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Nonlinear Interaction of a Small Cold Beam and a Plasma

### Abstract

Recently, a simple model was proposed for the nonlinear interaction of a low‐density monoenergetic electron beam and a relatively cold infinite homogeneous one‐dimensional plasma. The essential feature of this model is the observation that after several ‐foldings the bandwidth of the growing waves is so narrow that the electrons interact with a very nearly pure sinusoidal field. In terms of this single wave model, a properly scaled solution of the nonlinear beam‐plasma problem which depends analytically on all the basic parameters of the problem (i.e., plasma density, beam density, plasma thermal velocity, and beam drift velocity) is presented. This solution shows that the single wave grows exponentially at the linear growth rate until the beam electrons are trapped. At that time the wave amplitude stops growing and begins to oscillate about a mean value. During the trapping process the beam electrons are bunched in space and a power spectrum of the higher harmonics of the electric field is produced. Both the oscillation in wave amplitude and the power spectrum are given a simple physical interpretation.

© 1971 American Institute of Physics

Received 17 August 1970
Published online 08 August 2003

/content/aip/journal/pof1/14/6/10.1063/1.1693587

1.

1.The nonlinear interaction of a small monoenergetic beam and a plasma has also been considered by V. D. Shapiro, Zh. Eksp. Teor. Fiz. 44, 613 (1963)

1.[V. D. Shapiro, Sov. Phys. JETP 17, 416 (1963)]. However, he used quasilinear theory and thereby missed many of the effects we will discuss.

2.

2.W. E. Drummond and J. H. Malmberg, Bull. Am. Phys. Soc. 13, 1542 (1968);

2.W. E. Drummond, J. H. Malmberg, T. M. O’Neil, and J. R. Thompson, Phys. Fluids 13, 2422 (1970).

3.

3.The single wave model has been partially verified in beam plasma experiments and computer simulations: J. R. Apel, Phys. Fluids 12, 291 (1969);

3.R. L. Morse and C. W. Nielson, Phys. Fluids 12, 2418 (1969); , Phys. Fluids

3.and T. P. Armstrong and D. Montgomery, Phys. Fluids 12, 2094 (1969)., Phys. Fluids

4.

4.A recent example is the computer simulation by R. L. Morse and C. W. Nielson, Phys. Fluids 12, 2418 (1969).

5.

5.R. Z. Sagdeev and A. A. Galeev, Nonlinear Plasma Theory (Benjamin, New York, 1969), p. 37;

5.and J. H. Malmberg and C. B. Wharton, Phys. Rev. Letters 19, 775 (1967).

6.

6.J. H. Malmberg and C. B. Wharton, Phys. Fluids 12, 2600 (1969).

7.

7.The coefficients are time dependent but the equation is linear.

8.

8.L. D. Landau and E. M. Lifshitz, Fluid Mechanics (Addison‐Wesley, Reading, Mass. 1966), p. 102;

8.R. Z. Sagdeev and A. A. Galeev, Nonlinear Plasma Theory (Benjamin, New York, 1969), p. 24.

9.

9.W. M. Manheimer, Bull. Am. Phys. Soc. 14, 1041 (1969).

10.

10.J. R. Apel, Phys. Rev. Letters 19, 744 (1967);

10.J. R. Apel, Phys. Fluids 12, 640 (1969).

11.

11.J. Dawson, W. Kruer and R. Sudan, Bull. Am. Phys. Soc. 13, 1504 (1968);

11.M. V. Goldman and H. L. Berk, Bull. Am. Phys. Soc. 14, 1034 (1968)., Bull. Am. Phys. Soc.

12.

12.C. B. Wharton, J. H. Malmberg, and T. M. O’Neil, Phys. Fluids 11, 1761 (1968).

13.

13.This description is similar to the theory of A. Nordsieck, Proc. IRE 41, 630 (1953). However, for the traveling wave amplifier, the background dielectric represents a rigid circuit structure rather than a plasma, and the single wave grows in space rather than time. Also, the equations for the traveling wave amplifier were only integrated to the beginning of nonlinear saturation, which is the natural interest in an amplifier; whereas, our turbulence interests really only begin at the point of nonlinear saturation.

14.

14.T. M. O’Neil and J. H. Malmberg, Phys. Fluids 11, 1754 (1968).

15.

15.Actually, at the point where cannot be zero. It has a small imaginary part where ῡ is the plasma thermal velocity. Nevertheless, we can neglect this term in the following Taylor expansion provided it is much smaller than

16.

16.In other words, we compare the autocorrelation time to the nonlinear saturation time.

17.

17.Direct integration schemes may introduce systematic errors in the acceleration, because they do not account for changes in the orbit during a time step, whereas the predictor‐corrector scheme produces only random walk errors.

18.

18.Recall that we are working in a coordinate system moving with the initial beam velocity.

19.

19.During an e‐folding, the beam electrons move relative to the wave by about one‐tenth of a wavelength.

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