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 ψ [14]. This actually is a single nonzero component of well-known Hertz vector.

We proceed from equations for ψ and for the beam and the plasma currents j p, <sup>b</sup>.

$$\frac{\partial}{\partial t} \left( \Delta\_{\perp} + \frac{\partial^2}{\partial z^2} - \frac{1}{c^2} \frac{\partial^2}{\partial t^2} \right) \psi = -4\pi \left( j\_{lz} + j\_{pz} \right) \qquad ; \qquad E\_z = \frac{\partial^2 \psi}{\partial z^2} - \frac{1}{c^2} \frac{\partial^2 \psi}{\partial t^2} \tag{46}$$

$$\left( \frac{\partial}{\partial t} + u \frac{\partial}{\partial t} \right)^2 j\_{\flat} = \frac{\omega\_b^2 \gamma^{-3}}{4\pi} \frac{\partial}{\partial t} E\_z \qquad ; \qquad \left( \frac{\partial}{\partial t} + \nu \right) j\_p = \frac{\omega\_p^2}{4\pi} E\_z.$$

Here j bzð Þ¼ r⊥; z; t pbð Þ r<sup>⊥</sup> j <sup>b</sup>ð Þ z; t and j pzð Þ¼ r⊥; z; t ppð Þ r<sup>⊥</sup> j <sup>p</sup>ð Þ z; t are perturbations of the longitudinal current densities of the beam and the plasma. Functions pb, <sup>p</sup>ð Þ r<sup>⊥</sup> describe transverse density profiles for beam and plasma. For homogeneous beam/plasma pb, <sup>p</sup> � 1, for infinitesimal thin beam/plasma pb,p � δ r � rb, <sup>p</sup> � � (δ is Dirac function). Δ<sup>⊥</sup> is the Laplace operator over transverse coordinates, z is longitudinal coordinate, t is the time, c is speed of light, ωp, <sup>b</sup> are the Langmuir frequencies for plasma and beam respectively, ν – is the collision frequency in plasma, γ is the relativistic factors of the beam electrons, u is the beam velocity.

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

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

Figure 3. The function f xð Þgives the dependence of maximal growth rate on dissipation level.

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

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

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

4.1. Statement of the problem: the dispersion relation

34 Plasma Science and Technology - Basic Fundamentals and Modern Applications

detached from the beam in that they exist in beam absence.

Cherenkov Effect) and was named "Collective Cherenkov Effect" [14, 15].

system

$$
\Delta\_{\perp} \psi - \kappa^2 \left[ 1 - p\_p(\mathbf{r}\_{\perp}) \delta \varepsilon\_p - p\_b(\mathbf{r}\_{\perp}) \delta \varepsilon\_b \right] \ \psi = 0 \qquad ; \qquad \psi \vert\_{\Sigma} = 0 \tag{47}
$$

where ψ is the proper function of the problem, Σ means the surface of the waveguide (it is not specified yet).

$$
\kappa^2 = k^2 - \frac{\omega^2}{c^2} \qquad ; \qquad \delta\varepsilon\_p = \frac{\omega\_p^2}{\omega(\omega + i\nu)} \qquad ; \qquad \delta\varepsilon\_b = \frac{\omega\_b^2}{\gamma^3(\omega - k\nu)} . \tag{48}
$$

ω and kare the frequency and longitudinal wave vector, ν is the frequency of plasma collisions. As we have mentioned earlier, direct solution of the problem (47) presents considerable difficulties. However, in case of spatially separated beam and plasma that is, when pbð Þ r<sup>⊥</sup> ppð Þ¼ r<sup>⊥</sup> 0 and the integral describing the overlap of the fields (see below) is small, it is possible to apply perturbation theory. It assumes that in zero order approximation the beam and the plasma are independent and they may be described by two independent eigenvalue problems for plasma and beam respectively [14].

$$
\Delta\_{\mathsf{L}} \psi\_a - \kappa^2 \left[ 1 - p\_a(\mathbf{r}\_{\mathsf{L}}) \delta \varepsilon\_a \right] \psi\_a = 0 \qquad ; \qquad \psi\_a \big|\_{\Sigma} = 0 \qquad ; \qquad a = p, b \tag{49}
$$

4.2. The growth rates

where <sup>α</sup> <sup>¼</sup> <sup>ω</sup><sup>2</sup>

<sup>b</sup>=k 2

where <sup>q</sup> <sup>¼</sup> <sup>1</sup>=2γ<sup>2</sup> � � <sup>k</sup><sup>2</sup>

It depends on beam density as n

new γ<sup>2</sup>

where <sup>λ</sup> <sup>¼</sup> <sup>ν</sup><sup>=</sup> <sup>2</sup>δð Þ <sup>ν</sup>¼<sup>0</sup>

where x 0

growth is

The spectra of the beam waves are given by Db Eq. (52) and have following form

ffiffiffi α p γ �

section) β ¼ u=c. The beam-plasma interaction in the absence of dissipation leads to conventional beam instability that is caused by excitation of the system normal mode waves by the

of dissipation the conventional beam instability is gradually converted to that of dissipative

normal mode oscillations of the beam are neglected. The concept of the NEW is invoked only to explain the physical meaning of DSI. These results are valid only for the case of strong beam-plasma coupling. The decrease in beam-plasma coupling leads to exhibition of the beam's normal mode oscillation. In this case, the instability is caused by the excitation of the NEW. Specific features of weak beam-plasma interaction should appear themselves in solu-

with plasma wave corresponds to the condition q ¼ 0; however, the resonance between the beam slow wave and plasma wave (collective Cherenkov effect) corresponds to q ¼ �x�.The interaction of the beam and plasma waves leads to instability. Mathematically, it is due to corrections to the expression for NEW. Using the condition of collective Cherenkov resonance one can obtain

0

¼ x � x�. In the absence of dissipation the growth rate of instability caused by NEW

s

stable. Dissipation exhibits itself as additional factor that intensifies growth of the NEW. Eq. (56) gives following expression for the growth rate upon arbitrary level of the dissipation [8].

q

bility to that of dissipative type with increase in level of dissipation. This dependence on

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>1</sup> <sup>þ</sup> <sup>λ</sup><sup>2</sup> =4

� �. The expression (58) shows gradual transition of no dissipative insta-

ffiffiffiffiffiffiffiffiffiffi G ffiffiffi α p γ

tions of Eq. (51). If one looks them in the formω ¼ kuð Þ 1 þ x , then Eq. (51) becomes

q

<sup>⊥</sup><sup>b</sup>u<sup>2</sup>γ<sup>3</sup> is the parameter that determines the beam current value (see previous

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi β4

ð Þ <sup>x</sup> <sup>þ</sup> <sup>q</sup> <sup>þ</sup> <sup>i</sup>ν=ku ð Þ <sup>x</sup> � <sup>x</sup><sup>þ</sup> ð Þ¼ <sup>x</sup> � <sup>x</sup>� <sup>G</sup>α=2γ<sup>4</sup> (55)

� �. The usual Cherenkov resonance of the beam electrons

¼ � <sup>G</sup> ffiffiffi α p

γ<sup>2</sup>α þ 1

� <sup>β</sup><sup>2</sup> γ ffiffiffi <sup>α</sup> <sup>p</sup> � � (54)

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37

1=3

<sup>b</sup> . With increase in level

<sup>ν</sup> <sup>p</sup> . For these instabilities the

<sup>4</sup>γ<sup>3</sup> : (56)

: (57)

<sup>b</sup> . Under conventional Cherenkov resonance the system is

� � (58)

� λ=2

ω� ¼ kuð Þ 1 þ x� ; x� ¼

beam electrons. Its maximal growth rate depends on beam density as n

type. Its maximal growth rate depends on dissipation as � <sup>1</sup><sup>=</sup> ffiffiffi

⊥pu<sup>2</sup>γ<sup>2</sup>=ω<sup>2</sup>

<sup>p</sup> � 1

x 0 <sup>þ</sup> <sup>i</sup> <sup>ν</sup> <sup>2</sup>γ<sup>2</sup>ku � �<sup>x</sup>

1=4

δ λð Þ¼ <sup>δ</sup>ð Þ <sup>ν</sup>¼<sup>0</sup> new

δð Þ <sup>ν</sup>¼<sup>0</sup> new <sup>¼</sup> ku 2γ

Proper functions ψ<sup>p</sup> and ψbof these zero-order problems as well as the zero-order DR for the beam and the plasma are assumed to be known. If one applies perturbation theory to the zeroorder problems those are described by the DR

$$\{D\_p(\omega, k)\}\_{\omega = \omega\_0} = 0 \qquad ; \qquad \{D\_b(\omega, k)\}\_{\omega = \omega\_0} = 0 \tag{50}$$

$$k = k\_0 \tag{51}$$

(the point f g ω0; k<sup>0</sup> is the intersection point of the plasma and the beam curves) and search the solution of Eq. (47) in the form ψ ¼ Aψ<sup>p</sup> þ Bψb, A, B ¼ const, he can obtain in first order approximation the following DR

$$D\_p(\omega, k)D\_b(\omega, k) = G\left(\kappa^4 \delta \varepsilon\_p \delta \varepsilon\_b\right)\_{\omega = \omega\_0 \prime} \tag{51}$$
 
$$k = k\_0$$

where

k

$$D\_{p,b}(\omega,k) = k\_{\perp p,b}^2 - \kappa^2 \delta \varepsilon\_{p,b} = 0. \tag{52}$$

G is the coupling coefficient. It shows the efficiency of beam-plasma interaction, k⊥p, <sup>b</sup> are the actual transverse wavenumbers for the beam and the plasma respectively (see also [8])

$$G = \frac{\left(\iint\_{S\_{\nu}} p\_{p} \psi\_{p} \psi\_{b} d\mathbf{r}\_{\perp}\right) \left(\iint\_{S\_{\nu}} p\_{b} \psi\_{p} \psi\_{b} d\mathbf{r}\_{\perp}\right)}{\left(\iint\_{S\_{\nu}} p\_{p} \psi\_{p}^{2} d\mathbf{r}\_{\perp}\right) \left(\iint\_{S\_{\nu}} p\_{b} \psi\_{b}^{2} d\mathbf{r}\_{\perp}\right)} > 0\tag{53}$$

$$\mathcal{C}\_{\mathbf{r},b} = \left(\iint\_{S\_{\nu}} \left(\left(\nabla\_{\perp} \psi\_{p,b}\right)^{2} + \kappa^{2} \psi\_{p,b}^{2}\right) d\mathbf{r}\_{\perp}\right) \left(\iint\_{S\_{\nu}} p\_{p,b} \left(\mathbf{r}\_{\perp}\right) \psi\_{p,b}^{2} d\mathbf{r}\_{\perp}\right)^{-1}$$

Mathematically, G is expressed in terms of integrals those represent the overlap of the beam and the plasma fields. Physically, it determines as far the field of plasma wave penetrates into beam and vice versa. According to our consideration, G is small G < <1. One more condition of validity of presented consideration is homogeneity of the beam and the plasma inside the cross sections.

#### 4.2. The growth rates

difficulties. However, in case of spatially separated beam and plasma that is, when pbð Þ r<sup>⊥</sup> ppð Þ¼ r<sup>⊥</sup> 0 and the integral describing the overlap of the fields (see below) is small, it is possible to apply perturbation theory. It assumes that in zero order approximation the beam and the plasma are independent and they may be described by two independent eigenvalue

Proper functions ψ<sup>p</sup> and ψbof these zero-order problems as well as the zero-order DR for the beam and the plasma are assumed to be known. If one applies perturbation theory to the zero-

(the point f g ω0; k<sup>0</sup> is the intersection point of the plasma and the beam curves) and search the solution of Eq. (47) in the form ψ ¼ Aψ<sup>p</sup> þ Bψb, A, B ¼ const, he can obtain in first order

Dpð Þ <sup>ω</sup>; <sup>k</sup> Dbð Þ¼ <sup>ω</sup>; <sup>k</sup> <sup>G</sup> <sup>κ</sup><sup>4</sup>δεpδε<sup>b</sup>

2 <sup>⊥</sup>p, <sup>b</sup> � <sup>κ</sup><sup>2</sup>

actual transverse wavenumbers for the beam and the plasma respectively (see also [8])

ppψpψbdr<sup>⊥</sup>

ppψ<sup>2</sup> <sup>p</sup>dr<sup>⊥</sup>

<sup>2</sup> <sup>þ</sup> <sup>κ</sup><sup>2</sup> ψ2 p, b

� �

G is the coupling coefficient. It shows the efficiency of beam-plasma interaction, k⊥p, <sup>b</sup> are the

1 CA ðð

1 CA ðð

0 B@

0 B@

Sw

Sw

dr<sup>⊥</sup>

Mathematically, G is expressed in terms of integrals those represent the overlap of the beam and the plasma fields. Physically, it determines as far the field of plasma wave penetrates into beam and vice versa. According to our consideration, G is small G < <1. One more condition of validity of presented consideration is homogeneity of the beam and the plasma inside the cross

pbψ<sup>2</sup> <sup>b</sup>dr<sup>⊥</sup>

> 1 CA ðð

0 B@

Sw

pbψpψbdr<sup>⊥</sup>

Dp, <sup>b</sup>ð Þ¼ ω; k k

� �

k ¼ k<sup>0</sup>

ω ¼ ω<sup>0</sup> k ¼ k<sup>0</sup>

> 1 CA

pp, <sup>b</sup>ðr⊥Þψ<sup>2</sup>

p, <sup>b</sup>dr<sup>⊥</sup>

1 CA

�1

1 CA

<sup>¼</sup> <sup>0</sup> ; Df g <sup>b</sup>ð Þ <sup>ω</sup>; <sup>k</sup> <sup>ω</sup> <sup>¼</sup> <sup>ω</sup><sup>0</sup>

� �

<sup>Σ</sup> ¼ 0 ; α ¼ p, b (49)

¼ 0 (50)

, (51)

> 0 (53)

δεp, <sup>b</sup> ¼ 0: (52)

� �ψα <sup>¼</sup> <sup>0</sup> ; ψα

problems for plasma and beam respectively [14].

36 Plasma Science and Technology - Basic Fundamentals and Modern Applications

<sup>Δ</sup>⊥ψα � <sup>κ</sup><sup>2</sup> <sup>1</sup> � <sup>p</sup>αð Þ <sup>r</sup><sup>⊥</sup> δεα

order problems those are described by the DR

approximation the following DR

k 2 <sup>⊥</sup>p, <sup>b</sup> ¼

where

sections.

Dpð Þ <sup>ω</sup>; <sup>k</sup> � �

G ¼

ðð

0 B@

Sw

ðð

0 B@

Sw

0 B@

ð∇⊥ψp, <sup>b</sup>Þ

ðð

Sw

ω ¼ ω<sup>0</sup> k ¼ k<sup>0</sup>

The spectra of the beam waves are given by Db Eq. (52) and have following form

$$\omega\_{\pm} = k u (1 + \mathbf{x}\_{\pm}) \qquad ; \qquad \mathbf{x}\_{\pm} = \frac{\sqrt{\alpha}}{\mathcal{V}} \left( \pm \sqrt{\beta^{4} \gamma^{2} \alpha + 1} - \beta^{2} \gamma \sqrt{\alpha} \right) \tag{54}$$

where <sup>α</sup> <sup>¼</sup> <sup>ω</sup><sup>2</sup> <sup>b</sup>=k 2 <sup>⊥</sup><sup>b</sup>u<sup>2</sup>γ<sup>3</sup> is the parameter that determines the beam current value (see previous section) β ¼ u=c. The beam-plasma interaction in the absence of dissipation leads to conventional beam instability that is caused by excitation of the system normal mode waves by the

beam electrons. Its maximal growth rate depends on beam density as n 1=3 <sup>b</sup> . With increase in level of dissipation the conventional beam instability is gradually converted to that of dissipative type. Its maximal growth rate depends on dissipation as � <sup>1</sup><sup>=</sup> ffiffiffi <sup>ν</sup> <sup>p</sup> . For these instabilities the normal mode oscillations of the beam are neglected. The concept of the NEW is invoked only to explain the physical meaning of DSI. These results are valid only for the case of strong beam-plasma coupling. The decrease in beam-plasma coupling leads to exhibition of the beam's normal mode oscillation. In this case, the instability is caused by the excitation of the NEW. Specific features of weak beam-plasma interaction should appear themselves in solutions of Eq. (51). If one looks them in the formω ¼ kuð Þ 1 þ x , then Eq. (51) becomes

$$(\mathbf{x} + \mathbf{q} + i\mathbf{v}/k\mathbf{u})(\mathbf{x} - \mathbf{x}\_{+}) \ (\mathbf{x} - \mathbf{x}\_{-}) = G\alpha/2\gamma^{4} \tag{55}$$

where <sup>q</sup> <sup>¼</sup> <sup>1</sup>=2γ<sup>2</sup> � � <sup>k</sup><sup>2</sup> ⊥pu<sup>2</sup>γ<sup>2</sup>=ω<sup>2</sup> <sup>p</sup> � 1 � �. The usual Cherenkov resonance of the beam electrons with plasma wave corresponds to the condition q ¼ 0; however, the resonance between the beam slow wave and plasma wave (collective Cherenkov effect) corresponds to q ¼ �x�.The interaction of the beam and plasma waves leads to instability. Mathematically, it is due to corrections to the expression for NEW. Using the condition of collective Cherenkov resonance one can obtain

$$
\left(\mathbf{x}' + i\frac{\nu}{2\gamma^2 k u}\right)\mathbf{x}' = -\frac{G\sqrt{a}}{4\gamma^3}.\tag{56}
$$

where x 0 ¼ x � x�. In the absence of dissipation the growth rate of instability caused by NEW growth is

$$
\delta\_{\text{new}}^{(v=0)} = \frac{ku}{2\gamma} \sqrt{\frac{G\sqrt{a}}{\gamma}}.\tag{57}
$$

It depends on beam density as n 1=4 <sup>b</sup> . Under conventional Cherenkov resonance the system is stable. Dissipation exhibits itself as additional factor that intensifies growth of the NEW. Eq. (56) gives following expression for the growth rate upon arbitrary level of the dissipation [8].

$$\delta(\lambda) = \delta\_{\text{new}}^{(v=0)} \left\{ \sqrt{1 + \lambda^2/4} - \lambda/2 \right\} \tag{58}$$

where <sup>λ</sup> <sup>¼</sup> <sup>ν</sup><sup>=</sup> <sup>2</sup>δð Þ <sup>ν</sup>¼<sup>0</sup> new γ<sup>2</sup> � �. The expression (58) shows gradual transition of no dissipative instability to that of dissipative type with increase in level of dissipation. This dependence on dissipation coincides to that depicted in Figure 3. In the limit of strong dissipationλ >> 1, Eq. (58) becomes

$$\delta\left(\nu>>\delta\_{\rm NEW}^{(\nu=0)}\right) = \delta\_{\rm NEW}^{(\nu\rightarrow\infty)} \left[1 - \frac{G\gamma\sqrt{\alpha}(ku)^2}{\nu^2}\right] \tag{59}$$

For the instability under weak beam-plasma coupling the velocities of unstable perturbation vary through the range v<sup>p</sup> ≤ v ≤ vb, The character of the instability is determined by group velocities of plasma wave and the NEW. The statements on the character of the instability (convective or absolute) remain valid with account of replacements v0 ! v<sup>p</sup> and u ! vb. The place and the

velocity of the peak of the wave train can be obtained, as earlier, by solving the equation

<sup>∂</sup><sup>z</sup> exp <sup>χ</sup>ð Þ ss

ð Þ ν¼0

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

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

<sup>0</sup> ; λ

ffiffiffiffiffiffiffiffiffiffiffiffiffi λ 0 1 þ λ 0

s

The peak shifts to the front of wave train. For high-level dissipation, we have w0, Vð Þ ss

<sup>E</sup><sup>0</sup> � exp <sup>δ</sup>0<sup>t</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

that is, one can conclude: the group velocity of perturbation of the new DSI is equal to the group velocity of the NEW. This distinguishes the DSI under weak coupling from the DSI of

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

<sup>1</sup> <sup>þ</sup> <sup>λ</sup>0<sup>2</sup>=<sup>4</sup> <sup>p</sup> � <sup>λ</sup>

The wave train shortens. Only high velocity perturbations (at velocities in the range Vth < v < vb) develop. Herewith the behavior of the fields in the peak (and the place/velocity of the peak) may be obtained by analyzing Eq. (63). If one takes into account the dissipation,

0 <sup>¼</sup> <sup>ν</sup><sup>∗</sup> 2δ<sup>0</sup> � �<sup>2</sup>

v<sup>b</sup> � v<sup>p</sup> � �

9 = ;

<sup>ν</sup> gives us the dependence of the growth rate on dissipation of

� � (66)

0 =2 > wgs (65)

<sup>ν</sup> ¼ 0: (63)

The Behavior of Streaming Instabilities in Dissipative Plasma

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(64)

39

th ! v<sup>b</sup>

� �. The field value in the peak exponentially

NEW (57). In the absence of dissipation, the waveform

∂

moves at the average velocity wgs ¼ ð Þ 1=2 v<sup>b</sup> þ v<sup>p</sup>

is symmetric with respect to its peak at all instants.

Vð Þ ss th <sup>¼</sup> <sup>λ</sup> 0 v<sup>b</sup> þ v<sup>p</sup> 1 þ λ

<sup>w</sup><sup>0</sup> <sup>¼</sup> <sup>1</sup>

<sup>2</sup> <sup>v</sup><sup>b</sup> <sup>þ</sup> <sup>v</sup><sup>p</sup> <sup>þ</sup>

OB (where the velocity of perturbations was equal to the beam velocity).

arbitrary level. The field value in the peak depends on dissipation as

of third-order algebraic equation) prevent showing it obviously.

8 < :

increases and the growth rate is equal to δ

the solution of (63) yields z ¼ w0t, where

Substitution of Eq. (65) into χð Þ ss

subsection)

where

$$\delta\_{\rm NEW}^{(\nu \to \infty)} = \frac{2\gamma^2 \left(\delta\_{\rm NEW}^{(\nu=0)}\right)^2}{\nu} = \frac{G\sqrt{\alpha}}{2\gamma} \frac{(ku)^2}{\nu} \tag{60}$$

δ ð Þ ν!∞ NEW presents the maximal growth rate of the new type of dissipative instability, shown up in [8]. It also follows from Eq. (56) by neglecting first term in parentheses. The new type of dissipative beam-plasma instability is now substantiated for beam and plasma layers in waveguide. The cross-sections of the layers and the waveguide are arbitrary. The instability of new type results from the superposition of dissipation on the instability that is already caused by the growth of the NEW. The instability comes instead of the conventional DSI (with growth rate � <sup>1</sup><sup>=</sup> ffiffiffi <sup>ν</sup> <sup>p</sup> ) when beam-plasma coupling becomes small. The dependence on dissipation becomes more critical. The same instability can be substantiated in finite external magnetic field also [18].

#### 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:

$$E\left(\frac{\partial}{\partial t} + \mathbf{v}\_{\theta} \frac{\partial}{\partial z}\right) \left(\frac{\partial}{\partial t} + \mathbf{v}\_{\mathbb{P}} \frac{\partial}{\partial z} + \boldsymbol{\nu}^\*\right) E\_0(\mathbf{z}, t) = \delta\_0^2 E\_0(\mathbf{z}, t). \tag{61}$$

where δ<sup>0</sup> � δ ð Þ ν¼0 NEW (57), vp, <sup>b</sup> are group velocities of the plasma wave and the NEW of the beam, respectively, and <sup>ν</sup><sup>∗</sup> <sup>¼</sup> Im Dp <sup>∂</sup>Dp=∂<sup>ω</sup> � ��<sup>1</sup> is proportional to collision frequency <sup>ν</sup><sup>∗</sup> <sup>¼</sup> const � <sup>ν</sup>.

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 v<sup>b</sup> > vp) and, where it is needed, to interpret results according new denotations. For this, we first rewrite the analyzing expression in new denotations

$$
\chi\_{\nu}^{(\text{ovl})} \rightarrow \chi\_{\nu}^{(\text{ss})} = \frac{2\delta\_0}{\mathbf{v}\_\mathbf{b} - \mathbf{v}\_p} \sqrt{(\mathbf{z} - \mathbf{v}\_p t)(\mathbf{v}\_b t - \mathbf{z})} - \nu^\* \frac{\mathbf{v}\_b t - \mathbf{z}}{\mathbf{v}\_\mathbf{b} - \mathbf{v}\_p}. \tag{62}
$$

For the instability under weak beam-plasma coupling the velocities of unstable perturbation vary through the range v<sup>p</sup> ≤ v ≤ vb, The character of the instability is determined by group velocities of plasma wave and the NEW. The statements on the character of the instability (convective or absolute) remain valid with account of replacements v0 ! v<sup>p</sup> and u ! vb. The place and the velocity of the peak of the wave train can be obtained, as earlier, by solving the equation

dissipation coincides to that depicted in Figure 3. In the limit of strong dissipationλ >> 1,

ð Þ ν¼0 NEW � �<sup>2</sup>

NEW presents the maximal growth rate of the new type of dissipative instability, shown up in [8]. It also follows from Eq. (56) by neglecting first term in parentheses. The new type of dissipative beam-plasma instability is now substantiated for beam and plasma layers in waveguide. The cross-sections of the layers and the waveguide are arbitrary. The instability of new type results from the superposition of dissipation on the instability that is already caused by the growth of the NEW. The instability comes instead of the conventional DSI (with growth rate

<sup>ν</sup> <sup>p</sup> ) when beam-plasma coupling becomes small. The dependence on dissipation becomes more critical. The same instability can be substantiated in finite external magnetic field also [18].

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

� �

respectively, and <sup>ν</sup><sup>∗</sup> <sup>¼</sup> Im Dp <sup>∂</sup>Dp=∂<sup>ω</sup> � ��<sup>1</sup> is proportional to collision frequency <sup>ν</sup><sup>∗</sup> <sup>¼</sup> const � <sup>ν</sup>.

q

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 v<sup>b</sup> > vp) and, where it is needed, to interpret results according new denotations. For this, we first rewrite the analyzing

∂t þ vp ∂ ∂z <sup>þ</sup> <sup>ν</sup><sup>∗</sup>

<sup>ν</sup> <sup>¼</sup> <sup>2</sup>δ<sup>0</sup> vb � v<sup>p</sup>

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:

NEW <sup>1</sup> � <sup>G</sup><sup>γ</sup> ffiffiffi

<sup>ν</sup> <sup>¼</sup> <sup>G</sup> ffiffiffi

α p 2γ

<sup>E</sup>0ð Þ¼ <sup>z</sup>; <sup>t</sup> <sup>δ</sup><sup>2</sup>

NEW (57), vp, <sup>b</sup> are group velocities of the plasma wave and the NEW of the beam,

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>z</sup> � <sup>v</sup>pt � �ð Þ <sup>v</sup>bt � <sup>z</sup> � <sup>ν</sup><sup>∗</sup> <sup>v</sup>bt � <sup>z</sup> vb � v<sup>p</sup>

<sup>α</sup> <sup>p</sup> ð Þ ku <sup>2</sup> ν2 " #

ð Þ ku <sup>2</sup>

, (59)

<sup>ν</sup> (60)

<sup>0</sup>E0ð Þ z; t : (61)

: (62)

δ ν >> δ

38 Plasma Science and Technology - Basic Fundamentals and Modern Applications

δ ð Þ ν!∞ NEW ¼

∂ ∂t þ v<sup>b</sup> ∂ ∂z � � ∂

ð Þ ν¼0 NEW � �

¼ δ ð Þ ν!∞

2γ<sup>2</sup> δ

Eq. (58) becomes

where

δ ð Þ ν!∞

� <sup>1</sup><sup>=</sup> ffiffiffi

where δ<sup>0</sup> � δ

ð Þ ν¼0

expression in new denotations

χð Þ ovl <sup>ν</sup> ! <sup>χ</sup>ð Þ ss

$$\frac{\partial}{\partial z} \exp \chi\_{\nu}^{(ss)} = 0. \tag{63}$$

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 wgs ¼ ð Þ 1=2 v<sup>b</sup> þ v<sup>p</sup> � �. The field value in the peak exponentially increases and the growth rate is equal to δ ð Þ ν¼0 NEW (57). In the absence of dissipation, the waveform is symmetric with respect to its peak at all instants.

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

$$V\_{th}^{(ss)} = \frac{\boldsymbol{\lambda}^{\prime}\mathbf{v}\_b + \mathbf{v}\_p}{\mathbf{1} + \boldsymbol{\lambda}^{\prime}} \qquad ; \qquad \boldsymbol{\lambda}^{\prime} = \left(\frac{\boldsymbol{\nu}^\*}{2\delta\_0}\right)^2 \tag{64}$$

The wave train shortens. Only high velocity perturbations (at velocities in the range Vth < v < vb) develop. Herewith the behavior of the fields in the peak (and the place/velocity of the peak) may be obtained by analyzing Eq. (63). If one takes into account the dissipation, the solution of (63) yields z ¼ w0t, where

$$w\_0 = \frac{1}{2} \left\{ \mathbf{v}\_b + \mathbf{v}\_p + \sqrt{\frac{\boldsymbol{\lambda}'}{1 + \boldsymbol{\lambda}'}} (\mathbf{v}\_b - \mathbf{v}\_p) \right\} > w\_{\rm gs} \tag{65}$$

The peak shifts to the front of wave train. For high-level dissipation, we have w0, Vð Þ ss th ! v<sup>b</sup> that is, one can conclude: the group velocity of perturbation of the new DSI is equal to the group velocity of the NEW. This distinguishes the DSI under weak coupling from the DSI of OB (where the velocity of perturbations was equal to the beam velocity).

Substitution of Eq. (65) into χð Þ ss <sup>ν</sup> gives us the dependence of the growth rate on dissipation of arbitrary level. The field value in the peak depends on dissipation as

$$E\_0 \sim \exp \delta\_0 t \left(\sqrt{1 + \lambda^2/4} - \lambda^{'}/2\right) \tag{66}$$

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 � 1=ν. In spite of their different physical nature, these instabilities have similar mathematical description. The contribution of the OB in the DR is given by expression having first order pole. The DR of the systems with spatially separated beam and plasma also may be reduced to analogous form. For comparison: the contribution of underlimiting e-beam is given by an expression with second-order pole for all types beam instabilities (Cherenkov, cyclotron etc.). This leads to their similar behavior. However, a difference between these two DSI also exists. In system with OB dissipation shifts the velocities of unstable modes to the beam velocity u. In the second case, the velocities are approximately equal to group velocity of NEW.
