Long term analysis of splitting methods for charged-particle dynamics

In this paper, we rigorously analyze the energy, momentum and magnetic moment behaviours of two splitting methods for solving charged-particle dynamics. The near-conservations of these invariants are given for the system under constant magnetic field or quadratic electric potential. By the approach named as backward error analysis, we derive the modified equations and modified invariants of the splitting methods and based on which, the near-conservations over long times are proved. Some numerical experiments are presented to demonstrate these long time behaviours

Study of variational symplectic algorithms for the numerical integration of guiding center equations of motion

Questa tesi presenta una discussione dei più moderni algoritmi simplettici variazionali per l'integrazione delle equazioni del moto del centro di guida in particelle cariche in campi magnetici arbitrari statici, utili nello studio di plasmi debolmente collisionali. Differenti varianti degli algoritmi sono presentate, insieme a studi numerici ed analitici che ne evidenziano la stabilità numerica, o la relativa mancanza di essa

Reformation and ion dynamics of quasiperpendicular collisionless shocks

Collisionless shocks are ubiquitous throughout the Universe. Shocks are nonlinearly steepened waves which cause plasmas to undergo dramatic changes in their density, temperature, flow velocity, and magnetic field. A plasma is called collisionless when the frequency of collisions between charged particles is substantially lower than the natural frequencies of the plasma, such as the electron plasma frequency or the electron gyrofrequency. The presence of a magnetic field in collisionless plasmas leads to a rich collection of different shock types and furthermore a wide variety of wave phenomena and particle dynamics are supported at such shocks. The unique physical behaviors and properties of magnetized collisionless shocks, compared to those of the classical gasdynamic shock, make them one of the most extensively studied nonlinear classes of phenomena in plasmas. In this thesis we study several different but connected problems in collisionless shock physics. In Chapter 1 we present an overview of basic plasma theory, particularly of collisionless shocks in plasmas described using magnetohydrodynamic (MHD) theory. In Chapter 2 detailed quantitative evidence for shock reformation (a temporal variability of the shock structures) is presented from the Voyager 2 spacecraft's encounter with the Uranian bow shock. This evidence is based on finding very good agreement between Voyager observations of the Uranian bow shock and results from a standard 1-D hybrid simulation code run with similar plasma parameters. Specifically, the multiple large localized magnetic field enhancements B/B1 ~16 observed downstream by Voyager 2, where B1 = 0.19 nT is the upstream magnetic field, are quantitatively consistent with the reforming shock found in the simulation. Chapter 3 develops a new analytic model for the reflection and transmission of ions at quasiperpendicular (the angle between the shock normal and upstream magnetic field is between 45°-90° shocks. We show, using 1-D hybrid and test-particle simulations, that ions reflected at the shock ramp are decelerated by magnetic field effects, herein referred to as “magnetic deflection'', in addition to the cross-shock electrostatic potential. We quantify the contribution of magnetic deflection to ion reflection and in doing so resolve a discrepancy between the predicted ion reflection efficiency (based on the electrostatic potential jump alone) and numerical values calculated in our hybrid simulations. Moreover, an analytic expression for the reflection cutoff of ions reflected by the cross-shock potential and magnetic deflection is derived. We find excellent agreement between the simulations and the new model for ion reflection by the shock front, and show that both the electrostatic potential and magnetic field effects are vital in the ion reflection process. Chapter 4 proposes a new classification scheme at perpendicular shocks for the classes of ion trajectories across the different shock regions of the foot, ramp, overshoot, and downstream plasma. For each particle class we calculate its energization, fractional population, and source region in initial velocity phase space using multiple test-particle simulations. We find that particle energization is mostly sensitive to the thickness of the shock and the magnitude of the electrostatic potential. Mild energy gains are observed for shocks with an overshoot whereas including a foot structure only changes the source regions for certain particle classes. Furthermore, we develop and test two new analytic reflection conditions for: (1) particles that are initially reflected by the electrostatic potential, complete half a gyro-orbit upstream, and finally escape downstream, and (2) particles that are initially transmitted downstream but return to the shock as part of their downstream gyromotion, complete half a gyro-orbit upstream, and finally escape downstream. In the former case, we find excellent agreement between analytic and numerical results; however, in the latter case we find good agreement only in simulations with thin shocks

Arbitrarily high-order energy-preserving methods for simulating the gyrocenter dynamics of charged particles

Gyrocenter dynamics of charged particles plays a fundamental role in plasma physics. In particular, accuracy and conservation of energy are important features for correctly performing long-time simulations. For this purpose, we here propose arbitrarily high-order energy conserving methods for its simulation. The analysis and the efficient implementation of the methods are fully described, and some numerical tests are reported