetic
particles in an environment that is dominated by non-dispersive waves are available.
However, the effect of dispersive waves on particle transport is less well-understood, which
is partly due to the more difficult treatment of dispersive waves in theoretical models. The
theoretical approach to describe particle transport is usually based on the quasi-linear approximation, which assumes that resonant scattering processes can be described by diffusion
in phase space. Using particle-in-cell simulations again, the resonant interaction of energetic electrons and dispersive waves is studied. The particles are scattered off of the waves' electromagnetic fields, creating a specific resonance pattern in phase space. The simulation data is compared to analytical predictions, which can be obtained from a model originally based on magnetostatic quasi-linear theory and which has recently been enhanced in order to allow for the description of dispersive waves. While the model predictions and the simulation results gen-erally agree, it can be seen that the resonant interaction of energetic particles and a single wave does not lead to diffusion, but rather to trapping of the particles in the electromagnetic fields of the wave. Diffusion can only occur when several waves with different frequencies, wave lengths, or directions of propagation are present. Even though these simulations do not model particle transport in turbulence, they contribute to a better understanding of the micro-physical properties of the scattering processes which are responsible for the transport and acceleration of solar energetic particles.
Finally, kinetic turbulence is directly studied in simulations. A set of initially excited
whistler waves is used as a seed population for the development of a turbulent cascade.
Whistler waves are chosen because they allow for a continuation of the spectrum above the
proton cyclotron frequency into a regime where the interaction of the waves with electrons
becomes dominant. The simulations are analyzed especially with regard to the shape of
the energy spectrum, since very little is known about the typical spectral index. However,
no consistent picture of the dependence of the spectral shape on the physical parameters is
obtained, yet. Extended parameter studies, which might yield more conclusive results, are
hindered by the limited amount of computational resources available for this work. They
remain as an eligible task for future projects.
Despite the absence of a detailed picture of kinetic turbulence, the simulations support
the idea that the magnetic energy spectrum in the kinetic regime is steeper than in Alfvénic
turbulence. It can also be assumed that a spectral break is produced at the transition into
the dissipation range. The spectrum forms an even steeper power law after the break.
Choosing two similar setups for simulations of whistler turbulence as a basis, the transport
of energetic electrons in kinetic turbulence is investigated. The analysis shows that the steep
energy spectra in the kinetic regime lead to a particular dominance of waves at low wave
numbers. These waves carry most of the energy and, thus, are most important for the
interactions with the energetic particles. Although particles may resonate with waves at
higher wave numbers (in the dispersive or dissipative regime), these interactions do not
seem to contribute significantly to the transport mechanism.
Comparison with a theoretical model suggests that the turbulent spectrum can be approximated by the relatively at regime at low wave numbers, before the spectral break is
encountered. Although the model predictions are not very accurate, the basic features of
the pitch angle diffusion coefficient derived from the particle data can be recovered. This is
especially interesting, since the model is derived for Alfvénic turbulence