5. The behavior of the Buneman instability in dissipative plasma

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

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

$$1 - \frac{\omega\_{l\varepsilon}^2}{(\omega - \mathbf{k}\mathbf{u})(\omega + i\nu\_{\rm Bn} - \mathbf{k}\mathbf{u})} - \frac{\omega\_{l\varepsilon}^2}{\omega^2} = 0 \qquad ; \qquad \delta\_{\rm Bn}^{(m)} = \frac{\sqrt{3}}{2} \omega\_{l\varepsilon} \left(\frac{m}{2M}\right)^{1/3} \tag{67}$$

(u is the velocity of streaming electrons, ωLe and ωLi are Langmuir frequencies for electrons and. ions respectively, νBnis the frequency of collisions). The BI develops if ωLe ≥ ku, and the growth rate attains its maximum under ωLe ≈ ku.

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

$$D\_0(\omega, \mathbf{k}) + \Delta D = 0 \tag{68}$$

An initial perturbation arises and the instability begins to develop in point z ¼ 0 (electron stream propagates in the direction z > 0) at instant t ¼ 0. Our aim is to obtain the shape of the perturbation and investigate in detail the behavior of the BI. The procedure for obtaining the

k ¼ k<sup>0</sup>

Im δBn is the general form of the resonant growth rate of the low-frequency BI [1, 4] (compare to Eq. (67)); v0 is the group velocity of the resonant wave in the system. Here, it is equal to

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

exp χBnð Þ� z; t ν

0 z v0

n o <sup>e</sup>

2zτ<sup>2</sup> v0 � �<sup>1</sup> 3

2zτ<sup>2</sup> v0 � �<sup>1</sup>=<sup>3</sup>

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

<sup>0</sup> z v0

χBn � ν

� �

(compare to Eq. (16)) determines the peak's movement. In the absence of dissipation the peak disposes on 2/3 of the train's length from its front and moves at velocity v0=3. Substitution of z ¼ v0t=3 into Eq. (72) gives the field's behavior in the peak. It grows exponentially

( )

6v0zj j δBn

3

actually presents dissipation. In unbound plasma, the main

<sup>i</sup> <sup>χ</sup>Bn<sup>þ</sup> ffi 3 p k0z ffi 3 <sup>p</sup> �<sup>π</sup> 6 � �

; τ ¼ t � z=v0,

� ν <sup>0</sup> z v0

; <sup>ν</sup> <sup>¼</sup> Im <sup>D</sup><sup>0</sup> ∂D0=∂ω � � <sup>ω</sup> ! <sup>0</sup>

E0ð Þ z; t (70)

þ ik0v0 � ν

in this form to collision

0 þ ik0v0 41

k ¼ k<sup>0</sup>

The Behavior of Streaming Instabilities in Dissipative Plasma

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

0

<sup>3</sup> � �<sup>1</sup>=<sup>2</sup> (71)

: (72)

¼ 0 (73)

equation for SVA is known. Applying this procedure, we arrive to following Eq. [16]

∂D0=∂ω � � <sup>ω</sup> ! <sup>0</sup>

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

; v0 ¼ � <sup>∂</sup>D0=∂<sup>k</sup>

0

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

χBnð Þ¼ z; t

exp

5.2. Analysis of the Buneman instability behavior

cause of dissipation is collisions of plasma particles. Equality of the ν

ffiffiffiffiffiffi <sup>2</sup><sup>π</sup> <sup>p</sup>

> 3 ffiffiffi 3 p <sup>4</sup> j j <sup>δ</sup>Bn

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

3 ffiffiffi 3 p <sup>4</sup> j j <sup>δ</sup>Bn

> ∂ ∂z

velocity v0. The length of the induced wave train increases as l ≈ v0t. The condition

k ¼ k<sup>0</sup>

velocity of streaming electrons; ν

frequency is not obligatory.

j j δBn

<sup>3</sup> <sup>¼</sup> <sup>ω</sup><sup>2</sup> Li ∂D0=∂ω � � <sup>ω</sup> ! <sup>0</sup>

where <sup>Δ</sup><sup>D</sup> ¼ �ω<sup>2</sup> Li=ω<sup>2</sup> describes the contribution of ions in the DR, while <sup>D</sup>0ð Þ <sup>ω</sup>; <sup>k</sup> describes contribution of moving electrons as well as collisions/dissipation in the system. In following consideration, we do not specify the form of D0ð Þ ω; k . As ωLi << ωLe we have j j ΔD << j j D<sup>0</sup> and the ions in Eq. (68) play a role under small ω that is, ωLi >> ω ! 0. One can at once see imaginary roots of the Eq. (68). The system becomes unstable (low frequency instability) and the growth rate may be obtained from

$$\left|\left|\omega(k)\right|^{3} = \omega\_{Li}^{2}\left[\left(\frac{\partial D\_{0}(\omega,k)}{\partial\omega}\right)\_{\omega\to 0}\right]^{-1}\tag{69}$$

An initial perturbation arises and the instability begins to develop in point z ¼ 0 (electron stream propagates in the direction z > 0) at instant t ¼ 0. Our aim is to obtain the shape of the perturbation and investigate in detail the behavior of the BI. The procedure for obtaining the equation for SVA is known. Applying this procedure, we arrive to following Eq. [16]

$$\frac{\partial^2}{\partial t^2} \left( \frac{\partial}{\partial t} + \mathbf{v}\_0 \frac{\partial}{\partial z} + \nu \right) E\_0(z, t) = i \left| \delta\_{Bn} \right|^3 E\_0(z, t) \tag{70}$$

j j δBn <sup>3</sup> <sup>¼</sup> <sup>ω</sup><sup>2</sup> Li ∂D0=∂ω � � <sup>ω</sup> ! <sup>0</sup> k ¼ k<sup>0</sup> ; v0 ¼ � <sup>∂</sup>D0=∂<sup>k</sup> ∂D0=∂ω � � <sup>ω</sup> ! <sup>0</sup> k ¼ k<sup>0</sup> ; <sup>ν</sup> <sup>¼</sup> Im <sup>D</sup><sup>0</sup> ∂D0=∂ω � � <sup>ω</sup> ! <sup>0</sup> k ¼ k<sup>0</sup> þ ik0v0 � ν 0 þ ik0v0

Im δBn is the general form of the resonant growth rate of the low-frequency BI [1, 4] (compare to Eq. (67)); v0 is the group velocity of the resonant wave in the system. Here, it is equal to velocity of streaming electrons; ν 0 actually presents dissipation. In unbound plasma, the main cause of dissipation is collisions of plasma particles. Equality of the ν 0 in this form to collision frequency is not obligatory.

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

$$\begin{aligned} E\_0(z,t) &= \frac{I\_0}{\sqrt{2\pi}} \frac{\exp\left\{ \chi\_{B\text{tr}}(z,t) - \nu^\prime \frac{z}{\mathbf{v}\_0} \right\} \ e^{i\left(\frac{x\_{B\text{tr}} + \sqrt{3}\mathbf{v}\_0 \cdot \mathbf{z}}{\sqrt{3}}\right)}}{\left(6\mathbf{v}\_0 \mathbf{z} |\delta\_{B\text{tr}}|^3\right)^{1/2}} \\\\ \chi\_{B\text{tr}}(z,t) &= \frac{3\sqrt{3}}{4} |\delta\_{B\text{tr}}| \left\{ \frac{2\pi\tau^2}{\mathbf{v}\_0} \right\}^{\frac{1}{2}}; \tau = t - z/\mathbf{v}\_0 \end{aligned} \tag{71}$$

#### 5.2. Analysis of the Buneman instability behavior

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.

5. The behavior of the Buneman instability in dissipative plasma

the simplest case of the BI (cold e-stream, heavy ions, and accounting for collisions)

Li

(u is the velocity of streaming electrons, ωLe and ωLi are Langmuir frequencies for electrons and. ions respectively, νBnis the frequency of collisions). The BI develops if ωLe ≥ ku, and the growth

contribution of moving electrons as well as collisions/dissipation in the system. In following consideration, we do not specify the form of D0ð Þ ω; k . As ωLi << ωLe we have j j ΔD << j j D<sup>0</sup> and the ions in Eq. (68) play a role under small ω that is, ωLi >> ω ! 0. One can at once see imaginary roots of the Eq. (68). The system becomes unstable (low frequency instability) and

> ∂D0ð Þ ω; k ∂ω � �

� ��<sup>1</sup>

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

Li=ω<sup>2</sup> describes the contribution of ions in the DR, while <sup>D</sup>0ð Þ <sup>ω</sup>; <sup>k</sup> describes

ω!0

ð Þ m Bn ¼

D0ð Þþ ω; k ΔD ¼ 0 (68)

ffiffiffi 3 p <sup>2</sup> <sup>ω</sup>Le

m 2M � �<sup>1</sup>=<sup>3</sup>

(67)

(69)

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

5.1. Statement of the problem: the equation for SVA

40 Plasma Science and Technology - Basic Fundamentals and Modern Applications

Le ð Þ <sup>ω</sup> � ku ð Þ <sup>ω</sup> <sup>þ</sup> <sup>i</sup>νBn � ku � <sup>ω</sup><sup>2</sup>

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

j j <sup>ω</sup>ð Þ<sup>k</sup> <sup>3</sup> <sup>¼</sup> <sup>ω</sup><sup>2</sup>

Li

<sup>1</sup> � <sup>ω</sup><sup>2</sup>

rate attains its maximum under ωLe ≈ ku.

the growth rate may be obtained from

where <sup>Δ</sup><sup>D</sup> ¼ �ω<sup>2</sup>

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

$$\exp\left\{\frac{3\sqrt{3}}{4}|\delta\_{\mathrm{Bn}}|\left\{\frac{2z\pi^2}{\mathbf{v}\_0}\right\}^{1/3} - \nu\left(\frac{z}{\mathbf{v}\_0}\right).\tag{72}$$

In the absence of dissipation the velocities of unstable perturbations range from 0 to the group velocity v0. The length of the induced wave train increases as l ≈ v0t. The condition

$$\frac{\partial}{\partial z} \left( \chi\_{Bn} - \nu \frac{z}{\mathbf{v}\_0} \right) = 0 \tag{73}$$

(compare to Eq. (16)) determines the peak's movement. In the absence of dissipation the peak disposes on 2/3 of the train's length from its front and moves at velocity v0=3. Substitution of z ¼ v0t=3 into Eq. (72) gives the field's behavior in the peak. It grows exponentially <sup>E</sup><sup>0</sup> � exp ffiffiffi 3 <sup>p</sup> <sup>=</sup><sup>2</sup> � �j j <sup>δ</sup>Bn <sup>t</sup> � � and the growth rate is equal to the maximal growth rate of the BI obtained earlier as a result of initial problem (e.g., see [1, 4] and Eq. (67)). However, in contrary to this approach, the initial problem does not give the point of the maximal growth. This approach gives the point. In addition, it gives the rates of the field growth in every point of the wave train (in the presence of dissipation also).

Dissipation changes the fields' dynamics and mode structure. It is easily seen from Eq. (72) that dissipation suppresses fast perturbations. The threshold velocity vth can be obtained from the equation χBnð Þ¼ z; t νz=v0 and is equal

$$
\sigma\_{\rm th} = \frac{\mathbf{v}\_0}{1 + \lambda\_0^{3/2}} \qquad ; \qquad \lambda\_0 = \frac{2^{5/3}}{3^{-3/2}} \frac{\mathbf{v}'}{|\delta\_{\rm Bn}|} \tag{74}
$$

where the growth rate δν ¼

6. Conclusion

second/third order.

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi j j δBn 3 =2ν q <sup>0</sup>

<sup>E</sup>0ð Þ� <sup>z</sup> <sup>¼</sup> vt; <sup>t</sup> exp G vð Þt ; Gvð Þ¼ <sup>3</sup> ffiffiffi

type [1, 16, 17]. This once again justifies that high-level dissipation transforms the BI to DSI.

perturbation, moving at given velocity v and determine the rate of their growth

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

In addition, the expression for χBnð Þ z; t gives much other information on the character of BI development. For example, by substituting z ¼ vt one can investigate the behavior of the

Now, we can generalize the properties of the SI. Originated perturbations form a wave train, carrier frequency and wave vector of which are determined by resonant conditions. The expression for space–time distribution of the fields gives much information on the behavior of the instability in limit of comparatively large times. The solutions of conventional initial and boundary problems follow from the expression by itself. The growth rate in the peak is equal to maximal growth rate of resonant instability δ, which usually describes given instability. The initial value problem gives the same growth rate without specifying where the growth takes place. That is, the approach gives realistic picture of the SI development. Dissipation leads to shortening of the wave train. With increase in level of dissipation the SI gradually turns to dissipative type. In the limit ν >> δ (ν is the collision frequency) the growth of the fields takes place according to dissipative instability. The approach gives also information on the growth rate for arbitrary δ=ν. Obvious expression may be obtained by solving algebraic equation of

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

Actually the approach presents solution of the well-known problem of time evolution of initial perturbation in systems those undergo the instabilities of streaming type. The importance of the problem is doubtless. Its traditional solution is restricted by mathematical difficulties. Presented methods allows without any difficulties obtain result for various SI in spite of their different mathematical description (e.g., the description of Buneman instability differs from the instability in spatially separated beam-plasma system and from beam-plasma instabilities; herewith, the description various types of beam-plasma instabilities (Cherenkov, cyclotron, and other) also differs from each other). The approach by itself unified the differences. For beam-plasma instabilities results of the approach are unified even more and their usage is not

Presented approach obviously shows the transition to the new types of DSI.

3 p v0

is nothing else, as the growth rate of DSI of conventional

The Behavior of Streaming Instabilities in Dissipative Plasma

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j j δBn 2vð Þ v0 � v

<sup>2</sup> n o<sup>1</sup>=<sup>3</sup>

� ν <sup>0</sup> v v0

(78)

43

The wave train shortens. Actually the pulse slows down. Dissipation influences on the peak location/movement. Its place z ¼ zmax can be obtained from the equation

$$(\mathbf{v}\_0 t - \mathbf{3}\mathbf{z})^3 = (\mathbf{3}\lambda\_0)^3 \mathbf{z}^2 (\mathbf{v}\_0 t - \mathbf{z})\tag{75}$$

The solution of this third-order algebraic equation gives location and velocity of the peak under arbitrary ratio ν 0 =δBn. To avoid cumbersome expressions, we present here the solution only in the most interesting limit of high dissipation λ<sup>0</sup> ! ∞.

$$z = z\_{\text{max}} = \left(\frac{\mathfrak{Z}^{3/4}}{\mathfrak{Z}^{5/2}}\right) \left(\frac{|\delta\_{\text{Bn}}|}{\nu'}\right)^3 \mathbf{v}\_0 t \tag{76}$$

Substitution of this expression into χBnð Þ z; t gives the field's behavior in the peak under highlevel dissipation. The field's value increases exponentially

$$E\_0 \sim \exp\left\{\delta\_\nu \ t\right\} \tag{77}$$

Figure 4. The shapes of initial perturbation for various level of dissipation. The dimensionless distance ζ ¼ zδBn=v0, and the dimensionless field ε ¼ E0= Jð Þ <sup>0</sup>=ð Þ v0δBn are marked along the axes. Curve 1 corresponds to λ 0 ¼ ν=j j δBn ¼ 0; curve 2 – To λ 0 ¼ 0:5; curve 3 – To λ 0 ¼ 1:5; curve 4 – To λ 0 ¼ 3.

where the growth rate δν ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi j j δBn 3 =2ν q <sup>0</sup> is nothing else, as the growth rate of DSI of conventional type [1, 16, 17]. This once again justifies that high-level dissipation transforms the BI to DSI.

In addition, the expression for χBnð Þ z; t gives much other information on the character of BI development. For example, by substituting z ¼ vt one can investigate the behavior of the perturbation, moving at given velocity v and determine the rate of their growth

$$E\_0(\mathbf{z} = \boldsymbol{\upsilon}t, \mathbf{t}) \sim \exp \mathbf{G}(\boldsymbol{\upsilon})t \qquad ; \qquad \mathbf{G}(\boldsymbol{\upsilon}) = \frac{3\sqrt{3}}{\mathbf{v}\_0} |\delta\_{\mathrm{Bn}}| \left\{ 2\mathbf{v}(\mathbf{v}\_0 - \boldsymbol{\upsilon})^2 \right\}^{1/3} - \boldsymbol{\upsilon}' \frac{\boldsymbol{\upsilon}}{\mathbf{v}\_0} \tag{78}$$

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