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, 12–14]. 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

$$k\_{\perp}^2 + \left(k^2 - \frac{\omega^2}{c^2}\right)\left(1 - \frac{\omega\_p^2}{\omega(\omega + i\nu)} - \frac{\omega\_b^2}{\gamma^3(\omega - ku)^2}\right) = 0\tag{27}$$

δund ¼

δ ð Þν

overlimiting e-beam with the growth rate [6].

ffiffiffi 3 p 2 ω0 γ

und <sup>¼</sup> <sup>ω</sup>bω<sup>0</sup> 2γ<sup>3</sup>=<sup>2</sup>ω<sup>p</sup>

of exact Cherenkov resonance. The growth rate differs from Eq. (30) and is equal [13].

δ ð Þν ovl <sup>¼</sup> <sup>β</sup><sup>2</sup> γ ω2 b ω2 p ω2 0

dependence on ν is due to superposition of two factors those lead to NEW excitation.

the case of a cylindrical waveguide this condition leads to Ib ≥ 1, 4 mc<sup>3</sup>=e � �β<sup>3</sup>

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

∂t þ v0 ∂ ∂z þ ν � �E0ð Þ¼ <sup>z</sup>; <sup>t</sup> <sup>δ</sup><sup>2</sup>

have not been used in beam-plasma interaction experiments.

equation for SVA. Making use the condition of OB 2β<sup>2</sup>

∂ ∂t þ u ∂ ∂z � � ∂ <sup>δ</sup>ovl <sup>¼</sup> <sup>ω</sup>b<sup>β</sup>

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

We emphasize new dependence on ν, that is, actually we have new type of DSI. More critical

Higher values of parameter α (that is, α >> 1) correspond to very high currents. For example, in

the beam current is more than the limiting Pierce current. Until now such high currents beams

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 ω<sup>0</sup> and k<sup>0</sup> (see (29) and derive an

<sup>γ</sup>1=<sup>2</sup> <sup>1</sup> <sup>þ</sup> <sup>μ</sup> � �1=<sup>2</sup>

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

ω2 b

ω0 ν � �<sup>1</sup>=<sup>2</sup>

γ�<sup>2</sup> << α << 1, (32)

The Behavior of Streaming Instabilities in Dissipative Plasma

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<sup>ν</sup> (34)

γ<sup>2</sup>δ=k0u ≥ 1 [13], one can obtain [6, 12].

ovlE0ð Þ z; t , (35)

γ<sup>3</sup> and means that

(30)

31

(31)

(33)

2ω<sup>2</sup> <sup>0</sup> <sup>1</sup> <sup>þ</sup> <sup>μ</sup> � � !1=<sup>3</sup>

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

ω and k are the frequency and the longitudinal (along z axis) wave vector, k<sup>⊥</sup> ¼ μ0s=R. R is the waveguide's radius, μ0s are the roots of Bessel function J0:J<sup>0</sup> μ0<sup>s</sup> � � <sup>¼</sup> 0, s = 1,2,3…, <sup>ω</sup>p, <sup>b</sup> are the respective Langmuir frequencies for the beam and the plasma, u is the velocity of the beam, <sup>γ</sup> <sup>¼</sup> <sup>1</sup> � <sup>u</sup><sup>2</sup>=c<sup>2</sup> � ��1=<sup>2</sup> , c is speed of light. The DR (27) determines the growth rates of the beamplasma 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 ω ¼ ku þ δ, δ << ku. The DR (27) reduces to [1, 6].

$$\mathbf{x}^3 + i \frac{\nu}{\omega\_0} \frac{\omega\_p^2 \mathbf{v}\_0}{\nu \gamma^2 \omega\_\perp^2} \mathbf{x}^2 + \frac{\alpha \mathbf{v}\_0 \mu}{\gamma^2 c^2} \mathbf{x} = \frac{\alpha}{2 \gamma^4} \frac{\mathbf{v}\_0}{\mu} \tag{28}$$

where <sup>x</sup> <sup>¼</sup> <sup>δ</sup>=ku, <sup>α</sup> <sup>¼</sup> <sup>ω</sup><sup>2</sup> <sup>b</sup>=k 2 <sup>⊥</sup>u<sup>2</sup>γ3, <sup>β</sup> <sup>¼</sup> <sup>u</sup>=c, <sup>ω</sup><sup>2</sup> <sup>⊥</sup> ¼ k 2 <sup>⊥</sup>u<sup>2</sup>γ2, and v0 <sup>¼</sup> <sup>u</sup>μ<sup>=</sup> <sup>1</sup> <sup>þ</sup> <sup>μ</sup> � � is the group velocity of the resonant wave in "cold" system, <sup>μ</sup> <sup>¼</sup> <sup>γ</sup><sup>2</sup>ω<sup>2</sup> ⊥=ω<sup>2</sup> <sup>0</sup>; <sup>ω</sup><sup>0</sup> <sup>¼</sup> <sup>ω</sup><sup>2</sup> <sup>p</sup> � <sup>ω</sup><sup>2</sup> ⊥ � �<sup>1</sup>=<sup>2</sup> is the resonant frequency of the plasma waveguide that is, ω<sup>0</sup> satisfies following conditions

$$D\_0(\omega, k) = 0 \qquad ; \qquad \omega = k\omega \tag{29}$$

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�<sup>2</sup> ) to the ratio of the beam current to the limiting current in vacuum waveguide [14] <sup>I</sup><sup>0</sup> <sup>¼</sup> mu<sup>3</sup>γ=4e, that is, <sup>α</sup> <sup>¼</sup> ð Þ Ib=I<sup>0</sup> <sup>γ</sup>�<sup>2</sup> (Ib is the beam current). The values α << γ�2, correspond to underlimiting beam current I << I<sup>0</sup> and the instability in this case is caused by induced radiation of system proper waves by the beam electrons. Neglecting the second and third terms one can obtain the well-known growth rate of resonant beam instability in plasma waveguide

The Behavior of Streaming Instabilities in Dissipative Plasma http://dx.doi.org/10.5772/intechopen.79247 31

$$\delta\_{\rm und} = \frac{\sqrt{3}}{2} \frac{\omega\_0}{\gamma} \left( \frac{\omega\_b^2}{2\omega\_0^2 (1+\mu)} \right)^{\circ\_3} \tag{30}$$

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

Cherenkov resonance, (2) it is of nonradiative type, and (3) with increase in dissipation, it turns

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

> <sup>1</sup> � <sup>ω</sup><sup>2</sup> p ω ωð Þ <sup>þ</sup> <sup>i</sup><sup>ν</sup> � <sup>ω</sup><sup>2</sup>

ω and k are the frequency and the longitudinal (along z axis) wave vector, k<sup>⊥</sup> ¼ μ0s=R. R is the

respective Langmuir frequencies for the beam and the plasma, u is the velocity 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 ω ¼ ku þ δ,

> αv0u <sup>γ</sup><sup>2</sup>c<sup>2</sup> <sup>x</sup> <sup>¼</sup> <sup>α</sup>

<sup>⊥</sup> ¼ k 2

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 inter-

current in vacuum waveguide [14] <sup>I</sup><sup>0</sup> <sup>¼</sup> mu<sup>3</sup>γ=4e, that is, <sup>α</sup> <sup>¼</sup> ð Þ Ib=I<sup>0</sup> <sup>γ</sup>�<sup>2</sup> (Ib is the beam current). The values α << γ�2, correspond to underlimiting beam current I << I<sup>0</sup> and the instability in this case is caused by induced radiation of system proper waves by the beam electrons. Neglecting the second and third terms one can obtain the well-known growth rate of resonant

b <sup>γ</sup><sup>3</sup>ð Þ <sup>ω</sup> � ku <sup>2</sup>

¼ 0 (27)

� � <sup>¼</sup> 0, s = 1,2,3…, <sup>ω</sup>p, <sup>b</sup> are the

<sup>u</sup> (28)

is the reso-

<sup>⊥</sup>u<sup>2</sup>γ2, and v0 <sup>¼</sup> <sup>u</sup>μ<sup>=</sup> <sup>1</sup> <sup>þ</sup> <sup>μ</sup> � � is the group

) to the ratio of the beam current to the limiting

<sup>p</sup> � <sup>ω</sup><sup>2</sup> ⊥ � �<sup>1</sup>=<sup>2</sup>

<sup>0</sup>; <sup>ω</sup><sup>0</sup> <sup>¼</sup> <sup>ω</sup><sup>2</sup>

D0ð Þ¼ ω; k 0 ; ω ¼ ku (29)

!

, c is speed of light. The DR (27) determines the growth rates of the beam-

2γ<sup>4</sup> v0

⊥=ω<sup>2</sup>

to a new type of DSI [6, 14].

described by the following DR

<sup>γ</sup> <sup>¼</sup> <sup>1</sup> � <sup>u</sup><sup>2</sup>=c<sup>2</sup> � ��1=<sup>2</sup>

where <sup>x</sup> <sup>¼</sup> <sup>δ</sup>=ku, <sup>α</sup> <sup>¼</sup> <sup>ω</sup><sup>2</sup>

k 2 <sup>⊥</sup> þ k

δ << ku. The DR (27) reduces to [1, 6].

<sup>2</sup> � <sup>ω</sup><sup>2</sup> c2 � �

waveguide's radius, μ0s are the roots of Bessel function J0:J<sup>0</sup> μ0<sup>s</sup>

<sup>x</sup><sup>3</sup> <sup>þ</sup> <sup>i</sup> <sup>ν</sup> ω0

<sup>⊥</sup>u<sup>2</sup>γ3, <sup>β</sup> <sup>¼</sup> <sup>u</sup>=c, <sup>ω</sup><sup>2</sup>

<sup>b</sup>=k 2

action. It corresponds (correct to the factor γ�<sup>2</sup>

beam instability in plasma waveguide

velocity of the resonant wave in "cold" system, <sup>μ</sup> <sup>¼</sup> <sup>γ</sup><sup>2</sup>ω<sup>2</sup>

ω2 <sup>p</sup>v0 uγ<sup>2</sup>ω<sup>2</sup> ⊥ <sup>x</sup><sup>2</sup> <sup>þ</sup>

nant frequency of the plasma waveguide that is, ω<sup>0</sup> satisfies following conditions

3.1. Statement of the problem: analysis of the DR

30 Plasma Science and Technology - Basic Fundamentals and Modern Applications

$$
\delta\_{\rm und}^{(\nu)} = \frac{\alpha\_b \omega\_0}{2\gamma^{3/2} \omega\_p} \left(\frac{\omega\_0}{\nu}\right)^{1/2} \tag{31}
$$

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

$$
\gamma^{-2} \ll a \ll 1,\tag{32}
$$

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].

$$\delta\_{\rm ovl} = \frac{\omega\_b \beta}{\nu^{\nu\_2'} \left(1 + \mu\right)^{\psi\_2}} \tag{33}$$

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].

$$
\delta\_{\rm ovl}^{(\nu)} = \frac{\beta^2}{\mathcal{V}} \frac{\omega\_b^2}{\omega\_p^2} \frac{\omega\_0^2}{\nu} \tag{34}
$$

We emphasize new dependence on ν, that is, actually we have new type of DSI. More critical dependence on ν is due to superposition of two factors those lead to NEW excitation.

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 Ib ≥ 1, 4 mc<sup>3</sup>=e � �β<sup>3</sup> γ<sup>3</sup> and means that the beam current is more than the limiting Pierce current. Until now such high currents beams have not been used in beam-plasma interaction experiments.

#### 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 ω<sup>0</sup> and k<sup>0</sup> (see (29) and derive an equation for SVA. Making use the condition of OB 2β<sup>2</sup> γ<sup>2</sup>δ=k0u ≥ 1 [13], one can obtain [6, 12].

$$
\left(\frac{\partial}{\partial t} + \nu \frac{\partial}{\partial z}\right)\left(\frac{\partial}{\partial t} + \mathbf{v}\_0 \frac{\partial}{\partial z} + \nu\right)E\_0(z,t) = \delta\_{\text{ovl}}^2 E\_0(z,t),\tag{35}
$$

(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].

$$E\_0(\mathbf{z}, t) = -\frac{J\_0}{2\sqrt{\pi}} \frac{\exp \chi^{(\text{ovl})}(\mathbf{z}, t)}{(\mu - \mathbf{v}\_0)^\dagger \delta\_{\text{ovl}}^\natural (\mu t - \mathbf{z})^\natural} \tag{36}$$

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

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

In limit of high-level dissipation, we have

eter k ¼ ν=δovl k<sup>1</sup> ¼ 0, k<sup>2</sup> ¼ 1, k<sup>3</sup> ¼ 2, k<sup>4</sup> ¼ 4.

zm ¼ wgot 1 þ

8 < :

Substitution of zm into χð Þ ovl gives the maximal growth rate under arbitrary ν=δovl

level of dissipation are plotted in Figure 2. Figure 3 presents the curve f xð Þ.

<sup>E</sup>0ð Þ� <sup>z</sup> <sup>¼</sup> zm; <sup>t</sup> exp ð Þ <sup>δ</sup>ovl<sup>t</sup> � <sup>f</sup>ð Þ <sup>λ</sup> ; fxð Þ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffi

E � exp δ

transforms to the new type of DSI. The shapes of the waveform for OB instability for various

Figure 2. Shapes of the waveform versus longitudinal coordinate at fixed instant t ¼ 3=δovl for various values of param-

ð Þν

ovl is given by Eq. (34). That is, with increase in level of dissipation the instability of OB

<sup>z</sup> � wgot � �<sup>2</sup> � <sup>λ</sup>ð Þ ut � <sup>z</sup> ð Þ¼ <sup>z</sup> � v0<sup>t</sup> <sup>0</sup> (42)

9 = ;

1 þ x<sup>2</sup>

The Behavior of Streaming Instabilities in Dissipative Plasma

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ovlt (45)

<sup>p</sup> � <sup>x</sup> (44)

(43)

33

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

w<sup>2</sup> go

<sup>1</sup> <sup>þ</sup> <sup>λ</sup> <sup>1</sup> � <sup>u</sup>v0

vu ! ut

λ

equation

where δ

ð Þν

$$\chi^{(\text{ovl})}(z,t) = \chi\_0^{(\text{ovl})}(z,t) - \nu \frac{\text{ut} - z}{\text{u} - \text{v}\_0} \qquad ; \qquad \chi\_0^{(\text{ovl})}(z,t) = \frac{2\delta\_{\text{ovl}}}{\text{u} - \text{v}\_0} \{(z - \text{v}\_0 t)(\text{ut} - z)\}^{\circ\_2}$$

The analysis of the expression (36) is similar to previous case. It again reduces to the analysis of the exponent <sup>χ</sup>ð Þ ovl ð Þ <sup>z</sup>; <sup>t</sup> . The analysis shows that unstable perturbations vary through the same range from v0 to u. The analysis of the instability character (absolute/convective) fully coincides to that for underlimiting e-beams. However, in this case, the waveform is symmetric with respect to its peak. The peak places in the middle at all instants and moves at average velocity

$$
\omega\_{\mathcal{S}^o} = \mathbb{Y}\_2(\mathfrak{u} + \mathfrak{v}\_0) \tag{37}
$$

The field's value in the peak exponentially increases and the growth rate is equal to maximal growth rate for OB δovl (33) (or, the same, to solution of the initial problem).

At fixed point z the SVA attains its maximum � exp δovlz=ð Þ uv0 <sup>1</sup>=<sup>2</sup> at the instant <sup>t</sup> <sup>¼</sup> <sup>z</sup>=wa, where

$$
\omega\_{\text{av}} = \frac{2\mu \mathbf{v}\_0}{\mu + \mathbf{v}\_0} \tag{38}
$$

The expression δovl=ð Þ uv0 <sup>1</sup>=<sup>2</sup> is the maximal spatial growth rate at wave amplification by OB, and coincides to result of the boundary problem. The SVA depends on the perturbations' velocity v as

$$E\_0(z = vt, t) \sim \exp\left\{\Gamma\_0(v)t\right\} \qquad ; \qquad \Gamma\_0(v) = 2\delta\_{\text{vol}} \frac{\sqrt{(\mathbf{u} - \mathbf{v})(\mathbf{v} - \mathbf{v}\_0)}}{\mathbf{u} - \mathbf{v}\_0} \tag{39}$$

The character of the space growth depending on perturbations' velocity is � exp Γ0ð Þv z=ð Þ uv0 1=2 .

Dissipation fundamentally changes this picture of the instability. For given velocity v the dependence of the SVA on the dissipation level becomes

$$
\Gamma\_0(\upsilon) \to \Gamma\_\upsilon(\upsilon) = \Gamma\_0(\upsilon) - \nu \xrightarrow[\mu \to -\infty]{\mu \to \upsilon}] \tag{40}
$$

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

$$\mathbf{V}\_{th}^{(\text{ovl})} = \frac{\lambda \mu + \mathbf{v}\_0}{1 + \lambda} \qquad ; \qquad \lambda = \nu^2 / 4\delta\_{\text{ovl}}^2 \tag{41}$$

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

$$\left(\left(z - w\_{\mathcal{S}^0}t\right)^2 - \lambda(\mu t - z)(z - \mathbf{v}\_0t) = \mathbf{0}\right) \tag{42}$$

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

$$z\_m = w\_{\mathcal{g}o}t \left\{ 1 + \sqrt{\frac{\lambda}{1+\lambda} \left( 1 - \frac{\mu \mathbf{v}\_0}{w\_{\mathcal{g}o}^2} \right)} \right\} \tag{43}$$

Substitution of zm into χð Þ ovl gives the maximal growth rate under arbitrary ν=δovl

$$E\_0(z = z\_m, t) \sim \exp\left(\delta\_{\text{vol}} t \cdot f(\lambda)\right) \qquad ; \qquad f(\mathbf{x}) = \sqrt{1 + \mathbf{x}^2} - \mathbf{x} \tag{44}$$

In limit of high-level dissipation, we have

(the denotations coincide to those in (8)). The Eq. (35) for SVA may be solved by analogy to

ðu � v0Þ 1=2 δ 1=2 ovlðut � zÞ

; χ<sup>ð</sup>ovl<sup>Þ</sup>

The analysis of the expression (36) is similar to previous case. It again reduces to the analysis of the exponent <sup>χ</sup>ð Þ ovl ð Þ <sup>z</sup>; <sup>t</sup> . The analysis shows that unstable perturbations vary through the same range from v0 to u. The analysis of the instability character (absolute/convective) fully coincides to that for underlimiting e-beams. However, in this case, the waveform is symmetric with respect to its peak. The peak places in the middle at all instants and moves at average velocity

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

wao <sup>¼</sup> <sup>2</sup>uv0 u þ v0

and coincides to result of the boundary problem. The SVA depends on the perturbations'

The character of the space growth depending on perturbations' velocity is � exp Γ0ð Þv z=ð Þ uv0

Γ0ð Þ!v Γνð Þ¼ v Γ0ð Þ� v ν

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

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

<sup>1</sup> <sup>þ</sup> <sup>λ</sup> ; <sup>λ</sup> <sup>¼</sup> <sup>ν</sup><sup>2</sup>

growth rate for OB δovl (33) (or, the same, to solution of the initial problem).

E0ð Þ� z ¼ vt; t exp f g Γ0ð Þv t ; Γ0ð Þ¼ v 2δovl

At fixed point z the SVA attains its maximum � exp δovlz=ð Þ uv0

dependence of the SVA on the dissipation level becomes

Vð Þ ovl

th <sup>¼</sup> <sup>λ</sup><sup>u</sup> <sup>þ</sup> v0

exp χðovl<sup>Þ</sup>

ðz, tÞ

<sup>0</sup> <sup>ð</sup>z, tÞ ¼ <sup>2</sup>δovl

1=2

u � v0

wgo ¼ <sup>1</sup>=2ð Þ u þ v0 (37)

<sup>1</sup>=<sup>2</sup> is the maximal spatial growth rate at wave amplification by OB,

u � v u � v0

=4δ<sup>2</sup>

p

fð<sup>z</sup> � v0tÞðut � <sup>z</sup>Þg<sup>1</sup>=<sup>2</sup>

<sup>1</sup>=<sup>2</sup> at the instant <sup>t</sup> <sup>¼</sup> <sup>z</sup>=wa, where

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð Þ u � v ð Þ v � v0

ovl (41)

u � v0

(36)

(38)

(39)

1=2 .

(40)

solution of Eq. (8). Without delving into details, we present here the results [6, 12].

2 ffiffiffi <sup>π</sup> <sup>p</sup>

<sup>E</sup>0ðz, t޼� <sup>J</sup><sup>0</sup>

ut � z u � v0

χðovl<sup>Þ</sup>

<sup>ð</sup>z, tÞ ¼ <sup>χ</sup><sup>ð</sup>ovl<sup>Þ</sup>

The expression δovl=ð Þ uv0

threshold velocity is

velocity v as

<sup>0</sup> ðz, tÞ � ν

32 Plasma Science and Technology - Basic Fundamentals and Modern Applications

$$E \sim \exp \delta\_{\text{ovl}}^{\langle \nu \rangle} t \tag{45}$$

where δ ð Þν ovl is given by Eq. (34). That is, with increase in level of dissipation the instability of OB transforms to the new type of DSI. The shapes of the waveform for OB instability for various level of dissipation are plotted in Figure 2. Figure 3 presents the curve f xð Þ.

Figure 2. Shapes of the waveform versus longitudinal coordinate at fixed instant t ¼ 3=δovl for various values of parameter k ¼ ν=δovl k<sup>1</sup> ¼ 0, k<sup>2</sup> ¼ 1, k<sup>3</sup> ¼ 2, k<sup>4</sup> ¼ 4.

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

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.

bz þ j

Ez ;

pzð Þ¼ r⊥; z; t ppð Þ r<sup>⊥</sup> j

dinal 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 infinitesi-

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

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

h i <sup>ψ</sup> <sup>¼</sup> <sup>0</sup> ; <sup>ψ</sup><sup>j</sup>

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

p ω ωð Þ þ iν

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

plasma, γ is the relativistic factors of the beam electrons, u is the beam velocity.

pz � � ; Ez <sup>¼</sup> <sup>∂</sup><sup>2</sup>

∂ ∂t þ ν � �<sup>j</sup>

� � (δ is Dirac function). Δ<sup>⊥</sup> is the Laplace operator over

; δε<sup>b</sup> <sup>¼</sup> <sup>ω</sup><sup>2</sup>

b

p, <sup>b</sup>.

<sup>2</sup> (46)

35

<sup>Σ</sup> ¼ 0 (47)

<sup>γ</sup><sup>3</sup>ð Þ <sup>ω</sup> � ku : (48)

ψ <sup>∂</sup>z<sup>2</sup> � <sup>1</sup> c2 ∂2 ψ ∂t

The Behavior of Streaming Instabilities in Dissipative Plasma

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

<sup>p</sup>ð Þ z; t are perturbations of the longitu-

<sup>p</sup> <sup>¼</sup> <sup>ω</sup><sup>2</sup> p 4π Ez:

We proceed from equations for ψ and for the beam and the plasma currents j

∂ ∂t

prevents transversal motion of beam and plasma particles.

j <sup>b</sup> <sup>¼</sup> <sup>ω</sup><sup>2</sup> bγ�<sup>3</sup> 4π

<sup>b</sup>ð Þ z; t and j

may be easily adapted to systems of any cross-section.

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

<sup>Δ</sup>⊥<sup>ψ</sup> � <sup>κ</sup><sup>2</sup> <sup>1</sup> � ppð Þ <sup>r</sup><sup>⊥</sup> δε<sup>p</sup> � pbð Þ <sup>r</sup><sup>⊥</sup> δε<sup>b</sup>

<sup>c</sup><sup>2</sup> ; δε<sup>p</sup> <sup>¼</sup> <sup>ω</sup><sup>2</sup>

∂ ∂t

bzð Þ¼ r⊥; z; t pbð Þ r<sup>⊥</sup> j

Here j

specified yet).

<sup>κ</sup><sup>2</sup> <sup>¼</sup> <sup>k</sup>

<sup>2</sup> � <sup>ω</sup><sup>2</sup>

Δ<sup>⊥</sup> þ

mal thin beam/plasma pb,p � δ r � rb, <sup>p</sup>

∂ ∂t þ u ∂ ∂t � �<sup>2</sup>

∂2 <sup>∂</sup>z<sup>2</sup> � <sup>1</sup> c2 ∂2 ∂t 2 � �<sup>ψ</sup> ¼ �4<sup>π</sup> <sup>j</sup>

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