**1. Introduction**

Streaming instabilities (SI) occupy a prominent place among other plasma instabilities. They are caused by a motion of some plasma components relative to others. An example is the well-known beam–plasma instability [1]. With this instability, the directed motion of a group of fast electrons passing through the background plasma excites potential oscillations with a large growth rate near the plasma frequency. Particular attention to this instability is mainly due to the idea of creating sources of powerful electromagnetic radiation on its basis. At present, these sources have many advantages over the known vacuum sources [2, 3]. One more example (we mention these two only) is the Buneman instability [4], in which plasma electrons move relative to ions.

In the overwhelming majority of investigations beam–plasma interaction is considered without any noticeable dissipation. It, actually, was assumed that the dissipation is small and cannot have any noticeable effect on the physical processes. In this case, the development of instability leads to an increase in the amplitude of

electromagnetic oscillations in the plasma, as well as their energy at the expense of beam kinetic energy. In the absence of dissipation, the level of excited oscillations may be quite high, and their energy can even be comparable to the initial energy of the beam [5].

However, generally speaking, dissipation in the system (collisions between plasma particles, heating of metal surfaces due to their complex impedance, etc.) can play an essential role in plasma–beam interaction. It can become not only a decisive factor in limiting the spatial and temporal growth, determining the field amplitude and the mode structure and limits the growth rates. In addition to these properties, which are common to all systems, it is necessary to pay special attention to the unique role of dissipation in systems with a stream of charge particles: dissipation of high level does not suppress the SI completely. Strong dissipation transforms each SI to instability of other type – to dissipative streaming instability (DSI) [1]. This type of instabilities is due to the presence of the negative energy wave (NEW) in a stream of charge particles [6, 7]. In fact, dissipation serves as a channel for energy removal for excitation of this wave. This leads to instabilities of a new physical nature, to DSI. Dissipation is the cause of this instability.

The physical nature of SI is not as simple as it might seem at first glance. It takes a lot of effort<sup>1</sup> to understand it clearly. This is all the more so, if we are dealing with the transformation of SI into a DSI. The transformation (in general, the transformation of one type of instability into another) makes the behavior of SI in a system with dissipation especially interesting. In addition, there are other reasons that significantly increase interest in the study of problems associated with dissipation and the DSI caused by it. Some of them are as follows.

Modern high-frequency microwave electronics, both plasma and vacuum, have two basic trends of development: an increase in the frequency and power of the output radiation [2]. With increasing frequency, the thickness of the skin layer on the resonators' walls decreases. This, in turn, leads to an increase in active energy losses. Actual dissipation in the system increases.

The second trend – an increase in the power of output radiation – leads to the need to increase the beam current. The role of space charge phenomena increases also, as well as the role of the NEW. In these circumstances it becomes important to take into account all factors that also lead to the buildup of the same wave i.e. to dissipation. In a sense, dissipation becomes associated with the space charge phenomena. In addition with an increase in the beam current, the return current increases also. With account the decrease in the skin layer and the finite conductivity of metallic surfaces, this leads to an increase in the level of dissipation in the system. All this indicates that dissipation, along with the space charge of the beam plays an important role in microwave electronics. A detailed understanding of the role of all these phenomena is vital for many problems aimed at achieving highintensity beams and their applications.

Until recently, only one DSI was known in beam–plasma interaction theory [1]. Its maximal growth rate depends on collision frequency *ν* in plasma and on the beam density *nb* as ffiffiffiffiffiffiffiffiffi *nb=ν* p . All types of the beam–plasma instabilities (Cherenkov, cyclotron, etc.), with an increase in the level of dissipation, transform into it. This only known DSI has a number of specific features in comparison with other (no-dissipative) instabilities: relatively low level of excited oscillations, relatively small growth rate, etc. Many investigations have been devoted to its study. It was

<sup>1</sup> The instability of low density e-beam in plasma is a vivid example demonstrating this sense. It is discovered in 1948, experimentally proven in early sixties; however its physical meaning became finally clear in the middle of seventies (see [8]).

### *New Types of Dissipative Streaming Instabilities DOI: http://dx.doi.org/10.5772/intechopen.98901*

assumed that various phenomena in space plasma and in plasma of controlled fusion can be explained on the basis of this instability.

However, recent studies have shown that there are other DSI also [9–11]. The interaction of the stream with the background plasma critically depends on some basic parameters of the system and/or on its geometry. Their changes lead to new physics of the beam-plasma interaction and to previously unknown types of DSI. The parameters are: the level of correlation between the beam and the plasma fields and the value of the beam current.

Available methods of instability investigation do not allow getting complete information on the process of transformation of given instability into another type. Is known the most complete information on instability can be obtained by solving the problem of the evolution of fields in space and time during the development of an initial perturbation. This problem is classical in theory of instabilities [12]. Its results can clear up how the fields of given instability transform to the fields of another one along with many other accompanying details. The character of the space–time evolution of an initial perturbation is an important issue in many branches of physics. However, the results of this problem are hardly achievable. Ultimately its mathematical solution reduces to calculation of the integral with complete dispersion relation (DR) in the denominator of the integrand. For the result the DR should be specified and solved before integration. This sharply reduces generality of results. And even in the special cases, it is not always possible to carry out the integration. In [13] an approach is presented that allowed overcome difficulties and obtain analytical expression for the fields'space–time structure for all types of conventional beam-plasma instabilities. Results show that with increase in level of dissipation all types of beam-plasma instabilities transform to the only known type of DSI.

This review shows that the number of DSI is not limited by the above-mentioned DSI. Two new types of DSI are substantiated. They follow from solution of the same classical problem of initial perturbation development. One of the DSI manifests itself in the results of solving the problem in systems with weak beam-plasma coupling. Weak interaction realizes if the beam and the plasma are spatially separated by a considerable distance. Under weak coupling the beam actually is left to its own and its proper oscillation come into play. Moreover, among them is the NEW. Its interaction with plasma causes instability, the growth rate of which reaches maximum at resonance of the plasma wave with the NEW. This resonance of wave– wave type was called "Collective Cherenkov effect" [14]. An increase in the level of dissipation leads to a new DSI with the growth rate � ffiffiffiffiffi *nb* p *=ν*. Actually the new approach to solution of the classical problem has detected this new DSI.

The second new DSI appears in results of solving of the same problem in uniform cross-section beam-plasma waveguide with over-limiting e-beam. With an increase in the beam current the fields of its space charge affects more and more on the beam-plasma interaction. This manifests itself in two ways. Along with the increasing of the role of space charge oscillations, static fields of the beam space charge set an upper limit on the beam current that can pass through a given vacuum electro-dynamical system. The limit can be overcome by plasma filling. Plasma neutralizes the space charge of the beam. Plasma-filled waveguides can transmit e-beams with a current that is several times higher than the limiting current in vacuum waveguide. The fields of overlimiting e-beam space charge changes the character of its instability. The instability of over-limiting beams is not associated with any radiation mechanism [9, 14]. Its growth rate reaches maximum at the point of exact Cherenkov resonance and depends on the beam density as ffiffiffiffiffi *nb* p [9, 14, 15], With an increase in the level of dissipation, one more new type of DSI develops [9]. Its growth rate depends on the parameters as � *nb=ν*.

In present review special attention is paid to systems, the geometry of which is similar to geometry of plasma microwave sources. These devices are a cylindrical waveguide with thin annular plasma and spatially separated thin annular e-beam. In this geometry the new types of DSI manifest themselves also [10].

In order to dispel all possible doubts about the correctness of the results, both new DSI are also substantiated by conventional analysis of the corresponding DR. To obtain a geometry-independent result for weak beam-plasma coupling we use perturbation theory based on smallness of the coupling parameter.

## **2. The only known DSI and transition to it**

For the beginning we shortly present rezoning, from which follow: all types of beam-plasma instabilities (Cherenkov, cyclotron, beam instability in spatially periodical structure) transform to the only known DSI with the maximal growth rate � *<sup>ω</sup>b<sup>=</sup>* ffiffi *<sup>ν</sup>* <sup>p</sup> (*ω<sup>b</sup>* is the Langmuir frequency of the beam, *<sup>ν</sup>* is the collision frequency in plasma). The transition takes place with an increase in the level of dissipation. This help us to reveal a criterion for identification of DSI type.

In general, the dispersion relation (DR), describing a plasma system penetrating by an electron beam can be written as

$$D(o, \mathbf{k}) = D\_0(o, \mathbf{k}) + D\_b(o, \mathbf{k}) = \mathbf{0} \tag{1}$$

where *D*0ð Þ¼ *ω*, **k** Re *D*0ð Þþ *ω*, **k** *i*Im*D*0ð Þ *ω*, **k** describes the plasma (without beam), but *Db*ð Þ *ω*, **k** describes the beam contribution in the system dispersion

$$D\_b(\boldsymbol{\alpha}, \mathbf{k}) = -\frac{\alpha\_b^2 A(\boldsymbol{\alpha}, \mathbf{k})}{\gamma^3 (\boldsymbol{\alpha} - \mathbf{k} \mathbf{V}\_b - \boldsymbol{f})^2},\tag{2}$$

*ω* is the frequency, **k** is the wave vector of perturbations, *ω<sup>b</sup>* is Langmuir frequency of the e-beam, **V***<sup>b</sup>* is the velocity of the beam electrons (directed along *z* axis), *<sup>A</sup>*ð Þ *<sup>ω</sup>*, **<sup>k</sup>** is a polynomial with respect to *<sup>ω</sup>* and **<sup>k</sup>**, *<sup>γ</sup>* <sup>¼</sup> <sup>1</sup> � *<sup>V</sup>*<sup>2</sup> *b=c*<sup>2</sup> � ��1*=*<sup>2</sup> . It is assumed that Imj j *D*<sup>0</sup> < <j j Re *D*<sup>0</sup> and j j *Db*ð Þ *ω*, **k** < <j j *D*0ð Þ *ω*, **k** , *f* ¼ 0 with the Cherenkov interaction, with the cyclotron interaction *f* ¼ *n*Ω*=γ*, (Ω is the cyclotron frequency, *n* is the harmonic number), and *f* ¼ *k*cor*Vb* when e-beam interacts with the periodical structure, *k*cor ¼ 2*π=l*, *l* is the length of spatial period.

The beam electrons interact with the proper oscillations of the system and the interaction leads to instability. Developing instability manifests itself most effectively at frequencies and wavelengths close to the proper frequencies of the system in the absence of the beam, and, at the same time, close to the beam natural frequencies. In fact, along with (1) following condition is met

$$
\alpha - kV\_b - f = 0.\tag{3}
$$

All (conventional) beam-plasma instabilities, including DSI, follow from (1)–(3). With an increase in level of dissipation all types of no-dissipative instabilities (Cherenkov, cyclotron etc) transform into the well-known DSI. If one searches the solutions of DR (1) in the form *ω* ¼ *ω*<sup>0</sup> þ *δ* (*ω*<sup>0</sup> satisfies (1) and (3); this case called resonance instability) he arrives to the expression

$$\delta \left( \frac{\partial D\_0}{\partial \boldsymbol{\alpha}} \right) \boldsymbol{\omega} = \boldsymbol{\omega}\_{0,} \quad + i \text{Im} D\_0(\boldsymbol{\omega}\_0, \boldsymbol{k}\_0) = \frac{\left( \boldsymbol{\omega}\_b^2 / \boldsymbol{\gamma}^3 \right) \boldsymbol{A}(\boldsymbol{\omega}\_0, \boldsymbol{k}\_0)}{\delta^2}. \tag{4}$$
 
$$\boldsymbol{k} = \boldsymbol{k}\_0$$

*New Types of Dissipative Streaming Instabilities DOI: http://dx.doi.org/10.5772/intechopen.98901*

All types of no-dissipative instabilities follow the first and the right-hand side term. In this case the dissipative (second) term in (4) is small. The DSI follows from the second term (when it is greater than the first term) and the right-hand side term. The relation between the respective growth rates *δ*ð Þ *<sup>ν</sup>*¼<sup>0</sup> and *δ*ð Þ *<sup>ν</sup>*!<sup>∞</sup> is

$$\delta^{(\nu \to \infty)} = \sqrt{\frac{\left\{\delta^{(\nu=0)}\right\}^3}{2\text{Im}D\_0}} \frac{\partial D\_0}{\partial \nu} \sim \sqrt{\frac{\left\{\delta^{(\nu=0)}\right\}^3}{\nu}} \sim \sqrt{n\_b/\nu} \tag{5}$$

where the frequency of collisions in plasma *ν* is introduced (Im*D*<sup>0</sup> � *ν*). The expression (5) presents relation between the growth rates of no-dissipative and dissipative instabilities. Below we use (5) and its analogs as a criterion for identification of DSI type.
