## Abstract

An approach is presented that allows getting detailed information on the behavior of streaming instabilities (SI) from the dispersion relation (DR). The approach is based on general assumptions and does not refer to any particular model and/or type of the stream interaction with background system (Cherenkov, cyclotron, etc.). The basis of the approach is transformation of the DR to an equation for slowly varying amplitude of the developing waveform. The solution of the equation actually presents results of the important problem of time evolution of initial perturbation and gives detailed information on the instability behavior. Most of the information is unavailable by other methods. For particular SI, only two parameters should be specified. The expression for the fields’ structure shows that with increase in level of dissipation, SI gradually turns to dissipative streaming instability (DSI). Two new, previously unknown types of DSI are presented: DSI of overlimiting electron beam and DSI under weak beam-plasma coupling. Growth rates of these DSI depend on dissipation more critically than usual. Presented approach is valid for a large class of SI: beam-plasma instabilities of various types (Cherenkov, cyclotron, etc.) including over-limiting e-beam instabilities, the instability in spatially separated beam-plasma systems, Buneman instability, etc.

### Keywords

- streaming instability
- dissipative instability
- space–time evolution
- slowly varying amplitude
- transformation to dissipative instability

## 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. 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.

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, 7, 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:

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.

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 over-limiting 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.

## 2. The behavior beam-plasma instabilities in dissipative plasma

### 2.1. Equation for slowly varying amplitude

Consider an electrodynamical system of arbitrary geometry (plasma filling is not obligatory) and let a monoenergetic relativistic electron beam penetrate it. The general form of the dispersion relation (DR) of such system is

where

where

where

Let an initial perturbation arises in point *k*. Other arguments play no part in following. To avoid overburdening of the formulas below, they are omitted. The transversal structure of the fields may be obtained in regular way by expansion on series of eigenfunctions of given system.

The development of wave pulse in its linear stage obeys the DR (1). The beam instability reveals itself most effectively on frequencies, closely approximating to roots of the main part of Eq. (1) and simultaneously to the beam proper oscillations (e.g., space charge wave). This means that following two conditions must be satisfied:

Therefore, it would appear reasonable to assume that developing fields form a wave train of following type

where the carrier frequency

In such formulation, the problem of the instability evolution reduces to determination of the slowly varying amplitude (SVA)

Expanding the DR (1) in power series near

where

The Eq. (8) describes the evolution of SVA *E*_{0} (*z*, *t*) in space and time for all systems those may be described by the DR in form (1). Eq. (8) may be solved by using Fourier transformation with respect to spatial coordinate *z* and Laplace transformation with respect to time *t*. The corresponding equation for the transform

where the function

Here,

The second integration (over

As a result of the integration, we obtain following expression for the SVA [11].

and

### 2.2. Analysis of the fields’ dynamics

We have arrived to very complex expressions (14). However, the field’s structure (i.e., the instability behavior) may be determined by analyzing the factor

The information, which are available from the analysis are much more detailed and complete as compared to results of well-known initial and boundary problems. The analysis gives: growth rate(s), the velocities of unstable perturbations, the character of the instability and influence of the dissipation on it, etc. The expression (15) shows that along with exponential increasing the field covers more and more space. In the absence of dissipation, the velocities of unstable perturbations range from *z* at various instants

The peak (and the field’s properties in it) may be determined from the equation

Its solution in the absence of dissipation gives

That is, the peak places on 1/3 of the train’s length from the front and moves at the velocity

In a fixed point z, the field first increases and attains maximum at instant

Then, the field falls off and at the time *wa* is the group velocity of the resonant wave upon amplification with account of the beam contribution in the DR. For given *z,* the field’s maximum is

The exponent

The relations between characteristic velocities are

At fixed instant

Generally, the dependence of the perturbations’ amplitudes on their velocity

The character of spatial growth depending on

Presented above analysis is true if we neglect dissipation. Dissipation essentially changes the instability behavior. It suppresses slow perturbations. The threshold velocity is

Only perturbations moving at higher velocities

The dynamics of the field in the peak may be obtained by analyzing the Eq. (16). It takes following form

If ν << *δ*_{0}, this equation leads to small corrections to the expressions (17) and (18) for characteristic velocities and for the maximal growth rate in the peak. In the opposite case of high-level dissipation, only the perturbations are unstable, whose velocity is close to the beam velocity *u*. In this approximation, the solution of Eq. (25) is

and the expression for maximal growth rate takes the form

In general, by substitution of two parameters only: growth rate and the group velocity of resonant wave in “cold” system one can obtain the behavior of specific e-beam instability.

It is not superfluous to repeat once again that the expression (14) and resulting analysis is valid for all types of e-beam instabilities: Cherenkov, cyclotron, beam instability in periodical structures, etc. Also, the analysis does not depend on specific geometry, external fields, etc.

## 3. The behavior of overlimiting electron beam instability

The picture described above is valid for e-beams, instability of which is due to induced radiation of the system proper waves by the beam electrons. However, it is known that with increase in beam current the physical nature of e-beam instabilities changes [6, 7, 12, 13, 14]. This is a result of influence of the beam space charge. It sets a limit for the beam current in vacuum systems. The limit may be overcome, for example, in plasma filled waveguide. The instability of over-limiting e-beams (OB) is due either to aperiodical modulation of the beam density in media with negative dielectric constant or to excitation of the NEW. In this section, we consider behavior of the first type of OB instability. It develops, for example, in uniform cross-section magnetized beam-plasma waveguide. It is clear that the change of the physical nature of the instability affects on its behavior. This instability sharply differs from the instability of conventional (underlimiting) e-beams: (1) its growth rate attains maximum at the point of exact Cherenkov resonance, (2) it is of nonradiative type, and (3) with increase in dissipation, it turns to a new type of DSI [6, 14].

### 3.1. Statement of the problem: analysis of the DR

Mathematical description of OB is not so well-known as for underlimiting beams, and in order to catch the differences, we consider both cases simultaneously. Consider a cylindrical waveguide, fully filled by cold plasma. A monoenergetic relativistic electron beam penetrates it. The external longitudinal magnetic field is assumed to be strong enough to freeze transversal motion of the beam and the plasma electrons. For simplicity, we assume that the beam and plasma radii coincide to the waveguide’s radius and consider only the symmetrical *E*-modes with nonzero components *Er*, *Ez*, and *Bφ*. It is known [1] that the system under consideration is described by the following DR

*ω* and *k* are the frequency and the longitudinal (along *z* axis) wave vector, *R* is the waveguide’s radius*, μ0s* are the roots of Bessel function *s =* 1,2,3…, *u* is the velocity of the beam, *c* is speed of light. The DR (27) determines the growth rates of the beam-plasma instability. As we have mentioned earlier, the character of the beam-plasma interaction changes depending on the beam current value. This change must reveal itself in the solutions of the DR (27). In order to consider the solutions, we look them in the form

where

The solutions of Eq. (28) depend on the value of parameter α. This parameter actually serves as a parameter that determines the beam current value and the character of beam-plasma interaction. It corresponds (correct to the factor γ^{−2}) to the ratio of the beam current to the limiting current in vacuum waveguide [14] *Ib* is the beam current). The values

However, if dissipation exceeds growth rate, the instability turns to DSI with the growth rate

If the beam current increases and became higher than the limiting vacuum current that is,

the instability has the same nature as the instability in medium with negative dielectric constant. If the beam is underlimiting, this effect is slight and is not observed. But now, this effect is dominant. Its distinctive peculiarity is that this effect attains its maximum in the point of exact Cherenkov resonance. The growth rate differs from Eq. (30) and is equal [13].

The different dependence of the growth rates of Eqs. (30) and (33) on beam density should be noted.

If, along with the beam current, dissipation also increases the instability turns to DSI of overlimiting e-beam with the growth rate [6].

We emphasize new dependence on

Higher values of parameter α (that is, α >> 1) correspond to very high currents. For example, in the case of a cylindrical waveguide this condition leads to

### 3.2. Equation for SVA and its solution: transition to the new type of DSI

In order to consider the evolution of an initial perturbation in a magnetized plasma waveguide penetrated by an OB, we proceed from the DR (27). Our steps coincide to those for the case of underlimiting e-beams: expand the DR (27) in series near

(the denotations coincide to those in (8)). The Eq. (35) for SVA may be solved by analogy to solution of Eq. (8). Without delving into details, we present here the results [6, 12].

The analysis of the expression (36) is similar to previous case. It again reduces to the analysis of the exponent

The field’s value in the peak exponentially increases and the growth rate is equal to maximal growth rate for OB

At fixed point *z* the SVA attains its maximum ∼

The expression

The character of the space growth depending on perturbations’ velocity is

Dissipation fundamentally changes this picture of the instability. For given velocity

Dissipation suppresses slow perturbations. Only high-velocity perturbations can develop. The threshold velocity is

The dynamics of the peak in the presence of dissipation may be obtained by analyzing the equation

The solution of Eq. (42) presents the peak’s coordinate

Substitution of

In limit of high-level dissipation, we have

where

## 4. The behavior of the instability in spatially separated beam-plasma system

### 4.1. Statement of the problem: the dispersion relation

There is a factor which significantly influences on the physics of beam-plasma interaction. The factor is the level of overlap of the beam and the plasma fields. The well-known beam-plasma instability corresponds to full overlap of the beam and the plasma fields (strong beam-plasma coupling). In this case, physical nature of developing instability is due to induced radiation of the system’s normal mode oscillations by the beam electrons. The oscillations are determined by plasma alone, as its density is assumed much higher than the beam density. The beam oscillations are actually suppressed and do not reveal themselves. Excited fields are actually detached from the beam in that they exist in beam absence.

The opposite case when the beam and plasma fields are overlapped slightly is the case of weak beam-plasma coupling. It may be realized, for instance, if the beam and the plasma are spatially separated in transverse direction. This transverse geometry provides conditions for increasing the role of the beam’s normal mode oscillations. In this case, the beam-plasma interaction has other physical nature. Electron beam is actually left to its own. Its oscillations come into play. Account of the beam’s normal mode oscillation leads to substantially new effects. Moreover, there is NEW among beam proper waves. Its growth causes instability due to the sign of energy. The growth rate of this instability attains maximum in resonance of plasma wave with NEW. Resonance of this (wave–wave) type comes instead of wave-particle resonance (conventional Cherenkov Effect) and was named “Collective Cherenkov Effect” [14, 15].

Consider weak interaction of monoenergetic electron beam and plasma in waveguide in general form [8, 14]. The only assumption is following. The beam and plasma are separated spatially, which implies weak coupling of the beam and the plasma fields. For a start, we do not particularize the cross sections. The beam current is assumed to be less than the limiting vacuum current. Dissipation in the system is taken into account by introducing collisions in plasma. We restrict ourselves by the case of strong external longitudinal magnetic field that prevents transversal motion of beam and plasma particles.

In strong external magnetic field, perturbations in plasma and beam have longitudinal components only. In such system, it is expedient to describe perturbations by using polarization potential

We proceed from equations for

Here *z* is longitudinal coordinate, *t* is the time, *c* is speed of light,

In general, the analytical treatment of the problem may be developed in different ways. The traditional way is to consider a multilayer structure of given geometry. With increase in number of layers this way leads to a very cumbersome DR. However, in the case of weak coupling (namely when the integral describing the overlap of the beam and the plasma fields (see below) is small), the interaction may be considered by another approach. The approach is perturbation theory over wave coupling [14]. Parameter of weak beam-plasma coupling serves as a small parameter that underlies this approach. This way leads to a DR of much simpler form, which, in addition, clearly shows the interaction of the beam and the plasma waves. Also, the procedure is not associated with a specific shape/geometry; that is, obtained results may be easily adapted to systems of any cross-section.

The set of Eq. (46) reduces to following eigenvalue problem

where

Proper functions

(the point

where

Mathematically,

### 4.2. The growth rates

The spectra of the beam waves are given by

where

where

where

It depends on beam density as

where

where

### 4.3. The space–time dynamics of the instability in spatially separated beam and plasma

We have already obtained some properties of the instability in system with spatially separated beam and plasma. Consider now the behavior of this instability in detail. In so doing, we consider the evolution of an initial perturbation in system with spatially separated e-beam and plasma. We proceed from the DR (51). The successive steps are known: to derive the equation for SVA, solve it and analyze the solution. As a result, we have following equation for SVA:

where

The Eq. (61) is actually the same Eq. (35). This implies that the fields’ space–time evolution at the instability development in spatially separated beam-plasma system qualitatively coincides to that of over-limiting e-beam instability. It remains to repeat briefly the milestones of the analysis above for behavior of OB instability in new terms (assuming

For the instability under weak beam-plasma coupling the velocities of unstable perturbation vary through the range

In the absence of dissipation, the peak places in the middle of the train at all instants that is, it moves at the average velocity

Dissipation suppresses slow perturbations. The threshold velocity is (compare to previous subsection)

The wave train shortens. Only high velocity perturbations (at velocities in the range

The peak shifts to the front of wave train. For high-level dissipation, we have

Substitution of Eq. (65) into

This result agrees to Eq. (58). This coincidence actually serves as an additional proof of the correctness of the approach based on analysis of developing wave train (i.e., correctness of the initial assumptions, derived equation for SVA, its solution etc.). Analogous coincidence exists in case of underlimiting e-beams (see Section 2), but very cumbersome expressions (solutions of third-order algebraic equation) prevent showing it obviously.

In conclusion to present section, we can state that two various types of e-beam instabilities: (1) the OB instability and (2) the instability under weak beam-plasma coupling have similar behavior. Both these instabilities transform to dissipative instabilities with the maximal growth rate *u*. In the second case, the velocities are approximately equal to group velocity of NEW.

## 5. The behavior of the Buneman instability in dissipative plasma

### 5.1. Statement of the problem: the equation for SVA

The physical essence of the Buneman instability (BI) [4] is in the fact that the proper space charge oscillations of moving electrons due to the Doppler Effect experience red shift, and this greatly reduced frequency becomes close to the proper frequency of ions. Actually, the BI is due to resonance of the negative energy wave with the ion oscillations. For future interpretations and comparisons, we present the well-known [1, 4] DR and the maximal growth rate for the simplest case of the BI (cold e-stream, heavy ions, and accounting for collisions)

(

ions respectively,

Now consider a plasma system, the DR of which may be written as

where

An initial perturbation arises and the instability begins to develop in point

Eq. (70) may be solved in known manner: that is, by using the Fourier and Laplace transformations. The problem reduces to integration in the inverse transformation. All these steps are known. So as not to repeat, we at once present resulting expression for the SVA [16]

### 5.2. Analysis of the Buneman instability behavior

As earlier, the structure of the fields is basically determined by the factor [16].

In the absence of dissipation the velocities of unstable perturbations range from 0 to the group velocity

(compare to Eq. (16)) determines the peak’s movement. In the absence of dissipation the peak disposes on 2/3 of the train’s length from its front and moves at velocity

Dissipation changes the fields’ dynamics and mode structure. It is easily seen from Eq. (72) that dissipation suppresses fast perturbations. The threshold velocity

The wave train shortens. Actually the pulse slows down. Dissipation influences on the peak location/movement. Its place

The solution of this third-order algebraic equation gives location and velocity of the peak under arbitrary ratio

Substitution of this expression into

where the growth rate

In addition, the expression for

Figure 4 presents shapes of induced wave train for various levels of dissipation.

## 6. Conclusion

Now, we can generalize the properties of the SI. Originated perturbations form a wave train, carrier frequency and wave vector of which are determined by resonant conditions. The expression for space–time distribution of the fields gives much information on the behavior of the instability in limit of comparatively large times. The solutions of conventional initial and boundary problems follow from the expression by itself. The growth rate in the peak is equal to maximal growth rate of resonant instability

The approach justifies existence of two new, previously unknown types of DSI. For these DSI, the role of the beam’s space charge and/or proper oscillation becomes decisive. For both DSI, the growth rates have more critical dependence on dissipation as compared to conventional. Presented approach obviously shows the transition to the new types of DSI.

Actually the approach presents solution of the well-known problem of time evolution of initial perturbation in systems those undergo the instabilities of streaming type. The importance of the problem is doubtless. Its traditional solution is restricted by mathematical difficulties. Presented methods allows without any difficulties obtain result for various SI in spite of their different mathematical description (e.g., the description of Buneman instability differs from the instability in spatially separated beam-plasma system and from beam-plasma instabilities; herewith, the description various types of beam-plasma instabilities (Cherenkov, cyclotron, and other) also differs from each other). The approach by itself unified the differences. For beam-plasma instabilities results of the approach are unified even more and their usage is not more difficult than usage of the result of the initial and boundary problems (in spite of presented approach gives incomparably more data). In this sense, the approach can be used instead of the problems. It could seem that the procedure is a bit more difficult. However, this difficulty only seems.

The general character of presented approach should be emphasized once more. It is based on very general assumptions and does not refer on any particular model. The approach transforms the general form of the DR to an equation for SVA of the developing wave train. For a large class of beam-plasma instabilities (Cherenkov, cyclotron, etc.), the equation for SVA is actually the same. Its solution gives analytical expression describing evolution of initial perturbation. Various SI evolve in similar manner. This emphasizes identity of their physical nature (induced radiation of the system proper waves by the beam electrons). For given instability, one should specify two parameters only: the resonant growth rate and the group velocity of the resonant wave. Obtained expression gives detailed information on the instability. The information is: the shape of developing wave train (envelope), velocities of unstable perturbations, the type of given instability (absolute or convective), location of the peak and the character of its movement, the rate of field’s growth in the peak, temporal and spatial growth rates, the rate of growth for perturbation moving at given velocity. Most of these data are unavailable by other methods.

Validity limitations also should be mentioned. Obtained results may not be applied to the systems where beam instability is caused by finite longitudinal dimension, for example, Pierce instability.

Presented approach has neither inner contradictions, no contradictions to previous results of the beam-plasma interaction theory. Its results fully coincide to those obtained by direct analysis of the DR. In some cases, (e.g., for overlimiting e-beam instability and the instability in spatially separated beam-plasma system) obvious analysis is possible due to comparatively simple contribution of the beam in the DR (namely when the contribution has first (but not second) order pole).

The results of presented approach actually are continuation and further development of the results of the initial and boundary problems. In its turn, the results of the problems have been repeatedly tested and rechecked experimentally. This actually can serve as confirmation of validity of the approach.

In [19, 20] the nonlinear dynamics of the beam-plasma instability was investigated numerically at no stationary beam injection into plasma-filled systems. The results show that at the initial stage of instability development the field has a shape matching reasonably to presented results.

Obtained results on SI evolution help to understand how the instability transforms given equilibrium of background plasma, estimate the level and/or scale of originated irregularities clear up how the nonlinear stage arises and predict saturation mechanisms. The systems, to which this may be applied are numerous, as the SI are the most common instabilities: from the Earth ionosphere to current carrying plasma (where the Buneman instability plays important role). Not to mention relativistic microwave electronics etc.