**4. Uniform cross section beam-plasma waveguide. One more new type of DSI**

## **4.1 Evolution of the initial perturbation in plasma waveguide with over-limiting electron e-beam**

One more new DSI arises under consideration of the problem of the initial perturbation development for the instability of over-limiting beam (OEB) in uniform cross-section plasma waveguide.

Consider a cylindrical waveguide, fully filled with 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. We also assume that the beam and plasma radii coincide with the waveguide's radius and consider only the symmetrical *E*-modes with

nonzero components *Er*, *Ez*, and *Bφ*. The development of resonant instability in this system is described by the DR and resonant condition those are [1].

$$D\_0(\boldsymbol{\alpha}, \boldsymbol{k}) + D\_b(\boldsymbol{\alpha}, \boldsymbol{k}) = \mathbf{0}; \boldsymbol{\alpha} = kV\_b \tag{30}$$

$$D\_0 = \boldsymbol{k}\_\perp^2 + \kappa^2 \left( \mathbf{1} - \frac{\boldsymbol{\alpha}\_p^2}{\boldsymbol{\alpha}(\boldsymbol{\alpha} + i\boldsymbol{\nu})} \right); D\_b = -\kappa^2 \frac{\boldsymbol{\alpha}\_b^2 / \boldsymbol{\gamma}^3}{\left(\boldsymbol{\alpha} - kV\_b\right)^2}; \kappa^2 = \boldsymbol{k}^2 - \frac{\boldsymbol{\alpha}^2}{\boldsymbol{c}^2}$$

*ω* and *k* are the frequency and the longitudinal (along beam propagation direction that is *z* axis) wave vector, *k*<sup>⊥</sup> ¼ *μ*0*s=R*, *R* is the waveguide's radius, *μ*0*<sup>s</sup>* are the roots of Bessel function *J*0: *J*<sup>0</sup> *μ*0*<sup>s</sup>* ð Þ¼ 0, *s* = 1,2,3 … , *ωp*,*<sup>b</sup>* are the Langmuir frequencies for the plasma and the beam, *Vb* is the beam velocity, *<sup>γ</sup>* <sup>¼</sup> <sup>1</sup> � *<sup>V</sup>*<sup>2</sup> *b=c*<sup>2</sup> � ��1*=*<sup>2</sup> , *ν* is the frequency of collisions in plasma, *c* is the speed of light.

The character of the beam-plasma interaction changes depending on the beam current value. If the beam current is less than the limiting current in vacuum waveguide the instability is due to induced radiation of the system eigenwaves by the beam electrons. But, if the beam is over-limiting, its instability has the same nature as the instability in medium with negative dielectric constant [9, 14, 15]. We introduce a parameter *<sup>α</sup>* <sup>¼</sup> *<sup>ω</sup>*<sup>2</sup> *b=k*<sup>2</sup> ⊥*V*<sup>2</sup> *<sup>b</sup>γ*3, which represents the beam current value and the character of beam-plasma interaction. It corresponds (correct to the factor *γ*�2) to the ratio of the beam current to the limiting current in vacuum waveguide [14] *<sup>I</sup>*<sup>0</sup> <sup>¼</sup> *mV*<sup>3</sup> *<sup>b</sup>γ=*4*e*, i.e. *<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 *<sup>α</sup>*< <*γ*�<sup>2</sup> correspond to under-limiting beam currents *Ib* < <*I*0, but the values *γ*�<sup>2</sup> < <*α* < <1 correspond to over-limiting beam currents. This is possible under comparatively high values of the relativistic factor *γ*. Here we consider development of an initial perturbation in the system, when the beam current slightly exceeds the limiting vacuum value. In this case the instability is due to a-periodical modulation of the beam density in medium with negative dielectric constant. Its growth rate attaints maximum under exact Cherenkov resonance and is equal [15].

$$\delta\_{\rm ovl}^{(\nu=0)} = \frac{a\_b V\_b}{c \sqrt{\gamma (1+\mu)}}, \quad \mu = \gamma^2 \frac{k\_\perp^2 V\_b^2}{a\_p^2 - k\_\perp^2 V\_b^2 \gamma^2} \tag{31}$$

However, the resonant frequency, which is determined by the expressions (30), remains unchanged [15].

In order to show the variety of possible approaches to the solution of the problem of the initial perturbation development, in given case we solve it by other way. We turn to the set of origin equations, which describes e-beam instability in magnetized plasma waveguide

$$\frac{\partial E\_r}{\partial x} - \frac{\partial E\_x}{\partial r} = -\frac{1}{c} \frac{\partial B\_\mathbf{v}}{\partial t}; \quad \hat{L} \text{ v}\_\mathbf{b}^\prime = \frac{e}{m} E\_\mathbf{z}; \quad \frac{\partial \mathbf{v}\_\mathbf{p}^\prime}{\partial \mathbf{t}} = \frac{e}{m} E\_\mathbf{z} - \nu \,\mathbf{v}\_\mathbf{p}^\prime$$

$$\frac{\partial B\_\mathbf{v}}{\partial \mathbf{z}} = -\frac{1}{c} \frac{\partial E\_r}{\partial t}; \quad \hat{L} \mathbf{n}\_\mathbf{b}^\prime = -n\_0 \frac{\partial \mathbf{v}\_\mathbf{b}^\prime}{\partial \mathbf{t}}; \quad \frac{\partial n\_p^\prime}{\partial \mathbf{t}} = -n\_{p0} \frac{\partial \mathbf{v}\_\mathbf{p}^\prime}{\partial \mathbf{z}}\tag{32}$$

$$\frac{1}{r} \frac{\partial}{\partial r} r B\_\mathbf{v} = \frac{1}{c} \frac{\partial E\_r}{\partial t} + 4\pi e \left(n\_{p0} \mathbf{v}\_\mathbf{p}^\prime + n\_{b0} \mathbf{v}\_\mathbf{b}^\prime + n\_b^\prime \, V\_b\right); \quad \hat{L} \equiv \frac{\partial}{\partial t} + V\_b \frac{\partial}{\partial \mathbf{z}};$$

where *t* is time, *z* and *r* are the cylindrical coordinates, *Er*, *Ez* and *B<sup>φ</sup>* are the fields' components which are coupled with the beam, v<sup>0</sup> *b,p* and *n*<sup>0</sup> *b,p* are the perturbations of velocity and density for the beam and the plasma respectively, *n*<sup>0</sup> and *np*<sup>0</sup> are the unperturbed densities for beam and plasma respectively. In the process, we

*r*

### *Plasma Science and Technology*

interest only the longitudinal structure of the fields, i.e., their dependence on the longitudinal coordinate and time. The transverse structure of the fields can be obtained by expansion on series of the system's eigenfunctions. For given case those are the Bessel functions. We use the expansions

$$E\_{\mathbf{z}}(r,\mathbf{z},\mathbf{t}) = \sum\_{\mathfrak{s}} E\_{\mathbf{z}}^{(s)}(\mathbf{z},\mathbf{t}) I\_{0}(\mu\_{0\mathfrak{s}}r/\mathfrak{R}),\\B\_{\boldsymbol{\Phi}}(r,\mathbf{z},\mathbf{t}) = \sum\_{\mathfrak{s}} B\_{\boldsymbol{\Phi}}^{(s)}(r,\mathfrak{t}) I\_{1}(\mu\_{\mathfrak{s}}r/\mathfrak{R})\tag{33}$$

where *J*<sup>0</sup> and *J*<sup>1</sup> are the Bessel functions; *μos* and *μ*1*<sup>s</sup>* their roots in ascending order, *J*<sup>0</sup> *μ*0*<sup>s</sup>* ð Þ¼ 0, *J*<sup>1</sup> *μ*1*<sup>s</sup>* ð Þ¼ 0, *s* ¼ 1, 2, 3, … . The quantities v*p*,*<sup>b</sup>* and *np*,*<sup>b</sup>* should be expanded by analogy to *Ez*, but *Er* – by analogy to *Bφ*. From here on we deal with the expansion coefficients and mention arguments *z* and *t* only.

The fields' growth in the linear stage reveals itself most effectively on frequencies, closely approximating to roots of the DR and, simultaneously, to *kVb* (resonant instability). The conditions (30) hold. In this connection it is reasonable to assume that originated perturbations form a wave packet of following type (e.g. for *E*ð Þ*<sup>s</sup> <sup>z</sup>* ð Þ *z*, *t* ):

$$E\_{\mathbf{z}}^{(\varsigma)}(\mathbf{z},t) = E\_0(\mathbf{z},t) \exp\left(-i\alpha\_0 t + ik\_0 \mathbf{z}\right),\tag{34}$$

where the carrier frequency *ω*<sup>0</sup> and wave vector *k*<sup>0</sup> satisfy the conditions (30). We also assume that the amplitude of the wave train *E*0ð Þ *z*, *t* varies slowly in space and time as compared to *k*<sup>0</sup> and *ω*<sup>0</sup> that is

$$
\left|\frac{\partial E\_0}{\partial t}\right| < < |\alpha\_0 E\_0| \; ; \; \left|\frac{\partial E\_0}{\partial \mathbf{z}}\right| < < |k\_0 E\_0| \;. \tag{35}
$$

Thus, the problem of the initial pulse behavior reduces to determination of the slowly varying amplitude (SVA) *E*0ð Þ *z*, *t* . The equation that *E*0ð Þ *z*, *t* satisfies can be derived from the set of origin Eqs. (32). The expansions (33) reduce it to a set of the equation for the amplitudes of expansions. In its turn the resulting set can be reduced to one equation for *E*0ð Þ *z*, *t* . We write it in form similar to the DR

$$\left(\hat{\boldsymbol{\alpha}} - \hat{\boldsymbol{k}}\boldsymbol{V}\_{b}\right)^{2} D\_{0}\left(\hat{\boldsymbol{\alpha}}, \hat{\boldsymbol{k}}\right) E\_{\boldsymbol{x}}^{(\boldsymbol{s})}(\boldsymbol{z}, \boldsymbol{t}) = \boldsymbol{a}\_{b}^{2} \boldsymbol{\gamma}^{-3} \kappa^{2} E\_{\boldsymbol{x}}^{(\boldsymbol{s})}(\boldsymbol{z}, \boldsymbol{t})\tag{36}$$

where *ω*^ and ^ *<sup>k</sup>* are differential operators *<sup>ω</sup>*^ � *<sup>i</sup> <sup>∂</sup> ∂t* ; ^ *<sup>k</sup>* � �*<sup>i</sup> <sup>∂</sup> ∂z* . The DR in form (30) follows from (36). To derive the equation for *E*0ð Þ *z*, *t* one should expanding (36) in power series near resonant values of frequency *ω*<sup>0</sup> and wave vector *k*<sup>0</sup> by using the relations *<sup>ω</sup>*^ ! *<sup>ω</sup>*<sup>0</sup> <sup>þ</sup> *<sup>i</sup> <sup>∂</sup> <sup>∂</sup><sup>t</sup>* and ^ *<sup>k</sup>* ! *<sup>k</sup>*<sup>0</sup> � *<sup>i</sup> <sup>∂</sup> <sup>∂</sup><sup>z</sup>* with account of OEB existence condition. As a result we arrive to the following second-order partial differential equation for *E*0ð Þ *z*, *t*

$$E\left(\frac{\partial}{\partial t} + V\_b \frac{\partial}{\partial \mathbf{z}}\right) \left(\frac{\partial}{\partial t} + V\_\mathbf{p} \frac{\partial}{\partial \mathbf{z}} + \nu'\right) E\_0(\mathbf{z}, \mathbf{t}) = \delta\_{\text{ovl}}^2 E\_0(\mathbf{z}, \mathbf{t}) \tag{37}$$

$$\begin{aligned} \text{where } \nu' = \text{Im}D\_0(\partial D\_0/\partial \boldsymbol{\omega})^{-1}, \boldsymbol{V}\_0 = -\left\{ (\partial D\_0/\partial \boldsymbol{k})(\partial D\_0/\partial \boldsymbol{\omega})^{-1} \right\}\_{\boldsymbol{\omega} = \boldsymbol{\omega}\_0} \text{ and the } \\ \boldsymbol{k} = \boldsymbol{k}\_0 \end{aligned} $$

expression for *δ*ovl is obtained from the relation *δ*<sup>3</sup> ovl <sup>¼</sup> *<sup>κ</sup>*<sup>2</sup>*ω*<sup>2</sup> *<sup>b</sup>γ*�<sup>3</sup>ð Þ *<sup>∂</sup>D*0*=∂<sup>ω</sup>* �<sup>1</sup> accounting the condition for OEB. It is important to emphasize that this denotation (as well as *V*0) is introduced for reasons of simplicity of the resulting Eq. (36) only.

The solution of (37) is, actually, known. If one returns to the set (6) and transforms it (under *J z*ð Þ¼ , *t* 0) to one equation for *Ew*ð Þ *z*, *t* then the equation will completely coincide to (37). This means that we already have the solution of (36)

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

and its analysis. It only remains to rewrite the solution (12) in new denotations and, where needed, re-interpret results. This shows that the instability in uniform crosssection beam-plasma waveguide develops in space and time in the same manner as the instability in weakly coupled beam-plasma system, and *δ*ovl is its growth rate in limit *ν* ! 0, that is *δ*ovl � *δ* ð Þ *ν*¼0 ovl . However there is a very important quantitative difference. In present case the growth rate *δ* ð Þ *ν*¼0 ovl depends on the beam density as � *n* 1*=*2 *<sup>b</sup>* (for the case of weak beam-plasma coupling the dependence is � *n* 1*=*4 *<sup>b</sup>* (see (27)). The criterion for determining the type of DSI takes the form

$$
\delta\_{\rm ovl}^{(\nu \to \infty)} = \left[ \delta\_{\rm ovl}^{(\nu = 0)} \right]^2 / \nu' \sim o\_b^2 / \nu' \tag{38}
$$

Comparison of (38) with (22) indicates one more new type of DSI. It develops in uniform cross section beam-plasma waveguide under over-limiting beam current and high level of dissipation. Its growth rate depends on the beam density and collision frequency as � *nb=ν*<sup>0</sup> .

### **4.2 Substantiation of the second new DSI by conventional method**

Now we substantiate the second new DSI by solving the DR (30). We look for its roots in the form *ω* ¼ *kVb* þ *δ*, *δ*< <*kVb*. The DR (30) reduces to [1, 14].

$$\mathbf{x}^3 + i \frac{\nu}{a\nu\_0} \frac{a\nu\_p^2 \mathbf{v}\_0}{V\_b \gamma^2 a\_\perp^2} \mathbf{x}^2 + \frac{a\mathbf{v}\_0 V\_b}{\gamma^2 c^2} \mathbf{x} = \frac{a}{2\gamma^4} \frac{\mathbf{v}\_0}{V\_b} \tag{39}$$

where *<sup>x</sup>* <sup>¼</sup> *<sup>δ</sup>=kVb*, *<sup>α</sup>* <sup>¼</sup> *<sup>ω</sup>*<sup>2</sup> *b=k*<sup>2</sup> ⊥*V*<sup>2</sup> *<sup>b</sup>γ*3, *<sup>β</sup>* <sup>¼</sup> *Vb=c*, *<sup>ω</sup>*<sup>2</sup> <sup>⊥</sup> <sup>¼</sup> *<sup>k</sup>*<sup>2</sup> ⊥*V*<sup>2</sup> *<sup>b</sup>γ*2, v0 <sup>¼</sup> *<sup>μ</sup>Vb=*ð Þ <sup>1</sup> <sup>þ</sup> *<sup>μ</sup>* , is the group velocity of the resonant wave in the system without beam, *<sup>μ</sup>* <sup>¼</sup> *<sup>γ</sup>*<sup>2</sup>*ω*<sup>2</sup> ⊥*=ω*<sup>2</sup> 0; *<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.

The solutions of (39) depend on the beam current value that is on the value of parameter α. If *α* < < *γ*�<sup>2</sup> (under-limiting e-beams) one can obtain the growth rates of conventional instability under *ν* ¼ 0 (first and right-hand side terms) and in limit *ν*> >*δ*und i.e. DSI

$$\delta\_{\rm und} = \frac{\sqrt{3}}{2} \frac{o\nu\_0}{\gamma} \left(\frac{o\nu\_b^2}{2o\_0^2(1+\mu)}\right)^{\natural \circ}; \quad \delta\_{\rm und}^{(\nu)} = \frac{o\nu\_0^{3/2}}{2\gamma^{3/2}o\nu\_p} \sqrt{\frac{o\nu\_b}{\nu}}\tag{40}$$

If the beam current increases and become comparable or higher than the limiting vacuum current i.e. *γ*�<sup>2</sup> ≤*α* < < 1, the physical nature of the instability changes. It becomes due to a-periodical modulation of the beam density in medium with negative dielectric constant. The distinctive peculiarity of this instability is in following: its growth rate attains maximum under exact Cherenkov resonance and is equal to (31) [9, 11, 14, 15]. If, along with the beam current, dissipation also increases the instability turns to DSI of over-limiting beam with growth rate [9].

$$
\delta\_{\rm ovl}^{(\nu)} = \frac{\beta^2}{\gamma} \frac{\alpha\_b^2}{\alpha\_p^2} \frac{\alpha\_0^2}{\nu} \sim \frac{\alpha\_b^2}{\nu}. \tag{41}
$$

We emphasize new dependences on *ν* and on the beam density. This, actually, substantiates one more new type of DSI. It develops in uniform cross section beamplasma waveguide if the beam current is higher than the limiting vacuum current.
