**3.3 The influence of dissipation. New type of DSI**

Dissipation significantly influences on the presented picture of the instability development and changes it. First of all, it suppresses slow perturbations. The wave packet shortens. The threshold velocity *V*ð Þ wk *th* is determined from the condition *χ* ð Þ wk <sup>0</sup> <sup>¼</sup> *<sup>ν</sup>* <sup>∗</sup> ð Þ *Vbt* � *<sup>z</sup> <sup>=</sup>*ð Þ *Vb* � *<sup>V</sup>*<sup>0</sup> and is equal

$$\left(V\_{\rm th}^{(\rm wk)}\right) = \frac{\lambda'^2 V\_{\rm b} + V\_0}{1 + \lambda'^2} > V\_0; \lambda' = \nu^\* / \left(2\delta\_{\rm NEW}^{(\nu=0)}\right) \tag{17}$$

Only high-velocity perturbations (in the range *V*ð Þ *wk th* <*v*<*Vb*) grow. The change in the velocity of the trailing edge shortens the packet's length and can affects the nature of instability (convective/absolute) if the frame's velocity lies in the range *Vp* ≤ *v*≤*V*ð Þ *wk th* . Also, dissipation limits the growth rates of perturbations with velocity *v*. Substituting *z* ¼ *vt* we have for the field *E z*ð Þ� ¼ *vt*, *t* exp *G v*ð Þ*t*, where

$$G(\nu) = \frac{2\delta\_{\rm NEW}^{(\nu=0)}}{V\_b - V\_p} \sqrt{(V\_b - \nu)(\nu \cdot V\_p)} - \nu^\* \frac{V\_b - \nu}{V\_b - V\_p} \tag{18}$$

As expected, the growth rates fall down. Dissipation distorts the symmetry of the induced wave packet. In presence of dissipation the dynamics of the fields can be obtained from the same Eq. (14) accounting for dissipation. It has the form

$$\left(\mathbf{z} - \boldsymbol{w}\_{\mathbf{g}}\mathbf{t}\right)^{2} = \lambda^{2}(\mathbf{V}\_{\mathbf{b}}\mathbf{t} - \mathbf{z})(\mathbf{z} - \mathbf{V}\_{0}\mathbf{t}).\tag{19}$$

The solution of (18) gives the point of the field maximum *z* ð Þ*ν pk* <sup>¼</sup> *<sup>w</sup>*ð Þ*<sup>ν</sup> pk t*, where

$$w\_{pk}^{(\nu)} = \frac{1}{2} \left\{ (V\_b + V\_\mathbb{P}) + \sqrt{\frac{\lambda^2}{1 + \lambda^2}} (V\_b - V\_\mathbb{P}) \right\} > w\_{pk}^{(\nu=0)} \tag{20}$$

This expression shows that with an increase in the level of dissipation, the peak shifts more and more to the front of the wave packet. This takes place along with the decreasing of the wave packet's length. Substitution of *w*ð Þ*<sup>ν</sup> pk* into *χ* ð Þ wk *<sup>ν</sup>* gives the field value in the peak and shows the respective growth rate as the function on the level of dissipation

$$E\_0(\mathbf{z} = \mathbf{z}\_{pk}\mathbf{t}, \mathbf{t}) \sim \exp\left\{\delta^{(\nu)}\_{\mathrm{NEW}}\mathbf{t}\right\}; \delta^{(\nu)}\_{\mathrm{NEW}} = \delta^{(\nu=0)}\_{\mathrm{NEW}} f(\lambda^2); f(\mathbf{x}) = \sqrt{\mathbf{1} + \mathbf{x}} - \sqrt{\mathbf{x}} \quad \text{(21)}$$

The function *f x*ð Þ presents the dependence of the growth rate on the level of dissipation (see **Figures 2** and **<sup>3</sup>**). In the limit *<sup>ν</sup>* <sup>∗</sup> ! <sup>∞</sup> we have *<sup>E</sup>*<sup>0</sup> !� exp *<sup>δ</sup>* ð Þ *ν*!∞ *wk t* n o, where

$$
\delta\_{\nu\Bbbk}^{(\nu\longrightarrow\infty)} = \left[\delta\_{\text{NEW}}^{(\nu=0)}\right]^2 / \nu^\* \sim \sqrt{n\_b}/\nu^\* \tag{22}
$$

As a criterion for the type of DSI this relation between the growth rates of DSI *δ* ð Þ *ν*!∞ *NEW* and the growth rate of SI *δ* ð Þ *ν*¼0 *NEW* sharply differs from that for the conventional case (5). Actually the expression (22) shows that with an increase in level of dissipation in weakly coupled beam-plasma systems the instability, caused by the

### **Figure 2.**

*The function f x*ð Þ *presents the dependence of the growth rate of the instability, caused by NEW excitation on the level of dissipation. Here x* <sup>¼</sup> *<sup>ν</sup>=δ<sup>ν</sup>*¼<sup>0</sup> *NEW.*

### **Figure 3.**

*Shapes of developing waveform versus longitudinal coordinate at fixed instant* 3*=δ* ð Þ *ν*¼0 *NEW for various values of dissipation (parameter k* ¼ *ν=δ* ð Þ *ν*¼0 *NEW ): k*<sup>1</sup> ¼ 0*, k*<sup>2</sup> ¼ 1*, k*<sup>3</sup> ¼ 2*, k*<sup>4</sup> ¼ 4*.*

beam's NEW interaction with the plasma transforms to a new type of DSI. Its characteristic peculiarity is in new, previously unknown, *inverse proportional* dependence of the growth rate on dissipation. Below this result is confirmed by conventional electro-dynamical analysis of the DR for weakly coupled beam-plasma system.

### **3.4 Substantiation of the new DSI by conventional analysis of the DR**

From electro-dynamical point of view, a spatially separated beam-plasma system is nothing, but a multilayer structure. The traditional analytical consideration of such systems leads to a very cumbersome DR, which, in addition, is highly dependent on the geometry and greatly complicates with an increase in the number of layers. However, the importance of the problem and the need for its analytical investigation has led to development of specific methods. Here an approach is presented that allows avoiding abovementioned difficulties. Also, the approach has an important advantage: the procedure for obtaining the DR does not depend on specific shape/geometry. In other words, obtained results can be adapted to systems of any geometry. The approach considers the problem of weak beam-plasma interaction by perturbation theory. The small parameter, which underlies the theory, is

*New Types of Dissipative Streaming Instabilities DOI: http://dx.doi.org/10.5772/intechopen.98901*

the parameter of weak beam-plasma coupling. We briefly present here the basics of this approach accounting for dissipation [11].

Consider a system consisting of a mono-energetic rectilinear electron beam and cold plasma. To begin with, suppose the following: the plasma and the beam are weakly coupled (e.g. a consequence of a sufficiently large distance between them). We also assume their homogeneity in the cross section. The geometry of the system is not specified. It also is assumed that the beam current is less than the limiting current in the vacuum waveguide. Dissipation in the system is taken into account by the introduction the collisions in plasma. For simplicity, consideration is limited to the case of a strong external longitudinal (to the beam propagation direction) magnetic field, which prevents the transverse motion of the beam and plasma particles.

The small parameter underlying the perturbation theory is the parameter of weak coupling between the beam and the plasma (that is, the smallness of the integrals describing the overlap of beam and plasma fields). In the zero order approximation, the perturbation theory assumes independence of the beam and plasma. In the first-order approximation, the theory leads to the DR [11, 14].

$$D\_p(\boldsymbol{\alpha}, \boldsymbol{k}) D\_b(\boldsymbol{\alpha}, \boldsymbol{k}) = G \left(\boldsymbol{\kappa}^4 \delta \varepsilon\_p \delta \varepsilon\_b\right)\_{\boldsymbol{\alpha} = \boldsymbol{\alpha} \boldsymbol{0}, \boldsymbol{k} = \boldsymbol{k}\_0} \tag{23}$$

$$D\_{p,b}(\boldsymbol{\alpha}, \boldsymbol{k}) = \boldsymbol{k}\_{\perp p, b}^2 - \boldsymbol{\kappa}^2 \delta \varepsilon\_{p, b}; G < < \mathbf{1}$$

$$\boldsymbol{\kappa}^2 = \boldsymbol{k}^2 - \frac{\boldsymbol{\alpha}^2}{c^2}; \delta \varepsilon\_p = \frac{\boldsymbol{\alpha}\_p^2}{\boldsymbol{\alpha}(\boldsymbol{\alpha} + \boldsymbol{i}\boldsymbol{\nu})}; \delta \varepsilon\_b = \frac{\boldsymbol{\alpha}\_b^2}{\boldsymbol{\gamma}^3 (\boldsymbol{\alpha} - \boldsymbol{k} V\_b)^2},$$

*ω* and *k* are the frequency and longitudinal component of the wave vector, *ω<sup>p</sup>*,*<sup>b</sup>* are Langmuir frequencies for the plasma and the beam respectively, *ν* is the collision frequency in the plasma, *Vb* is the velocity of the beam electrons, *γ* ¼ <sup>1</sup> � *<sup>V</sup>*<sup>2</sup> *b=c*<sup>2</sup> � ��1*=*<sup>2</sup> , *c* is speed of light, *G* is the coupling parameter, the point f g *ω*0, *k*<sup>0</sup> is the intersection point of the beam and the plasma dispersion curves, the values *k*⊥*<sup>p</sup>* and *k*⊥*<sup>b</sup>* play role of transverse wave numbers. Analytically, *G* as well as *k*⊥*<sup>p</sup>* and *k*⊥*<sup>b</sup>* are expressed through the integrals of eigenfunctions of the zero order problem [11, 14]. The integral for *G* represents overlap of the beam and plasma fields. It shows how far the plasma field penetrates the beam and vice versa. The specific expressions for *k*⊥*<sup>p</sup>*, *k*⊥*<sup>b</sup>* and *G* are not essential for the subsequent presentation and are not presented here (see [11, 14]).

The expressions *Dp*,*<sup>b</sup>*ð Þ¼ *ω*, *k* 0 are the zero order DR for the plasma and the beam respectively. Their solutions are assumed to be known. The form of the DR (23) is comparatively simple. It shows the interaction of beam and plasma waves. Using (23) with small *G*, it is easy to describe instabilities in given system. The main result of a decrease in the beam–plasma coupling is in the increase in role of the beam NEW. Its interaction with plasma leads to instability. The spectra of slow (�) and fast (+) beam waves follow from the roots of *Db*ð Þ¼ *ω*, *k* 0. If one searches them in form *ω*� ¼ *kVb*ð Þ 1 þ *x*� , j j *x*� < <1, the roots become [11, 14].

$$\propto = \pm \left(\sqrt{a}/\gamma\right) \left(\sqrt{\beta^4 \gamma^2 a + 1} + \beta^2 \gamma \sqrt{a}\right),\tag{24}$$

where *<sup>α</sup>* <sup>¼</sup> *<sup>ω</sup>*<sup>2</sup> *b=k*<sup>2</sup> ⊥*bV*<sup>2</sup> *<sup>b</sup>γ*3, *<sup>β</sup>* <sup>¼</sup> *Vb=c*. The interaction of the NEW (*x*�) with the plasma leads to instability. If one looks for the solutions of (23) in the form *ω* ¼ *kVb*ð Þ 1 þ *x* , (j j *x* < < 1) it becomes [11].

$$(\varkappa + q + i\nu/kV\_b)(\varkappa - \varkappa\_+)(\varkappa - \varkappa\_-) = Ga/2\gamma^4\tag{25}$$

where *<sup>q</sup>* <sup>¼</sup> <sup>2</sup>*γ*<sup>2</sup> ð Þ�<sup>1</sup> *<sup>k</sup>*<sup>2</sup> ⊥*pV*<sup>2</sup> *bγ*2*=ω*<sup>2</sup> *<sup>p</sup>* � 1 � �. Mathematically, the instability is due to corrections to the expression for the slow beam wave *x* ¼ *x*� þ *x*<sup>0</sup> . Under collective Cherenkov resonance *q* ¼ �*x*� [11], the equation for *x*<sup>0</sup> is

$$(\mathbf{x'} + i\nu/(2\gamma^2 k V\_b))\mathbf{x'} = -\mathbf{G}\sqrt{a}/(4\gamma^3) \tag{26}$$

In absence of dissipation the instability is due to NEW interaction with the plasma. Its growth rate is

$$
\delta\_{\rm NEW}^{(\nu=0)} = k V\_b \text{Im}\mathbf{x}' = (k V\_b / 2\gamma) \sqrt{(\mathbf{G}\sqrt{a})/\gamma}.\tag{27}
$$

We emphasize unusual dependence on the beam density as *n* 1*=*4 *<sup>b</sup>* (for strong coupling this dependence is � *n* 1*=*3 *<sup>b</sup>* ). With ordinary Cherenkov resonance the system is stable. Under collective Cherenkov resonance dissipation manifested itself as an additional factor that enhances NEW growth and the instability gradually transforms to that of dissipative type. The Eq. (26) gives an expression for the growth rate as a function on level of dissipation

$$\delta(\lambda) = \delta\_{\text{NEW}}^{(\nu=0)} f(\lambda^2); \quad \lambda = \left(\mathbf{1}/2\boldsymbol{\gamma}^2\right) \left(\boldsymbol{\nu}/\delta\_{\text{NEW}}^{(\nu=0)}\right). \tag{28}$$

where *f x*ð Þ is the function given in (21). The dependence of the growth rate on the level of dissipation in (28) coincides to that in (21). In limit *λ* ! 0 (28) coincides to (27). In the opposite limit of strong dissipation *λ* ! ∞ (28) represents the growth rate of the new type of DSI (it also follows from (26) by neglecting the first term in brackets)

$$\delta\_{\rm NEW}^{(\nu \to \infty)} = \frac{2\gamma^2 \left(\delta\_{\rm NEW}^{(\nu = 0)}\right)^2}{\nu} = \frac{\mathcal{G}\sqrt{a}}{2\gamma} \frac{\left(kV\_b\right)^2}{\nu} \sim \frac{a\nu\_b}{\nu} \tag{29}$$

We arrive to the same new type of DSI presented in (22). The expression (28) shows a gradual transition of the growth rate of no-dissipative instability caused by NEW interaction with plasma into the growth rate of new type of DSI. It develops under weak coupling and differs from the conventional DSI (with an growth rate � *ωb=* ffiffi *ν* p ). In [21] the same new DSI is substantiated in a finite external magnetic field.
