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

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

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

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

where the carrier frequency ω<sup>0</sup> and k<sup>0</sup> satisfy the conditions (4). We also assume that the

In such formulation, the problem of the instability evolution reduces to determination of the slowly varying amplitude (SVA) E0ð Þ z; t . As the fields vary near ω<sup>0</sup> and k0, one can use

Expanding the DR (1) in power series near ω<sup>0</sup> and k0, one can obtain the equation for SVA

<sup>∂</sup>D0ð Þ <sup>ω</sup>;<sup>k</sup> =<sup>∂</sup><sup>ω</sup> � �

Im δ<sup>0</sup> is the maximal growth rate of the beam instability, ν describes dissipation in the system (its coincidence to collision frequency is not obligatory), and v0 is the group velocity of

The Eq. (8) describes the evolution of SVA E<sup>0</sup> (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

ðω � kv0 þ iνÞ � j j δ<sup>0</sup>

ω ¼ ω<sup>0</sup> k ¼ k<sup>0</sup>

<sup>3</sup> n oE0ð Þ¼ <sup>ω</sup>; <sup>k</sup> <sup>J</sup>ð Þ <sup>ω</sup>; <sup>k</sup> (9)

<sup>∂</sup><sup>t</sup> ; k ! <sup>k</sup><sup>0</sup> � <sup>i</sup>

∂E<sup>0</sup> ∂z � � � �

� � � �

D0ð Þ¼ ω; k 0 ; ω � ku � f ¼ 0: (4)

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25

E zð Þ¼ ; t E0ð Þ z; t exp f g �iω0t þ ik0z , (5)

∂ ∂z

3

; v0 ¼ � <sup>∂</sup>D0ð Þ <sup>ω</sup>;<sup>k</sup> <sup>=</sup><sup>∂</sup><sup>k</sup>

<sup>∂</sup>D0ð Þ <sup>ω</sup>;<sup>k</sup> =<sup>∂</sup><sup>ω</sup> � �

<< k0E<sup>0</sup> (6)

, (7)

E0ð Þ z; t (8)

ω ¼ ω<sup>0</sup> k ¼ k<sup>0</sup>

means that following two conditions must be satisfied:

amplitude E0ð Þ z; t is slowly varying as compared to ω<sup>0</sup> and k<sup>0</sup> that is,

<< ω0E<sup>0</sup> ;

∂

∂E<sup>0</sup> ∂t � � � �

following formal substitutions to derive an equation for SVA

∂ ∂t þ u ∂ ∂z � �<sup>2</sup> ∂

> ω ¼ ω<sup>0</sup> k ¼ k<sup>0</sup>

corresponding equation for the transform E0ð Þ ω; k is

ð Þ <sup>ω</sup> � ku <sup>2</sup>

ω ! ω<sup>0</sup> þ i

∂t þ v0 ∂ ∂z þ ν � �E0ð Þ¼ <sup>z</sup>; <sup>t</sup> <sup>i</sup>j j <sup>δ</sup><sup>0</sup>

; <sup>ν</sup> <sup>¼</sup> Im <sup>D</sup><sup>0</sup>

� � � �

following type

where

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

<sup>b</sup>Að Þ ω; k <sup>∂</sup>D0=<sup>∂</sup><sup>ω</sup> � �

resonant wave in the "cold" system.

where ω is the frequency of perturbations and k is the wave vector. D0ð Þ¼ ω; k 0 is the "cold" DR describing proper frequencies of the systems in the absence of the beam (its main part), and Dbð Þ ω; k is the beam contribution. We also assume that the beam density is small enough to satisfy the condition j j Dbð Þ ω; k << j j D0ð Þ ω; k . In following consideration, we will not specify the form of D0ð Þ ω; k . Beam electrons interact with proper oscillations of background system and this interaction leads to instability. The interaction may be of various types: Cherenkov, cyclotron, interaction with periodical structure, etc. The general form of Dbð Þ ω; k may be written as

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

where u is the velocity of the beam electrons, ω<sup>b</sup> is the Langmuir frequency of streaming electrons, γ is the relativistic factor of the beam electrons, and Að Þ ω; k is a polynomial with respect to ω and k of degree no higher than two. The expression for f depends on the type of the beam interaction with plasma:

$$f = \begin{cases} 0, & \text{if the interaction is of Cherenkov type} \\ n\Omega/\gamma, & \text{if the interaction is of cyclotron type} \\\\ k\_{\text{cor}}\mu\_{\prime} & \text{if the beam interacts with periodic structure} \end{cases} \tag{3}$$

where Ω is the cyclotron frequency, n ¼ 1, 2, 3…, kcor ¼ 2π=l<sup>0</sup> l<sup>0</sup> is the spatial period of the structure. Below, we will show that properties of the instabilities follow from the general form (2) and do not depend on the expression for f.

Let an initial perturbation arises in point z ¼ 0 (electron stream propagates in the direction z > 0) at instant t ¼ 0 and the instability begins developing. Our aim is to obtain shape of the perturbation (i.e., space–time structure of the fields) at arbitrary instant t and based on the expression, investigate the behavior of the instability. In following consideration, we interest in longitudinal structure of the field (their dependence on z and t only). We single out two arguments: the frequency ω and longitudinal wavelength 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:

$$D\_0(\omega, k) = 0 \qquad ; \qquad \omega - k\omega - f = 0. \tag{4}$$

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

$$E(z,t) = E\_0(z,t) \exp\left\{-i\omega\_0 t + i k\_0 z\right\},\tag{5}$$

where the carrier frequency ω<sup>0</sup> and k<sup>0</sup> satisfy the conditions (4). We also assume that the amplitude E0ð Þ z; t is slowly varying as compared to ω<sup>0</sup> and k<sup>0</sup> that is,

$$\left|\frac{\partial E\_0}{\partial t}\right| \ll \omega\_0 E\_0 \qquad ; \qquad \left|\frac{\partial E\_0}{\partial z}\right| \ll k\_0 E\_0 \tag{6}$$

In such formulation, the problem of the instability evolution reduces to determination of the slowly varying amplitude (SVA) E0ð Þ z; t . As the fields vary near ω<sup>0</sup> and k0, one can use following formal substitutions to derive an equation for SVA

$$
\omega \to \omega\_0 + i \frac{\partial}{\partial t} \qquad ; \qquad k \to k\_0 - i \frac{\partial}{\partial z'} \tag{7}
$$

Expanding the DR (1) in power series near ω<sup>0</sup> and k0, one can obtain the equation for SVA

$$\left(\frac{\partial}{\partial t} + \mu \frac{\partial}{\partial z}\right)^2 \left(\frac{\partial}{\partial t} + \mathbf{v}\_0 \frac{\partial}{\partial z} + \nu\right) E\_0(z, t) = i|\delta\_0|^3 E\_0(z, t) \tag{8}$$

where

2. The behavior beam-plasma instabilities in dissipative plasma

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 disper-

where ω is the frequency of perturbations and k is the wave vector. D0ð Þ¼ ω; k 0 is the "cold" DR describing proper frequencies of the systems in the absence of the beam (its main part), and Dbð Þ ω; k is the beam contribution. We also assume that the beam density is small enough to satisfy the condition j j Dbð Þ ω; k << j j D0ð Þ ω; k . In following consideration, we will not specify the form of D0ð Þ ω; k . Beam electrons interact with proper oscillations of background system and this interaction leads to instability. The interaction may be of various types: Cherenkov, cyclotron, interaction with periodical structure, etc. The general form of Dbð Þ ω; k may be

Dbð Þ¼� <sup>ω</sup>; <sup>k</sup> <sup>ω</sup><sup>2</sup>

where u is the velocity of the beam electrons, ω<sup>b</sup> is the Langmuir frequency of streaming electrons, γ is the relativistic factor of the beam electrons, and Að Þ ω; k is a polynomial with respect to ω and k of degree no higher than two. The expression for f depends on the type of

0, if the interaction is of Cherenkov type

nΩ=γ, if the interaction is of cyclotron type

kcoru, if the beam interacts with periodical structure

where Ω is the cyclotron frequency, n ¼ 1, 2, 3…, kcor ¼ 2π=l<sup>0</sup> l<sup>0</sup> is the spatial period of the structure. Below, we will show that properties of the instabilities follow from the general form

Let an initial perturbation arises in point z ¼ 0 (electron stream propagates in the direction z > 0) at instant t ¼ 0 and the instability begins developing. Our aim is to obtain shape of the perturbation (i.e., space–time structure of the fields) at arbitrary instant t and based on the expression, investigate the behavior of the instability. In following consideration, we interest in longitudinal structure of the field (their dependence on z and t only). We single out two arguments: the frequency ω and longitudinal wavelength 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

<sup>b</sup>Að Þ ω; k

D0ð Þþ ω; k Dbð Þ¼ ω; k 0 (1)

<sup>γ</sup><sup>3</sup>ð Þ <sup>ω</sup> � ku � <sup>f</sup> <sup>2</sup> (2)

,

(3)

2.1. Equation for slowly varying amplitude

24 Plasma Science and Technology - Basic Fundamentals and Modern Applications

sion relation (DR) of such system is

the beam interaction with plasma:

f ¼

of given system.

8 >>><

>>>:

(2) and do not depend on the expression for f.

written as

$$\delta\_0 = \left\{ \frac{\omega\_b^2 A(\omega, k)}{\mathrm{d}D\_0 \langle\_{\partial \omega}} \right\}\_{k=k\_0}; \quad \nu = \left\{ \frac{\mathrm{Im} \, D\_0}{\mathrm{d} \mathbf{D}\_0(\omega k) \langle\_{\partial \omega}} \right\}\_{k=k\_0}; \quad \mathbf{v}\_0 = -\left\{ \frac{\mathrm{d} \mathbf{D}\_0(\omega, k) \langle\_{\partial \omega}}{\mathrm{d} \mathbf{D}\_0(\omega, k) \langle\_{\partial \omega}} \right\}\_{k=k\_0}$$

Im δ<sup>0</sup> is the maximal growth rate of the beam instability, ν describes dissipation in the system (its coincidence to collision frequency is not obligatory), and v0 is the group velocity of resonant wave in the "cold" system.

The Eq. (8) describes the evolution of SVA E<sup>0</sup> (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 E0ð Þ ω; k is

$$\left\{ (\omega - k\mathfrak{u})^2 (\omega - k\mathfrak{v}\_0 + i\mathfrak{v}) - \left| \mathfrak{d}\_0 \right|^3 \right\} \mathcal{E}\_0(\omega, k) = \mathcal{J}(\omega, k) \tag{9}$$

$$E\_0(\omega, k) = \bigcap\_{0}^{\infty} \text{d}z \bigcap\_{-\infty}^{\infty} \text{E}\_0(z, t) \exp\left(i\omega t - i\text{k}z\right) :\tag{10}$$

2.2. Analysis of the fields' dynamics

instability behavior) may be determined by analyzing the factor

moves at velocity u, but the back edge moves at velocity v0 < u).

Its solution in the absence of dissipation gives z ¼ wgt, where

Figure 1. Asymptotic shapes of beam instability <sup>ε</sup> <sup>¼</sup> exp <sup>χ</sup>ð Þ und

various instants τ<sup>1</sup> ¼ δ0t<sup>1</sup> < τ<sup>2</sup> ¼ δ0t<sup>2</sup> < τ<sup>3</sup> ¼ δ0t3.

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

∂ ∂z χð Þ und

That is, the peak places on 1/3 of the train's length from the front and moves at the velocity wg. Actually, wg represents group velocity of the generated wave, with account of the beam contribution in the DR. The field's value in the peak exponentially increases and the growth

We have arrived to very complex expressions (14). However, the field's structure (i.e., the

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 v0 to u. The length of the wave train increases depending on time l � ð Þ u � v0 t. One can easily see convective character of streaming instabilities in laboratory frame, as well as in other frames moving at velocities v < v0 and v > u. If the observer's velocity is within the range v0 < v < u, the instability is absolute (see Figure 1, where the dependence of the SVA on z at various instants t1, t<sup>2</sup> and t<sup>3</sup> is presented; the leading edge

<sup>ν</sup> ð Þ z; t (15)

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<sup>ν</sup> ¼ 0 (16)

<sup>0</sup> ð Þ z; t depending on longitudinal coordinate ζ ¼ zδ0=u at

wg ¼ ð Þ 1=3 ð Þ 2u þ v0 (17)

exp χð Þ und

where the function Jð Þ ω; k is determined by initial conditions. Its power with respect to ω and k is no higher than the power of the origin equation. The specific form of this function is not essential for following. It is only necessary that Jð Þ ω; k be smooth and not equal to zero identically. The amplitude of the wave train can be obtained by inverse transformation

$$E\_0(z,t) = \frac{1}{\left(2\pi\right)^2} \int\_{\mathcal{C}(\omega)} d\omega \int\_{-\infty}^{\infty} \frac{d\mathbf{k} \{\omega, \mathbf{k}\} \exp\left(-i\omega t + i\mathbf{k}z\right)}{\left(\omega - k\nu\right)^2 \left(\omega - k\mathbf{v}\_0 + i\nu\right) - \left|\delta\_{\mathrm{Bn}}\right|^3} \tag{11}$$

Here, Cð Þ ω is the contour of integration over ω. For given case, it is a straight line that lies in the upper half plane of the complex plane ω ¼ Reω þ iIm ω and passes above all singularities of the integrand. Thus, the problem has been reduced to the integration in Eq. (11). It is convenient to transform the variables ω and k to another pair ω and ω<sup>0</sup> ¼ ω � ku. The first integration (over ω) may be carried out by residue method and the integration contour must be closed in the lower half plane. The pole is

$$
\omega\_1 \left(\omega'\right) = \left(1 - \frac{\mathbf{v}\_0}{\mu}\right)^{-1} \left(\frac{|\delta\_0|^3}{\omega'^2} - \omega'\frac{\mathbf{v}\_0}{\mu} - \dot{\mathbf{n}}\prime\right) \tag{12}
$$

The second integration (over ω<sup>0</sup> ) cannot be carried out exactly, and we are forced to restrict ourselves by approximate, steepest descend method. That is, Eq. (11) will be worked out in asymptotic limit of comparatively large t. In this case, the integration contour should be deformed in order to pass through the saddle point in needed direction. The saddle point is

$$
\omega\_s' = \delta\_0 \left\{ \frac{2(\mathbf{u}t - \mathbf{z})}{(z - \mathbf{v}\_0 t)} \right\}^{1/3} \exp\left(2\pi i/3\right) \tag{13}
$$

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

$$E\_0(z,t) = \frac{J\_0}{2\sqrt{\pi}} \frac{\exp\left\{\chi\_\nu^{(\text{und})}(z,t)\right\}}{\sqrt{(u-v\_0)f(z,t)}} \exp\left\{i\varphi(z,t)\right\} \tag{14}$$

$$\chi\_{\nu}^{(\text{und})}(z,t) = \chi\_0^{(\text{und})}(z,t) - \nu \frac{z - \mathbf{v}\_0 t}{u - \mathbf{v}\_0} \qquad ; \qquad \chi\_0^{(\text{und})}(z,t) = \frac{3\sqrt{3}}{4} \frac{\delta\_0}{u - \mathbf{v}\_0} \left\{ 2(ut - z)(z - \mathbf{v}\_0 t)^2 \right\}^{1/3}$$

$$f(z,t) = 3\delta\_0^3 (ut - z) \qquad ; \qquad \varphi(z,t) = \frac{\chi(z,t)}{\sqrt{3}} + \frac{\pi}{4}$$

and <sup>J</sup><sup>0</sup> is the value of <sup>J</sup> <sup>ω</sup>; <sup>ω</sup> � �<sup>0</sup> at the points. ω ¼ ω<sup>1</sup> ω<sup>0</sup> s � �, ω<sup>0</sup> ¼ ω<sup>0</sup> s.

#### 2.2. Analysis of the fields' dynamics

E0ð Þ¼ ω; k

26 Plasma Science and Technology - Basic Fundamentals and Modern Applications

1 ð Þ 2π 2 ð

nient to transform the variables ω and k to another pair ω and ω<sup>0</sup>

<sup>ω</sup><sup>1</sup> <sup>ω</sup><sup>0</sup> � �

ω0 <sup>s</sup> ¼ δ<sup>0</sup>

<sup>E</sup>0ð Þ¼ <sup>z</sup>; <sup>t</sup> <sup>J</sup><sup>0</sup>

z � v0t u � v0

f zð Þ¼ ; <sup>t</sup> <sup>3</sup>δ<sup>3</sup>

Cð Þ ω

dω ð ∞

<sup>¼</sup> <sup>1</sup> � v0 u � ��<sup>1</sup> j j <sup>δ</sup><sup>0</sup>

�∞

E0ð Þ¼ z; t

the lower half plane. The pole is

The second integration (over ω<sup>0</sup>

χð Þ und

<sup>ν</sup> ð Þ¼ <sup>z</sup>; <sup>t</sup> <sup>χ</sup>ð Þ und

and <sup>J</sup><sup>0</sup> is the value of <sup>J</sup> <sup>ω</sup>; <sup>ω</sup> � �<sup>0</sup>

<sup>0</sup> ð Þ� z; t ν

ð ∞

0 dt ð ∞

�∞

where the function Jð Þ ω; k is determined by initial conditions. Its power with respect to ω and k is no higher than the power of the origin equation. The specific form of this function is not essential for following. It is only necessary that Jð Þ ω; k be smooth and not equal to zero identically. The amplitude of the wave train can be obtained by inverse transformation

ð Þ <sup>ω</sup> � ku <sup>2</sup>

Here, Cð Þ ω is the contour of integration over ω. For given case, it is a straight line that lies in the upper half plane of the complex plane ω ¼ Reω þ iIm ω and passes above all singularities of the integrand. Thus, the problem has been reduced to the integration in Eq. (11). It is conve-

(over ω) may be carried out by residue method and the integration contour must be closed in

ourselves by approximate, steepest descend method. That is, Eq. (11) will be worked out in asymptotic limit of comparatively large t. In this case, the integration contour should be deformed in order to pass through the saddle point in needed direction. The saddle point is

> 2ð Þ ut � z ð Þ z � v0t � �<sup>1</sup>=<sup>3</sup>

> > exp χð Þ und

; χð Þ und

at the points. ω ¼ ω<sup>1</sup> ω<sup>0</sup>

<sup>ν</sup> ð Þ z; t n o

<sup>0</sup> ð Þ¼ z; t

s � �, ω<sup>0</sup>

<sup>0</sup>ð Þ ut � <sup>z</sup> ; <sup>φ</sup>ð Þ¼ <sup>z</sup>; <sup>t</sup> <sup>χ</sup>ð Þ <sup>z</sup>; <sup>t</sup>

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

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

2 ffiffiffi <sup>π</sup> <sup>p</sup>

3 <sup>ω</sup>0<sup>2</sup> � <sup>ω</sup><sup>0</sup> v0

!

dz E0ð Þ z; t exp ð Þ iωt � ikz : (10)

dkJð Þ ω; k exp ð Þ �iωt þ ikz

ðω � kv0 þ iνÞ � j j δBn

<sup>u</sup> � <sup>i</sup><sup>ν</sup>

) cannot be carried out exactly, and we are forced to restrict

exp 2ð Þ πi=3 (13)

<sup>2</sup>ð Þ ut � <sup>z</sup> ð Þ <sup>z</sup> � v0<sup>t</sup> <sup>2</sup> n o<sup>1</sup>=<sup>3</sup>

ð Þ <sup>u</sup> � v0 f zð Þ ; <sup>t</sup> <sup>p</sup> exp f g <sup>i</sup>φð Þ <sup>z</sup>; <sup>t</sup> (14)

δ0 u � v0

ffiffiffi 3 p þ π 4

3 ffiffiffi 3 p 4

¼ ω<sup>0</sup> s. <sup>3</sup> (11)

(12)

¼ ω � ku. The first integration

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

$$\exp \chi\_{\nu}^{(\text{und})}(z, t) \tag{15}$$

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 v0 to u. The length of the wave train increases depending on time l � ð Þ u � v0 t. One can easily see convective character of streaming instabilities in laboratory frame, as well as in other frames moving at velocities v < v0 and v > u. If the observer's velocity is within the range v0 < v < u, the instability is absolute (see Figure 1, where the dependence of the SVA on z at various instants t1, t<sup>2</sup> and t<sup>3</sup> is presented; the leading edge moves at velocity u, but the back edge moves at velocity v0 < u).

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

$$\frac{\partial}{\partial \mathbf{z}} \chi\_{\nu}^{(\text{und})} = 0 \tag{16}$$

Its solution in the absence of dissipation gives z ¼ wgt, where

$$w\_{\mathcal{S}} = (1/3)(2\mu + \mathbf{v}\_0) \tag{17}$$

That is, the peak places on 1/3 of the train's length from the front and moves at the velocity wg. Actually, wg represents group velocity of the generated wave, with account of the beam contribution in the DR. The field's value in the peak exponentially increases and the growth

Figure 1. Asymptotic shapes of beam instability <sup>ε</sup> <sup>¼</sup> exp <sup>χ</sup>ð Þ und <sup>0</sup> ð Þ z; t depending on longitudinal coordinate ζ ¼ zδ0=u at various instants τ<sup>1</sup> ¼ δ0t<sup>1</sup> < τ<sup>2</sup> ¼ δ0t<sup>2</sup> < τ<sup>3</sup> ¼ δ0t3.

rate is equal to <sup>δ</sup><sup>m</sup> <sup>¼</sup> ffiffiffi 3 <sup>p</sup> <sup>=</sup><sup>2</sup> � �j j <sup>δ</sup><sup>0</sup> that is, coincides to solution of the initial problem. However, the initial problem can not specify the point, where the maximal growth occurs. The advantage of this approach is evident.

In a fixed point z, the field first increases and attains maximum at instant t ¼ z=wa where

$$
\omega\_{\mathfrak{u}} = \frac{\mathfrak{Z}\mathfrak{u}\mathfrak{v}\_{0}}{\mathfrak{u} + \mathfrak{2}\mathfrak{v}\_{0}} \tag{18}
$$

Only perturbations moving at higher velocities v > Vthr develop. The wave train shortens.

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

If v << δ0, 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.

λ�<sup>1</sup>

case corresponds to dissipative streaming instability (DSI). The same expression for Γ<sup>ν</sup>!<sup>∞</sup> can be obtained from Eq. (1) by direct usage of the initial problem [1]. If one specifies δm, he can obtain the growth rate of DSI in unbound beam-plasma system, in magnetized beam-plasma

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 struc-

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

tures, etc. Also, the analysis does not depend on specific geometry, external fields, etc.

u � v u � v0

<sup>2</sup> <sup>¼</sup> <sup>1</sup>=λ<sup>2</sup> <sup>z</sup> � wgt <sup>3</sup> (25)

The Behavior of Streaming Instabilities in Dissipative Plasma

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ð Þ u � v0 , (26)

<sup>m</sup>=<sup>ν</sup> <sup>1</sup>=<sup>2</sup>

. Obviously, this

(24)

29

Γð Þ!v Γνð Þ¼ v Γð Þ� v ν

ð Þ z � v0t ð Þ ut � z

<sup>Δ</sup><sup>u</sup> <sup>¼</sup> <sup>3</sup>�3=<sup>2</sup>

In this approximation, the solution of Eq. (25) is z ¼ u � Δu where

and the expression for maximal growth rate takes the form <sup>Γ</sup><sup>ν</sup>!<sup>∞</sup> <sup>¼</sup> <sup>δ</sup><sup>3</sup>

3. The behavior of overlimiting electron beam instability

Dissipation decreases the field growth

following form

waveguide, etc.

Then, the field falls off and at the time t ≥ z=v0 the train passes the considered point. The velocity 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

$$E\_0 \sim \exp \delta\_m z / \left(\mu^2 \mathbf{v}\_0\right)^{1/3} \tag{19}$$

The exponent δm= u2v0 � �<sup>1</sup>=<sup>3</sup> coincides to solution of the boundary problem as it is the maximal spatial growth rate. The coincidence to the results of well-known initial and boundary problems testifies presented approach. It may appear that this way of instability analysis is a bit more complicate. However, it must be admitted that along with growth rates we have obtained much other information. The information obviously clarifies the picture of the instability and makes it realistic. One can easily see the merits of presented approach.

The relations between characteristic velocities are

$$\mathbf{v}\_0 < \mathbf{w}\_t < \mathbf{w}\_\xi < \mathfrak{u} \tag{20}$$

At fixed instant t, perturbations exist only at distances v0t ≤ z ≤ ut. The wave train passes given point z during the time z=u ≤ t ≤ z=v0. In a fixed point, the amplitude attains maximum at the instant, when the peak has already passed it (see Figure 1). The reason is that the perturbations with smaller velocities reach considered point in longer time, and they grow more efficiently. Perturbations with velocity wa are the most efficiently enhanced perturbations.

Generally, the dependence of the perturbations' amplitudes on their velocity v has a form E � exp Γð Þv t, where

$$\Gamma(\mathbf{v}) = \frac{3}{2^{2/3}} \frac{\delta\_{\mathbf{m}}}{\mathbf{u} - \mathbf{v}\_0} \left\{ (\mathbf{v} - \mathbf{v}\_0)^2 (\mathbf{u} - \mathbf{v}) \right\}^{1/3} \tag{21}$$

The character of spatial growth depending on v is

$$E \sim \exp \Gamma(v) z / v \tag{22}$$

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

$$V\_{thr} = \frac{\lambda \mu + \mathbf{v}\_0}{(1 + \lambda)} \qquad ; \qquad \lambda = \frac{2^{5/2}}{3^{9/4}} \left(\frac{\nu}{|\delta\_0|}\right)^{\frac{2}{3}} \tag{23}$$

Only perturbations moving at higher velocities v > Vthr develop. The wave train shortens. Dissipation decreases the field growth

rate is equal to <sup>δ</sup><sup>m</sup> <sup>¼</sup> ffiffiffi

of this approach is evident.

The exponent δm= u2v0

E � exp Γð Þv t, where

3

28 Plasma Science and Technology - Basic Fundamentals and Modern Applications

The relations between characteristic velocities are

beam contribution in the DR. For given z, the field's maximum is

<sup>p</sup> <sup>=</sup><sup>2</sup> � �j j <sup>δ</sup><sup>0</sup> that is, coincides to solution of the initial problem. However,

(18)

(21)

(23)

the initial problem can not specify the point, where the maximal growth occurs. The advantage

In a fixed point z, the field first increases and attains maximum at instant t ¼ z=wa where

wa <sup>¼</sup> <sup>3</sup>uv0 u þ 2v0

Then, the field falls off and at the time t ≥ z=v0 the train passes the considered point. The velocity wa is the group velocity of the resonant wave upon amplification with account of the

spatial growth rate. The coincidence to the results of well-known initial and boundary problems testifies presented approach. It may appear that this way of instability analysis is a bit more complicate. However, it must be admitted that along with growth rates we have obtained much other information. The information obviously clarifies the picture of the insta-

At fixed instant t, perturbations exist only at distances v0t ≤ z ≤ ut. The wave train passes given point z during the time z=u ≤ t ≤ z=v0. In a fixed point, the amplitude attains maximum at the instant, when the peak has already passed it (see Figure 1). The reason is that the perturbations with smaller velocities reach considered point in longer time, and they grow more efficiently.

Generally, the dependence of the perturbations' amplitudes on their velocity v has a form

Presented above analysis is true if we neglect dissipation. Dissipation essentially changes the

ð Þ <sup>1</sup> <sup>þ</sup> <sup>λ</sup> ; <sup>λ</sup> <sup>¼</sup> <sup>25</sup>=<sup>2</sup>

39=<sup>4</sup>

ð Þ v � v<sup>0</sup> 2 ð Þ u � v n o<sup>1</sup>=<sup>3</sup>

v0

� �<sup>1</sup>=<sup>3</sup> coincides to solution of the boundary problem as it is the maximal

� �<sup>1</sup>=<sup>3</sup> (19)

v0 < wa < wg < u (20)

E � exp Γð Þv z=v (22)

ν j j δ<sup>0</sup> � �<sup>2</sup> 3

<sup>E</sup><sup>0</sup> � exp <sup>δ</sup>mz<sup>=</sup> <sup>u</sup><sup>2</sup>

bility and makes it realistic. One can easily see the merits of presented approach.

Perturbations with velocity wa are the most efficiently enhanced perturbations.

δm u � v0

instability behavior. It suppresses slow perturbations. The threshold velocity is

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

3 22=<sup>3</sup>

Γð Þ¼ v

The character of spatial growth depending on v is

$$
\Gamma(\upsilon) \to \Gamma\_\upsilon(\upsilon) = \Gamma(\upsilon) - \nu \frac{\mu - \upsilon}{\mu - \mathbf{v}\_0} \tag{24}
$$

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

$$(z - \mathbf{v}\_0 t)(\mu t - z)^2 = \left(1/\lambda^2\right) \left(z - w\_\\$ t\right)^3 \tag{25}$$

If v << δ0, 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 z ¼ u � Δu where

$$
\Delta \mu = \mathfrak{Z}^{-3/2} \lambda^{-1} (\mu - \mathbf{v}\_0),
\tag{26}
$$

and the expression for maximal growth rate takes the form <sup>Γ</sup><sup>ν</sup>!<sup>∞</sup> <sup>¼</sup> <sup>δ</sup><sup>3</sup> <sup>m</sup>=<sup>ν</sup> <sup>1</sup>=<sup>2</sup> . Obviously, this case corresponds to dissipative streaming instability (DSI). The same expression for Γ<sup>ν</sup>!<sup>∞</sup> can be obtained from Eq. (1) by direct usage of the initial problem [1]. If one specifies δm, he can obtain the growth rate of DSI in unbound beam-plasma system, in magnetized beam-plasma waveguide, etc.

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.
