1. Introduction

Plasma is rich in instabilities. Many of them are a result of relative motion of plasma components. These, streaming instabilities (SI) are the most common in space and laboratory plasmas.

© 2016 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and eproduction in any medium, provided the original work is properly cited. © 2018 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

A well-known example is the beam-plasma instability [1], in which the directed motion of a small group of fast electrons passing through the background plasma excites potential oscillations with high growth rate near the plasma frequency. Close attention to this instability is due mainly to design of high power sources of electromagnetic radiation based on this instability. The sources have many advantages as compared to well-known vacuum devices [2, 3]. Another example (we mention these two only) is the Buneman instability [4], in which plasma electrons move with respect to ions. The instability plays an important role in many scenarios in space physics and geophysics. A striking example of plasma with relative electron-ion motion is current-carrying plasma. This object is often considered in plasma physics. The instabilities which are due to relative electron-ion motion play an important role in physics of controlled fusion also.

The character of space–time evolution of given instability is an important issue in many branches of physics. In plasma physics, we firstly note theory of amplifiers and oscillators in the microwave range based on interaction of e-beam with wave, where obvious progress is achieved [2, 3]. These studies are also important for research on plasma instabilities associated

The Behavior of Streaming Instabilities in Dissipative Plasma

http://dx.doi.org/10.5772/intechopen.79247

23

The mathematical solution of the problem of initial perturbation evolution reduces to calculation of the integral with a complete dispersion relation (DR) in the denominator of the integrand. An overall view on the character of the instability may be obtained by investigation of the asymptotic behavior of the Green's function. In order to derive analytical expression for the fields' space–time distribution, the DR should be specified and solved before integration. In this way, essential difficulties appear which usually cannot be overcome. One must apply approximate methods to obtain results. Presented here (see also [11]) approach is similar to traditional approach in many respects, but, in the same time, advantageously differs from it. Representation of the fields in form of wave train with slowly varying amplitude (SVA) allowed to overcome the difficulties and to obtain the space–time structure of the fields without reference on any particular model. Thereby, the approach singles out intrinsic peculiarities of various types of SI. The results show that all types of the beam-plasma instabilities (Cherenkov, cyclotron, etc.) have similar dynamics of development. By specifying only two parameters in the unified expression one can investigate given particular case of beam insta-

This review considers all these aspects: getting detailed information on SI, their space–time evolution and transformation to DSI. Presented approach shows that the DR which usually describes given SI can serve not only for solution of the well-known (and very simplified) initial and boundary problems. Its application is much wider. It can give much more information on the instability. Namely, it actually gives the solution of the well-known (and very important [9]) problem of time evolution of initial perturbation. The DR can give space–time structure of the fields at the instability development. In its turn, the fields' structure contains complete information on the instability. Most of this information is unavailable by other methods. The expressions for fields' evolution also show in detail the transformation of SI to

Large variety of SI characterize by various types of the interaction with background systems (plasma-filled or not), various values of streaming currents, etc. From this follows various types of their DR and ensuing equation for SVA. They are considered separately. In Section 2, the evolution of various types of beam instabilities (Cherenkov, cyclotron, and the instability in periodical structure) are considered. All they characterize by small contribution of the beam in DR and this fact allowed generalizing the consideration. Section 3 gives the evolution of overlimiting e-beam instability. Due to influence of the beam space charge, the instability of such beams has other physical nature as compared to instability of conventional e-beams. In Sections 4 and 5, the instability in spatially separated beam-plasma system and the Buneman

with research on nuclear fusion, astrophysics, etc.

bility. With increase in level of dissipation all SI gradually turn to DSI.

dissipative type. Two new, previously unknown types of DSI are presented.

instability are considered. The peculiarity of last case is in the role of plasma ions.

A clear understanding of physical nature of the SI, their role and influence on various processes in plasma requires substantial efforts. Physics of interaction of plasma components moving relatively to each other is essentially based on the concept of negative energy wave (NEW) [5]. This requires account of all factors which lead to NEW growth. Among them, dissipation plays an important role. Dissipation leads to energy losses for the growth of NEW. Influence of dissipation on the instabilities of streaming type is unique. Dissipation never suppresses the instabilities completely regardless on its level. Dissipation of high-level transforms the SI to dissipative streaming instability (DSI) [1]. These instabilities have a number of features: comparatively low growth rate, comparatively low level of excited oscillations, etc. For a few decades, DSI have been widely discussed, and it is supposed that they can be applied to explain various phenomena in space and laboratory plasma. Up to recently only one type of DSI was known, and it was believed that all types of electron stream instabilities (e.g., Cherenkov type, cyclotron type etc.) transform to the single known type of DSI. However, it turned out that other types of DSI also exist [6–8]. Changes in some basic physical parameters and/or system geometry lead to significant changes in physical nature of e-stream interaction with plasma. This changes result in two new, previously unknown types of DSI: DSI of over-limiting electron beam and DSI under weak coupling of the stream with the plasma. In both cases, the growth rate depends on dissipation more critically: 1=ν instead of conventional 1= ffiffiffi <sup>ν</sup> <sup>p</sup> (here <sup>ν</sup> is the frequency of the collisions).

The transformation of the SI to dissipative type makes their behavior in the presence of dissipation of particular interest. In order to understand how instability turns to another type, it is necessary to investigate the evolution of its fields in space and time [9, 10]. Simultaneously, the expressions for fields' evolution give all available information on the SI: growth rates (spatial and temporal) under arbitrary level of dissipation, character of the instability (absolute/convective), range of unstable perturbations' velocities, influence of dissipation on the instability, etc. These details help to understand how the instability turns to DSI, how it transforms given equilibrium of background plasma, predict the level and/or scale of the changes, how nonlinear phenomena arise as well as predict possible saturation mechanisms, etc. In general, the character of the fields' development in space and time is one of the most important aspects of every instability.

The character of space–time evolution of given instability is an important issue in many branches of physics. In plasma physics, we firstly note theory of amplifiers and oscillators in the microwave range based on interaction of e-beam with wave, where obvious progress is achieved [2, 3]. These studies are also important for research on plasma instabilities associated with research on nuclear fusion, astrophysics, etc.

A well-known example is the beam-plasma instability [1], in which the directed motion of a small group of fast electrons passing through the background plasma excites potential oscillations with high growth rate near the plasma frequency. Close attention to this instability is due mainly to design of high power sources of electromagnetic radiation based on this instability. The sources have many advantages as compared to well-known vacuum devices [2, 3]. Another example (we mention these two only) is the Buneman instability [4], in which plasma electrons move with respect to ions. The instability plays an important role in many scenarios in space physics and geophysics. A striking example of plasma with relative electron-ion motion is current-carrying plasma. This object is often considered in plasma physics. The instabilities which are due to relative electron-ion motion play an important role in physics of

22 Plasma Science and Technology - Basic Fundamentals and Modern Applications

A clear understanding of physical nature of the SI, their role and influence on various processes in plasma requires substantial efforts. Physics of interaction of plasma components moving relatively to each other is essentially based on the concept of negative energy wave (NEW) [5]. This requires account of all factors which lead to NEW growth. Among them, dissipation plays an important role. Dissipation leads to energy losses for the growth of NEW. Influence of dissipation on the instabilities of streaming type is unique. Dissipation never suppresses the instabilities completely regardless on its level. Dissipation of high-level transforms the SI to dissipative streaming instability (DSI) [1]. These instabilities have a number of features: comparatively low growth rate, comparatively low level of excited oscillations, etc. For a few decades, DSI have been widely discussed, and it is supposed that they can be applied to explain various phenomena in space and laboratory plasma. Up to recently only one type of DSI was known, and it was believed that all types of electron stream instabilities (e.g., Cherenkov type, cyclotron type etc.) transform to the single known type of DSI. However, it turned out that other types of DSI also exist [6–8]. Changes in some basic physical parameters and/or system geometry lead to significant changes in physical nature of e-stream interaction with plasma. This changes result in two new, previously unknown types of DSI: DSI of over-limiting electron beam and DSI under weak coupling of the stream with the plasma. In both cases, the growth rate depends on dissipation more critically: 1=ν instead of

<sup>ν</sup> <sup>p</sup> (here <sup>ν</sup> is the frequency of the collisions).

opment in space and time is one of the most important aspects of every instability.

The transformation of the SI to dissipative type makes their behavior in the presence of dissipation of particular interest. In order to understand how instability turns to another type, it is necessary to investigate the evolution of its fields in space and time [9, 10]. Simultaneously, the expressions for fields' evolution give all available information on the SI: growth rates (spatial and temporal) under arbitrary level of dissipation, character of the instability (absolute/convective), range of unstable perturbations' velocities, influence of dissipation on the instability, etc. These details help to understand how the instability turns to DSI, how it transforms given equilibrium of background plasma, predict the level and/or scale of the changes, how nonlinear phenomena arise as well as predict possible saturation mechanisms, etc. In general, the character of the fields' devel-

controlled fusion also.

conventional 1= ffiffiffi

The mathematical solution of the problem of initial perturbation evolution reduces to calculation of the integral with a complete dispersion relation (DR) in the denominator of the integrand. An overall view on the character of the instability may be obtained by investigation of the asymptotic behavior of the Green's function. In order to derive analytical expression for the fields' space–time distribution, the DR should be specified and solved before integration. In this way, essential difficulties appear which usually cannot be overcome. One must apply approximate methods to obtain results. Presented here (see also [11]) approach is similar to traditional approach in many respects, but, in the same time, advantageously differs from it. Representation of the fields in form of wave train with slowly varying amplitude (SVA) allowed to overcome the difficulties and to obtain the space–time structure of the fields without reference on any particular model. Thereby, the approach singles out intrinsic peculiarities of various types of SI. The results show that all types of the beam-plasma instabilities (Cherenkov, cyclotron, etc.) have similar dynamics of development. By specifying only two parameters in the unified expression one can investigate given particular case of beam instability. With increase in level of dissipation all SI gradually turn to DSI.

This review considers all these aspects: getting detailed information on SI, their space–time evolution and transformation to DSI. Presented approach shows that the DR which usually describes given SI can serve not only for solution of the well-known (and very simplified) initial and boundary problems. Its application is much wider. It can give much more information on the instability. Namely, it actually gives the solution of the well-known (and very important [9]) problem of time evolution of initial perturbation. The DR can give space–time structure of the fields at the instability development. In its turn, the fields' structure contains complete information on the instability. Most of this information is unavailable by other methods. The expressions for fields' evolution also show in detail the transformation of SI to dissipative type. Two new, previously unknown types of DSI are presented.

Large variety of SI characterize by various types of the interaction with background systems (plasma-filled or not), various values of streaming currents, etc. From this follows various types of their DR and ensuing equation for SVA. They are considered separately. In Section 2, the evolution of various types of beam instabilities (Cherenkov, cyclotron, and the instability in periodical structure) are considered. All they characterize by small contribution of the beam in DR and this fact allowed generalizing the consideration. Section 3 gives the evolution of overlimiting e-beam instability. Due to influence of the beam space charge, the instability of such beams has other physical nature as compared to instability of conventional e-beams. In Sections 4 and 5, the instability in spatially separated beam-plasma system and the Buneman instability are considered. The peculiarity of last case is in the role of plasma ions.
