6. Conclusion

<sup>E</sup><sup>0</sup> � exp ffiffiffi

3

under arbitrary ratio ν

To λ 0 ¼ 0:5; curve 3 – To λ

0

the wave train (in the presence of dissipation also).

42 Plasma Science and Technology - Basic Fundamentals and Modern Applications

vth <sup>¼</sup> v0

<sup>1</sup> <sup>þ</sup> <sup>λ</sup><sup>3</sup>=<sup>2</sup> 0

location/movement. Its place z ¼ zmax can be obtained from the equation

ð Þ v0t � 3z

<sup>z</sup> <sup>¼</sup> <sup>z</sup>max <sup>¼</sup> <sup>33</sup>=<sup>4</sup>

equation χBnð Þ¼ z; t νz=v0 and is equal

0

only in the most interesting limit of high dissipation λ<sup>0</sup> ! ∞.

level dissipation. The field's value increases exponentially

<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

; <sup>λ</sup><sup>0</sup> <sup>¼</sup> <sup>25</sup>=<sup>3</sup>

3�3=<sup>2</sup>

=δBn. To avoid cumbersome expressions, we present here the solution

E<sup>0</sup> � exp f g δν t (77)

j j δBn ν 0 � �<sup>3</sup>

ν 0 j j δBn

ð Þ v0t � z (75)

v0t (76)

0

¼ ν=j j δBn ¼ 0; curve 2 –

(74)

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

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

The wave train shortens. Actually the pulse slows down. Dissipation influences on the peak

<sup>3</sup> <sup>¼</sup> ð Þ <sup>3</sup>λ<sup>0</sup>

The solution of this third-order algebraic equation gives location and velocity of the peak

25=<sup>2</sup> !

Substitution of this expression into χBnð Þ z; t gives the field's behavior in the peak under high-

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 3.

¼ 1:5; curve 4 – To λ

3 z2

> 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 second/third order.

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

> 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

more difficult than usage of the result of the initial and boundary problems (in spite of presented approach gives incomparably more data). In this sense, the approach can be used instead of the problems. It could seem that the procedure is a bit more difficult. However, this difficulty only seems.

Author details

Armenia

References

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[10] Kuzelev MV. Plasma Physics Report. 2006;32:572 [11] Rostomyan EV. Physics of Plasmas. 2000;7:1595

Soviet Physics Uspekhi. 1987;30:507

[16] Rostomyan EV. Physics of Plasmas. 2017;24:102102 [17] Rostomyan EV. Physics of Plasmas. 2016;23:102115

[18] Rostomyan EV. Physics Letters A. 2009;373:2581

[7] Rostomyan EV. EPL. 2007;77:45001

Eduard V. Rostomyan

Address all correspondence to: eduard\_rostomyan@mail.ru

In: Rukhadze AA, editor. Berlin: Springer-Verlag; 1984. 490pp [2] Kuzelev MV, Rukhadze AA. Plasma Physics Report. 2000;26:231

[6] Rostomyan EV. IEEE Transactions on Plasma Sciences. 2003;31(6):1278

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[13] Aizatski NI. Soviet Journal of Plasma Physics. 1980;6:597

Institute of Radiophysics and Electronics Armenian National Academy of Sciences, Astarack,

The Behavior of Streaming Instabilities in Dissipative Plasma

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

45

[1] Alexandrov AF, Bogdankevich LS, Rukhadze AA. Principles of Plasma Electrodynamics.

[3] Kuzelev MV, Loza OT, Rukhadze AA, Strelkov PS, Shkvarunets AG. Plasma Physics

[5] Briggs RJ. Electron-Stream interaction with Plasma. Cambridge, Mass: M.I.T Press; 1964. 187pp

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[14] Kuzelev MV, Rukhadze AA. Plasma Free Electron Lasers. Paris: Edition Frontier; 1995 [15] Kuzelev MV, Rukhadze AA. Behavior of Streaming Instabilities in Dissipative Plasma.

[20] Vedenin PS, Roukhlin VG, Tarakanov VP. Soviet Journal of Plasma Physics. 1989;15:1246

The general character of presented approach should be emphasized once more. It is based on very general assumptions and does not refer on any particular model. The approach transforms the general form of the DR to an equation for SVA of the developing wave train. For a large class of beam-plasma instabilities (Cherenkov, cyclotron, etc.), the equation for SVA is actually the same. Its solution gives analytical expression describing evolution of initial perturbation. Various SI evolve in similar manner. This emphasizes identity of their physical nature (induced radiation of the system proper waves by the beam electrons). For given instability, one should specify two parameters only: the resonant growth rate and the group velocity of the resonant wave. Obtained expression gives detailed information on the instability. The information is: the shape of developing wave train (envelope), velocities of unstable perturbations, the type of given instability (absolute or convective), location of the peak and the character of its movement, the rate of field's growth in the peak, temporal and spatial growth rates, the rate of growth for perturbation moving at given velocity. Most of these data are unavailable by other methods.

Validity limitations also should be mentioned. Obtained results may not be applied to the systems where beam instability is caused by finite longitudinal dimension, for example, Pierce instability.

Presented approach has neither inner contradictions, no contradictions to previous results of the beam-plasma interaction theory. Its results fully coincide to those obtained by direct analysis of the DR. In some cases, (e.g., for overlimiting e-beam instability and the instability in spatially separated beam-plasma system) obvious analysis is possible due to comparatively simple contribution of the beam in the DR (namely when the contribution has first (but not second) order pole).

The results of presented approach actually are continuation and further development of the results of the initial and boundary problems. In its turn, the results of the problems have been repeatedly tested and rechecked experimentally. This actually can serve as confirmation of validity of the approach.

In [19, 20] the nonlinear dynamics of the beam-plasma instability was investigated numerically at no stationary beam injection into plasma-filled systems. The results show that at the initial stage of instability development the field has a shape matching reasonably to presented results.

Obtained results on SI evolution help to understand how the instability transforms given equilibrium of background plasma, estimate the level and/or scale of originated irregularities clear up how the nonlinear stage arises and predict saturation mechanisms. The systems, to which this may be applied are numerous, as the SI are the most common instabilities: from the Earth ionosphere to current carrying plasma (where the Buneman instability plays important role). Not to mention relativistic microwave electronics etc.
