**3. Weak beam-plasma coupling. New type of DSI**

## **3.1 Solution of the problem of initial perturbation development under weak beam-plasma coupling**

The best way to study an instability in detail and its possible transformation to that of other type is the solving of the problem of initial perturbation development. The information obtained by other ways is insufficient and does not give any details. Here we present general (geometry independent) solution of the problem for weakly coupled beam-plasma systems.

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. in a consequence of a sufficiently large distance between them). Let an initial perturbation arises at a point *z* ¼ 0 (the electron beam propagates in the direction *z*>0) at the instant *t* ¼ 0 and the instability begins developing. Our goal is to obtain the fields'space–time distribution at an arbitrary instant *t*>0 and investigate in detail the instability behavior by analyzing obtained expression. In the process, we interest only the longitudinal structure of the fields, i.e., their dependence on the longitudinal coordinate *z* and time *t*. The transverse structure of the fields can be obtained by expanding in terms of the system's eigenfunctions. In accordance with this, only two arguments are highlighted below: frequency and longitudinal component of the wave vector. Other arguments are irrelevant in the consideration below. To avoid overburdening the formulas, they are omitted.

In given case of weak beam-plasma coupling the instability is the result of the interaction of the beam negative energy wave (NEW) and the slowed down wave in the plasma. The interaction is of Collective Cherenkov type. We proceed from the theory of wave interaction in plasma [16]. In terms of this theory the problem of the initial perturbation evolution under instability development in non-equilibrium plasma can be considered based on the set of partial differential equations for the amplitudes of the interacting waves: beam charge density wave *Eb*ð Þ *z*, *t* and the slowed down electromagnetic wave *Ew*ð Þ *z*, *t* in the plasma

$$\left(\frac{\partial}{\partial t} + V\_{\rm b} \frac{\partial}{\partial \mathbf{z}}\right) E\_{b}(\mathbf{z}, t) - i \delta^{2} E\_{w}(\mathbf{z}, t) = J(\mathbf{z}, t) \tag{6}$$

$$\left(\frac{\partial}{\partial t} + V\_{p} \frac{\partial}{\partial \mathbf{z}} + \nu^{\*}\right) E\_{w}(\mathbf{z}, t) - i E\_{b}(\mathbf{z}, t) = \mathbf{0}$$

where *t* is the time, *z* is the coordinate along the beam propagation direction, *J z*ð Þ , *t* is a function determined by the initial conditions, *V*<sup>b</sup> is the directed velocity of the beam, *Vp* is the group velocity of the resonant wave in plasma, *V*<sup>b</sup> >*Vp*, *ν* <sup>∗</sup> describes dissipation in plasma and is proportional to the frequency of collisions in it. The meaning of the denotation *δ* will be cleared up below. Note, the set (6) is meaningful irrespective of the problem of development of any instability. Generally, it describes resonant interactions between two waves in unstable medium. One only condition should be satisfied: the growth rate attains maximum under Collective Cherenkov Effect. If the maximum is attained under conventional Cherenkov Effect, as for conventional beam-plasma instabilities, the interaction should be described by other set of Equations [16].

The solution of the set (6) gives the dependence of the field's amplitude on longitudinal coordinate and time under instability development. Applying the Laplace transformation with respect to time *t* and the Fourier transformation with respect to the spatial coordinate *z*, we obtain following expressions for the transform *Ew*ð Þ *ω*, *k* :

$$E\_w(o\nu, k) = \frac{f(o\nu, k)}{D(o\nu, k)}$$

$$D(o\nu, k) = (o - kV\_b) \left(o - kV\_p + i\nu^\*\right) + \delta^2 \tag{7}$$

The field's amplitude *Ew*ð Þ *z*, *t* can be found by inverse transformation

$$E\_w(z,t) = \frac{1}{\left(2\pi\right)^2} \int\_{C(o)} d\boldsymbol{\alpha} \int\_{-\infty}^{\infty} \frac{dk \,\, J(\boldsymbol{\alpha},k) \exp\left(-i\boldsymbol{\alpha}\cdot\boldsymbol{t} + ik\boldsymbol{z}\right)}{\left(\boldsymbol{\alpha} - kV\_b\right)\left(\boldsymbol{\alpha} - kV\_p + i\boldsymbol{\nu}^\*\right) + \delta^2} \tag{8}$$

where *C*ð Þ *ω* is the contour of integration with respect to *ω*. For given case it is a straight line that lies in the upper half plane of the complex plane *ω* ¼ Re*ω* þ *i*Im*ω* and passes above all singularities of the integrand.

Thus, the problem has been reduced to the problem of integration in (8). It is somewhat simpler in comparison to the integral, which represents classical solution. Instead of full DR its analog stands. The analog is determined by interaction of the waves, participating in the instability development. This replacement simplifies integration. However, it remains difficult and many authors use roundabout methods carry out an expression for possible estimation of the fields behavior [17, 18]. Presented here method easily leads to the desired result i.e. to expression for space–time distribution of the fields. We merely transform the variables *ω* and *k* to another pair *ω* and *ω*<sup>0</sup> ¼ *ω* � *kVb*. The first integration (over *ω*) may be carried out by the residue method and the integration contour must be closed in the lower half-plane. The first order pole is

$$\boldsymbol{\alpha} = -\left(\mathbf{1} - \mathbf{V}\_p/\mathbf{V}\_b\right)^{-1} \left\{ \delta^2/\alpha' + \dot{\mathbf{u}} + \alpha' \mathbf{V}\_p/\mathbf{V}\_b \right\} \tag{9}$$

The second integration (over *ω*<sup>0</sup> ) cannot be carried out exactly, and we are forced to restrict ourselves to the approximate steepest descent method [19]. This method gives result in the limit of relatively large *t*. According to this method, the contour of integration should be deformed to pass through the saddle point in the direction of the steepest descent. The saddle point is found from the condition

$$\frac{d}{d\alpha'} (\alpha(\alpha')\mathbf{t} + i\alpha'\mathbf{z}/V\_b) = \mathbf{0} \tag{10}$$

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

and is equal to

$$\alpha\_s' = \mathrm{i}\delta\left\{ (\mathbf{V\_b t} - \mathbf{z})/(\mathbf{z} - \mathbf{V\_p t}) \right\}^{1/2} \tag{11}$$

As a result we arrive to the following expression for the field's space time structure under development of the instability in spatially separated beam-plasma system

$$E\_w(\mathbf{z}, t) = -\frac{J\_0}{2\sqrt{\pi}} \frac{\exp \chi\_\nu^{(wk)}(\mathbf{z}, t)}{\left(V\_b - V\_p\right)^{\natural\_2} \delta^{\natural\_2} (V\_b t - \mathbf{z})^{\natural\_2}} \tag{12}$$

$$\chi\_\nu^{(\text{wk})} = \chi\_0^{(\text{wk})} - \nu^\* \frac{V\_b t - \mathbf{z}}{V\_b - V\_p}; \chi\_0^{(\text{wk})} = \frac{2\delta}{V\_b - V\_p} \sqrt{(\mathbf{z} - V\_p t)(V\_b t - \mathbf{z})}$$

$$J\_0 = J\left(\boldsymbol{\alpha} = \boldsymbol{\alpha}(\boldsymbol{\alpha}'), \boldsymbol{\alpha}' = \boldsymbol{\alpha}\_s'\right)$$

## **3.2 Analysis of the instability development**

The expression (12) looks very complicate. At first glance it is impossible to extract any information on the instability behavior from it. However, it turned out, the expression may be easily analyzing. Moreover, the results are obtained from scratch, i.e. they are not based on prior research. Substantial part of the information is unavailable by other way. In particular, the analysis clearly shows that with increase in level of dissipation the no-dissipative instability turns to a new type of DSI and provides detailed information on both instabilities.

The properties of the instability is determined mainly by the exponential factor

$$\exp \chi\_{\nu}^{(wk)}(z,t) = \exp \left\{ \frac{2\delta}{V\_b - V\_p} \sqrt{(z - V\_p t)(V\_b t - z)} - \nu^\* \frac{V\_b t - z}{V\_b - V\_p} \right\},\tag{13}$$

which provides many information: the temporal and the spatial growth rates, the spread of the unstable perturbations' velocities, the nature of the instability (absolute or convective), the effect of dissipation on instability, etc.

First consider some general properties of the instability, which follow from (13).

It is easily seen that in the absence of dissipation unstable perturbations have velocities in the range from *Vp* to *Vb*. The wave packet moves in the beam propagation direction and, along with exponential growth of the fields, expands. Its length increases over time *l* � *Vb* � *Vp* � � *t*. The knowledge of the boundary velocities of unstable perturbations allows at once determining the nature of the instability (convective/absolute) based on the definition only, without reference to additional studies (we mean the Sturrock's laws [20]). It is clearly seen that the instability is convective in the laboratory frame and other frames moving at velocities *V* >*Vb* and *V* <*Vp*. However, if the observer's speed is within the range *Vp* <*V* <*Vb*, then the same instability is absolute (see **Figure 1**).

Now we turn to determination of the meaning of the denotation *δ* in (6). For this we consider case *ν* ¼ 0 and find the point of the field's maximum from expression

$$\frac{\partial \chi\_0^{(\text{wk})}(\mathbf{z}, t)}{\partial \mathbf{z}} = \mathbf{0} \tag{14}$$

Its root is *zm* <sup>¼</sup> *<sup>w</sup>*ð Þ *<sup>ν</sup>*¼<sup>0</sup> *pk t* i.e. the point of the field's maximum moves at velocity

$$
\omega\_{pk}^{(\nu=0)} = (\mathbf{1}/2)(V\_b + V\_p). \tag{15}
$$

In the wave theory the velocity (15) is called convective velocity. It characterizes the spatial convection of the fastest growing perturbations. (15) shows that the peak of the wave packet disposes in its middle. The packet is symmetric with respect to its peak. Substitution of *zm* into the *χ* ð Þ *wk* <sup>0</sup> ð Þ *z*, *t* determines the field's behavior in the maximum as *E*0ð Þ� *zm*, *t* exp ð Þ *δ t* , i.e. *δ* represents the maximal growth rate of the instability, which develops in absence of dissipation in systems with weak beamplasma coupling. At the point *zm* <sup>¼</sup> *<sup>w</sup>*ð Þ *<sup>ν</sup>*¼<sup>0</sup> *pk t* the peak forms, because here the growth rate of perturbations is maximal.

The meaning of the parameter *δ* may also be determined from the DR (7) only, bypassing the results of integration (12). The general expression for the group velocity *Vgr*ð Þ *ω*, *k* obtained from DR (7) has the limit (15) under *k* ¼ 0. The same limit (note that *<sup>ν</sup>* <sup>¼</sup> 0) leads to DR in form *<sup>ω</sup>*<sup>2</sup> <sup>þ</sup> *<sup>δ</sup>*<sup>2</sup> <sup>¼</sup> 0, i.e. the parameter *<sup>δ</sup>* is the imaginary part of complex frequency (the growth rate). In this case (absence of dissipation) the instability is due to interaction of the NEW with the plasma. To emphasize the important role of *δ* we add the respective indexes *δ* � *δ* ð Þ *ν*¼0 *NEW* . Its dependence on specific parameters is found out below.

At a fixed point *z* the field first grows up to the value � exp *δ* ð Þ *ν*¼0 *NEW z=*ð Þ *VbV*<sup>0</sup> <sup>1</sup>*=*<sup>2</sup> n o that is reached at the instant *t* ¼ *z=wa* where

$$
\omega\_a = \mathcal{D} V\_b V\_p / (V\_b + V\_p). \tag{16}
$$

Then the field decreases, and at the time *t*≥ *z=V*<sup>0</sup> the wave packet completely passes given point. The exponent *δ* ð Þ *ν*¼0 *NEW z=*ð Þ *VbV*<sup>0</sup> <sup>1</sup>*=*<sup>2</sup> is, in fact, the maximal spatial growth rate. At a given point, the field reaches its maximum at the moment when the peak has already passed it (see **Figure 1**). The reason is that perturbations moving at lower velocities reach the point for a longer time, and they have time to grow more. *wa* is the velocity of the most effectively amplified perturbations.

Thus, the solution of the problem of initial perturbation development along with other detailed information, gave results of conventional initial and boundary problems. This coincidence confirms correctness of developed approach (initial assumptions, mathematics, etc.). An additional advantage of the approach is in its geometry-independence. At first glance, the presented approach seems more complicated than traditional approaches, but this complexity is only apparent.

### **Figure 1.**

*Asymptotic shapes of the instability development under weak beam-plasma coupling vs. longitudinal coordinate z at instants t*<sup>1</sup> <sup>¼</sup> 0, 5*=δ<sup>ν</sup>*¼<sup>0</sup> *NEW <sup>t</sup>*<sup>2</sup> <sup>¼</sup> 0, 9*=δ<sup>ν</sup>*¼<sup>0</sup> *NEW <sup>t</sup>*<sup>3</sup> <sup>¼</sup> 1, 2*=δ<sup>ν</sup>*¼<sup>0</sup> *NEW. The dotted line gives the shape of the wave packet for strong beam-plasma coupling.*
