NUMERICAL SIMULATIONS ͳ APPLICATIONS, EXAMPLES AND THEORY Edited by Prof. Lutz Angermann Numerical Simulations - Applications, Examples and Theory Edited by Prof. Lutz Angermann Published by InTech Janeza Trdine 9, 51000 Rijeka, Croatia Copyright © 2011 InTech All chapters are Open Access articles distributed under the Creative Commons Non Commercial Share Alike Attribution 3.0 license, which permits to copy, distribute, transmit, and adapt the work in any medium, so long as the original work is properly cited. After this work has been published by InTech, authors have the right to republish it, in whole or part, in any publication of which they are the author, and to make other personal use of the work. Any republication, referencing or personal use of the work must explicitly identify the original source. Statements and opinions expressed in the chapters are these of the individual contributors and not necessarily those of the editors or publisher. No responsibility is accepted for the accuracy of information contained in the published articles. The publisher assumes no responsibility for any damage or injury to persons or property arising out of the use of any materials, instructions, methods or ideas contained in the book. Publishing Process Manager Jelena Marusic Technical Editor Teodora Smiljanic Cover Designer Martina Sirotic Image Copyright agsandrew, 2010. Used under license from Shutterstock.com First published January, 2011 Printed in India A free online edition of this book is available at www.intechopen.com Additional hard copies can be obtained from orders@intechweb.org Numerical Simulations - Applications, Examples and Theory, Edited by Prof. Lutz Angermann p. cm. ISBN 978-953-307-440-5 free online editions of InTech Books and Journals can be found at www.intechopen.com [...]... study for the long-time evolution of the Vlasov- 4 Numerical Simulations - Applications, Examples and Theory Poisson system for the problem of the bump-on-tail instability, for the case when the beam density is about 10% of the total density, which provides a more vigorous beam-plasma interaction and important wave-particle and trapped particles effects In this case the instability and trapping oscillations... Physics and Optics 1 Numerical Simulation of the Bump-on-Tail Instability Magdi Shoucri Institut de recherche Hydro-Québec (IREQ), Varennes, Québec J3X1S1, Canada 1 Introduction Wave-particle interaction is among the most important and extensively studied problems in plasma physics Langmuir waves and their Landau damping or growth are fundamental examples of wave-particle interaction The bump-on-tail... 2980 15 Numerical Simulation of the Bump-on-Tail Instability Fig 10 Ion distribution function at t = 2980 (a) (c) Fig 11 (a) Same as Fig.(9) (concentrates on the tail) (b) Same as Fig.(9) (concentrates on the bulk) (c) Same as Fig.(9) (concentrates on the top) (d) Contour plot for the distribution in Fig.(11c) (b) (d) 16 Numerical Simulations - Applications, Examples and Theory Figs.(12a-19a,2 0-2 2) show... plot of the distribution function, t=1960 13 Numerical Simulation of the Bump-on-Tail Instability (m) (n) (o) Fig 7 (m) Contour plot of the distribution function, t=2000, (n) Contour plot of the distribution function, t=2200, (o) Contour plot of the distribution function, t=2980 14 Numerical Simulations - Applications, Examples and Theory Fig 8 Three-dimensional view of the results in Fig.7o distribution... mode with k = 0.15 , n = 4 in 24 Numerical Simulations - Applications, Examples and Theory (a) (b) (c) (d) Fig 27 (a) Contour plot of the distribution function, t = 60 (b) Contour plot of the distribution function, t = 200 (c) Contour plot of the distribution function, t = 400 (d) Contour plot of the distribution function, t = 600 25 Numerical Simulation of the Bump-on-Tail Instability (e) (f) (g) (h)... t=1120 11 Numerical Simulation of the Bump-on-Tail Instability (e) (f) (g) (h) Fig 7 (e) Contour plot of the distribution function, t=1140, (f) Contour plot of the distribution function, t=1400, (g) Contour plot of the distribution function, t=1600, (h) Contour plot of the distribution function, t=1800 12 Numerical Simulations - Applications, Examples and Theory (i) (j) (k) (l) Fig 7 (i) Contour plot... 0.9 and for the electron beam density nb = 0.1 for a total density of 1 This high beam density will cause a strong beam-plasma instability to develop We take ni = 1 , Te / Ti = 1 , me / mi = 1 / 1836 In our normalized units υthi = Ti me Te mi We use a time-step Δt = 0.002 The length of the system in the present simulations is L = 80x2π / 3 = 167.552 6 Numerical Simulations - Applications, Examples. .. vortex centered around x ≈ 30 Fig.(6a) shows the phase- space at t=780 Between Fig.(5a) and Fig.(6a), there is a time delay of 20, in which the structure moves a distance of about 3.42x20 ≈ 68 The small vortex centered at x ≈ 30 in Figs.(5a,b) has now moved to the position x ≈ 98 in Fig.(6a) 8 Numerical Simulations - Applications, Examples and Theory Fig 3 Spatially averaged distribution function... the Bump-on-Tail Instability (a) (b) Fig 18 (a) Time evolution of the Fourier mode with k=0.2625, (b) Spectrum of the Fourier mode k=0.2625 (a) (b) Fig 19 (a) Time evolution of the Fourier mode with k=0.3, (b) Spectrum of the Fourier mode k=0.3 (from t=100 to t=755.36), (c) Spectrum of the Fourier mode k=0.3 (from t=2344 to t=3000) (c) 20 Numerical Simulations - Applications, Examples and Theory Fig... 3.7 , and another one around υ ≈ 4.8 Fig.(11b) shows on a logarithmic scale a plot of the 10 Numerical Simulations - Applications, Examples and Theory (a) (b) (c) (d) Fig 7 (a) Contour plot of the distribution function, t=800, (b) Contour plot of the distribution function, t=1040, (c) Contour plot of the distribution function, t=1100, (d) Contour plot of the distribution function, t=1120 11 Numerical . NUMERICAL SIMULATIONS ͳ APPLICATIONS, EXAMPLES AND THEORY Edited by Prof. Lutz Angermann Numerical Simulations - Applications, Examples and Theory Edited by Prof. Lutz. orders@intechweb.org Numerical Simulations - Applications, Examples and Theory, Edited by Prof. Lutz Angermann p. cm. ISBN 97 8-9 5 3-3 0 7-4 4 0-5 free online editions of InTech Books and Journals can. for the long-time evolution of the Vlasov- Numerical Simulations - Applications, Examples and Theory 4 Poisson system for the problem of the bump-on-tail instability, for the case when the