By Mitsuo Kono, Milos Skoric

A nonlinearity is among the most crucial notions in sleek physics. A plasma is wealthy in nonlinearities and offers various behaviors inherent to instabilities, coherent wave buildings and turbulence. The e-book covers the fundamental innovations and mathematical equipment, essential to understand nonlinear difficulties broadly encountered in modern plasmas, but additionally in different fields of physics and present study on self-organized constructions and magnetized plasma turbulence. The analyses utilize strongly nonlinear versions solved by means of analytical suggestions sponsored through wide simulations and on hand experiments. The textual content is written for senior undergraduates, graduate scholars, teachers and researchers in laboratory, area and fusion plasmas.

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**Extra info for Nonlinear Physics of Plasmas**

**Example text**

Pe in the limit k ! 0. When a dispersion relation is given by ! 0 t . 79) to be a wave equation, p has to be even and is equal to or larger than 2. In this case, the dispersion is said strong. =dk is large in the long wave limit. Waves with this property are called strongly dispersive waves. 80) For a plasma with n0 1012 cm 3 , the electron plasma frequency is in a range of microwave frequencies, fpe 9 109 Œ1=sec D 9ŒGHz. pi d2 C C k 2 v2Ti dt 2 k 2 C kD2 ! k; t/ D 0; where the nonlinear term is neglected.

Therefore, the net flow of energy from the wave to the particles is driven 22 1 Basic Properties of High Temperature Plasmas to lead the wave damping. =k When the main body of the particles resonates with the wave, in other words, the thermal velocity is equal to the phase velocity, the wave number equals to the Debye wave number. Therefore, the collective motion is absorbed by the thermal motion and the wave cannot be excited. 2 Kinetic Instability Since a way of exchanging energy between particles and waves depends on the derivative of the particle velocity distribution function at the phase velocity, waves can grow for velocity distributions with a positive slope at the phase velocity.

C kk vk C k? v? / @ @ F˛0 ; C kk @v? / k? ' m˛ Â/ where ˝˛ D e˛ B0 =m˛ c. '/ defined by i˝˛ @ C k? v? 57) ` where Z˛ D k? v? =˝˛ . 58) for which the following identities have been used. Z˛ / m˛ m˝˛ .! kk vk / v? @v? Z˛ / 2 2 k m m˝˛ .! kk vk / @ m˝˛ @ F˛0 dv: C kk v? @v? 64), the resonance condition is given by vk D ! 65) For m D 0, the resonance gives the Landau damping. =kk /=j˝˛ j D m=kk to show that the particle moves along the magnetic field during one cyclotron period to come back to the same phase of the wave, leading to the wave damping.